Estimating the Morse index of free boundary minimal hypersurfaces through covering arguments
-
Santiago Cordero-Misteli
and
Giada Franz
Abstract
Given a compact Riemannian manifold, of dimension between 3 and 7, with boundary, we adapt Song’s covering method to the free boundary case to show that the Morse index of a free boundary minimal hypersurface grows linearly with the sum of its Betti numbers, where the constant of growth depends on an upper bound on the area of the free boundary minimal hypersurface in question.
1 Introduction
The goal of this paper is to prove a quantitative bound on the sum of the Betti numbers of a free boundary minimal hypersurface in a compact Riemannian manifold
The question whether, and how, the Morse index of a minimal hypersurface bounds its topology goes back already to the classical theory of complete minimal surfaces in
In the case of a boundaryless compact ambient manifold
for every closed minimal surface
In the case with boundary, the assumption corresponding to “positive Ricci curvature” is that the ambient manifold
In the case of a boundaryless compact three-dimensional ambient manifold
Observe that, in absence of any curvature assumption, we cannot hope for a bound on the topology of a minimal surface solely in terms of its index (see [8, Example 1.16]). However, given a boundaryless compact ambient manifold
Theorem 1.1.
Suppose that
For every smooth, compact, properly embedded, free boundary minimal hypersurface
Moreover, for
Remark 1.2.
We state the theorem with the assumption that Σ is properly embedded, namely
Indeed, the only point in which we use effectively the definition of index is in the curvature estimates Theorem 4.7, which holds also for almost properly embedded free boundary minimal hypersurface by [20, Theorem 1.4].
Remark 1.3.
In the case
Remark 1.4.
In the case of orientable hypersurfaces we can use Poincaré–Lefschetz duality [21, Theorem 3.43] and the exactness of the long exact sequence in cohomology associated to the pair
We decided to state the theorem this way so that it also applies to nonorientable hypersurfaces and to highlight the fact that the Betti numbers of the boundary are also controlled by the index (and the area). However, we emphasize that
Remark 1.5.
Note that, in the case when
Indeed, we have that
1.1 Idea of the proof
A recurrent method for obtaining quantitative estimates relating index and topology of a minimal hypersurface is to rely on the use of harmonic one-forms to obtain negative directions for the second derivative of the area.
More precisely, given a minimal hypersurface
In order to deal with any closed ambient manifold
As anticipated, in this paper we take this same approach and we generalize it to the free boundary setting. Let us describe the structure of the proof of Theorem 1.1 in some more detail.
First, we consider the cover
(see Definition 5.1). The constants
Heuristic picture of the stability balls around a catenoidal neck.
Now, the idea is to find a subset
which would conclude the proof.
In order to find such subset
The general idea is similar. Heuristically (see also Figure 2), the index concentrates around a finite number of points, and away from those points the curvature and the topology are controlled (cf. [32, Theorem 2.3]). Thus, we would like to deal with the two regions separately.
Heuristic picture of the almost conical region
We define (in Section 7) an almost conical region
The advantage is that (a suitable subset of) the stability balls centered in
The one described above is a very general outline and it is essentially the same as in the closed case (see [34, Section 1] for more details).
However, there are unfortunately many technical details to take care of.
The main further subtleties in dealing with the free boundary setting regard the construction of the almost conical region. Indeed, in our case with boundary, the almost conical region consists of annuli contained in the interior of M (which behave as in the closed case) but also “half-annuli” centered at the boundary
Case $n+1=3$.
As stated in Theorem 1.1, in the case
1.2 Higher-dimensional results
In [34], Song also proves an estimate on the size of the singular set of
codimension one stationary integral varifolds given that this set is already a priori small (see [34, Theorem 2]).
