Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces

Haizhong Li; Ruixuan Li; Changwei Xiong

doi:10.1515/anona-2025-0068

Artikel Open Access

Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces

Haizhong Li , Ruixuan Li und Changwei Xiong

Veröffentlicht/Copyright: 4. März 2025

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Abstract

We derive various sharp upper bounds for the p -capacity of a smooth compact set K in the hyperbolic space H n and the Euclidean space R n . First, by using the inverse mean curvature flow, for the mean convex and star-shaped set K in H n , we obtain sharp upper bounds for the p -capacity Cap p ( K ) in three cases: (1) n ≥ 2 and p = 2 , (2) n = 2 and p ≥ 3 , and (3) n = 3 and 1 < p ≤ 3 ; using the unit-speed normal flow, we prove a sharp upper bound for Cap p ( K ) of a convex set K in H n for n ≥ 2 and p > 1 . Second, for the compact set K in R 3 , using the weak inverse mean curvature flow, we obtain a sharp upper bound for the p -capacity ( 1 < p < 3 ) of the set K with connected boundary; by using the inverse anisotropic mean curvature flow, we deduce a sharp upper bound for the anisotropic p -capacity ( 1 < p < 3 ) of an F -mean convex and star-shaped set K in R 3 .

Keywords: hyperbolic space; capacity; inverse mean curvature flow; modified Hawking mass

MSC 2010: 31B15; 53C21; 74G65; 49Q10

1 Introduction

The capacity of compact sets in a Riemannian manifold is an important geometric quantity and admits many crucial applications in the topics such as parabolicity criteria, eigenvalue estimates, and heat kernel estimates. See [9,10] for an overview of the applications.

For the problem to estimate the p -capacity in the Euclidean space, the asymptotically flat Riemannian manifold and some other Riemannian manifolds (mainly modelled by the Euclidean space), there have been extensive investigations; see the previous works [1–4,6,13,18–21,29–32]. In contrast, we find there are few results in the hyperbolic space, partially because even for the model set, the geodesic ball B r ⊂ H n , the p -capacity Cap p ( B r ¯ ) does not have a simple expression. See [16] for some limit results on the p -capacity of geodesic balls in an asymptotically hyperbolic manifold. To fill this gap in the literature, our first main focus in this article is to derive various sharp upper bounds for the p -capacity of compact sets in the hyperbolic space.

Let K ⊂ H n be a compact set in the hyperbolic space. For p ≥ 1 , define the p -capacity of K as follows:

Cap p ( K ) = inf ∫ H n ∣ ∇ f ∣ p d v ,

where the infimum is taken over all smooth functions f with compact support satisfying f = 1 on K . For the p -capacity in a general Riemannian manifold, see [9,10].

Remark 1

Besides the aforementioned standard definition, the p -capacity Cap p ( K ) enjoys another characterization; see Proposition 1.1 in [33].

Let p > 1 . Assume U ≔ H n \ K is foliated by a family of hypersurfaces M t with M 0 = ∂ K . It has been derived in [9] that

(1.1) Cap p ( K ) ≤ ∫ 0 ∞ T p − 1 ⁄ ( p − 1 ) ( t ) d t 1 − p ,

where the function T p ( t ) is defined as follows:

T p ( t ) = ∫ M t ∣ D ψ ∣ p − 1 d μ t ,

and ψ : H n \ K → R is the level-set function determined by the hypersurfaces M t

ψ ( x ) = t , when x ∈ M t .

Remark 2

The 1-capacity Cap 1 ( K ) admits the following characterization [20, page 149]

(1.2) Cap 1 ( K ) = inf ∣ ∂ Ω ∣ ,

where the infimum is taken over all bounded open sets Ω ⊂ H n with smooth boundary containing K .

In the first part of the article, we will utilize the estimate (1.1) to obtain four kinds of upper bounds for the p -capacity of smooth compact sets K in H n in terms of certain geometric quantities of K (Theorems 3–7).

