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Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities

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Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 19 Issue 1

Abstract

We consider a CMC hypersurface with an isolated singular point at which the tangent cone is regular, and such that, in a neighbourhood of said point, the hypersurface is the boundary of a Caccioppoli set that minimises the standard prescribed-mean-curvature functional. We prove that in a ball centred at the singularity there exists a sequence of smooth CMC hypersurfaces, with the same prescribed mean curvature, that converge to the given one. Moreover, these hypersurfaces arise as boundaries of minimisers. In ambient dimension 8 the condition on the cone is redundant. (When the mean curvature vanishes identically, the result is the well-known Hardt–Simon approximation theorem.

Keywords: Constant-mean-curvature; isolated singularity; Hardt–Simon foliation; smooth approximation
MSC 2020: 53A10; 49Q20; 35J93

1 Introduction

It is well known that variational constructions for area-type functionals may lead to singularity formation. Already in the widely studied case of area minimisation for hypersurfaces, if the ambient dimension is 8 or higher, solutions cannot be expected to be completely smooth. The case of volume-constrained perimeter minimisation, which leads to isoperimetric regions, is analogous: in ℝ n + 1 , or more generally in an ( n + 1 ) -dimensional Riemannian manifold, such regions have boundaries that are smoothly embedded away from a possible singular set of dimension at most ( n - 7 ) ; when n = 7 , the singular set is made more precisely of isolated points. The phenomenon arises yet again in the case of minimax constructions for prescribed-mean-curvature functionals.

Examples show that this singular set is in general unavoidable. The well-known minimal cone C 4 , 4 = { ( x , y ) ∈ ℝ 4 × ℝ 4 ≡ ℝ 8 : | x | 2 = | y | 2 } (shown to be stable by Simons [27]) is smooth away from the isolated singularity at the origin, and is area-minimising, e.g. in any ball B ⊂ ℝ 8 , with respect to the boundary condition C 4 , 4 ∩ ∂ ⁡ B . This was proved by Bombieri, De Giorgi and Giusti ([6], see also a more straightforward proof in [13]). This cone is in fact the unique minimiser for said boundary condition. An isoperimetric region with two isolated singular points in an 8-dimensional Riemannian manifold was recently constructed in [23].

On the other hand, it is fruitful to ask whether the appearance of singularities is a generic phenomenon. This question led to very important progress already in the 80s and has received renewed attention in recent years. The fundamental work by Hardt and Simon [19] shows an instance of generic regularity for solutions to the Plateau problem, in the following sense. Let a 7-dimensional area minimiser in ℝ 8 be given, with (prescribed) 6-dimensional smooth boundary Γ, and with an isolated singular point; then a slight perturbation of Γ yields a minimiser that is completely smooth. This type of result lends itself to geometric applications, by shifting the genericity condition onto the Riemannian metric, as exemplified by Smale’s proof of generic regularity of area-minimisers in any non-zero homology class [28]. (We also refer the reader to [9, 21].) Very recently, the question of generic regularity for area minimisers has found affirmative answer in ambient dimension 9 and 10, in the work by Chodosh, Mantoulidis and Schulze [10], making progress on a long-standing conjecture [28].[1] We also refer the reader to [8] and references therein, for generic regularity in the setting of mean curvature flow.

Our main goal here is to prove a (local) smooth approximation result in the constant-mean-curvature (CMC) case, establishing a generic regularity result for the CMC Plateau problem analogous to the one proven in [19] (in particular, if the mean curvature vanishes identically, the Hardt–Simon theorem gives the result). The variational setting for CMC hypersurfaces involves an energy that we will denote by J λ , where λ ∈ ℝ is the prescribed constant value of the scalar mean curvature. Roughly speaking, J λ evaluates the n-dimensional area of the hypersurface, from which it subtracts λ times the ( n + 1 ) -volume enclosed by it. A natural way to formalise this is by working with boundaries of sets with finite perimeter. We briefly recall the relevant notions (with more details in Section 2 below).

Let E ⊂ U be a set with locally finite perimeter in a bounded open set U ⊂ ℝ n + 1 , and let λ ∈ ℝ . We denote by J λ the functional (defined on any set D ⊂ U with locally finite perimeter in U),

J λ ⁢ ( D ) = Per U ⁡ ( D ) - λ ⁢ | D | ,

where the notation | D | stands for ℒ n + 1 ⁢ ( D ) . Given W ⊂ ⊂ U , the set E is said to be a minimiser of J λ in W ⊂ U if it attains the following infimum:

inf ⁡ { J λ ⁢ ( D ) : D ∩ ( U ∖ W ) = E ∩ ( U ∖ W ) } .

In other words, the class of competitors for E is that of sets (with locally finite perimeter in U) that coincide with E outside W. Equalities and inclusions between sets of locally finite perimeter are always understood to hold in the ℒ n + 1 -a.e. sense. Prescribing the set in U ∖ W amounts to fixing the boundary condition for the Plateau problem in W (as customary in the setting of Caccioppoli sets).

If E is a minimiser of J λ in W ⊂ ⊂ U , it is well known (see e.g. [22, 18, 4]) that there exists a set Σ ⊂ W with dim ℋ ⁡ ( Σ ) ≤ n - 7 , such that ( ∂ * ⁡ E ¯ ∩ W ′ ) ∖ Σ is smoothly embedded in W ′ for every open set W ′ ⊂ ⊂ W , and that ( ∂ * ⁡ E ¯ ∩ W ′ ) ∖ Σ has constant scalar mean curvature equal to λ. Here ∂ * ⁡ E denotes the reduced boundary of the set E. (More precisely, the mean curvature vector is λ ⁢ ν E , where ν E is the unit normal pointing into E.)

The most immediate instance of our result states the following.

Theorem 1.

Let E be a set with locally finite perimeter in an open set U ⊂ R 8 , and assume that E minimises J λ in a ball B ^ ⊂ ⊂ U , for a given λ ∈ R . There exists a ball B ⊂ B ^ , with the same centre, and a sequence of hypersurfaces T j smoothly embedded in B, with scalar mean curvature λ, and with T j → ∂ * ⁡ E in B. (The convergence holds in the sense of currents, in the sense of varifolds, as well as in the Hausdorff distance sense.) Moreover, T j = ∂ * ⁡ E j , where each E j is a set with finite perimeter in B and ∂ * ⁡ E j stands for the reduced boundary of E j in B, and we have E j ⊂ E and E j → E in B.

We remark that the significance of Theorem 1 lies in the fact that the centre p of B ^ may be a singular point of ∂ * ⁡ E ¯ .

In ambient dimension 8, as in Theorem 1, isolated singular points are the only type of interior singularities that ∂ * ⁡ E ¯ may possess. This is no longer the case when the ambient dimension is higher. Just as in [19], we can remove the dimensional restriction in Theorem 1 by (strongly) restricting the singular behaviour of E (Theorem 2 below). We work in a neighbourhood of an isolated (interior) singular point p of ∂ * ⁡ E ¯ , with the further property that the multiplicity-1 varifold associated to ∂ * ⁡ E , denoted by | ∂ * ⁡ E | , admits a tangent cone at p that is regular. We recall that a cone is regular when it is smooth away from the vertex, and the multiplicity is 1 on the smooth part.

Theorem 2.

Let E be a set with locally finite perimeter in an open set U ⊂ R n + 1 , with n ≥ 7 , and assume that E minimises J λ in a ball B ^ ⊂ ⊂ U , for a given λ ∈ R . Assume furthermore that the centre p of B ^ is an isolated singularity of | ∂ * ⁡ E | and that | ∂ * ⁡ E | admits a tangent cone at p that is regular (in the sense of varifolds). There exists a ball B ⊂ B ^ , with the same centre p, and a sequence of hypersurfaces T j smoothly embedded in B, with scalar mean curvature λ, and with T j → ∂ * ⁡ E in B. (The convergence holds in the sense of currents, in the sense of varifolds, as well as in the Hausdorff distance sense.) Moreover, T j = ∂ * ⁡ E j , where each E j is a set with finite perimeter in B and ∂ * ⁡ E j stands for the reduced boundary of E j in B, and we have E j ⊂ E and E j → E in B.

Remark 1.1.

By construction, for each j the set E j is a minimiser, more precisely, it is given by E ^ j ∩ B for a set with finite perimeter E ^ j ⊂ B ^ that minimises J λ in B ⊂ B ^ (among sets that coincide with E ^ j in B ^ ∖ B ). The mean curvature vector of | ∂ * ⁡ E j | in B is given by λ ⁢ ν E j , where ν E j is the inward pointing unit normal.

Remark 1.2.

The regularity theory for n = 7 implies not only that the singular set is made of isolated points, but also that any varifold tangent cone (at a singular point) must be regular, via a standard dimension reduction argument. Therefore Theorem 1 follows from Theorem 2.

Remark 1.3.

In the special case λ = 0 Theorems 1 and 2 were proved in [19] (see also [10]). Our proof relies on the result for λ = 0 .

Remark 1.4.

In both Theorems 1 and 2, the convergence T j → ∂ * ⁡ E is strong (graphical and C 2 ) in B ∖ { p } , thanks to Allard’s regularity theorem and standard elliptic PDE theory.

Remark 1.5.

Theorems 1 and 2 lend themselves applications in geometry, such as the surgery procedure in [3] (where a generic existence result for smooth CMC closed hypersurfaces in compact Riemannian 8-dimensional manifolds is proved).

In proving Theorem 2 (which we will do in Section 5, see Theorem 5) we establish a result of independent interest on the existence and regularity of minimisers of J λ , for the CMC Plateau problem. We present here a simplified version (sufficient for its scope within the proof of Theorem 2). The more general result requires some notation and will be given in Theorem 4 of Section 2.

Theorem 3.

Let E 0 be a set with finite perimeter in U = B R n + 1 ⁢ ( p ) . Let λ ∈ ( 0 , ∞ ) and r ∈ ( 0 , n λ ) , with r < R . Assume that ∂ ⁡ E 0 is smooth in a neighbourhood of ∂ ⁡ B r n + 1 ⁢ ( p ) and that it intersects ∂ ⁡ B r n + 1 ⁢ ( p ) transversely; let T 0 denote (the ( n - 1 ) -dimensional submanifold) ∂ ⁡ E 0 ∩ ∂ ⁡ B r n + 1 ⁢ ( p ) . There exists a set E, with finite perimeter in B R n + 1 ⁢ ( p ) , that coincides a.e. with E 0 in B R n + 1 ⁢ ( p ) ∖ B r n + 1 ⁢ ( p ) , that is a minimiser of J λ in B r n + 1 ⁢ ( p ) ⊂ B R n + 1 ⁢ ( p ) , and with the following properties:

  1. There exists Σ ⊂ B r n + 1 ⁢ ( p ) , closed in B r n + 1 ⁢ ( p ) , with dim ℋ ⁡ ( Σ ) ≤ n - 7 such that ( ∂ * ⁡ E ¯ ∩ B r n + 1 ⁢ ( p ) ) ∖ Σ is a smoothly embedded hypersurface with mean curvature λ ⁢ ν E , where ν E is the inward unit normal to E ; more precisely, Σ = ∅ if n ≤ 6 , and Σ is discrete if n = 7 .

  2. ∂ * ⁡ E ¯ ∩ ∂ ⁡ B r n + 1 ⁢ ( p ) = T 0 .

In the more general formulation that we will provide with Theorem 4, both smoothness and transversality conditions will be removed (see also Remark 2.9).

The “boundary condition” in Theorem 3 is set by prescribing the coincidence a.e. with a reference set E 0 (the condition r < R provides an annulus in which E 0 is non-trivial). The submanifold T 0 acts as prescribed boundary condition for the hypersurface that we seek. The last conclusion of the theorem states that the solution does not touch ∂ ⁡ B r n + 1 ⁢ ( p ) except at T 0 . So ∂ * ⁡ E ¯ ∩ B r n + 1 ⁢ ( p ) ∖ Σ is a smooth hypersurface with boundary in the open set B R n + 1 ⁢ ( p ) ∖ Σ . (Since T 0 is smooth, Σ does not accumulate onto T 0 by Allard’s boundary regularity theorem, [2]; this property is not needed in our forthcoming arguments.)

While the existence of a minimiser follows for any λ, the condition λ < n r is essential for the last conclusion of Theorem 3, as well as for the verification of the prescribed mean curvature condition. We will discuss this with examples in Remark 2.8; when λ > n r , the hypersurface may in fact touch ∂ ⁡ B r n + 1 ⁢ ( 0 ) away from its boundary T 0 .

Theorem 3 (and Theorem 4 below) and its proof are close in spirit to the results in Duzaar and Fuchs [15] (and Duzaar [14]). We highlight that our last conclusion in Theorem 3 is sharper than the corresponding statement in [14, 15], since we are able to rule out any interior touching of the solution with the “obstacle” ∂ ⁡ B r n + 1 ⁢ ( p ) in which the boundary condition T 0 lies (the only touching is the necessary one at T 0 itself). The results in [14, 15], while establishing the validity of the CMC condition, would only prevent touching of the solution with larger spheres. The sharper conclusion we obtain is ultimately due to our use of the regularity theory for stable CMC hypersurfaces developed in [4, 5] (with the sheeting theorem therein being the key ingredient in our proof). The same reasoning that we employ to that end (see Lemma 2.4 and the discussion preceding it) can be applied to sharpen the corresponding conclusion in [15] (where the relevant class is that of integral currents, rather than boundaries of Caccioppoli sets).

We are now ready to present an outline of the proof of Theorem 2, setting p = 0 . By fairly standard arguments, there exists a sufficiently small ball centred at 0, which we denote by B 2 ⁢ R ⁢ ( 0 ) , such that E is the unique minimiser of J λ in B R ⁢ ( 0 ) ⊂ B 2 ⁢ R ⁢ ( 0 ) , and with the further requirements that λ < n R and that ∂ ⁡ E meets ∂ ⁡ B R ⁢ ( 0 ) smoothly and transversely.

Then we perturb E towards its interior (keeping it fixed outside an annulus that contains ∂ ⁡ B R ⁢ ( 0 ) ) and use the resulting set as ‘boundary condition’ in B 2 ⁢ R ⁢ ( 0 ) ∖ B R ⁢ ( 0 ) for a CMC Plateau problem. The perturbation is indexed on j and tends to the identity as j → ∞ , and we denote the deformed set by E j ⊂ E . For each j we find a minimiser of J λ with said boundary condition; note that Theorem 3 applies here. Theorem 2 follows by showing the existence of a sufficiently small ball centred at 0 in which, for all sufficiently large j, ∂ * ⁡ E j ¯ are smooth. Arguing by contradiction, we assume the existence of singular points p j ∈ ∂ * ⁡ E j ¯ , p j → 0 . If the condition p j ≠ 0 is valid (for all sufficiently large j) then we dilate E j around 0 by scaling B R ⁢ ( 0 ) to B R | p j | ⁢ ( 0 ) . Using [19], we check that the limit of these rescalings of ∂ * ⁡ E j ¯ has to be either one of the leaves of the Hardt–Simon foliation, or the tangent cone C to ∂ * ⁡ E ¯ at 0: in either case we find a contradiction to the smoothness respectively of the leaves, or of the cone (at points at distance 1 from the origin).

Therefore we have to establish the condition p j ≠ 0 . By construction E j ⊊ E and both boundaries are hypersurfaces with the same scalar mean curvature, and with mean curvature vectors both pointing inwards. We thus show that the inclusion is strict everywhere by proving an instance of a singular maximum principle for CMC hypersurfaces, see Proposition 1 below. Its proof (by contradiction) relies on a linearisation argument that yields a non-trivial Jacobi field on the cone C (an analogous argument appears in [19] in the minimal case), combined with Simon’s result [24], which gives a quantitative decay of ∂ * ⁡ E ¯ towards C at small scales. The resulting behaviour of the Jacobi field is in contradiction with the ones that are known [7] to be permitted by the stability of the cone (stability follows from the minimising condition for E).

2 Prescribed CMC Plateau problem

In the following we denote by B R the open ball B R n + 1 ⁢ ( 0 ) ⊂ ℝ n + 1 . Let E 0 be a set of finite perimeter in B 2 , that is, E 0 ⊂ B 2 is measurable and the perimeter of E 0 in B 2 is finite,

Per B 2 ⁡ ( E 0 ) = sup ⁡ { ∫ E 0 div ⁡ T ⁢ d ⁢ ℒ n + 1 : T ∈ C c 1 ⁢ ( B 2 ; ℝ n + 1 ) , sup ⁡ | T | ≤ 1 } < ∞ ,

where ℒ n + 1 denotes the Lebesgue measure on ℝ n + 1 . This is equivalent to the requirement that the characteristic function χ E 0 ∈ BV ⁢ ( B 2 ) , that is, the distributional gradient D ⁢ χ E 0 is a vector-valued Radon measure with finite total variation in B 2 .