More precisely, if
where
One of the main tools in lower dimensions was the use of curvature estimates, which are not available in the higher-dimensional setting. Instead, Song used a version of [24, Theorem 1.6] but assuming stability (and a priori smallness of the singular set and bounded area) instead of area-minimizing. The main consequence is that
where the definition of
A generalization of [24, Theorem 1.6] for free boundary area-minimizing codimension one integral currents was proven by Edelen in [13, Theorem 2.2] (see also Theorem 0.5 therein). We expect that one can substitute the hypothesis of free boundary area-minimizing by assuming an a priori smallness of the size of the singular set, an area bound and stability. This would allow us to get an estimate like (1.3) in the free boundary case. Having such a result, we think that it should be possible to prove a result analogous to (1.2) but in the free boundary case, following the same lines as in [34, Section 5.2]. This problem might be addressed elsewhere.
1.3 Structure of the article
We provide an outline of the contents of this article by means of the diagram below.
2 Notation
Let us denote by
We write
When we write
We put two capital letters as subscript for better reference to the several constants that appear in the paper. We decided to do so, because this allows us to track the dependence of
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3 Preliminaries from differential geometry
In this section, we collect some preliminary definitions and results from differential geometry.
3.1 Convexity radius
Let us give the definition of convexity radius that we use in this paper in place of the injectivity radius, which is not universally defined in manifolds with boundary.
Let us assume that
Definition 3.1.
We say that
Remark 3.2.
Here, by length-minimizing curve connecting
Remark 3.3.
Note that every convex subset is simply connected, because of the uniqueness assumption in Definition 3.1.
Definition 3.4.
Given any point
Moreover, we define the convexity radius
Proposition 3.5.
The convexity radius on
3.2 Almost acyclic covers and Betti numbers
In this subsection we define the notion of almost acyclic cover and we recall a topological lemma relating an almost acyclic cover to the Betti numbers of a manifold.
Definition 3.6.
Let
Moreover, we say that the overlap of a cover
Lemma 3.7 (cf. [34, Lemma 26]).
Given
3.3 Besicovitch theorem
In this subsection we recall Besicovitch theorem, which is useful to extract subcovers with controlled overlap.
Theorem 3.8 (Besicovitch theorem).
Let
Let
Proof.
Manifolds with boundary have the property of local uniqueness of length-minimizing curves (see [1, Theorem 5]) and their metric spheres are compact. These two facts imply the property of being directionally limited as defined in [14, Section 2.8.9]. Hence, the theorem follows from [14, Theorem 2.8.14]. ∎
3.4 Layered family of balls
In this subsection, we assume that
Definition 3.9.
Let
For any subset
Definition 3.10.
Let
We say that
We say that
Lemma 3.11.
Given
Let B be a ball of radius at most
Let
Proof.
Let
This means that b is contained in the ball centered at
and hence certainly in
We also know that
where we used that the balls in
If we consider the density
we know that, if
From that and the previous estimates, we get
which implies
In case ii, where every ball in
Therefore, it suffices to take
Using this lemma we show that, given a finite disjoint family, we can extract a layered family in the sense of Definition 3.10, with size bounded below by a definite fraction of the size of the initial family.
Proposition 3.12.
Given
Proof.
Let
So it suffices to take
Finally, we recall a couple of lemmas about layered
Lemma 3.13.
There exists a constant
is bounded above by
Lemma 3.14.
Given
is bounded above by
4 Preliminaries on free boundary minimal hypersurfaces
In this section, we assume that
4.1 First and second variation formulas
We denote by
where
Definition 4.1.
We say
From now on, let us assume that
with
where
Definition 4.2.
We define the Morse index
Under the above assumptions, the Morse index of Σ is a finite number equal to the number of negative eigenvalues of the following elliptic problem on
Definition 4.3.
We call
4.2 Monotonicity formula and a density estimate
In this subsection, we recall the monotonicity formula for free boundary minimal hypersurfaces.
Theorem 4.4 (Monotonicity formula, cf. [33, Section 17] and [19, Theorem 3.4]).
Let
If
Let us also recall a consequence of the monotonicity formula (see also [3, Corollary 16]), which will be useful in what follows.