Theorem 3

Let K ⊂ H n ( n ≥ 2 ) be a compact set with smooth, mean convex, and star-shaped boundary. Then

Cap 2 ( K ) ≤ n ( n − 1 ) ∫ 0 ∞ e n − 2 n − 1 t W 2 ( K ) + n − 2 n ( n − 1 ) ∫ 0 t e − n − 2 n − 1 τ I ( e τ ∣ ∂ K ∣ ) d τ + 1 n I ( e t ∣ ∂ K ∣ ) − 1 d t − 1 ,

where the second quermassintegral W 2 ( K ) is given by

W 2 ( K ) ≔ 1 n ( n − 1 ) ∫ ∂ K σ 1 d μ − 1 n ∣ K ∣ ,

with σ 1 being the mean curvature of the boundary ∂ K (the summation of principal curvatures of ∂ K ), and I : R + → R + is the isoperimetric function in H n , i.e., I is defined by ∣ B r ∣ = I ( ∣ ∂ B r ∣ ) ( r > 0 ). In particular, when n = 2 , we obtain

Cap 2 ( K ) ≤ ∫ 0 ∞ ( 4 π 2 + e 2 t ∣ ∂ K ∣ 2 ) − 1 ⁄ 2 d t − 1 = 2 π arsinh ( 2 π ⁄ ∣ ∂ K ∣ ) .

Moreover, the equalities hold if and only if K is a geodesic ball.

Theorem 4

Let K ⊂ H 2 be a compact set with smooth convex boundary M = ∂ K and p ≥ 3 . Then

Cap p ( K ) ≤ ∫ 0 ∞ e p − 2 p − 1 t ∣ M ∣ p − 2 p − 1 e 2 t ∣ M ∣ 2 + ( ∣ M ∣ p − 2 T p ( 0 ) ) 2 p − 1 − ∣ M ∣ 2 − 1 2 d t 1 − p ,

where T p ( 0 ) = ∫ M σ 1 p − 1 d μ . Moreover, the equality holds if and only if K is a geodesic ball.

Theorem 5

Let K ⊂ H 3 be a compact set with smooth, mean convex, and star-shaped boundary M = ∂ K and 1 < p ≤ 3 . Then

Cap p ( K ) ≤ ∫ 0 ∞ ∫ M σ 1 2 d μ − 4 ∣ M ∣ − 16 π e − t + 4 ∣ M ∣ e t + 16 π − 1 2 e − 3 − p 2 ( p − 1 ) t d t 1 − p ∣ M ∣ 3 − p 2 .

Moreover, the equality holds if and only if K is a geodesic ball.

Remark 6

In Theorem 5, taking p → 1 + , by a direct computation, we may see

Cap 1 ( K ) ≤ ∣ M ∣ .

Theorem 7

Let K ⊂ H n ( n ≥ 2 ) be a compact set with smooth convex boundary M = ∂ K and p > 1 . Then

Cap p ( K ) ≤ ∫ 0 ∞ ∫ M ∑ i = 0 n − 1 cosh n − 1 − i t ⋅ sinh i t ⋅ σ i d μ − 1 p − 1 d t 1 − p ,

where σ i denotes the ith mean curvature of the boundary M. Moreover, the equality holds if K is a geodesic ball.

Remark 8

As in the last remark, taking p → 1 + in Theorem 7, by a direct calculation, we may see

Cap 1 ( K ) ≤ ∣ M ∣ .

For the proofs of Theorems 3, 4, and 5, we use the inverse mean curvature flow [8] to generate the foliation M t ; while for the proof of Theorem 7, we use the unit-speed normal flow to obtain the foliation M t .

Next, we come to the second part of the article, the cases in the Euclidean space R n . Different from in the hyperbolic space, in R n , we may consider the more general anisotropic p -capacity for a compact set. That is, given a Minkowski norm F on R n , we define the anisotropic p -capacity ( 1 < p < n ) of a compact set K ⊂ R n by

Cap F , p ( K ) = inf ∫ R n F p ( D v ) d x : v ∈ C c ∞ ( R n ) , v ≥ 1 on K ,

where C c ∞ ( R n ) is the set of smooth functions with compact support in R n . See Section 2.3 for more details on the definitions. In particular, when F is the Euclidean norm ( F ( ξ ) = ∣ ξ ∣ ), the anisotropic p -capacity reduces to the ordinary p -capacity in R n .