For λ ≥ 0 we will be interested in the following energy, defined on the class of sets of finite perimeter in B 2 that coincide with the given E 0 in B 2 ∖ B 1 :

J λ ⁢ ( E ) = Per B 2 ⁡ ( E ) - λ ⁢ | E | ,

where | E | = ℒ n + 1 ⁢ ( E ) = ℋ n + 1 ⁢ ( E ) is the ( n + 1 ) -volume of the Caccioppoli set E ⊂ B 2 . (The Lebesgue measure ℒ n + 1 agrees with the Hausdorff measure ℋ n + 1 in ℝ n + 1 .) This class is non-empty, since E 0 is one such set, and J λ ⁢ ( E 0 ) < ∞ , hence it makes sense to seek a minimiser of J λ in this class.

Lemma 2.1.

There exists a minimiser F of J λ in the class of sets with finite perimeter that coincide with the given E 0 in B 2 ∖ B 1 .

Proof.

We will use the direct method. Let E j , for j ∈ ℕ ∖ { 0 } , be a minimising sequence (of sets in the admissible class), that is

lim j → ∞ ⁡ J λ ⁢ ( E j ) = inf ⁡ { J λ ⁢ ( E ) : χ E ∈ BV ⁢ ( B 2 ) , χ E | B 2 ∖ B 1 = χ E 0 | B 2 ∖ B 1 } .

For all sufficiently large j we must then have

J λ ⁢ ( E j ) = Per B 2 ⁡ ( E j ) - λ ⁢ | E j | ≤ J λ ⁢ ( E 0 ) + 1 = Per B 2 ⁡ ( E 0 ) - λ ⁢ | E 0 | + 1 ,

from which

Per B 2 ⁡ ( E j ) ≤ Per B 2 ⁡ ( E 0 ) - λ ⁢ | E 0 | + λ ⁢ | E j | + 1 ≤ Per B 2 ⁡ ( E 0 ) + λ ⁢ | B 2 | + 1 .

Therefore Per B 2 ⁡ ( E j ) are uniformly bounded above and there exist (by BV compactness) a set of finite perimeter F in B 2 and a subsequence (that we do not relabel) E j such that χ E j → χ F in BV ⁢ ( B 2 ) . In particular, χ E j → χ F in L 1 ⁢ ( B 2 ) , so that | E j | → | F | ; moreover, by the hypothesis that E j = E 0 on B 2 ∖ B 1 , we have also that F = E 0 on B 2 ∖ B 1 . The lower semi-continuity of perimeters then gives J λ ⁢ ( F ) ≤ lim inf j → ∞ ⁡ J λ ⁢ ( E j ) , therefore F minimises J λ in the admissible class. ∎

The energy J λ is relevant in many variational problems. The geometric significance of J λ lies in the fact that it should select, as its critical points, sets whose boundary is a hypersurface with constant mean curvature λ. With the set up above, we are using E 0 to prescribe a boundary condition (in the sense of the Plateau problem). If ∂ ⁡ E 0 is smooth and intersects ∂ ⁡ B 1 transversely, then the set up amounts to fixing ∂ ⁡ E 0 ∩ ∂ ⁡ B 1 as ( n - 1 ) -dimensional boundary data, and looking for a (n-dimensional) CMC hypersurface-with-boundary, with mean curvature λ, and whose boundary is ∂ ⁡ E 0 ∩ ∂ ⁡ B 1 . The hope is to obtain this hypersurface-with-boundary as ∂ ⁡ F ∖ ( ∂ ⁡ E 0 ∩ ( B 2 ∖ B 1 ¯ ) ) (if ∂ ⁡ F is smooth).

Remark 2.6.

If λ < 0 and F is a minimiser of J | λ | in B 1 ⊂ B 2 , then U ∖ F is a minimiser of J λ in B 1 ⊂ B 2 (and vice versa), so we only treat the case λ ≥ 0 (and all results extend in a straightforward manner to λ < 0 ). This follows from the fact that complementary sets have the same perimeter (in an open set).

A well-known consequence of the minimising property is that the integral varifold V (in B 2 ) defined by

V = | ∂ * ⁡ F ∖ ( ∂ * ⁡ E 0 ∩ ( B 2 ∖ B 1 ¯ ) ) |

(the notation | ⋅ | denotes the multiplicity-1 varifold associated to a rectifiable set) has first variation in B 1 represented by the vector-valued measure

λ ⁢ ( ℋ n ⁢ ⌞ ⁢ ( ∂ * ⁡ F ∩ B 1 ) ) ⁢ ν F ,

where ν F is the (measure theoretic) inward unit normal ( ℋ n -a.e. well-defined on ∂ * ⁡ F ). Indeed, given any vector field X ∈ C c 1 ⁢ ( B 1 ; ℝ n + 1 ) , we can consider, for δ > 0 sufficiently small, the one-parameter family of diffeomorphisms Φ t = Id + t ⁢ X for t ∈ ( - δ , δ ) . For every such t, we have Φ t = Id on B 2 ∖ B 1 and therefore the set Φ t ⁢ ( F ) remains in the admissible class for every t. The image of V under Φ t is | ∂ * ⁡ Φ t ⁢ ( F ) ∖ ( ∂ * ⁡ E 0 ∩ ( B 2 ∖ B 1 ¯ ) ) | .

This permits to write the stationarity condition for V with respect to the energy J λ , which gives (see e.g. [22, Chapters 17 and 19])

∫ div ∂ * ⁡ F ⁡ X ⁢ d ⁢ V + λ ⁢ ∫ ( ν F ⋅ X ) ⁢ 𝑑 V = 0

and the desired conclusion. The candidate V thus has the correct mean curvature in B 1 .

Remark 2.7.

The notation | ⋅ | has been (and will be) employed to denote the ( n + 1 ) -volume when the argument is a Caccioppoli set (as in | E | above), and to denote the multiplicity-1 (n-dimensional) varifold associated to an n-dimensional rectifiable set (as for V above). The context and the different character of the argument should avoid any confusion.

Next we are going to examine when it is possible to conclude this same condition away from the prescribed boundary: the missing analysis at this stage is the behaviour at points that potentially lie on ∂ ⁡ B 1 but are not part of the prescribed boundary. We begin by pointing out that, if the vector field X is non-zero somewhere on ∂ ⁡ B 1 , then the above argument breaks down, since a one-parameter family of diffeomorphisms with initial speed X may map F to a set that is not in the admissible class (no matter how small δ is). In fact, the minimiser may just fail to have mean curvature λ when λ > n , as the following examples show.

Remark 2.8.

Let H be the half-space { x n + 1 < 0 } and E 0 = H ∩ B 2 . Then for any given λ > n the minimisation procedure fails to produce a set whose boundary is a CMC hypersurface-with-boundary with mean curvature λ and boundary condition ∂ ⁡ H ∩ ∂ ⁡ B 1 . (In fact, the unique minimiser F is given by E 0 ∪ B 1 for all λ ≥ n .) To see that, we observe that, for any given possible value v ∈ [ | B 1 | 2 , | B 1 | ] , the (unique) perimeter-minimiser with volume v in B 1 , that coincides with E 0 in B 2 ∖ B 1 , is given by the set E 0 ∪ E v , where E v is the ball of radius r centred at the point ( 0 , … , 0 , - r 2 - 1 ) , where r ≥ 1 is chosen so that | E v ∩ B 1 | = v . Similarly, for any given possible value v ∈ | E ∩ B 1 | ∈ [ 0 , | B 1 | 2 ] , the perimeter-minimiser with volume v in B 1 , and that coincides with E 0 in B 2 ∖ B 1 , is given by the set E 0 ∖ E ~ v , where E ~ v is the ball of radius r centred at the point ( 0 , … , 0 , r 2 - 1 ) , where r ≥ 1 is chosen so that | E ~ v ∩ B 1 | = | B 1 | - v . The minimisation property just claimed is checked by a calibration argument, using the fact that ∂ ⁡ E v ∩ B 1 (and, similarly, ∂ ⁡ E ~ v ∩ B 1 ) is a CMC graph on B 1 n ⊂ ℝ n ≡ ℝ n × { 0 } . (See e.g. [4, Appendix B].) With this understood, the minimiser of J λ (for any λ) has to be one of the minimising sets that have been exhibited for each possible value of v. Each of these minimisers has scalar mean curvature in [ - n , n ] (away from B 2 ∖ B 1 ¯ ). Hence for any λ > n the minimisation procedure will not produce the desired CMC hypersurface of mean curvature λ. (By direct computation, one can check that the lowest value of J λ for λ > n is attained by E 0 ∪ B 1 .)

In the case λ = n + 1 one can alternatively see that the minimiser is E 0 ∪ B 1 by arguing as follows. Given any Caccioppoli set D that coincides with E 0 in B 2 ∖ B 1 , consider the ( n + 1 ) -current C = [ [ E 0 ∪ B 1 ] ] - [ [ D ] ] . Denoting by ι T the interior product with T, we define the n-form β = ι T ⁢ ( d ⁢ x 1 ∧ … ∧ d ⁢ x n + 1 ) , with T = ( x 1 , … , x n + 1 ) . Then

d ⁢ β = ( div ⁡ T ) ⁢ d ⁢ x 1 ∧ … ∧ d ⁢ x n + 1 = ( n + 1 ) ⁢ d ⁢ x 1 ∧ … ∧ d ⁢ x n + 1 .

We note that C is supported in B 1 ¯ , so it can act on d ⁢ β (by introducing a cut off function that is 1 on B 1 ¯ and vanishes outside B 2 ). Then the equality C ⁢ ( d ⁢ β ) = ( ∂ ⁡ C ) ⁢ ( β ) gives ∂ ⁡ [ [ E 0 ∪ B 1 ] ] ⁡ ( β ) - ( n + 1 ) ⁢ | E 0 ∪ B 1 | = ∂ ⁡ [ [ D ] ] ⁡ ( β ) - ( n + 1 ) ⁢ | D | . Finally, we note that

∂ ⁡ [ [ E 0 ∪ B 1 ] ] ⁡ ( β ) = Per B 2 ⁡ ( E 0 ∪ B 1 ) - Per B 2 ∖ B 1 ¯ ⁡ H + ℋ n ⁢ ( ∂ ⁡ B 2 ) ,

while

∂ ⁡ [ [ D ] ] ⁡ ( β ) ≤ Per B 2 ⁡ ( D ) - Per B 2 ∖ B 1 ¯ ⁡ ( H ) + ℋ n ⁢ ( ∂ ⁡ B 2 ) ,

which gives that J n + 1 ⁢ ( E 0 ∪ B 1 ) ≤ J n + 1 ⁢ ( D ) , that is, E 0 ∪ B 1 is a minimiser. In fact, the inequality is not strict if and only if ∂ * ⁡ D ∖ ( B 2 ∖ B 1 ¯ ) is a.e. orthogonal to T and contained in ∂ ⁡ B 1 , which shows that E 0 ∪ B 1 is the unique minimiser.

Before proceeding further we set up some notation. The integral ( n + 1 ) -current [ [ E 0 ] ] in B 2 admits a well-defined (outer) slice 〈 [ [ E 0 ] ] , | x | = 1 + 〉 = - ∂ [ [ E 0 ∩ ( B 2 ∖ B 1 ¯ ) ] ] + ( ∂ [ [ E 0 ] ] ) ⌞ ( B 2 ∖ B 1 ¯ ) . (See e.g. [16, Section 2.5].) This (outer) slice also coincides with 〈 [ [ F ] ] , | x | = 1 + 〉 . Let T 0 denote the ( n - 1 ) -dimensional current

T 0 = - ∂ ⁡ 〈 [ [ E 0 ] ] , | x | = 1 + 〉 = - ∂ ⁡ ( ( ∂ ⁡ [ [ E 0 ] ] ) ⁢ ⌞ ⁢ ( B 2 ∖ B 1 ¯ ) ) .

Then the Plateau problem under consideration seeks an integral n-current with boundary T 0 . Note that ∂ ⁡ [ [ F ] ] = ∂ ⁡ [ [ F ∩ B 1 ] ] + ∂ ⁡ [ [ E 0 ∩ ( B 2 ∖ B 1 ) ] ] so

S := ∂ [ [ F ∩ B 1 ] ] - 〈 [ [ F ] ] , | x | = 1 + 〉 = ∂ [ [ F ] ] - ( ∂ [ [ E 0 ] ] ) ⌞ ( B 2 ∖ B 1 ¯ )

has boundary ∂ ⁡ S = T 0 . The integral n-current S is our candidate (hypersurface-with-boundary) solution to the Plateau problem. We let

𝒮 = ∂ * ⁡ F ∖ ( ∂ * ⁡ E 0 ∩ ( B 2 ∖ B 1 ¯ ) ) .

Then S = ( 𝒮 , 1 , - ⋆ ν F ) , where ⋆ is the Hodge star (so ν F ∧ ⋆ ν F gives the positive orientation of ℝ n + 1 ) and ν F is the unit inward (measure theoretic) normal for F on its reduced boundary. Also note that V = v ¯ ⁢ ( 𝒮 , 1 ) is the associated varifold (with notation from [25]).

We turn our attention to the analysis of the first variation (with respect to J λ ) of V on B 2 ∖ spt ⁡ T 0 . Combining Lemma 2.1 with Lemmas 2.2, 2.3, 2.4 below, we will in particular prove the following overall result.

Theorem 4.

With the above setting and notation, let λ ∈ ( 0 , n ) . In the class of sets with finite perimeter that coincide with the given E 0 in B 2 ∖ B 1 there exists a minimiser F of J λ , and there exists a set Σ ⊂ B 1 with dim H ⁡ Σ ≤ n - 7 , such that ( spt ⁡ V ∖ spt ⁡ T 0 ) ∖ Σ is a smoothly embedded CMC hypersurface with mean curvature vector λ ⁢ ν F . If n = 7 , more precisely, Σ is made of isolated points (possibly accumulating onto spt ⁡ T 0 ). Moreover, spt ⁡ V ∖ spt ⁡ T 0 ⊂ B 1 .

Remark 2.9.

By scaling and translating, the theorem can be stated replacing B 1 , B 2 and ( 0 , n ) respectively with B r n + 1 ⁢ ( p ) , B 2 ⁢ r n + 1 ⁢ ( p ) , ( 0 , n r ) . Moreover, the role of B 2 ⁢ r n + 1 ⁢ ( p ) is only to provide an annulus in which E 0 is non-trivial, so 2 ⁢ r can be replaced by any radius R > r . Theorem 3 is thus a special case of Theorem 4, and in the case of Theorem 3 the accumulation of Σ onto T 0 is ruled out by [2]. We also recall that, as well as the varifold V, we can associate to the minimiser F an integral n-current S such that ∂ ⁡ S = T 0 (see above for the definition of S).

Our first result on the first variation (with respect to J λ ), Lemma 2.2, is valid for any λ and yields a sign condition and an upper bound. The analysis needs to be carried out only in a neighbourhood of an arbitrary p ∈ ∂ ⁡ B 1 ∖ spt ⁡ T 0 (since spt ⁡ V ⊂ B 1 ¯ and we have established that the first variation is 0 in B 1 ). This result is the analogue of [15, Theorem 4.1]. Here we keep using the notation introduced above (e.g. ν F , V, 𝒮 , T 0 ).

Lemma 2.2.

Let X ∈ C c 1 ⁢ ( B 2 ∖ spt ⁡ T 0 ; R n + 1 ) . Then the first variation with respect to J λ of V evaluated on the vector field X (equal to the left-hand-side of the following expression) satisfies

∫ div 𝒮 ⁡ X ⁢ d ⁢ ℋ n ⁢ ⌞ ⁢ 𝒮 + λ ⁢ ∫ ( ν F ⋅ X ) ⁢ 𝑑 ℋ n ⁢ ⌞ ⁢ 𝒮 = ∫ ( X ⋅ N ) ⁢ 𝑑 ℳ ,

where M is a positive Radon measure supported in ∂ ⁡ B 1 and N = - x | x | (for x ≠ 0 ). Moreover,

ℳ ≤ ( div 𝒮 ⁡ N + λ ⁢ ( ν F ⋅ N ) ) ⁢ d ⁢ ℋ n ⁢ ⌞ ⁢ ( 𝒮 ∩ ∂ ⁡ B 1 )

(as measures).