Corollary 4.5.
Let
Proof.
The proof is essentially a corollary of the monotonicity formula Theorem 4.4. Observe that the behavior of the “corrected” density
as
For the lower bound, note that, if
In this case, depending on whether
For the upper bound note that, if
If
So it is clear how big
Remark 4.6.
We are also interested in applying the lower bound of this density estimate for hypersurfaces of the type
4.3 Curvature estimates
We finally recall the curvature estimates for stable free boundary minimal hypersurfaces in ambient manifolds of dimension
Theorem 4.7 (Curvature estimates).
Assume that the dimension of the ambient manifold
Moreover, if
Proof.
First note that, since
5 Stability balls
In this section, let us assume again that
5.1 Stability radius
Here we recall the definition of stability radius with a couple of properties.
Definition 5.1.
Let
where, by convention, we assume that Σ is stable in
Lemma 5.2.
The stability radius is
Proof.
Assume
By symmetry, we get that
The fact that
Lemma 5.3.
Assume that
Proof.
Fix any
Therefore, we get that
Let
If
Note that this can easily be done for any
either
or
which proves that
Definition 5.4.
We denote by
Remark 5.5.
Note that
The following lemma is a consequence of the curvature estimates Theorem 4.7 and says that the second fundamental form of a free boundary minimal hypersurface is controlled at the radius of stability.
Lemma 5.6.
For any
Proof.
By the curvature estimates Theorem 4.7 we have that, for any
On the other hand, the following lemma shows that, passed the radius of stability, there exists one point where the second fundamental for of the free boundary minimal hypersurface is sufficiently large.
Lemma 5.7.
We can choose
for some constant
Proof.
This is the analogue in the free boundary case of [34, (8)], but the proof is the same. Let us sketch it for completeness.
Taking
5.2 Controlled topology at the radius of stability
In this subsection, we prove that (suitably choosing
Proposition 5.8.
We can choose the constants
For all
For all
For every
Proof.
The proof is fairly standard, so we only give the main ingredients for completeness.
First note that, if the ball
Let us then consider a ball
In particular, we can assume that every ball
the Euclidean second fundamental form of
Note that here we are using that
Thanks to Lemma 5.6, for any
Thus, given that the corresponding Euclidean quantities are comparable and choosing ε (universally) small enough, we can apply [12, Lemma 2.4] to obtain that any connected component
Furthermore, since B is close to the corresponding Euclidean half-ball as described above,
The same argument proves that
From this observation, the proof of the proposition follows easily. Indeed, we have that
Thanks to the previous proposition about the behavior of Σ on a ball of stability, we obtain the following two corollaries.
Corollary 5.9.
We can choose the parameters
Proof.
Let the constants
5.3 Bounding the number of sheets in balls of stability
Together with the area bound, the consequence of the monotonicity formula, Corollary 4.5, implies that Σ cannot have arbitrarily many sheets in the balls of the cover obtained in Corollary 5.9.
Proposition 5.10.
There exist some constants
Proof.
Let the point
We let
where we used that
6 Setting and choice of parameters
From now on, if not otherwise stated,
Case $n+1=3$.
Observe that we write explicitly the dependence on the constant
Observe that, setting
for all
7 The almost conical region
In this section we define the almost conical region in the setting with boundary, following [34, Section 2.2]. In the setting of Section 6, the idea is to define a region in M where Σ has well-controlled curvature and topology. This region looks like a union of annuli where Σ is close to Euclidean cones.
7.1 Choice of cones
Let us start by identifying a set of cones that will play the role of models for our free boundary minimal hypersurface in the “almost conical region” that we are about to define. In this subsection we define these Euclidean cones we are interested in and we prove some properties.
Given constants
Remark 7.1.
Observe that the only nonproperly embedded free boundary minimal cone in
Remark 7.2.
Observe that the definition is the analogue in the free boundary setting to the definition of
Case $n+1=3$.