First, we obtain a sharp upper bound for the (ordinary) p -capacity of a compact set K ⊂ R 3 with smooth and connected boundary Σ = ∂ K . In R 3 , we use H for σ 1 to denote the mean curvature of a surface Σ ⊂ R 3 .

Theorem 9

Let K ⊂ R 3 be a compact set with smooth connected boundary Σ = ∂ K and 1 < p < 3 . Then we have two cases: (1) ∫ Σ H 2 d μ = 16 π , Σ is a round sphere with radius r 0 > 0 and

Cap p ( K ) = 3 − p p − 1 p − 1 4 π r 0 3 − p ;

(2) ∫ Σ H 2 d μ > 16 π and

Cap p ( K ) < 3 − p p − 1 p − 1 ( 4 π ) p − 1 2 ∣ Σ ∣ 3 − p 2 s 3 − p 2 θ 1 − p ,

where

s = ∫ Σ H 2 d μ 16 π − 1 and θ = ∫ 0 s 3 − p 2 ( p − 1 ) 1 + r 2 ( p − 1 ) 3 − p − 1 2 d r .

Remark 10

For a closed surface Σ ⊂ R 3 , the Willmore energy ∫ Σ H 2 d μ satisfies ∫ Σ H 2 d μ ≥ 16 π with the equality if and only if Σ is a round sphere. See, e.g., [5,22,27].

Remark 11

Theorem 9 with p = 2 was proved by the third author in [34] in order to obtain sharp estimates for an exterior Steklov eigenvalue problem. Besides, Theorem 9 improves on partial results in [4,30]. Precisely, only in the special case of Euclidean spaces, the result in Theorem 9 improves on Theorem 1.1 (i) in Xiao’s [30] and Corollary 2 in Bray and Miao’s [4], which may be verified by an elementary comparison. Note the general results in [4,30] hold for a much wider class of asymptotically flat Riemannian manifolds.

Remark 12

By direct computations, we have the limiting cases in Theorem 9

Cap 1 ( K ) ≤ ∣ Σ ∣ and Cap 3 ( K ) = 0 .

Second, we obtain a sharp upper bound for the anisotropic p -capacity of compact sets in the Euclidean space R 3 . For the Wulff shape ∂ W and anisotropic quantities in Theorem 13, we refer to Section 2.3.

Theorem 13

Let K ⊂ R 3 be a compact set with smooth, anisotropically mean convex and star-shaped boundary Σ = ∂ K and 1 < p < 3 . Then we have two cases: (1) ∫ Σ H F 2 d μ F = 4 ∣ ∂ W ∣ F , the surface Σ is a translated scaled Wulff shape r 0 ∂ W + x 0 ( r 0 > 0 , x 0 ∈ R 3 ) and

Cap F , p ( K ) = 3 − p p − 1 p − 1 ∣ ∂ W ∣ F r 0 3 − p ;

(2) ∫ Σ H F 2 d μ F > 4 ∣ ∂ W ∣ F and

Cap F , p ( K ) < 3 − p p − 1 p − 1 ∣ ∂ W ∣ F p − 1 2 ∣ Σ ∣ F 3 − p 2 s 3 − p 2 θ 1 − p ,

where

s = ∫ Σ H F 2 d μ F 4 ∣ ∂ W ∣ F − 1 and θ = ∫ 0 s 3 − p 2 ( p − 1 ) 1 + r 2 ( p − 1 ) 3 − p − 1 2 d r .

Remark 14

Similarly, by direct computations, we have the limiting cases in Theorem 13

Cap F , 1 ( K ) ≤ ∣ Σ ∣ F and Cap F , 3 ( K ) = 0 .

When p = 2 , we obtain the following corollary.

Corollary 15

Let K ⊂ R 3 be a compact set with smooth, anisotropically mean convex and star-shaped boundary Σ = ∂ K . Then we have two cases: (1) ∫ Σ H F 2 d μ F = 4 ∣ ∂ W ∣ F , the surface Σ is a translated scaled Wulff shape r 0 ∂ W + x 0 ( r 0 > 0 , x 0 ∈ R 3 ) and

Cap F , 2 ( K ) = ∣ ∂ W ∣ F r 0 ;

(2) ∫ Σ H F 2 d μ F > 4 ∣ ∂ W ∣ F and

(1.3) Cap F , 2 ( K ) < ∣ ∂ W ∣ F ∣ Σ ∣ F s arsinh s ,

where

s = ∫ Σ H F 2 d μ F 4 ∣ ∂ W ∣ F − 1 .