Proof.

Let p ∈ ∂ ⁡ B 1 ∖ spt ⁡ T 0 and consider B r ⁢ ( p ) ⊂ B 5 4 ∖ spt ⁡ T 0 . In the first part of the proof, we analyse the action of the first variation on a vector field of the type η ⁢ N , where η ∈ C c 1 ⁢ ( B r ⁢ ( p ) ) , η ≥ 0 . Let d ⁢ ( ⋅ ) = dist ⁡ ( ⋅ , ∂ ⁡ B 1 ) , where dist is the signed distance, taken to be positive in B 1 and negative in B 2 ∖ B 1 ¯ . Note that in any tubular neighbourhood of ∂ ⁡ B 1 we have that d is smooth and its gradient is N. Given ϵ > 0 , let f ϵ : ℝ → ℝ be a C 1 function such that f ϵ ≡ 0 on [ 2 ⁢ ϵ , ∞ ) , f ϵ ≡ 1 on ( - ∞ , ϵ ] and f ′ ≤ 0 . We consider the following one-sided ( s ∈ [ 0 , s 0 ] , with s 0 > 0 sufficiently small, depending on ϵ) one-parameter family of diffeomorphisms:

ϕ s ⁢ ( z ) = z + s ⁢ η ⁢ ( z ) ⁢ ( f ϵ ∘ d ) ⁢ ( z ) ⁢ N ⁢ ( z ) .

The reason for the one-sided restriction, s ≥ 0 , is that we need to ensure that we stay in the admissible class when deforming via ϕ s , which we check next.

Since ∂ ⁡ S = T 0 , and spt ⁡ S ⊂ B 1 ¯ , by the conditions on ϕ s we also have ∂ ( ϕ s ) ♯ S = T 0 and spt ( ϕ s ) ♯ S ⊂ B 1 ¯ . On one hand we have S + 〈 [ [ F ] ] , | x | = 1 + 〉 = ∂ [ [ F ∩ B 1 ] ] , therefore (for any σ ∈ [ 0 , s 0 ] )

( ϕ σ ) ♯ S + ( ϕ σ ) ♯ 〈 [ [ F ] ] , | x | = 1 + 〉 = ∂ ( ϕ σ ) ♯ [ [ F ∩ B 1 ] ] .

On the other hand, letting Φ ⁢ ( s , z ) = ϕ s ⁢ ( z ) for s ∈ [ 0 , σ ] (this is a homotopy between the identity ϕ 0 and ϕ σ on B 2 ) we obtain, from the homotopy formula,

( ϕ σ ) ♯ 〈 [ [ F ] ] , | x | = 1 + 〉 - 〈 [ [ F ] ] , | x | = 1 + 〉 = ∂ ( Φ ♯ ( [ 0 , σ ] × 〈 [ [ F ] ] , | x | = 1 + 〉 ) ) .

Next we check that - Φ ♯ ( [ 0 , σ ] × 〈 [ [ F ] ] , | x | = 1 + 〉 ) is a Caccioppoli set. Note that Φ ⁢ ( s , ⋅ ) only acts on z ∈ ∂ ⁡ B 1 in this case. The map Φ | [ 0 , σ ] × ∂ ⁡ B 1 : [ 0 , σ ] × ∂ ⁡ B 1 → B 1 is Lipschitz and orientation-reversing wherever its differential is injective, moreover it is injective on the set where its differential is non-degenerate. Therefore, since [ 0 , σ ] × 〈 [ [ F ] ] , | x | = 1 + 〉 is a Caccioppoli set in ℝ × ∂ ⁡ B 1 , so is its negative pushforward (e.g. by employing the image formula for integral currents, see e.g. [16, p. 149] or [25, 26.21(2)]). We finally note that - Φ ♯ ( [ 0 , σ ] × 〈 [ [ F ] ] , | x | = 1 + 〉 ) is disjoint from ( ϕ σ ) ♯ ⁢ [ [ F ∩ B 1 ] ] . Indeed, Φ ⁢ ( [ 0 , σ ] × ∂ ⁡ B 1 ) is contained in { x ∈ B 1 : | x - x | x | | ≤ σ ⁢ η ⁢ ( x | x | ) } , while the image ϕ σ ⁢ ( B 1 ) is contained in { x ∈ B 1 : | x - x | x | | > σ ⁢ η ⁢ ( x | x | ) } . We can therefore conclude that

( ϕ σ ) ♯ S + 〈 [ [ F ] ] , | x | = 1 + 〉 = ∂ [ [ F ~ σ ] ] ,

where F ~ σ is the Caccioppoli set

F ~ σ = ( ϕ σ ) ♯ [ [ F ∩ B 1 ] ] - Φ ♯ ( [ 0 , σ ] × 〈 [ [ F ] ] , | x | = 1 + 〉 ) .

Recalling that F and E 0 agree in B 2 ∖ B 1 , and since F ~ σ ⊂ B 1 , we set

F σ = F ~ σ ∪ ( F ∩ ( B 2 ∖ B 1 ) )

and conclude that (the following is an identity between currents in B 2 )

( ϕ σ ) ♯ ⁢ S + ( ∂ ⁡ [ [ E 0 ] ] ) ⁢ ⌞ ⁢ ( B 2 ∖ B 1 ) = ∂ ⁡ [ [ F σ ] ] ,

with F σ a set of finite perimeter in B 2 that coincides with E 0 in B 2 ∖ B 1 (that is, it is in the admissible class).

The previous conclusion permits to use the minimising property of F, as we are allowed to compare the energy with that of F σ (for any σ ∈ [ 0 , s 0 ] , s 0 depends on ϵ). For ϵ > 0 fixed, we can write (from the minimising property)

(2.1) 0 ≤ lim σ → 0 + ⁡ J λ ⁢ ( F σ ) - J λ ⁢ ( F ) σ = ∫ 𝒮 div 𝒮 ⁡ ( η ⁢ ( f ϵ ∘ d ) ⁢ N ) ⁢ 𝑑 ℋ n + λ ⁢ ∫ 𝒮 ν F ⋅ ( η ⁢ ( f ϵ ∘ d ) ⁢ N ) ⁢ 𝑑 ℋ n .

This equality is justified as follows. First, as by construction

Per B 2 ⁡ ( F σ ) - Per B 2 ⁡ ( F ) = 𝕄 ⁢ ( ( ϕ σ ) ♯ ⁢ S ) - 𝕄 ⁢ ( S ) ,

we can use the well-known formula for the first variation of n-area, which gives the first term on the right-hand-side of (2.1). Next we observe that, denoting by dx the ( n + 1 ) -form d ⁢ x 1 ∧ … ∧ d ⁢ x n + 1 and by x = ( x 1 , … , x n + 1 ) , and since (by Cartan’s formula, denoting by 𝔏 the Lie derivative) d ⁢ ( ι x ⁢ d ⁢ x ) = 𝔏 x ⁢ d ⁢ x = ( n + 1 ) ⁢ d ⁢ x , we have

| F σ | - | F | = ( [ [ F σ ] ] - [ [ F ] ] ) ⁢ ( d ⁢ x ) = 1 n + 1 ⁢ ∂ ⁡ ( [ [ F σ ] ] - [ [ F ] ] ) ⁡ ( ι x ⁢ d ⁢ x ) = 1 n + 1 ⁢ ( ( ϕ σ ) ♯ ⁢ S - S ) ⁢ ( ι x ⁢ d ⁢ x )

= 1 n + 1 ⁢ ∂ ⁡ ( Φ ♯ ⁢ ( [ 0 , σ ] × S ) ) ⁡ ( ι x ⁢ d ⁢ x ) = ( Φ ♯ ⁢ ( [ 0 , σ ] × S ) ) ⁢ ( d ⁢ x ) .

Then by direct computation (using the image formula [16, p. 149], [25, 26.21(2)], together with the fundamental theorem of calculus)

d d ⁢ σ | σ = 0 + ⁢ ( Φ ♯ ⁢ ( [ 0 , σ ] × S ) ) ⁢ ( d ⁢ x ) = ( Φ ♯ ⁢ ( { 0 } × S ) ) ⁢ ( ι d ⁢ Φ ⁢ ( ∂ ∂ ⁡ s ) ⁢ d ⁢ x )
= S ⁢ ( ι η ⁢ ( f ϵ ∘ d ) ⁢ N ⁢ d ⁢ x ) = - ∫ 𝒮 ν F ⋅ ( η ⁢ ( f ϵ ∘ d ) ⁢ N ) ⁢ 𝑑 ℋ n ,

which completes the proof of (2.1).

The next argument follows [15, Theorem 4.1] verbatim. We check that the right-hand-side of (2.1) is independent of ϵ. Indeed, for ϵ ′ < ϵ we consider

ψ s ⁢ ( z ) = z + s ⁢ η ⁢ ( z ) ⁢ ( ( f ϵ ∘ d ) ⁢ ( z ) - ( f ϵ ′ ∘ d ) ⁢ ( z ) ) ⁢ N ⁢ ( z ) .

This is (for s ∈ ( - δ , δ ) with δ > 0 sufficiently small, depending on ϵ ′ ) a (two-sided) one-parameter family of diffeomorphisms, equal to the identity in a neighbourhood of ∂ ⁡ B 1 . We can then use the vanishing of the first variation under the deformation induced by ψ s , that is,

∫ 𝒮 div 𝒮 ⁡ ( η ⁢ ( z ) ⁢ ( ( f ϵ ∘ d ) ⁢ ( z ) - ( f ϵ ′ ∘ d ) ⁢ ( z ) ) ⁢ N ⁢ ( z ) ) ⁢ 𝑑 ℋ n ⁢ ( z ) + λ ⁢ ∫ 𝒮 ν F ⁢ ( z ) ⋅ ( η ⁢ ( z ) ⁢ ( ( f ϵ ∘ d ) ⁢ ( z ) - ( f ϵ ′ ∘ d ) ⁢ ( z ) ) ⁢ N ⁢ ( z ) ) ⁢ 𝑑 ℋ n ⁢ ( z ) = 0 .

The linearity of divergence, scalar product and integration then implies that the right-hand-side of (2.1) is independent of ϵ.

By the sign condition in (2.1), and viewing the right-hand-side of (2.1) as the action of a distribution on C c 1 , there exists a (positive) Radon measure ℳ in B 2 such that the right-hand-side of (2.1) is given by ∫ η ⁢ 𝑑 ℳ . (A priori this distribution should depend on ϵ, however we have proved that the action is independent of ϵ.)

On the other hand, sending ϵ → 0 on the right-hand-side of (2.1) (denoting by ∇ 𝒮 = proj T ⁢ 𝒮 ⁢ ∇ the gradient on 𝒮 , a.e. well-defined), we obtain

∫ 𝒮 ( f ϵ ∘ d ) ⁢ ∇ 𝒮 ⁡ η ⋅ N ⁢ d ⁢ ℋ n → ∫ 𝒮 ∩ ∂ ⁡ B 1 ∇ 𝒮 ⁡ η ⋅ N ⁢ d ⁢ ℋ n = 0 ,

where the last equality follows from the fact that ∇ 𝒮 ⁡ η ⋅ N = 0 a.e. on 𝒮 ∩ ∂ ⁡ B 1 ; and

∫ 𝒮 ( f ϵ ∘ d ) ⁢ η ⁢ div 𝒮 ⁡ N ⁢ d ⁢ ℋ n → ∫ 𝒮 ∩ ∂ ⁡ B 1 η ⁢ div 𝒮 ⁡ N ⁢ d ⁢ ℋ n ;

and

∫ 𝒮 η ⁢ ∇ 𝒮 ⁡ ( f ϵ ∘ d ) ⋅ N ⁢ 𝑑 ℋ n = ∫ 𝒮 η ⁢ ( f ϵ ′ ∘ d ) ⁢ | ∇ 𝒮 ⁡ d | 2 ⁢ 𝑑 ℋ n ≤ 0 ,

where we used ∇ ⁡ d = N on the support of f ϵ ; and

∫ 𝒮 ν F ⋅ ( η ⁢ ( f ϵ ∘ d ) ⁢ N ) ⁢ 𝑑 ℋ n → ∫ 𝒮 ∩ ∂ ⁡ B 1 η ⁢ ν F ⋅ N ⁢ 𝑑 ℋ n .

These imply (expanding the divergence in (2.1))

∫ η ⁢ 𝑑 ℳ ≤ ∫ 𝒮 ∩ ∂ ⁡ B 1 η ⁢ div 𝒮 ⁡ N ⁢ d ⁢ ℋ n + λ ⁢ ∫ 𝒮 ∩ ∂ ⁡ B 1 η ⁢ ( ν F ⋅ N ) ⁢ 𝑑 ℋ n

is valid for all η ∈ C c 1 ⁢ ( B r ⁢ ( p ) ) , η ≥ 0 , hence

ℳ ^ = ( div 𝒮 ⁡ N + λ ⁢ ( ν F ⋅ N ) ) ⁢ d ⁢ ℋ n ⁢ ⌞ ⁢ ( 𝒮 ∩ ∂ ⁡ B 1 )

is a (positive) Radon measure. The first variation of V (with respect to J λ ) computed on the test vector field η ⁢ N can be decomposed as the sum of the first variation computed on η ⁢ ( f ϵ ∘ d ) ⁢ N and on η ⁢ ( 1 - ( f ϵ ∘ d ) ) ⁢ N . The latter contribution gives 0 since η ⁢ ( 1 - ( f ϵ ∘ d ) ) ⁢ N ∈ C c 1 ⁢ ( B 1 ; ℝ n + 1 ) . Therefore the first variation of V on η ⁢ N gives just (2.1), that is, is given by ∫ η ⁢ 𝑑 ℳ , and we have seen that 0 ≤ ℳ ≤ ℳ ^ .

In the first part of the proof we analysed the action of the first variation of V (with respect to J λ ) on a vector field of the form η ⁢ N , for η ∈ C c 1 ⁢ ( B r ⁢ ( p ) ) , η ≥ 0 . Now, in the second part of the proof, we consider instead the action on a vector field Y ∈ C c 1 ⁢ ( B r ⁢ ( p ) ; ℝ n + 1 ) such that Y ⋅ N = 0 . We note that in this case we are able to consider a two-sided deformation induced by Y, which will lead to a vanishing condition, see (2.2) below, rather than an inequality as in (2.1) (where we only had a one-sided deformation at our disposal).

Let ψ s be the flow of Y, that is, the one-parameter (two-sided) family of diffeomorphisms obtained by solving the ODE for each trajectory, d d ⁢ s ⁢ Ψ ⁢ ( s , x ) = Y ⁢ ( x ) , with initial condition Ψ ⁢ ( 0 , x ) = x , and setting ψ s ⁢ ( x ) = Ψ ⁢ ( s , x ) . Then ψ s ⁢ ( B 1 ) ⊂ B 1 and we consider F ~ s = ψ s ⁢ ( F ∩ B 1 ) . These are Caccioppoli sets with support in B 1 ¯ and such that ∂ * ⁡ F ~ s = ψ s ⁢ ( ∂ * ⁡ F ) is a.e. contained in B 1 . The Caccioppoli set F s = F ~ s ∪ ( F ∩ ( B 2 ∖ B 1 ) ) is in the admissible class. We need to show that its boundary (as a current) is ( ψ s ) ♯ ⁢ S + ( ∂ ⁡ [ [ E 0 ] ] ) ⁢ ⌞ ⁢ ( B 2 ∖ B 1 ) . The immediate expression for this boundary is ( ψ s ) ♯ ⁢ ( ∂ ⁡ [ [ F ∩ B 1 ] ] ) + ∂ ⁡ [ [ E 0 ∩ ( B 2 ∖ B 1 ) ] ] . Recalling that S = ∂ [ [ F ∩ B 1 ] ] - 〈 [ [ E 0 ] ] , | x | = 1 + 〉 we arrive at

( ψ s ) ♯ S + ( ψ s ) ♯ 〈 [ [ E 0 ] ] , | x | = 1 + 〉 + ( ∂ [ [ E 0 ] ] ) ⌞ ( B 2 ∖ B 1 ) - 〈 [ [ E 0 ] ] , | x | = 1 + 〉 .