Note that, when
Lemma 7.3.
There exists
Then there exist two stability balls
Case $n+1=3$.
Observe that, as remarked above, there are no such cones for
Proof.
Assume that
to conclude the proof. Note that
Definition 7.4.
Given a free boundary minimal hypersurface
such that
there is a (possibly nonproperly embedded) free boundary minimal cone
Three different prototypical cases of an annulus δ-close to a cone (gray region in figure).
The following analogue of [34, Theorem 8] at the boundary holds.
Theorem 7.5.
Let
Let g be a metric on
there is
the n - volume of Σ is at most
Then we have:
either there are two stability balls
or
Case $n+1=3$.
When
Theorem 7.5 for $n+1=3$.
Let
Let g be a metric on
there is
Then we have:
either there are two stability balls
or
Proof.
If Σ is a minimal hypersurface in
and such that (i) and (ii) do not hold.
The fact that, by assumption, (i) does not hold implies that the stability radius of
Thus, in
Thus, by the rigidity part of the monotonicity formula Theorem 4.4, the hypersurface
In fact, we have that
As a result, we obtain the existence of Ω, ε,
Proof of Theorem 7.5 for
n
+
1
=
3
.
If Σ is a minimal surface in
for some constant
7.2 Construction of the almost conical region
Let us assume to be in the setting described in Section 6. Moreover, assume that
Definition 7.6.
Given
either
Moreover, given another parameter
and
Definition 7.7.
We say that
and
The following proposition says that, by suitably choosing δ and K, we can cover the union of the annuli in
Proposition 7.8.
In the setting described in Section 6, there exist
There exists a finite family
Moreover, we have:
for any
for any
are topological n - discs and
for any
there exists
Case $n+1=3$.
In the case
The proof of the proposition relies on the following lemmas.
Lemma 7.9.
There exist
for all
Proof.
The proof is exactly the same as the one of [34, Section 2.2.3, equation (9)], given Lemma 5.7 (in place of [34, (8)]). ∎
Lemma 7.10 (cf. [34, Lemma 5]).
For any
Then either
Proof.
Assume by contradiction that
However, this contradicts that
Now, assume that
Therefore,
Proof of Proposition 7.8.
The proof is the same as in [34, Lemma 7] using Lemma 7.10 instead of [34, Lemma 5] since this prevents telescopes of boundary type to interact with interior ones. The additional properties (ii) and (iii) can be proven in a similar way to how [34, Lemma 7 (ii) and (iv)] are proven, given Proposition 5.8 in place of [34, (6)], Lemma 7.9 in place of [34, (9)] and Lemma 7.10 in place of [34, Lemma 5].
Property (iv) is proven similarly to Proposition 5.10, see also [34, paragraph after (20)].
To prove (i) one can argue as in [34, Lemma 7 (i)] to show that the diffeomorphism type of a connected component of
Thanks to the previous result, we now have all the tools to define the almost conical region.
Definition 7.11.
In the setting of Section 6, consider the constants
where
Case $n+1=3$.
Again, when
Remark 7.12.
In [34], when
8 The concentration region
Assume to be in the setting of Section 6, and assume that the almost conical region
8.1 Dichotomy property in the concentration region
The idea why we can control the size of a disjoint family of stability balls centered in the concentration region is that there cannot be a chain of stability balls intersecting each other and becoming smaller and smaller, or, if there is, we can “substitute” this chain with two stability balls that are sufficiently disjoint. The following lemma is a precise statement of what we roughly described in words.
Lemma 8.1.
Let us assume to be in the setting described in Section 6. Moreover, let us assume that the almost conical region
Then there are two stability balls
Case $n+1=3$.
When
Proof.
Set
Indeed, if we prove the existence of
Now, let x be the center of
Observe that A implies that
do not intersect
Hence, let us assume that B holds. Let
where
for some
for all
where we used Theorem 4.4 and Corollary 4.5 in the last two inequalities, respectively.