Remark 16

For the isotropic case (i.e., F is the Euclidean norm F ( ξ ) = ∣ ξ ∣ ), Theorem 13 is contained in Theorem 9. In addition, Theorem 13 and Corollary 15 improve on the corresponding results in [18]. For example, under the same conditions, the bound (1.7) in [18] reads (if Σ is not a Wulff shape)

(1.4) Cap F , 2 ( K ) < 1 2 ∣ ∂W ∣ F ∣ Σ ∣ F 1 + 1 4 ∣ ∂W ∣ F ∫ Σ H F 2 d μ F .

By an elementary computation, we see that the bound (1.3) is better than the bound (1.4).

For the proof of Theorem 9, we use the weak inverse mean curvature flow due to Huisken and Ilmanen [14]; while for that of Theorem 13, we employ the inverse anisotropic mean curvature flow [28]. Under these flows, we can foliate the domain R n \ K and apply the classical approach as in [21,24]. This approach has also been applied in other works, and the most relevant papers to Theorems 9 and 13 are [4,18,30,34] as mentioned earlier.

This article is organized as follows. In Section 2, we provide some preliminaries and backgrounds, mainly on the inverse mean curvature flow in the hyperbolic space, the weak inverse mean curvature flow in the Euclidean space, the anisotropic geometry in the Euclidean space, and the related inverse anisotropic mean curvature flow. In Section 3, we employ the inverse mean curvature flow in H n to prove Theorems 3, 4, and 5. In Section 4 we use the unit-speed normal flow to prove Theorem 7. In Sections 5 and 6, we discuss the problems in R 3 . In Section 5, we use the weak inverse mean curvature flow to prove Theorem 9. In Section 6 we apply the inverse anisotropic mean curvature flow to prove Theorem 13.

2 Preliminaries

In this section, we recall some backgrounds and tools, which will be used later in the article. Throughout the article, ∣ ⋅ ∣ denotes the volume of compact sets with nonempty interior or the area of hypersurfaces, σ i ( 1 ≤ i ≤ n − 1 ) denotes the i th mean curvature of a hypersurface M n − 1 in an n -dimensional Riemannian manifold (occasionally, we will use H for σ 1 ), and other standard notations will also be used without further indication whenever no confusion arises.

2.1 The quermassintegrals and the inverse mean curvature flow in H n

For a compact set K ⊂ H n in the hyperbolic space, the quermassintegrals are important geometric quantities. In this article, we shall use the first three of them as follows:

W 0 ( K ) ≔ ∣ K ∣ , W 1 ( K ) ≔ 1 n ∣ ∂ K ∣ , W 2 ( K ) ≔ 1 n ( n − 1 ) ∫ ∂ K σ 1 d μ − 1 n ∣ K ∣ .

See, e.g., [26] for more details on the quermassintegrals.

Next we briefly introduce the inverse mean curvature flow in the hyperbolic space due to Gerhardt [8]. Consider the inverse mean curvature flow for a mean convex hypersurface in H n ( n ≥ 2 ), i.e., the parabolic evolution equation

(2.1) ∂ t X = 1 σ 1 ν ,

where X : N × [ 0 , T ) → H n is a smooth family of embeddings from a closed manifold N to H n and ν denotes the outward unit normal vector on the flow hypersurface.

Given an initial mean convex and star-shaped hypersurface M 0 = X ( N , 0 ) , Gerhardt [8] proved the long-time existence and the smooth and exponential convergence of the flow (2.1) as follows.

Theorem 17

(Gerhardt [8]) The flow (2.1) with an initial mean convex and star-shaped hypersurface M 0 = X ( N , 0 ) exists for all time t ∈ [ 0 , ∞ ) . The flow hypersurfaces M t converge to infinity and become strictly convex exponentially fast and also more and more totally umbilic. In fact, there holds

∣ h j i − δ j i ∣ ≤ c e − t ⁄ ( n − 1 ) ,

i.e., the principal curvatures are uniformly bounded and converge exponentially fast to 1.