As Ψ ⁢ ( t , z ) , for ( t , z ) ∈ [ 0 , s ] × B 2 is a homotopy joining the identity ψ 0 to ψ s , we will use the homotopy formula. We note that Ψ ⁢ ( t , z ) = z in a neighbourhood of T 0 = - ∂ ⁡ 〈 [ [ E 0 ] ] , | x | = 1 + 〉 , so that Ψ ♯ ⁢ ( [ 0 , s ] × ∂ ⁡ 〈 [ [ E 0 ] ] , | x | = 1 + 〉 ) = 0 . Moreover, Ψ ⁢ ( [ 0 , s ] × ∂ ⁡ B 1 ) ⊂ ∂ ⁡ B 1 , so that Ψ ♯ ( [ 0 , s ] × 〈 [ [ E 0 ] ] , | x | = 1 + 〉 ) = 0 (as an ( n + 1 ) -current). The homotopy formula then gives ( ψ s ) ♯ 〈 [ [ E 0 ] ] , | x | = 1 + 〉 = 〈 [ [ E 0 ] ] , | x | = 1 + 〉 and therefore (the following is an identity between currents in B 2 )

∂ ⁡ [ [ F s ] ] = ( ψ s ) ♯ ⁢ S + ( ∂ ⁡ [ [ E 0 ] ] ) ⁢ ⌞ ⁢ ( B 2 ∖ B 1 ) .

We can therefore use the minimising condition to write the standard condition for the vanishing of the first variation (with respect to J λ ) as

(2.2) ∫ 𝒮 div 𝒮 ⁡ Y ⁢ d ⁢ ℋ n + λ ⁢ ∫ 𝒮 ν F ⋅ Y ⁢ 𝑑 ℋ n = 0 .

For the third (and final) part of the proof, given an arbitrary vector field X ∈ C c 1 ⁢ ( B r ⁢ ( p ) ; ℝ n + 1 ) we write the orthogonal decomposition X = X T + X N , where X N = ( X ⋅ N ) ⁢ N and both X T and X N are C c 1 ⁢ ( B r ⁢ ( p ) ; ℝ n + 1 ) . Then the first variation of J λ on X is given by the sum of the two actions on X T and X N . For the former, in view of (2.2) the action is 0. For the latter, we have that X N = η + ⁢ N - η - ⁢ N , where η + , η - ≥ 0 and η + = ( X ⋅ N ) + , η - = ( X ⋅ N ) - . By the conclusion in the first part (applied separately to η + ⁢ N and η - ⁢ N , using the linearity of the first variation), we then have that the action is given by ∫ ( η + - η - ) ⁢ 𝑑 ℳ = ∫ ( X ⋅ N ) ⁢ 𝑑 ℳ . ∎

As remarked in the example given in Remark 2.8, for λ > n one may actually have ℳ ≠ 0 . If λ ≤ n , on the other hand, we obtain the following result (this is analogous to [15, Theorem 7.1]).

Lemma 2.3.

Let λ ≤ n . Then M = 0 , that is, V is stationary (with respect to J λ ) in B 2 ∖ spt ⁡ T 0 .

Proof.

We have N = ν F a.e. on 𝒮 ∩ ∂ ⁡ B 1 and div 𝒮 ⁡ N = div ∂ ⁡ B 1 ⁡ N a.e. on 𝒮 ∩ ∂ ⁡ B 1 . By explicit computation, we obtain div ∂ ⁡ B 1 ⁡ N = - n (where n is the mean curvature of ∂ ⁡ B 1 ). Then the inequality

0 ≤ ℳ ≤ ( div 𝒮 ⁡ N + λ ⁢ ( ν F ⋅ N ) ) ⁢ d ⁢ ℋ n ⁢ ⌞ ⁢ ( 𝒮 ∩ ∂ ⁡ B 1 )

obtained in Lemma 2.2 becomes

0 ≤ ℳ ≤ ( λ - n ) ⁢ d ⁢ ℋ n ⁢ ⌞ ⁢ ( 𝒮 ∩ ∂ ⁡ B 1 ) .

Thus with λ ≤ n we must have ℳ = 0 (and if λ < n also ℋ n ⁢ ( 𝒮 ∩ ∂ ⁡ B 1 ) = 0 ). ∎

Having established this stationarity property, in order to obtain Theorem 4 we move on to the regularity of the minimiser, focusing on the case λ < n . We note immediately that, while the regularity in B 1 follows from the theory of minimisers, we may a priori have that spt ⁡ V ∩ ∂ ⁡ B 1 ≠ ∅ , and said theory is not applicable at these points. We will instead employ the regularity theory for stable CMC (or prescribed-mean-curvature) hypersurfaces [4, 5], in view of which we recall some relevant notions.

We say that p ∈ spt ⁡ V is a classical singularity of an integral n-varifold V in ℝ n + 1 when there exists an open ball B r n + 1 ⁢ ( p ) such that spt ⁡ V ∩ B r n + 1 ⁢ ( p ) is equal to the union of three or more hypersurfaces-with-boundary, all having a common boundary, all having C 1 , α -regularity up to the boundary, and with p in the common boundary, and with at least two of the hypersurfaces-with-boundary meeting transversely at p.

Given an integral n-varifold V in ℝ n + 1 , we denote by gen - reg ⁡ V the set of points p for which there exists an open ball B r n + 1 ⁢ ( p ) such that spt ⁡ V ∩ B r n + 1 ⁢ ( p ) is either a single C 2 embedded disc, or the union of two (distinct) C 2 -embedded discs that lie on one side of each other and whose intersection contains p.

Lemma 2.4.

Let λ < n and V, F as above. Then spt ⁡ V ∖ spt ⁡ T 0 ⊂ B 1 . Moreover, there exists Σ ⊂ B 1 with dim H ⁡ Σ ≤ n - 7 such that ( spt ⁡ V ∖ spt ⁡ T 0 ) ∖ Σ is a smoothly embedded CMC hypersurface (with mean curvature vector λ ⁢ ν F ). If n = 7 , more precisely, Σ is made of isolated points (possibly accumulating onto spt ⁡ T 0 ).

Proof.

If p ∈ ∂ ⁡ B 1 ∩ spt ⁡ V ∖ spt ⁡ T 0 is a point in gen - reg ⁡ V , then by definition there exists an embedded disc D ⊂ spt ⁡ V ∖ spt ⁡ T 0 ⊂ B 1 ¯ of class C 2 with p ∈ D . The C 2 regularity of D and the stationarity of V with respect to J λ (Lemma 2.3) imply that D is CMC with mean curvature λ. (We remark that, by Allard’s regularity theorem [1] and standard elliptic PDE regularity, there exists a dense open subset of spt ⁡ V that is smoothly embedded with mean curvature λ. It follows that, in the case in which the local structure of spt ⁡ V around p is the union of two distinct C 2 embedded discs, the C 2 regularity of each disc implies that both discs have mean curvature λ.) The maximum principle gives a contradiction if λ < n (since n is the mean curvature of ∂ ⁡ B 1 with respect to the inward normal to B 1 ). This means that if λ < n then gen - reg ⁡ V ∩ ( ∂ ⁡ B 1 ∖ spt ⁡ T 0 ) = ∅ .

In other words, gen - reg ⁡ V ∖ spt ⁡ T 0 ⊂ B 1 . In the (open) ball B 1 we are able to use the minimising assumption to further conclude that gen - reg ⁡ V ∖ spt ⁡ T 0 is a C 2 embedded hypersurface (that is, only the first occurrence in the definition of gen - reg ⁡ V can happen). This follows e.g. from density estimates (see e.g. [22, Theorem 21.11]). The minimising assumption also implies that gen - reg ⁡ V ∖ spt ⁡ T 0 (as a C 2 embedded hypersurface) is stable with respect to J λ .

We further note that for p ∈ ∂ ⁡ B 1 ∩ spt ⁡ V ∖ spt ⁡ T 0 the varifold V has a unique tangent cone at p, given by the hyperplane that is tangent to ∂ ⁡ B 1 at p, possibly counted with integer multiplicity. The existence of tangent cones, and the fact that any such cone is a stationary varifold, both follow from the monotonicity-type formula for the mass, valid thanks to the stationarity with respect to J λ . Since spt ⁡ V ⊂ B 1 ¯ , any such tangent cone must be contained in a half-space (whose boundary is the tangent to ∂ ⁡ B 1 at p), and thus it has to be supported on that tangent hyperplane itself (see e.g. [25]), from which the claim follows (thanks to the constancy theorem [25]).

Finally, we note the absence of classical singularities in spt ⁡ V ∖ spt ⁡ T 0 . In B 1 , this is a consequence of the minimising property, while at any p ∈ ∂ ⁡ B 1 ∩ spt ⁡ V ∖ spt ⁡ T 0 we have proved that the tangent has to be supported on a hyperplane (which rules out that p could be a classical singularity).

Having checked all hypotheses, we can now apply the sheeting results from [4] or [5], namely [4, Theorems 3.1 and 3.3] or [5, Theorems 6.2 and 6.4]. We conclude that, if p ∈ spt ⁡ V ∩ ∂ ⁡ B 1 ∖ spt ⁡ T 0 , then spt ⁡ V is, in a suitable coordinate system in a neighbourhood of p, given by the union of (finitely many) ordered C 2 graphs (each giving an embedded C 2 disc with constant mean curvature λ), and in particular p ∈ gen - reg ⁡ V , contradicting the earlier conclusion that gen - reg ⁡ V ∖ spt ⁡ T 0 ⊂ B 1 . (Alternatively, one may directly use the maximum principle, the fact that ∂ ⁡ B 1 has mean curvature n, and the condition λ < n , to find a contradiction.)

We thus conclude (in a first instance) that spt ⁡ V ∖ spt ⁡ T 0 ⊂ B 1 . At this stage one may either use the standard regularity theory for minimisers (e.g. [22, Theorem 21.8] in conjunction with standard elliptic regularity) or alternatively [4, Corollary 2.1] or [5, Corollary 1.1], for the remaining conclusions. ∎

Remark 2.10.

We expect that the same regularity conclusions should hold for λ = n , albeit with the possibility that open subsets of ∂ ⁡ B 1 may be contained in spt ⁡ V ∖ spt ⁡ T 0 , as in the example of Remark 2.8.

3 Regular minimal cones, graphs, Jacobi operator

In Section 4 we will prove Proposition 1, an instance of a singular maximum principle for CMC hypersurfaces, which will then be needed in Section 5. In this section we collect some preliminaries on stable minimal cones and their Jacobi fields that will be needed in Section 4.

In what follows let C be a regular cone that is also minimal. We recall that the notion of regular cone means that C = { r ⁢ y : r ≥ 0 , y ∈ Σ } , where Σ (the link of C) is a smooth embedded compact ( n - 1 ) -dimensional submanifold of the unit sphere S n . The minimality condition is the vanishing of the mean curvature of C ∖ { 0 } (as a submanifold of ℝ n + 1 ). (This requirement is equivalent to the minimality of Σ as a submanifold of S n , see [27]). We first recall some facts about graphs over C and their mean curvature operator.

Let C = ∂ ⁡ [ [ E ] ] , for a set[2] of locally finite perimeter E ⊂ ℝ n + 1 . The graph of u ∈ C 2 ⁢ ( C 1 ; ℝ ) over C 1 = ( C ∖ { 0 } ) ∩ B 1 is defined to be

gr C ⁡ u = { x + u ⁢ ( x ) ⁢ N ⁢ ( x ) : x ∈ C 1 } ,

where N is the inward pointing unit normal on C ∖ { 0 } . We will be interested in functions u that satisfy the following radial decay:

(3.1) | u ⁢ ( x ) | | x | + | ∇ ⁡ u ⁢ ( x ) | + | x | ⁢ | ∇ 2 ⁡ u ⁢ ( x ) | → | x | → 0 0 ,

where ∇ denotes the Levi-Civita connection on C ∖ { 0 } with respect to the Riemannian metric induced on C ∖ { 0 } by the Euclidean one in ℝ n + 1 , and | ⋅ | is taken with respect to the Euclidean inner product.

We remark that there exists M = M Σ such that, if

(3.2) | u ⁢ ( x ) | | x | + | ∇ ⁡ u ⁢ ( x ) | ≤ M

is valid for all x ∈ C 1 , then gr C ⁡ u is an embedded hypersurface, with { 0 } = ( gr C ⁡ u ¯ ∖ gr C ⁡ u ) ∩ B 1 an isolated singularity when C is not a hyperplane. We will assume in this section that (3.2) is satisfied on C 1 . We further note that (3.1) implies the validity of (3.2) for all 0 < | x | < r for sufficiently small r, and therefore, after rescaling, u ~ ⁢ ( x ) = u ⁢ ( x r ) satisfies

| u ~ ⁢ ( x ) | | x | + | ∇ ⁡ u ~ ⁢ ( x ) | ≤ M

on C 1 . (This fact will be implicitly used in Section 4.)

Assume now that the associated current to gr C ⁡ u is of the form ∂ ⁡ [ [ F ] ] ⁡ ⌞ ⁢ B 1 ,[3] where F is a set of finite perimeter and that F is a critical point of J λ thus in particular we have that

d d ⁢ t | t = 0 ⁢ J λ ⁢ ( F t ) = 0 ,

where F t is the set of finite perimeter whose boundary is gr C ⁡ ( u + t ⁢ v ) and v ∈ C c 2 ⁢ ( C 1 ; ℝ ) . We recall that the mean curvature operator ℳ C of the cone is defined as follows, by defining in duality its action on u ∈ C 2 ⁢ ( C 1 ; ℝ ) :

d d ⁢ t | t = 0 ⁢ ℋ n ⁢ ( gr C ⁡ ( u + t ⁢ v ) ) = - 〈 ℳ C ⁢ u , v 〉 L 2 ,

where 〈 ⋅ , ⋅ 〉 L 2 denotes the L 2 -inner product on C ∖ { 0 } and v ∈ C c 2 ⁢ ( C 1 ; ℝ ) . The PDE that the function u satisfies is given in terms of ℳ C as we prove in the following:

Lemma 3.1.

Let u and gr C ⁡ u be as above. Then

(3.3) ℳ C ⁢ u = λ ⁢ det ⁡ ( Id - u ⁢ A C ) ,

where A C denotes the second fundamental form of C 1 .

Proof.

Let G ⁢ ( x ) = x + u ⁢ ( x ) ⁢ N ⁢ ( x ) and consider an extension N ^ of N (defined in an open cone over a tubular neighbourhood of Σ in 𝕊 n ). Then for any v ∈ C c 2 ⁢ ( C 1 ; ℝ ) we have that

0 = d d ⁢ t | t = 0 ⁢ J λ ⁢ ( F t ) = - 〈 ℳ C ⁢ u , v 〉 L 2 + λ ⁢ ∫ gr C ⁢ u v ⁢ N ^ ⋅ ν ^ ⁢ 𝑑 ℋ n ,

where F t is the associated set to gr C ⁡ ( u + t ⁢ v ) , ν ^ the inward pointing unit normal of gr C ⁡ u and the last term is the derivative of the volume term. Using the area formula the latter can be written as ∫ C v ⁢ N ^ ⋅ ν ^ ⁢ | J G | ⁢ 𝑑 ℋ n , where | J G | denotes the Jacobian of G. Thus it suffices to compute N ^ ⋅ ν ^ ⁢ | J G | . Let ( τ i ) be an orthonormal basis of C 1 then

D τ i ⁢ G ⋅ τ j = δ i ⁢ j - u ⁢ A i ⁢ j ,
D τ i ⁢ G ⋅ N ^ = D τ i ⁢ u ,

where D τ i ⁢ G denotes the differential of G in the direction of τ i and ( A i ⁢ j ) is the matrix that corresponds to the second fundamental form of C 1 with respect to the chosen basis. Consider the matrix

B = ( D τ 1 ⁢ G ⋅ N ^ D τ 1 ⁢ G ⋅ τ 1 … D τ 1 ⁢ G ⋅ τ n ⋮ ⋮ ⋱ ⋮ D τ n ⁢ G ⋅ N ^ D τ n ⁢ G ⋅ τ 1 … D τ n ⁢ G ⋅ τ n ) .

Let B ( k ) denote the n × n minor of the matrix B for 2 ≤ k ≤ n + 1 obtained by erasing the k-th column of the matrix B. Then

ν ^ = ( det ⁡ ( Id - u ⁢ A C ) ⁢ N ^ + ∑ k = 2 n + 1 ( - 1 ) k - 1 ⁢ B ( k ) ⁢ τ k - 1 ) ⁢ | J G | - 1 .