Therefore, there exists
which is smaller than Ω by choosing
By Lemma 5.2, observe that
Moreover, denoting by y the center of
Finally, note that
Recall that we chose m sufficiently large such that at scale
either there are two stability balls
or
If case ii occurs, then
As a result, i holds and we obtain two stability balls
To conclude the proof, we then just need to check (iii).
By using that
which concludes the proof. ∎
Proof of Lemma 8.1 for
n
+
1
=
3
.
The proof is simpler than the higher-dimensional case. Indeed, it works the same but we do not have to ensure the density estimate (8.2) and the area bound (8.5) to apply Theorem 7.5 in dimension
Thanks to the previous lemma, we are now able to show that any suitable (i.e. layered
Definition 8.2.
Let
either the size of the set
is bounded above by J,
or there are
is bounded above by J.
Definition 8.3 ([34, Section 2.5.1]).
Let
for all
Proposition 8.4.
Let us assume to be in the setting described in Section 6. Moreover, let us assume that the almost conical region
Let
Case $n+1=3$.
When
Proof.
The proof is similar to the proof of [34, Proposition 10], we report it (with some changes and rearrangements) for completeness.
Choose
As a result, we can assume that all balls
Let
Consider m given by Lemma 8.1.
Assume that there are two distinct balls
is bounded above by
Thus, we may assume that for each number
If
8.2 Index–topology estimate for the concentration region
Thanks to the previous results, we are now able to prove the aforementioned estimate relating topology and index of a free boundary minimal hypersurface. The corresponding result in the closed case can be found in [34, Corollary 12].
Theorem 8.5.
Let us assume to be in the setting described in Section 6. Moreover, assume that the almost conical region
Case $n+1=3$.
In this case, the same result hold with
Proof.
First, we classify the balls of
As a result, we now just need to bound
This suffices to conclude the proof of the theorem. Indeed, since
Together with (8.6), this implies that
So it suffices to take
The proof of the existence of the family
First, thanks to Lemma 3.13 and Lemma 3.14, we can extract a layered
where
Then one can use the property P[J] as a compensating procedure for a greedy algorithm trying to find the family
9 Proof of the main theorem
The aim of this section is proving Theorem 1.1. Hence, let us assume to be in the setting described in Section 6. Moreover, assume that the almost conical region
Applying Theorem 3.8 to the cover
where
by compactness. Without loss of generality we can assume that
Now, observe that
is an α-almost acyclic cover of Σ with overlap at most
Analogously, we have that
is a controlled cover of
Now observe that, by Proposition 5.8 ii, and Proposition 5.10,
Moreover,
Therefore, we get
where we used Theorem 8.5 in the last inequality. Therefore, by putting together the estimates, we get
where
Observe that, in dimension
Funding source: European Research Council
Award Identifier / Grant number: 947923
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2104229
Funding statement: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 947923), and was also partially supported by the NSF award DMS-2104229 and by Simons Foundation International, LTD.
Acknowledgements
This project originated from the Master thesis of the first author under the supervision of Alessandro Carlotto, whom we would like to thank. We would also like to thank Antoine Song for many helpful discussions and in particular for pointing out a mistake in the first version of this paper. Moreover, we thank Nick Edelen for kindly answering our questions, and the anonymous referee for helpful comments and suggestions.
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- 𝔸1-connected components and characterisation of 𝔸2
- New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows
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Articles in the same Issue
- Frontmatter
- Singular set and curvature blow-up rate of the level set flow
- Counting integral points of bounded height on varieties with large fundamental group
- 𝔸1-connected components and characterisation of 𝔸2
- New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows
- Torus counting and self-joinings of Kleinian groups
- Estimating the Morse index of free boundary minimal hypersurfaces through covering arguments
- On the integral Hodge conjecture for real abelian threefolds
- Weakly Kähler hyperbolic manifolds and the Green–Griffiths–Lang conjecture