In addition, we need the evolution equations for the area element d μ t and the mean curvature σ 1 , which is standard and may be found in, e.g., Proposition 3 in [17].

Proposition 18

Along the flow (2.1), there holds

∂ ∂ t d μ t = d μ t , ∂ t σ 1 = − Δ 1 σ 1 − 1 σ 1 ∣ h ∣ 2 + n − 1 σ 1 ,

where Δ is the Laplacian on the flow hypersurface M t and ∣ h ∣ 2 denotes the squared norm of the second fundamental form of M t . In particular, ∣ M t ∣ = e t ∣ M ∣ .

2.2 The weak inverse mean curvature flow in R 3

In this subsection, we introduce the weak inverse mean curvature flow for the compact set K ⊂ R 3 with smooth and connected boundary Σ = ∂ K , which is due to Huisken and Ilmanen [14]. By [14], there exists a proper, Lipschitz function ϕ ≥ 0 on U ¯ (here, U ≔ R 3 \ K ), called the solution to the weak inverse mean curvature flow with the initial surface Σ , satisfying the following properties:

The function ϕ takes value ϕ ∣ Σ = 0 and lim x → ∞ ϕ ( x ) = ∞ . For t > 0 , the sets Σ t = ∂ { ϕ ≥ t } and Σ t ′ = ∂ { ϕ > t } define two increasing families of C 1 , α surfaces.
For t > 0 , the surfaces Σ t ( Σ t ′ , resp.) minimize (strictly minimize, resp.) area among surfaces homologous to Σ t in the region { ϕ ≥ t } . The surface Σ ′ = ∂ { ϕ > 0 } strictly minimizes area among surfaces homologous to Σ in U .
For almost all t > 0 , the weak mean curvature H of Σ t is well defined and equals ∣ ∇ ϕ ∣ , which is positive for almost all x ∈ Σ t .
For each t > 0 , the area ∣ Σ t ∣ = e t ∣ Σ ′ ∣ ; and ∣ Σ t ∣ = e t ∣ Σ ∣ if Σ is outer-minimizing (i.e., Σ minimizes area among all surfaces homologous to Σ in U ).
All the surfaces Σ t ( t > 0 ) remain connected. The Hawking mass
(2.2) m H ( Σ t ) = ∣ Σ t ∣ 16 π 1 − 1 16 π ∫ Σ t H 2 d μ t
satisfies lim t → 0 + m H ( Σ t ) ≥ m H ( Σ ′ ) and its right-lower derivative satisfies (see (5.24) in [14]; but note the misprints on some coefficients there, and the correct coefficients are as in (5.22) in [14])
D ̲ + m H ( Σ t ) ≔ liminf s → t + m H ( Σ s ) − m H ( Σ t ) s − t ≥ ∣ Σ t ∣ 16 π 1 16 π 8 π − 4 π χ ( Σ t ) + ∫ Σ t 2 ∣ ∇ log H ∣ 2 + 1 2 ( λ 1 − λ 2 ) 2 d μ t ,
where λ 1 and λ 2 are the weak principal curvatures of Σ t and χ ( Σ t ) is the Euler characteristic of Σ t . See [14, Section 5] for more details.

Furthermore, in [34], the third author defines a modified Hawking mass

(2.3) m ˜ H ( Σ t ) ≔ ∣ Σ t ∣ 16 π m H ( Σ t ) = ∣ Σ t ∣ 16 π 1 − 1 16 π ∫ Σ t H 2 d μ t ,

and finds lim t → 0 + m ˜ H ( Σ t ) ≥ m ˜ H ( Σ ′ ) and

D ̲ + m ˜ H ( Σ t ) = ∣ Σ t ∣ 16 π 1 2 m H ( Σ t ) + ∣ Σ t ∣ 16 π D ̲ + m H ( Σ t ) ≥ ∣ Σ t ∣ ( 16 π ) 2 8 π − 1 2 ∫ Σ t H 2 d μ t + 8 π − 4 π χ ( Σ t ) + ∫ Σ t 2 ∣ ∇ log H ∣ 2 + 1 2 ( λ 1 − λ 2 ) 2 d μ t = ∣ Σ t ∣ ( 16 π ) 2 16 π − 4 π χ ( Σ t ) + ∫ Σ t ( 2 ∣ ∇ log H ∣ 2 − 2 λ 1 λ 2 ) d μ t = ∣ Σ t ∣ ( 16 π ) 2 16 π − 8 π χ ( Σ t ) + ∫ Σ t 2 ∣ ∇ log H ∣ 2 d μ t ≥ 0 ,