In particular, N ^ ⋅ ν ^ ⁢ | J G | = det ⁡ ( Id - u ⁢ A C ) and this finishes the proof. ∎

In view of (3.3), we recall some properties of the operator ℳ C , referring to [7, (2.1)] and [7, Section 3], whose notation we adopt here. We also refer to [12, Lemma 2.26] for a proof, and to [19], and note that due to (3.2) the form established for ℳ C in [12] is the same as in [7] or [19]. The operator ℳ C has the form

ℳ C ⁢ u = L C ⁢ u + N ⁢ ( x , u | x | , ∇ ⁡ u ) ⋅ ∇ 2 ⁡ u ⁢ ( x ) + 1 | x | ⁢ P ⁢ ( x , u | x | , ∇ ⁡ u ⁢ ( x ) ) ,

where L C ⁢ u = Δ C ⁢ u + | A C | 2 ⁢ u is the Jacobi field operator of the cone C ∖ { 0 } , N is a symmetric bilinear form, ( ⋅ ) is to be understood as the trace of the linear transformations on T x ⁢ C 1 associated to the bilinear forms N , ∇ 2 ⁡ u (equivalently, using an orthonormal basis of T x ⁢ C 1 to write the associated matrices N i ⁢ j and ∇ i ⁢ j ⁡ u , this is N i ⁢ j ⁢ ∇ i ⁢ j ⁡ u with summation over repeated indices) and both N , P have a C 2 -dependency on the arguments ( x , z , p ) ∈ C 1 × ℝ × T ⁢ C 1 . Moreover, ℳ C is a quasilinear elliptic operator of order two, and for | z | , | p | ≤ 1 we have the following inequalities at ( x , z , p ) :

(3.4)

| N ⁢ ( x , z , p ) | ≤ M Σ ⁢ ( | z | + | p | ) ,
| P ⁢ ( x , z , p ) | ≤ M Σ ⁢ ( | z | + | p | ) 2 ,
| P z | + | P p | + | x | ⁢ ( | P x ⁢ z | + | P x ⁢ p | ) ≤ M Σ ⁢ ( | z | + | p | ) ,
| x | ⁢ ( | N x | + | P x | + | N x ⁢ z | + | N x ⁢ p | ) + | N z | + | N p | + | N z ⁢ z | + | N z ⁢ p | + | N p ⁢ p | + | P z ⁢ z | + | P z ⁢ p | + | P p ⁢ p | ≤ M Σ ,

where the subscripts denote partial differentiation and M Σ is a constant that depends on the dimension n and the link Σ of the cone.

The estimates in (3.4) along with the radial decay assumption (3.1) allow us to prove that the linearisation of the PDE (3.3) has the following form:

Lemma 3.2.

Let u , v ∈ C 2 ⁢ ( C 1 ; R ) satisfy (3.1) and M C ⁢ u = λ ⁢ det ⁡ ( Id - u ⁢ A C ) , M C ⁢ v = λ ⁢ det ⁡ ( Id - v ⁢ A C ) . Then h = v - u satisfies the following linear PDE:

(3.5) L C ⁢ h = A 1 ⋅ ∇ 2 ⁡ h + 1 | x | ⁢ A 2 ⋅ ∇ ⁡ h + 1 | x | 2 ⁢ A 3 ⁢ h ,

where A 1 : C 1 → End ⁡ ( T ⁢ C 1 ) , A 2 : C 1 → T ⁢ C 1 , A 3 : C 1 → R and A 1 , A 2 , A 3 → 0 as | x | → 0 . Moreover, if u , v ∈ C 3 ⁢ ( C 1 ; R ) , then the coefficients of the PDE are in C 0 , α ⁢ ( U ; R ) for some α ∈ ( 0 , 1 ) and any U ⊂ ⊂ C 1 .

Proof.

We first compute the operator ℒ such that ℒ ⁢ h = ℳ C ⁢ v - ℳ C ⁢ u . We introduce the notation

N ⁢ ( u ) = N ⁢ ( x , u | x | , ∇ ⁡ u )   and   P ⁢ ( u ) = P ⁢ ( x , u | x | , ∇ ⁡ u )

(for N , P introduced above). Then, since L C is linear,

ℳ C ⁢ v - ℳ C ⁢ u = L C ⁢ h + N ⁢ ( v ) ⋅ ∇ 2 ⁡ v - N ⁢ ( u ) ⋅ ∇ 2 ⁡ u + 1 | x | ⁢ ( P ⁢ ( v ) - P ⁢ ( u ) ) .

We recall the standard method to rewrite N ⁢ ( v ) ⋅ ∇ 2 ⁡ v - N ⁢ ( u ) ⋅ ∇ 2 ⁡ u . We denote by N i ⁢ j the components of the matrix associated to the operator N (in an orthonormal basis of T x ⁢ C 1 ) and compute (with implicit summation on repeated indices)

N i ⁢ j ( x , v | x | , ∇ v ) ∇ i ⁢ j v - N i ⁢ j ( x , u | x | , ∇ u ) ∇ i ⁢ j u = = ∫ 0 1 d d ⁢ t ( N i ⁢ j ( t , u , v ) ( ∇ i ⁢ j u + t ( ∇ i ⁢ j v - ∇ i ⁢ j u ) ) d t ,

with the notation N i ⁢ j ⁢ ( t , u , v ) = N i ⁢ j ⁢ ( x , u | x | + t ⁢ ( v - u ) | x | , ∇ ⁡ u + t ⁢ ( ∇ ⁡ v - ∇ ⁡ u ) ) . Differentiating with respect to t, we get the following expression:

( ∫ 0 1 N i ⁢ j ⁢ ( t , u , v ) ⁢ 𝑑 t ) ⁢ ∇ i ⁢ j ⁡ h + ( ∫ 0 1 | x | ⁢ N i ⁢ j , z ⁢ ( t , u , v ) ⁢ ( ∇ i ⁢ j ⁡ u + t ⁢ ∇ i ⁢ j ⁡ h ) ⁢ 𝑑 t ) ⁢ h | x | 2 + ( ∫ 0 1 | x | ⁢ N i ⁢ j , p ⁢ ( t , u , v ) ⁢ ( ∇ i ⁢ j ⁡ u + t ⁢ ∇ i ⁢ j ⁡ h ) ⁢ 𝑑 t ) ⋅ ∇ ⁡ h | x | ,

where N i ⁢ j , z , N i ⁢ j , p denote partial differentiation of N i ⁢ j (with respect to z and p respectively). A similar computation gives that

1 | x | ⁢ ( P ⁢ ( v ) - P ⁢ ( u ) ) = ( ∫ 0 1 P z ⁢ ( t , u , v ) ⁢ 𝑑 t ) ⁢ h | x | 2 + ( ∫ 0 1 P p ⁢ ( t , u , v ) ⁢ 𝑑 t ) ⋅ ∇ ⁡ h | x | ,

where again we use the notation P ⁢ ( t , u , v ) = P ⁢ ( x , u | x | + t ⁢ ( v - u ) | x | , ∇ ⁡ u + t ⁢ ( ∇ ⁡ v - ∇ ⁡ u ) ) and P z , P p denote partial differentiation as above. Putting everything together, we get that

ℳ C ⁢ v - ℳ C ⁢ u = L C ⁢ h + A ¯ 1 ⋅ ∇ 2 ⁡ h + 1 | x | ⁢ A ¯ 2 ⋅ ∇ ⁡ h + 1 | x | 2 ⁢ A ¯ 3 ⁢ h ,

where

A ¯ 1 = ∫ 0 1 N i ⁢ j ⁢ ( t , u , v ) ⁢ 𝑑 t ,
A ¯ 2 = ∫ 0 1 | x | ⁢ N i ⁢ j , p ⁢ ( t , u , v ) ⁢ ( ∇ i ⁢ j ⁡ u + t ⁢ ∇ i ⁢ j ⁡ h ) ⁢ 𝑑 t + ∫ 0 1 P p ⁢ ( t , u , v ) ⁢ 𝑑 t ,
A ¯ 3 = ∫ 0 1 | x | ⁢ N i ⁢ j , z ⁢ ( t , u , v ) ⁢ ( ∇ i ⁢ j ⁡ u + t ⁢ ∇ i ⁢ j ⁡ h ) ⁢ 𝑑 t + ∫ 0 1 P z ⁢ ( t , u , v ) ⁢ 𝑑 t .

Using the estimates in (3.4), we have that

| A ¯ 1 | ≤ M Σ ⁢ ( | u | | x | + | h | | x | + | ∇ ⁡ u | + | ∇ ⁡ h | ) ,
| A ¯ 2 | ≤ M Σ ⁢ ( | u | | x | + | h | | x | + | x | ⁢ | ∇ 2 ⁡ u | + | x | ⁢ | ∇ 2 ⁡ h | ) ,
| A ¯ 3 | ≤ M Σ ⁢ ( | u | | x | + | h | | x | + | x | ⁢ | ∇ 2 ⁡ u | + | x | ⁢ | ∇ 2 ⁡ h | ) ,

where M Σ denotes a constant that depends on the link Σ of the cone C. Thus from (3.1) we have that A ¯ 1 , A ¯ 2 , A ¯ 3 → 0 as | x | → 0 .

In a similar way, we now compute (using the Jacobi formula for the derivative of the determinant)

det ⁡ ( Id - v ⁢ A C ) - det ⁡ ( Id - u ⁢ A C ) = ∫ 0 1 d d ⁢ t ⁢ det ⁡ ( Id - ( u + t ⁢ h ) ⁢ A C ) ⁢ 𝑑 t
= - ∫ 0 1 det ⁡ ( Id - ( u + t ⁢ h ) ⁢ A C ) ⁢ tr ⁡ ( ( Id - ( u + t ⁢ h ) ⁢ A C ) - 1 ⁢ h ⁢ A C ) ⁢ 𝑑 t
= - ∫ 0 1 h ⁢ det ⁡ ( Id - ( u + t ⁢ h ) ⁢ A C ) ⁢ tr ⁡ ( ( Id - ( u + t ⁢ h ) ⁢ A C ) - 1 ⁢ A C ) ⁢ 𝑑 t = 1 | x | 2 ⁢ A ¯ 4 ⁢ h ,

where

A ¯ 4 = - ∫ 0 1 | x | 2 ⁢ det ⁡ ( Id - ( u + t ⁢ h ) ⁢ A C ) ⁢ tr ⁡ ( ( Id - ( u + t ⁢ h ) ⁢ A C ) - 1 ⁢ A C ) ⁢ 𝑑 t .

From (3.1) we have that Id - ( u + t ⁢ h ) ⁢ A C → Id as | x | → 0 thus A ¯ 4 converges to 0 as | x | → 0 as well.

The statement follows by setting A 1 = - A ¯ 1 , A 2 = - A ¯ 2 and A 3 = A ¯ 4 - A ¯ 3 . ∎

From (3.5) we see that L C becomes the leading term of the PDE as | x | → 0 . This crucial fact will allow us, in Proposition 1 below, to construct a non-trivial positive Jacobi field of C. In view of that, we recall some well-known properties of the Jacobi operator L C .

For x ∈ C ∖ { 0 } let r = | x | and ω = x | x | ∈ Σ denote spherical coordinates on C. Then the metric of the cone is given by g = d ⁢ r 2 + r 2 ⁢ g Σ where g Σ is the pull-back on Σ of the round metric on S n (via the inclusion map). The operator L C is expressed in spherical coordinates as

(3.6) L C ⁢ f = r - 2 ⁢ L Σ ⁢ f + r 1 - n ⁢ ∂ r ⁡ ( r n - 1 ⁢ ∂ r ⁡ f ) ,

where L Σ = Δ Σ + | A Σ | 2 and A Σ is the second fundamental form of Σ in S n . Since L Σ is a linear elliptic operator on a smooth compact manifold, we consider the spectrum λ 1 < λ 2 ≤ ⋯ → + ∞ of - L Σ .

The first eigenvalue λ 1 is simple and it is known from [7] that C is stable if and only if

max ⁡ { - λ 1 , 0 } ≤ ( n - 2 ) 2 4 .

In particular, if C is stable (which will be the case in forthcoming sections), we define γ ± = n - 2 2 ± ( n - 2 ) 2 4 + λ 1 and we have γ + ≥ γ - ≥ 0 .

Remark 3.11.

Unless C is a hyperplane, one always has γ - > 0 . Indeed, if γ - = 0 , then λ 1 = 0 and from the variational characterisation of the first eigenvalue of L Σ , if we take as a test function a constant function, we get that | A Σ | ≡ 0 thus | A C | ≡ 0 and C is a plane.

Any positive solution of L C ⁢ f = 0 is of the form (see e.g. [19, p. 105], and Lemma A.2 below)

(3.7) f ⁢ ( r ⁢ ω ) = ( c 1 + r γ + + c 1 - r γ - ) ⁢ ϕ 1 ⁢ ( ω ) ,

where ϕ 1 > 0 is the first eigenfunction of L Σ , that is L Σ ⁢ ϕ 1 = - λ 1 ⁢ ϕ 1 and c 1 + , c 1 - are non-negative constants.

4 A singular maximum principle

We first state and prove the following fact regarding the convergence of minimisers of J λ . Analogous results hold (with similar arguments that require building competitors) for area-minimising currents (see e.g. [25, Chapter 7, Theorem 2.4]) and for perimeter minimisers or almost-minimisers (see e.g. [22, Theorem 21.14]).

Lemma 4.1.

For j ∈ N , let E j be sets with finite perimeter in B 2 , and let λ j , λ ∈ [ 0 , ∞ ) , with lim j ⁡ λ j = λ . For each j we assume that E j minimises J λ j among sets that coincide with E j in B 2 ∖ B 1 . Let E be a set with finite perimeter in B 2 and assume that [ [ E j ] ] → [ [ E ] ] (as currents) in B 2 . Then E minimises J λ among sets that coincide with E in B 2 ∖ B 1 . Moreover, | ∂ * ⁡ E j | → | ∂ * ⁡ E | in B 1 (as varifolds).

Remark 4.12.

Let D be a set with finite perimeter in B 2 . The outer and inner slices 〈 [ [ D ] ] , | x | = 1 + 〉 and 〈 [ [ D ] ] , | x | = 1 - 〉 are n-dimensional integral currents supported in ∂ ⁡ B 1 (which is n-dimensional), therefore there exist integer-valued BV -functions θ D + and θ D - such that 〈 [ [ D ] ] , | x | = 1 + 〉 = θ D + ( ℋ n ⌞ ∂ B 1 ) ξ → and 〈 [ [ D ] ] , | x | = 1 - 〉 = θ D - ( ℋ n ⌞ ∂ B 1 ) ξ → , where ξ → is the orientation of ∂ ⁡ B 1 corresponding (in Hodge duality) to the choice of outward pointing unit normal. In fact, θ D + , θ D - are { 0 , 1 } -valued ( ℋ n -a.e. on ∂ ⁡ B 1 ), since [ [ D ] ] is the current of integration on a Caccioppoli set.

Proof.

We remark that 〈 [ [ E j ] ] , | x | = 1 + 〉 → 〈 [ [ E ] ] , | x | = 1 + 〉 as currents (since by definition we have 〈 [ [ E j ] ] , | x | = 1 + 〉 = - ∂ [ [ E j ∩ ( B 2 ∖ B 1 ¯ ) ] ] + ( ∂ [ [ E j ] ] ) ⌞ ( B 2 ∖ B 1 ¯ ) , and [ [ E j ] ] → [ [ E ] ] in B 2 by assumption).

Let F be a set with finite perimeter that coincides with E in B 2 ∖ B 1 . Set

F j = ( F ∩ B 1 ) ∪ ( E j ∩ ( B 2 ∖ B 1 ) ) .

Then F j → F as sets of finite perimeter (when F j , F are sets with finite perimeter, the convergence F j → F as sets with finite perimeter is equivalent to [ [ F j ] ] → [ [ F ] ] as currents). Moreover, 〈 [ [ F j ] ] , | x | = 1 + 〉 = 〈 [ [ E j ] ] , | x | = 1 + 〉 by the definition of F j , and 〈 [ [ F ] ] , | x | = 1 + 〉 = 〈 [ [ E ] ] , | x | = 1 + 〉 by definition of F.

With notation as in Remark 4.12, we remark that θ E + = θ F + , θ F j - = θ F - and θ F j + = θ E j + . Using Lemma A.1 with E j , F j in place of D, we rewrite the minimising condition J λ j ⁢ ( E j ) ≤ J λ j ⁢ ( F j ) in the form,

Per B 1 E j + 𝕄 ( 〈 [ [ E j ] ] , | x | = 1 + 〉 - 〈 [ [ E j ] ] , | x | = 1 - 〉 ) - λ j ℋ n + 1 ( E j )
≤ Per B 1 F + 𝕄 ( 〈 [ [ F j ] ] , | x | = 1 + 〉 - 〈 [ [ F j ] ] , | x | = 1 - 〉 ) - λ j ℋ n + 1 ( F j ) .

(We have used Per B 1 ⁡ F = Per B 1 ⁡ F j and Per B 2 ∖ B 1 ¯ ⁡ E j = Per B 2 ∖ B 1 ¯ ⁡ F j .) The second term on the right-hand-side is written as

∫ ∂ ⁡ B 1 | θ F j + - θ F j - | = ∫ ∂ ⁡ B 1 | θ E j + - θ F - | .

Since ∂ ⁡ B 1 is compact, | θ E j + - θ F - | ≤ 1 , and θ E j + → θ E + = θ F + pointwise ℋ n -a.e. in ∂ ⁡ B 1 (by the hypothesis 〈 [ [ E j ] ] , | x | = 1 + 〉 → 〈 [ [ E ] ] , | x | = 1 + 〉 ), we conclude that (by dominated convergence) ∫ ∂ ⁡ B 1 | θ E j + - θ F - | → ∫ ∂ ⁡ B 1 | θ F + - θ F - | . The latter is 𝕄 ( 〈 [ [ F ] ] , | x | = 1 + 〉 - 〈 [ [ F ] ] , | x | = 1 - 〉 ) . Sending j → ∞ and using the lower-semi-continuity of mass and perimeter on the left-hand-side, as well as ℋ n + 1 ⁢ ( E j ) → ℋ n + 1 ⁢ ( E ) , ℋ n + 1 ⁢ ( F j ) → ℋ n + 1 ⁢ ( F ) (implied by E j → E , F j → F ), we find

Per B 1 E + 𝕄 ( 〈 [ [ E ] ] , | x | = 1 + 〉 - 〈 [ [ E ] ] , | x | = 1 - 〉 ) - λ ℋ n + 1 ( E )
≤ Per B 1 F + 𝕄 ( 〈 [ [ F ] ] , | x | = 1 + 〉 - 〈 [ [ F ] ] , | x | = 1 - 〉 ) - λ ℋ n + 1 ( F ) .

Adding Per B 2 ∖ B 1 ¯ ⁡ E = Per B 2 ∖ B 1 ¯ ⁡ F to both sides, and using Lemma A.1 again (with E , F in place of D), the inequality obtained becomes J λ ⁢ ( E ) ≤ J λ ⁢ ( F ) . Therefore E minimises J λ (among sets that coincide with E in B 2 ∖ B 1 ¯ ).

Repeating the above argument with E in place of F shows that we must have Per B 1 ⁡ E = lim j → ∞ ⁡ Per B 1 ⁡ E j , therefore ∥ ∂ * ⁡ E j ∥ → ∥ ∂ * ⁡ E ∥ as Radon measures in B 1 (and, by Allard’s compactness for integral varifolds, | ∂ * ⁡ E j | → | ∂ * ⁡ E | in B 1 ). ∎

Remark 4.13.

Assume that E minimises J λ ¯ in an open set U. Then, given a point x ∈ ∂ * ⁡ E ¯ and a sequence of dilations η x , r j ⁢ ( y ) = y - x r j , r j ↘ 0 , consider the sequence of Caccioppoli sets E j = η x , r j ⁢ ( E ) (blow up sequence). At the same time, we may consider the sequence of varifolds | ∂ * ⁡ E j | = η x , r j ♯ ⁢ | ∂ * ⁡ E | . Standard theory (respectively of Caccioppoli sets and of varifolds, see e.g. [22, 25]) guarantees that both sequences subsequentially converge. Any limit in the sense of varifolds is a so-called varifold tangent cone of | ∂ * ⁡ E | at x. Lemma 4.1 implies that any varifold tangent cone is of the form | ∂ * ⁡ E ∞ | , where E ∞ is a Caccioppoli set obtained as a (subsequential) limit of E j . This follows by passing to a subsequence (still denoted by r j ) for which we have convergence to a varifold tangent cone, and by using Lemma 4.1 (with λ j , λ therein replaced by r j ⁢ λ ¯ and 0 respectively), noting that the dilated set E j is a minimiser of J r j ⁢ λ ¯ , and letting E ∞ be the Caccioppoli set to which E j converges. In particular, any varifold tangent cone has multiplicity 1 on its regular part.

Remark 4.14.

If λ < n , then for a minimiser such as E (similarly for E j if λ j < n ) in Lemma 4.1, one has that ℋ n ⁢ ( ∂ * ⁡ E ∩ ∂ ⁡ B 1 ) = 0 (see Lemma 2.3). Therefore ∂ ⁡ [ [ E ] ] ⁡ ⌞ ⁢ ∂ ⁡ B 1 = 0 and 〈 [ [ E ] ] , | x | = 1 + 〉 = 〈 [ [ E ] ] , | x | = 1 - 〉 by (A.1) (therefore the standard slice 〈 [ [ E ] ] , | x | = 1 〉 exists).

We are now ready to prove the main result of this section, an instance of maximum principle for CMC hypersurfaces with isolated singularities.

Proposition 1.

Let E and F be sets with finite perimeter in B 2 that minimise J λ with respect to their own boundary condition, assumed in B 2 ∖ B 1 . Assume that ∂ * ⁡ E ¯ ∩ ( B 1 ∖ { 0 } ) is smoothly embedded, 0 ∈ ∂ * ⁡ E ¯ , and that a tangent cone to | ∂ * ⁡ E | at 0 is regular (which means, it is smooth away from 0 and has multiplicity 1 on its regular part). Assume further that F ⊂ E and that 0 ∈ ∂ * ⁡ F ¯ . Then E ∩ B 1 = F ∩ B 1 .

Remark 4.15.

Under the assumed condition on a tangent cone, by L. Simon’s renowned result [24], | ∂ * ⁡ E | possesses a unique tangent cone at 0 (which has to be the one about which the regularity and multiplicity hypotheses are made).

Proof.

We divide the proof into four steps.

Step 1. We begin by proving that ∂ * ⁡ F ¯ is smooth in B r ∖ { 0 } for some r > 0 . Let Σ ⊂ ∂ * ⁡ F ¯ denote the singular set of ∂ * ⁡ F ¯ . Arguing by contradiction, assume that x i → 0 , x i ∈ Σ . Letting ρ i = | x i | , we consider the sequence of dilations x ↦ x ρ i and take a blow up of F at 0 by setting F 0 , ρ i = F ρ i and taking a subsequential limit F 0 of F 0 , ρ i . By the assumption that F ⊂ E we have that F 0 ⊂ E 0 , where E 0 is the blow up of E at 0 obtained by taking the limit for said subsequence of dilations (as remarked above, the blow up for E at 0 is independent of the sequence of dilations). The stationarity property of F with respect to J λ translates into stationarity of F 0 , ρ i with respect to J ρ i ⁢ λ , which implies that F 0 is stationary for the perimeter (equivalently, J 0 ). Similarly, E 0 is perimeter-stationary, that is, both | ∂ * ⁡ E 0 | and | ∂ * ⁡ F 0 | are stationary varifolds in ℝ n + 1 . (We remark that both | ∂ * ⁡ E 0 | and | ∂ * ⁡ F 0 | are non-zero, since the origin is in the support of both | ∂ * ⁡ E | and | ∂ * ⁡ F | and thus both densities are ≥ 1 by the monotonicity formula.)

More precisely, by Lemma 4.1, E 0 and F 0 are perimeter minimisers in any compact set K ⊂ ℝ n + 1 , for their own boundary condition (assumed in the complement of K). Clearly, 0 ∈ spt ⁡ | ∂ * ⁡ E 0 | ∩ spt ⁡ | ∂ * ⁡ F 0 | . Then the singular maximum principle [20, Theorem A (iii)] implies that spt ⁡ | ∂ * ⁡ E 0 | = spt ⁡ | ∂ * ⁡ F 0 | , and thus | ∂ * ⁡ E 0 | = | ∂ * ⁡ F 0 | . (Alternatively, one may use the maximum principle in the form given in [26].)

Lemma 4.1 (see Remark 4.13) also gives that | ∂ * ⁡ F 0 , ρ i | converge (as varifolds) to | ∂ * ⁡ F 0 | . By the choice of dilations, and by Allard’s interior regularity theorem, see [1], the points x i ρ i lie in ∂ ⁡ B 1 and have density Θ ⁢ ( ∥ ∂ * ⁡ F 0 , ρ i ∥ , x i ρ i ) ≥ 1 + ϵ 0 , where ϵ 0 > 0 is the dimensional constant in Allard’s regularity theorem. This contradicts the hypothesis that the density of | ∂ * ⁡ E 0 | = | ∂ * ⁡ F 0 | is 1 at any point distinct from 0 (since | ∂ * ⁡ E 0 | is a regular cone by assumption). We have therefore established the smoothness of ∂ * ⁡ F ¯ in B r ∖ { 0 } for some r > 0 .

Step 2. As remarked above, | ∂ * ⁡ E 0 | is the unique tangent cone to | ∂ * ⁡ E | at 0. This also implies that | ∂ * ⁡ F 0 | = | ∂ * ⁡ E 0 | is the unique tangent cone for | ∂ * ⁡ F | at 0 (since, given any blow up sequence, the resulting blow up of F at 0 is contained in E 0 , and the maximum principle implies, as above, that the two blow up sets must coincide). In particular (see [24, Section 7]), we are able to write ∂ ⁡ E ∩ ( B δ ∖ { 0 } ) and ∂ ⁡ F ∩ ( B δ ∖ { 0 } ) , for sufficiently small δ > 0 , as graphs of C 2 functions over the common cone C δ = C ∩ ( B δ ∖ { 0 } ) , where C = ∂ * ⁡ E 0 ¯ , as follows:

(4.1)

∂ ⁡ E ∩ ( B δ ∖ { 0 } ) = gr C δ ⁡ u with  ⁢ u ∈ C 2 ⁢ ( C δ ; ℝ ) ,
∂ ⁡ F ∩ ( B δ ∖ { 0 } ) = gr C δ ⁡ v with  ⁢ v ∈ C 2 ⁢ ( C δ ; ℝ ) ,
lim | x | → 0 ⁡ ( | u ⁢ ( x ) | | x | + | ∇ ⁡ u ⁢ ( x ) | ) = 0 ,
lim | x | → 0 ⁡ ( | v ⁢ ( x ) | | x | + | ∇ ⁡ v ⁢ ( x ) | ) = 0 .

Taking the identification of ℝ with ( T ⁢ C δ ) ⊥ so that the orientation is inward (for E 0 ), we have, in view of E ⊂ F and the fact that | ∂ * ⁡ E | and | ∂ * ⁡ F | are stationary for J λ ,

u ≤ v   and   ℳ C ⁢ u = λ ⁢ det ⁡ ( Id - u ⁢ A C ) , ℳ C ⁢ v = λ ⁢ det ⁡ ( Id - v ⁢ A C ) .

Note that due to (4.1) the PDE for u and v satisfies the estimates (3.4) in C δ and from standard elliptic estimates, see also [24, Section 1], we deduce that | x | ⁢ | ∇ 2 ⁡ u ⁢ ( x ) | + | x | ⁢ | ∇ 2 ⁡ v ⁢ ( x ) | → 0 as | x | → 0 hence the radial decay (3.1) is satisfied. In particular, we may consider h = v - u ≥ 0 and from (3.5) we have that h satisfies the linear PDE

L C ⁢ h = A 1 ⋅ ∇ 2 ⁡ h + 1 | x | ⁢ A 2 ⋅ ∇ ⁡ h + 1 | x | 2 ⁢ A 3 ⁢ h ,

where A 1 , A 2 , A 3 → | x | → 0 0 . Thus for any K ⊂ ⊂ C δ we can apply the Harnack inequality to get that

sup K ⁡ h ≤ C K ⁢ inf K ⁡ h .

Hence either h > 0 on K ¯ or h ≡ 0 . Since K is arbitrary, we must have either h ≡ 0 on C δ , or h > 0 on C δ (and h = 0 at 0). We will next rule out the second occurrence.

Step 3. The minimising property of E 0 implies that C is a stable minimal cone and thus all positive Jacobi fields are of the form (3.7). To prove that u ≡ v , we will construct a non-existent positive Jacobi field on C ∖ { 0 } under the contradiction assumption that h > 0 on C δ . We argue as in [19, Lemma 1.20].

From the property that h → 0 as | x | → 0 we can construct a sequence of ρ j ′ ↘ 0 such that

sup C ρ j + 1 ′ ⁡ h < sup C ρ j ′ ⁡ h .

Let x j be the points where sup C ρ j ′ ⁡ h is achieved and set r j = | x j | . Then we have r j ↘ 0 (since r j ∈ ( ρ j + 1 ′ , ρ j ′ ) ) and sup C r j ⁡ h = sup ∂ ⁡ C r j ⁡ h . We define

h j ⁢ ( x ) = h ⁢ ( r j ⁢ x )

for x ∈ C δ r j and we have that

sup C 1 ⁡ h j = sup ∂ ⁡ C 1 ⁡ h j .

Let x j ′ ∈ ∂ ⁡ C 1 where sup C 1 ⁡ h j is achieved and set

f j ⁢ ( x ) = h j ⁢ ( x ) M j

for x ∈ C δ r j , where M j = h j ⁢ ( x j ′ ) . From the PDE for h we have that f j satisfies the following PDE:

L C ⁢ f j = A ~ j ( 1 ) ⋅ ∇ 2 ⁡ f j + 1 | x | ⁢ A ~ j ( 2 ) ⋅ ∇ ⁡ f j + 1 | x | 2 ⁢ A ~ j ( 3 ) ⁢ f j ,

where A ~ j ( i ) ⁢ ( x ) = A i ⁢ ( r j ⁢ x ) for x ∈ C δ r j and i = 1 , 2 , 3 .

Fix a set K ⊂ ⊂ C ∖ { 0 } and let K ′ be another set with K ⊂ ⊂ K ′ ⊂ ⊂ C ∖ { 0 } and x j ′ ∈ K ′ . Notice that, from the standard regularity theory for CMC hypersurfaces, we have that u , v ∈ C ∞ thus the coefficients A i , for i = 1 , 2 , 3 of the PDE are in C 0 , α ⁢ ( K ′ ) and since A ~ j ( i ) are rescalings of A i we have that [ A ~ j ( i ) ] α ; K ′ ≤ M 1 ⁢ r j α , where M 1 is a constant independent of j and [ ⋅ ] α ; K ′ is the Hölder semi-norm in K ′ with exponent α. In particular, if we combine with (3.1), we conclude that ∥ A ~ j ( i ) ∥ 0 , α ; K ′ → 0 , as j → ∞ for i = 1 , 2 , 3 , where ∥ f ∥ l , α ; K ′ = ∥ f ∥ l ; K ′ + max | β | = l [ D β f ] α ; K ′ denotes the Hölder norm in C l , α . Thus from the C 2 , α -Schauder estimates, see [17, Theorem 6.1], we get that

∥ f j ∥ 2 , α ; K ≤ M 3 ⁢ ∥ f j ∥ 0 ; K ′ ,

where M 3 is a constant independent of j.

From the Harnack inequality on K ′ and since x j ∈ K ′ and f j ⁢ ( x j ) = 1 we have that ∥ f j ∥ 0 ; K ′ ≤ C K ′ ⁢ inf K ′ ⁡ f j ≤ C K ′ where C K ′ is a constant that depends on K ′ . Putting everything together we get that

∥ f j ∥ 2 , α ; K ≤ M 4 ,

where M 4 is a constant independent of j (and depending on K ′ ). From Arzelà-Ascoli theorem, after a diagonal argument and passing to a subsequence that we still index with j, we have that

f j → C loc 2 ⁢ ( C ∖ { 0 } ) f ∈ C 2 , α ⁢ ( C ∖ { 0 } ) .

From the uniform convergence of A ~ j ( i ) on compact sets to zero, for i = 1 , 2 , 3 , we get that L C ⁢ f = 0 in C ∖ { 0 } . Furthermore, (again passing to a subsequence) we have that x j ′ → x 0 ∈ ∂ ⁡ C 1 and so f ⁢ ( x 0 ) = 1 . Thus from Harnack’s inequality f > 0 .

In conclusion, we have constructed a positive solution of L C ⁢ f = 0 , defined on C ∖ { 0 } for a stable minimal cone C of ℝ n + 1 , and satisfying

sup C 1 ⁡ f = sup ∂ ⁡ C 1 ⁡ f .

The latter contradicts (3.7) and thus proves that ∂ ⁡ E ∩ B δ = ∂ ⁡ F ∩ B δ .

Step 4. Finally, we show that E ∩ B 1 = F ∩ B 1 . Let

r 0 = sup ⁡ { r : ∂ ⁡ E ∩ B r = ∂ ⁡ F ∩ B r }

and note that the set over which we take the supremum is non-empty due to the existence of δ, from the previous step, and it is in fact a maximum. Assume for the contrary that r 0 < 1 and let x 0 ∈ ∂ ⁡ B r 0 ∩ ∂ ⁡ F ∩ ∂ ⁡ E . Then by virtue of Remark 4.13, we can consider a varifold tangent cone for | ∂ * ⁡ F | at x 0 , of the form | ∂ * ⁡ G | , with | ∂ * ⁡ G | stationary (for the perimeter functional), and with spt ⁡ | ∂ * ⁡ G | contained in a half space thanks to the condition F ⊂ E (more precisely, the half space whose boundary is the tangent plane to | ∂ * ⁡ E | at x 0 ). Then from [25, Theorem 36.5] we have that | ∂ * ⁡ G | is a plane hence the regularity theory implies that we can find a neighbourhood B ρ ′ ⁢ ( x 0 ) where ∂ ⁡ F is smooth and ∂ ⁡ F , ∂ ⁡ E meet tangentially at x 0 . Since F ⊂ E and due to the variational equations satisfied by J λ the mean curvature vectors point in the same direction at x 0 thus the standard maximum principle implies that ∂ ⁡ E ∩ B ρ ′ ⁢ ( x 0 ) coincides with ∂ ⁡ F ∩ B ρ ′ ⁢ ( x 0 ) . In particular, since x 0 is arbitrary and ∂ ⁡ B r ∩ ∂ ⁡ F is compact we can find ϵ > 0 such that ∂ ⁡ E ∩ B r 0 + ϵ = ∂ ⁡ F ∩ B r 0 + ϵ contradicting the choice of r 0 . Thus r 0 = 1 and we conclude that E ∩ B 1 = F ∩ B 1 . ∎

5 Approximation

Lemma 5.1 and Theorem 5 below will establish in particular the approximation results stated in the introduction, Theorems 1 and 2. (One should identify B ^ - p in Theorem 2 with the ball B R below.)

We assume that E ⊂ ℝ n + 1 satisfies the following properties. The topological boundary agrees with ∂ * ⁡ E ¯ and T = ∂ ⁡ E contains 0, the hypersurface ( T ∖ { 0 } ) ∩ B R is smooth for some R > 0 (so the origin is an isolated singularity for T), E minimises J λ in B R among Caccioppoli sets that coincide with E in B 2 ⁢ R ∖ B R , a tangent cone to | ∂ * ⁡ E | at 0 is regular (which means, it is smooth away from 0 and has multiplicity 1 on its regular part). In view of Remark 4.15, | ∂ * ⁡ E | thus possesses a unique tangent cone at 0.

Remark 5.16.

We note that if n = 7 , these properties can be fulfilled whenever we have a Caccioppoli set that minimises J λ locally. To begin with, one chooses a system of coordinates centred at a singular point, and R smaller than the distance of this to any other singular point (which is possible thanks to the interior regularity theory for minimisers). Moreover, (again by the regularity theory) any tangent cone must be smooth away from the origin (for otherwise, the radial invariance would give a singular set of dimension at least one). Finally, any tangent cone must have multiplicity 1 on its regular part since the rescaled varifolds | ∂ ⁡ E ρ i , 0 | converge as varifolds to | ∂ * ⁡ E 0 | (see Remark 4.13).

It may not be true, in the above situation, that E is the unique minimiser of J λ , among Caccioppoli sets that coincide with E in B 2 ⁢ R ∖ B R . However, by taking a slightly smaller R (which preserves all the assumptions above), we can ensure said uniqueness, thanks to a standard argument that we now recall.

Lemma 5.1.

Let E , T be as above. If R ′ < R , then E is the unique minimiser of J λ among sets that coincide with E in B 2 ⁢ R ∖ B R ′ (and therefore also among sets that coincide with E in B 2 ⁢ R ′ ∖ B R ′ ).

Proof.

Let R ′ < R . Clearly, E minimises J λ in B R ′ among Caccioppoli sets that coincide with E in B 2 ⁢ R ∖ B R ′ . Assume that there exists a Caccioppoli set E ′ ≠ E that minimises J λ in B R ′ among Caccioppoli sets that coincide with E in B 2 ⁢ R ∖ B R ′ . In particular, E ′ coincides with E in B 2 ⁢ R ∖ B R , and on E ′ the energy J λ attains the same value as it does on E. Therefore E ′ is a minimiser of J λ in B R , among Caccioppoli sets that coincide with E in B 2 ⁢ R ∖ B R . As such, its reduced boundary must enjoy the optimal regularity of minimisers, that is, ∂ * ⁡ E ′ ¯ ∩ B R is a smooth hypersurface (with mean curvature λ) away from a set Σ ⊂ ∂ * ⁡ E ′ ¯ ∩ B R with dim ℋ ⁡ Σ ≤ n - 7 . We aim to prove that ∂ * ⁡ E ′ ¯ coincides with ∂ * ⁡ E ¯ (which is in contradiction with E ′ ≠ E and E ′ = E in B 2 ⁢ R ∖ B R ′ ).

We define r ≤ R ′ by

r = inf ⁡ { t : ∂ * ⁡ E ′ ¯ = ∂ * ⁡ E ¯ ⁢  in  ⁢ B 2 ⁢ R ∖ B t }

and note that this is a minimum. The conclusion will follow upon establishing that r = 0 . Assume r > 0 . We remark that for p ∈ ∂ ⁡ B r ∩ ∂ * ⁡ E ¯ we must have that there exists a unique tangent cone to | ∂ * ⁡ E ′ | at p, and it must coincide with the hyperplane that is tangent to ∂ ⁡ E at p. (This follows from ∂ * ⁡ E ′ ¯ = ∂ * ⁡ E ¯ in B 2 ⁢ R ∖ B r and the smoothness of ∂ * ⁡ E ¯ around p.) The regularity theory implies that ∂ * ⁡ E ′ ¯ is smooth in an open ball B ρ n + 1 ⁢ ( p ) for some ρ > 0 . Recall however that

∂ * ⁡ E ′ ¯ = ( ∂ * ⁡ E ′ ¯ ∩ B r ) ∪ ( ∂ * ⁡ E ¯ ∩ ( B 2 ⁢ R ∖ B r ) ) ,

and we have established that this is smooth in B ρ ⁢ ( p ) . Unique continuation implies that ∂ * ⁡ E ′ ¯ ∩ B r coincides with ∂ * ⁡ E ¯ ∩ B r in B ρ ⁢ ( p ) .

As p ∈ ∂ ⁡ B r ∩ ∂ * ⁡ E ¯ is arbitrary and ∂ ⁡ B r ∩ ∂ * ⁡ E ¯ is compact, it follows that ∂ * ⁡ E ′ ¯ coincides with ∂ * ⁡ E ¯ in B 2 ⁢ R ∖ B r - δ for some δ > 0 , contradicting the choice of r. Hence r = 0 and E ′ = E in B 2 ⁢ R . ∎

Remark 5.17.

By taking R ′ sufficiently small we also ensure that λ < n R ′ . Therefore, upon dilating B 2 ⁢ R ′ to B 2 , we have that the working assumptions stated in the next theorem are fulfilled.

Theorem 5.

Let E be a set of finite perimeter in B 2 . Assume that T = ∂ ⁡ E = ∂ * ⁡ E ¯ contains 0, the hypersurface T ∩ ( B 2 ∖ { 0 } ) is smooth, E is the unique minimiser for J λ in B 2 among Caccioppoli sets that coincide with E in B 2 ∖ B 1 , λ < n . Given r ∈ ( 0 , 1 ) , there exists a sequence of sets E j that have finite perimeter in B 2 , such that ∂ ⁡ E j ∩ B r is smooth for each j, it has constant mean curvature λ ⁢ ν E j , where ν E j is the inward unit normal to E j , E j ⊂ E , E j → E and ∂ ⁡ E j converge to ∂ ⁡ E smoothly on any Ω ⊂ ⊂ B r ∖ { 0 } .

Remark 5.18.

We point out that the sequence E j will be constructed without any dependence on r; however, we will only prove that the boundaries ∂ ⁡ E j ∩ B r are smooth for sufficiently large j, with dependence on r.

Proof.

We divide the proof into three steps.

Step 1. The first step is to perturb the boundary condition E inwards, and then use this new boundary condition to define E j . The vector field ν E is smooth in ( B 2 ∖ { 0 } ) ∩ ∂ ⁡ E . Let d ⁢ ( ⋅ ) = dist ⁡ ( ⋅ , ∂ ⁡ E ) be the signed distance function to ∂ ⁡ E (taken to be positive in E and negative in its complement) and consider a tubular neighbourhood 𝒩 ρ of size ρ > 0 around ∂ ⁡ E ∩ ( B 3 2 ∖ B 1 2 ) . Then the gradient of d is a smooth extension of ν E to 𝒩 ρ . Let χ be a smooth function on B 2 that is equal to 1 in ( B 5 4 ∖ B 3 4 ) ∩ 𝒩 ρ 2 and with support contained in ( B 3 2 ∖ B 1 2 ) ∩ { | d | < 3 4 ρ } . Let X = χ ⁢ ∇ ⁡ d . Then X extends ν E and we may consider the flow ϕ t ⁢ ( x ) of X. (We view X as a vector field in B 2 .) For any t ∈ [ 0 , δ ) , with δ > 0 sufficiently small, ϕ t ⁢ ( E ) ⊂ E . By construction ϕ t ⁢ ( ∂ ⁡ E ∩ ∂ ⁡ B 1 ) is disjoint from ∂ ⁡ E ∩ ∂ ⁡ B 1 for all t ∈ ( 0 , δ ) , and ϕ t ⁢ ( E ) ∩ ( B 5 4 ∖ B 3 4 ) is strictly contained in E ∩ ( B 5 4 ∖ B 3 4 ) .

The sequence E j in the statement is built with the boundary condition E j = ϕ t j ⁢ ( E ) in B 2 ∖ B 1 , for a sequence t j → 0 . Namely, from Theorem 4 we may define E j to be a minimiser of J λ for said boundary condition; furthermore, we have that ∂ * ⁡ E j ¯ ∩ B 1 ¯ intersects ∂ ⁡ B 1 only at its boundary, and is a smoothly embedded CMC hypersurface-with-boundary away from a codimension 7 set.

Step 2. We show first that E j → E as j → ∞ as sets of finite perimeter (therefore [ [ E j ] ] → [ [ E ] ] as currents, hence ∂ ⁡ [ [ E j ] ] → ∂ ⁡ [ [ E ] ] as well). This follows from the uniqueness property of E, as we now show. To begin with, we have J λ ⁢ ( E j ) ≤ J λ ⁢ ( ϕ t j ⁢ ( E ) ) (by the minimising property of E j ). By smoothness of X, using the area formula we find that J λ ⁢ ( ϕ t j ⁢ ( E ) ) → J λ ⁢ ( E ) as j → ∞ . In particular, there exists a uniform upper bound for J λ ⁢ ( ϕ t j ⁢ ( E ) ) , and thus (since | E j | ≤ | B 2 | ) a uniform upper bound for Per B 2 ⁡ ( E j ) . Standard BV-compactness then gives the existence of a subsequential limit E j → D with | E j | → | D | and (by lower semi-continuity of perimeter) J λ ⁢ ( D ) ≤ lim inf j → ∞ ⁡ J λ ⁢ ( E j ) . Recalling the previous considerations, J λ ⁢ ( D ) ≤ lim inf j → ∞ ⁡ J λ ⁢ ( E j ) ≤ J λ ⁢ ( E ) . Finally, noting that E j ∩ ( B 2 ∖ B 1 ¯ ) = ϕ t j ⁢ ( E ) ∩ ( B 2 ∖ B 1 ¯ ) → E ∩ ( B 2 ∖ B 1 ¯ ) , we obtain that D = E in B 2 ∖ B 1 ¯ and therefore D is a minimiser (among sets with finite perimeter that coincide with E in B 2 ∖ B 1 ¯ ). The uniqueness hypothesis on E gives E = D .

Next we will prove that E j ⊂ E , for each given j. Considering the sets with finite perimeter E j ∩ E and E j ∪ E , we have [ [ E j ∩ E ] ] + [ [ E j ∪ E ] ] = [ [ E j ] ] + [ [ E ] ] , so that ∂ ⁡ [ [ E j ∩ E ] ] + ∂ ⁡ [ [ E j ∪ E ] ] = ∂ ⁡ [ [ E j ] ] + ∂ ⁡ [ [ E ] ] . Clearly we also have E j ∩ E ⊂ E j ∪ E . This implies that at ℋ n -a.e. x ∈ ∂ * ⁡ ( E j ∩ E ) ∩ ∂ * ⁡ ( E j ∪ E ) one must obtain the same half-space as the unique blow up at x for both sets E j ∩ E and E j ∪ E , and therefore the measure-theoretic outer normals are the same at x for both sets. The common orientation ℋ n -a.e. gives the equality

𝕄 ⁢ ( ∂ ⁡ [ [ E j ∩ E ] ] ) + 𝕄 ⁢ ( ∂ ⁡ [ [ E j ∪ E ] ] ) = 𝕄 ⁢ ( ∂ ⁡ [ [ E j ∩ E ] ] + ∂ ⁡ [ [ E j ∪ E ] ] ) ,

and therefore

𝕄 ⁢ ( ∂ ⁡ [ [ E j ∩ E ] ] ) + 𝕄 ⁢ ( ∂ ⁡ [ [ E j ∪ E ] ] ) = 𝕄 ⁢ ( ∂ ⁡ [ [ E j ] ] + ∂ ⁡ [ [ E ] ] ) ≤ 𝕄 ⁢ ( ∂ ⁡ [ [ E j ] ] ) + 𝕄 ⁢ ( ∂ ⁡ [ [ E ] ] ) .

Noting that | E j ∩ E | + | E j ∪ E | = | E j | + | E | , we conclude that

J λ ⁢ ( E j ∩ E ) + J λ ⁢ ( E j ∪ E ) ≤ J λ ⁢ ( E j ) + J λ ⁢ ( E ) .

On the other hand, since ϕ t j ⁢ ( E ) ⊂ E and E j agrees with ϕ t j ⁢ ( E ) in B 2 ∖ B 1 we conclude that ( E j ∩ E ) ∩ ( B 2 ∖ B 1 ) = E j ∩ ( B 2 ∖ B 1 ) and ( E j ∪ E ) ∩ ( B 2 ∖ B 1 ) = E ∩ ( B 2 ∖ B 1 ) thus the minimising properties of E j and E imply respectively that

J λ ⁢ ( E j ∩ E ) ≥ J λ ⁢ ( E j ) , J λ ⁢ ( E j ∪ E ) ≥ J λ ⁢ ( E ) .

Combining the inequalities obtained, we find that equalities must hold throughout, and therefore E j ∪ E is a minimiser of J λ (among sets with finite perimeter that coincide with E in B 2 ∖ B 1 ¯ ), so that the uniqueness of E gives E j ∪ E = E , that is, E j ⊂ E .[4]

Step 3. We conclude the proof of Theorem 5 by showing that, given any r < 1 , the sequence ∂ ⁡ E j ∩ B r is smooth for large j (depending on r). To that end we will use the Hardt–Simon foliation provided by [19, Theorem 2.1]. First note that as a consequence of Allard’s interior regularity theorem, and of the smoothness of ∂ ⁡ E away from the origin, we must then have that, for any r < 1 and σ ∈ ( 0 , r ) , there is C 1 , α convergence of ∂ ⁡ E j to ∂ ⁡ E in B r ∖ B σ . By elliptic regularity, the convergence is in fact smooth, and ∂ ⁡ E j ∩ ( B r ∖ B σ ) is smooth for all sufficiently large j, depending on the choice of σ , r .

Let Σ j denote the singular set of ∂ * ⁡ E j ¯ in B 1 (which is of dimension at most n - 7 ). Let r 0 < 1 be fixed and let p j ∈ Σ j ∩ B r 0 . In view of the previous conclusion, we must have p j → 0 as j → ∞ . Also we remark that, by Proposition 1, we must have 0 ∉ ∂ * ⁡ E j ¯ for all j, so p j ≠ 0 for all sufficiently large j. We will dilate E j around 0 by the homothety η j ⁢ ( x ) = x | p j | . Then E ~ j = η j ⁢ ( E j ) is a Caccioppoli set in B 1 / | p j | , in particular in B 2 for all sufficiently large j; moreover, the point p ~ j = p j | p j | is singular for ∂ * ⁡ E ~ j ¯ and lies on ∂ ⁡ B 1 . Upon extracting a subsequence that we do not relabel, we can assume that E ~ j → Ω and | ∂ * ⁡ E ~ j | converge to the (stationary) integral varifold | ∂ * ⁡ Ω | in B 2 . The minimising property of E j with respect to J λ implies that Ω minimises perimeter in any compact set. Moreover, as E j ⊂ E , we have Ω ⊂ E 0 , where E 0 is the blow up of E at 0 obtained from η j . Then [19, Theorem 2.1] (specifically, its final assertion) implies that either Ω = E 0 , or Ω belongs to the “Hardt–Simon family” of sets G s = η 0 , s ⁢ ( G ) , where η 0 , s ⁢ ( ⋅ ) = ⋅ s , s > 0 , and G ⊊ E 0 has smooth minimising boundary. On the other hand, the presence of a sequence of singular points p j ∈ ∂ ⁡ B 1 implies, by Allard’s interior regularity theorem, that a subsequential limit p ∈ ∂ ⁡ B 1 of p j must occur with density ≥ 1 + ϵ 0 in | ∂ * ⁡ Ω | , contradicting the smoothness and unit density of ∂ ⁡ E 0 and of ∂ ⁡ G s (regardless of s) in a tubular neighbourhood of ∂ ⁡ B 1 . The contradiction shows that Σ j ∩ B r 0 = ∅ for all sufficiently large j, so that ∂ ⁡ E j ∩ B r 0 is a smooth hypersurface (for all sufficiently large j). ∎

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1928930

Funding statement: The final stages of this work were completed while Costante Bellettini was in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley, California, during the Fall 2024 semester, supported by the National Science Foundation under Grant No. DMS-1928930.

A Auxiliary results

We give a proof of the following general property.

Lemma A.1.

Let D be a set with finite perimeter in B 2 . Then

(A.1) ∂ [ [ D ] ] ⌞ ∂ B 1 = 〈 [ [ D ] ] , | x | = 1 - 〉 - 〈 [ [ D ] ] , | x | = 1 + 〉

and

(A.2) Per B 2 D = Per B 1 D + Per B 2 ∖ B 1 ¯ D + 𝕄 ( 〈 [ [ D ] ] , | x | = 1 + 〉 - 〈 [ [ D ] ] , | x | = 1 - 〉 ) .

Proof.

To check this, we begin by recalling that for an open set U ⊂ B 2 , one has Per U ⁡ D = 𝕄 ⁢ ( ∂ ⁡ [ [ D ] ] ⁡ ⌞ ⁢ U ) , and 𝕄 ⁢ ( ∂ ⁡ [ [ D ] ] ) = 𝕄 ⁢ ( ∂ ⁡ [ [ D ] ] ⁡ ⌞ ⁢ B 1 ) + 𝕄 ⁢ ( ∂ ⁡ [ [ D ] ] ⁡ ⌞ ⁢ ( B 2 ∖ B 1 ¯ ) ) + 𝕄 ⁢ ( ∂ ⁡ [ [ D ] ] ⁡ ⌞ ⁢ ∂ ⁡ B 1 ) . Therefore (A.2) follows from (A.1).

We recall that the restriction of ∂ ⁡ [ [ D ] ] to ∂ ⁡ B 1 is well-defined (since the current is normal) via the limit, for any n-form ω with compact support in B 2 ,

( ∂ ⁡ [ [ D ] ] ⁡ ⌞ ⁢ ∂ ⁡ B 1 ) ⁢ ( ω ) = lim h → 0 ⁡ ( ∂ ⁡ [ [ D ] ] ) ⁢ ( γ h ⁢ ( | x | - 1 ) ⁢ ω ) ,

where γ h : ( - ∞ , ∞ ) → ℝ is C 1 , is identically 1 on ( - h , h ) , vanishes on ( - ∞ , - 2 ⁢ h ) ∪ ( 2 ⁢ h , ∞ ) , and γ ′ ∈ [ - 2 h , 0 ] on ( 0 , ∞ ) and γ ′ ∈ [ 0 , 2 h ] on ( - ∞ , 0 ) . Then

(A.3)

( ∂ ⁡ [ [ D ] ] ⁡ ⌞ ⁢ ∂ ⁡ B 1 ) ⁢ ( ω ) = lim h → 0 ⁡ ( [ [ D ] ] ) ⁢ ( γ h ′ ⁢ ( | x | - 1 ) ⁢ d ⁢ | x | ∧ ω ) + lim h → 0 ⁡ ( [ [ D ] ] ) ⁢ ( γ h ⁢ ( | x | - 1 ) ⁢ d ⁢ ω ⏟ →  0 ⁢  as  ⁢ h ⁣ →  0 )
= lim h → 0 ⁡ ( [ [ D ] ] ) ⁢ ( γ h ′ ⁢ ( | x | - 1 ) ⁢ d ⁢ | x | ∧ ω ) .

On the other hand, let γ h + : ( - ∞ , ∞ ) → ℝ be C 1 , identically 0 on ( - ∞ , 0 ) , and equal to 1 - γ h on [ 0 , ∞ ) . Let γ h - : ( - ∞ , ∞ ) → ℝ be defined by γ h + ⁢ ( s ) = γ h - ⁢ ( - s ) . Note that γ h + + γ h - + γ h = 1 . Then

〈 [ [ D ] ] , | x | = 1 + 〉 ( ω ) = - ∂ ( [ [ D ] ] ⌞ { | x | > 1 } ) ( ω ) + ( ∂ [ [ D ] ] ⌞ { | x | > 1 } ) ( ω )
= - lim h → 0 [ [ D ] ] ( γ h + ( | x | - 1 ) d ω ) + lim h → 0 [ [ D ] ] ( d ( γ h + ( | x | - 1 ) ω ) ) = lim h → 0 [ [ D ] ] ( ( γ h + ) ′ ( | x | - 1 ) d | x | ∧ ω ) )

and similarly

〈 [ [ D ] ] , | x | = 1 - 〉 ( ω ) = - lim h → 0 [ [ D ] ] ( ( γ h - ) ′ ( | x | - 1 ) d | x | ∧ ω ) ) .

Therefore

( 〈 [ [ D ] ] , | x | = 1 + 〉 - 〈 [ [ D ] ] , | x | = 1 - 〉 ) ( ω ) = - lim h → 0 [ [ D ] ] ( ( γ h ) ′ ( | x | - 1 ) d | x | ∧ ω ) ) ,

which, jointly with (A.3), gives (A.1). ∎

We provide the details regarding the positive solutions to the linear elliptic PDE L C ⁢ f = 0 , which is crucial in the proof of Proposition 1.

Lemma A.2.

Let C be a regular stable minimal n-cone in R n + 1 . Then every positive solution of L C ⁢ f = 0 is of the form

f ⁢ ( r ⁢ ω ) = ( c 1 + r γ + + c 1 - r γ - ) ⁢ ϕ 1 ⁢ ( ω ) ,

where ϕ 1 > 0 is the first eigenfunction of L Σ , and c 1 + , c 1 - are non-negative constants.

Proof.

We assume first that the cone is strictly stable thus γ + > γ - . Consider the eigenvalues of the operator - L Σ ,

λ 1 < λ 2 ≤ λ 3 ⁢ … → ∞

and let ( ϕ j ) be an orthonormal basis of L 2 ⁢ ( Σ ) such that ϕ j is an eigenfunction of λ j . Recall that ϕ 1 > 0 and λ 1 is a simple eigenvalue.

For any r > 0 the function f ⁢ ( r , ⋅ ) (on Σ) is of the form ∑ j = 1 ∞ a j ⁢ ( r ) ⁢ ϕ j ⁢ ( ω ) . Thus in order to solve L C ⁢ f = 0 we write L C in spherical coordinates and from (3.6) we get, after solving the corresponding ODE for a j , that

a j ⁢ ( r ) = c j + ⁢ r - γ j + + c j - ⁢ r - γ j - ,

where γ j ± = n - 2 2 ± ( n - 2 ) 2 4 + λ j and c j ± are constants. Thus

f ⁢ ( r ⁢ ω ) = ∑ j = 1 ∞ c j ± ⁢ r - γ j ± ⁢ ϕ j ⁢ ( ω ) .

Let us prove that c j ± = 0 for all j ≥ 2 . As L C ⁢ f = 0 and f > 0 from Harnack’s inequality on K 1 = C ∩ ( B 2 ¯ ∖ B 1 2 ) , [17, Corollary 8.21], we have that sup K 1 ⁡ f ≤ C K 1 ⁢ inf K 1 ⁡ f , where C K 1 is a constant that depends on K 1 and the operator L C . Let now K s = C ∩ ( B 2 ⁢ s ¯ ∖ B s / 2 ) , for some s > 0 to be fixed later. Notice that if we rescale f s ⁢ ( x ) = f ⁢ ( s ⁢ x ) , then the scale invariance of the operator L C implies that

sup K s ⁡ f ≤ C K 1 ⁢ inf K s ⁡ f .

We want to evaluate the L 2 -norm of f on K s with respect to the cone metric g C = d ⁢ r 2 + r 2 ⁢ g Σ . First note that

∥ f ∥ L 2 ⁢ ( K s ) ≤ ( ℋ n ⁢ ( C ∩ K 1 ) ⁢ s n ) 1 2 ⁢ sup K s ⁡ f = C ( K 1 , n , Σ ) ⁢ s n 2 ⁢ sup K s ⁡ f ,

where C ( K 1 , n , Σ ) denotes a constant that depends on K 1 , n , Σ that may vary from line to line. On the other hand, since ϕ j is an orthonormal basis of L 2 ⁢ ( Σ ) , we have

∥ f ∥ L 2 ⁢ ( K s ) = ( ∫ s / 2 2 ⁢ s ∑ j = 1 ∞ ( c j ± ) 2 ⁢ r - 2 ⁢ γ j ± ⁢ r n - 1 ⁢ d ⁢ r ) 1 2
= ( ∑ j = 1 ∞ ( c j ± ) 2 ⁢ s n - 2 ⁢ γ j ± ⁢ ( 2 n - 2 ⁢ γ j ± - 2 2 ⁢ γ j ± - n n - 2 ⁢ γ j ± ) ) 1 2 ,

and since 2 x - 2 - x x ≥ 1 for any x ∈ ℝ ∖ { 0 } , we conclude that

∥ f ∥ L 2 ⁢ ( K s ) ≥ s n 2 ⁢ ( ∑ j = 1 ∞ ( c j ± ) 2 ⁢ s - 2 ⁢ γ j ± ) 1 2 .

The three inequalities thus give

C ( K 1 , n , Σ ) ⁢ ( ∑ j = 1 ∞ ( c j ± ) 2 ⁢ s - 2 ⁢ γ j ± ) 1 2 ≤ inf K s ⁡ f ≤ f ⁢ ( r , ω )

for all r ∈ [ s 2 , s ] and ω ∈ Σ . Multiplying the latter with ϕ 1 , and integrating over Σ, we get

C ( K 1 , n , Σ ) ⁢ ( ∑ j = 1 ∞ ( c j ± ) 2 ⁢ s - 2 ⁢ γ j ± ) 1 2 ≤ c 1 + ⁢ r - γ 1 + + c 1 - ⁢ r - γ 1 -

for all r ∈ [ s 2 , 2 ⁢ s ] . Thus we may take r = s and get that

(A.4)

C ( K 1 , n , Σ ) ⁢ ( ∑ j = 1 ∞ ( c j ± ) 2 ⁢ s - 2 ⁢ γ j ± ) 1 2 ≤ c 1 + ⁢ s - γ 1 + + c 1 - ⁢ s - γ 1 - .

Multiplying now (A.4) by s γ 1 + , we have that

C ( K 1 , n , Σ ) ⁢ ( ∑ j = 1 ∞ ( c j ± ) 2 ⁢ s 2 ⁢ γ 1 + - 2 ⁢ γ j ± ) 1 2 ≤ c 1 + + c 1 - ⁢ s γ 1 + - γ 1 - .

In order to prove that c j + = 0 for all j ≥ 2 first note that ∑ j = 2 ∞ ( c j + ) 2 < ∞ (by Parseval’s identity it is bounded by ∥ f ∥ L 2 ⁢ ( Σ ) ), and recall that γ j - ≤ γ 2 - < γ 1 - < γ 1 + < γ 2 + ≤ γ j + for all j ≥ 2 . Thus for any E > 0 there exists s 0 > 0 such that s ≤ s 0 implies s 2 ⁢ γ 1 + - 2 ⁢ γ j + > E 2 for every j, and moreover s γ 1 + - γ 1 - < 1 | c 1 - | thus we obtain

E 2 ⁢ ∑ j = 2 ∞ ( c j + ) 2 ≤ C ( K 1 , n , Σ ) ⁢ ( c 1 + + 1 ) 2 ,

which gives a contradiction for sufficiently large E unless c j + = 0 for all j ≥ 2 . If we instead multiply (A.4) by s γ 1 - and choose s sufficiently large, a similar argument leads to a contradiction unless c j - = 0 for all j ≥ 2 .

It remains to show that c 1 + , c 1 - ≥ 0 . Assume for the contrary that c 1 + < 0 . Then

r γ + ⁢ f = c 1 + ⁢ ϕ 1 + c 1 - ⁢ r γ + - γ - ⁢ ϕ 1 ,

and letting r → 0 we get a contradiction. A similar argument gives c 1 - ≥ 0 .

In case the cone is not strictly stable, thus γ + = γ - = n - 2 2 , then the expression of the function f is given by

f ⁢ ( r ⁢ ω ) = c 1 + ⁢ r - γ ⁢ ϕ 1 ⁢ ( ω ) + c 1 - ⁢ log ⁡ r ⁢ ϕ 1 ⁢ ( ω ) + ∑ j = 2 ∞ c j ± ⁢ r - γ j ± ⁢ ϕ j ⁢ ( ω )

and repeating the same computations as above we will get that c 1 - = c j ± = 0 for all j ≥ 2 , thus f ⁢ ( r ⁢ ω ) = c 1 + ⁢ r - γ ⁢ ϕ 1 ⁢ ( ω ) , where c 1 + is a non-negative constant. This concludes the proof of Lemma A.2. ∎

Acknowledgements

The material in this work overlaps partly with the PhD Thesis of Konstantinos Leskas, who would thus like to thank University College London for the stimulating environment provided during the years spent there as a graduate student. Konstantinos Leskas would also like to thank Kobe Marshall-Stevens for many enlightening discussions. The authors wish to thank the anonymous referees for constructive comments and suggestions.

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Received: 2024-12-16
Revised: 2025-08-30
Accepted: 2025-08-31
Published Online: 2025-10-01
Published in Print: 2026-01-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Frontmatter
  2. On the second eigenvalue of the infinity Laplacian with Robin boundary conditions
  3. Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities
  4. The infinity-potential in the square
  5. A simple proof of the 1-dimensional flat chain conjecture
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https://doi.org/10.1515/acv-2024-0132
Keywords for this article
Constant-mean-curvature; isolated singularity; Hardt–Simon foliation; smooth approximation
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Articles in the same Issue

  1. Frontmatter
  2. On the second eigenvalue of the infinity Laplacian with Robin boundary conditions
  3. Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities
  4. The infinity-potential in the square
  5. A simple proof of the 1-dimensional flat chain conjecture
  6. Anisotropic energies for the modeling of cavitation in nonlinear elasticity
  7. On the Plateau problem for surfaces with minimum moment of inertia
  8. Existence theory for linear-growth variational integrals with signed measure data
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