where the last equality holds because of the weak Gauss–Bonnet formula (page 403 in [14]), and the last inequality is due to the fact that the surfaces Σ t remains connected.

Remark 19

The introduction of the modified Hawking mass is inspired by the work [15].

2.3 The anisotropic p -capacity and the inverse anisotropic mean curvature flow in R 3

For the anisotropic p -capacity, see previous studies [20, Section 2.2] and [13, Chapters 2 and 5]. First, we introduce the Minkowski norm on R n .

Definition 2.1

A function F ∈ C ∞ ( R n \ { 0 } ) ∩ C ( R n ) is called a Minkowski norm if

F is a convex, even, 1-homogeneous function, and F ( ξ ) > 0 if ξ ≠ 0 ;
F satisfies the uniformly elliptic condition, i.e., Hess R n ( F 2 ) is positive definite in R n \ { 0 } .

Let K ⊂ R n be a compact set. For n ≥ 2 and 1 < p < n , the anisotropic p-capacity of K is defined as follows:

Cap F , p ( K ) = inf ∫ R n F p ( D v ) d x : v ∈ C c ∞ ( R n ) , v ≥ 1 on K ,

where C c ∞ ( R n ) is the set of smooth functions with compact support in R n .

Remark 20

The limiting case p → 1 + reads

Cap F , 1 ( K ) = inf ∣ ∂ Ω ∣ F ,

where the infimum is taken over all bounded open sets Ω ⊂ R n with smooth boundary containing K ; see [20, page 149]. And the limiting case p → n − reads

Cap F , n ( K ) = 0 ;

cf. the results in [20, Section 2.2.4].

Next we review the anisotropic geometry of hypersurfaces in the Euclidean space, which is classical in the differential geometry.

Let F be a Minkowski norm on R n . We can define its dual norm F 0 as follows.

Definition 2.2

The dual norm F 0 of F is defined as follows:

F 0 ( x ) = sup ξ ≠ 0 ⟨ ξ , x ⟩ F ( ξ ) .

It is known that F 0 is also a Minkowski norm.

Recall that F and F 0 satisfy the following properties, which are very useful when we want to understand the relationship between the unit normal ν and the anisotropic unit normal ν F of a hypersurface below.

Proposition 21

F ( D F 0 ( x ) ) = 1 , F 0 ( D F ( ξ ) ) = 1 .
F 0 ( x ) D F ( D F 0 ( x ) ) = x , F ( ξ ) D F 0 ( D F ( ξ ) ) = ξ .

Next we define the Wulff ball and the Wulff shape determined by F .

Definition 2.3

The Wulff ball W centered at the origin is defined as follows:

W ≔ { x ∈ R n : F 0 ( x ) < 1 } .

Its boundary ∂ W is called the Wulff shape.

Given a Wulff ball W , we can recover F as the support function of W , namely,

F ( ξ ) = sup X ∈ W ⟨ ξ , X ⟩ , ξ ∈ S n − 1 .

Now we introduce the anisotropic area of a smooth oriented hypersurface X : N → M ⊂ R n .

Definition 2.4

Let M ⊂ R n be a smooth oriented hypersurface and ν be its unit normal vector. We define the anisotropic area of M as ∣ M ∣ F ≔ ∫ M F ( ν ) d μ . Denote by d μ F = F ( ν ) d μ the anisotropic area element of M .

Remark 22

For M = ∂ W , we can check by the divergence theorem that

∣ ∂ W ∣ F ≔ ∫ ∂ W F ( ν ) d μ = ∫ ∂ W ⟨ X , ν ⟩ d μ = ∫ W div X d x = n ∣ W ∣ .

Next we introduce the anisotropic Gauss map for an oriented hypersurface in R n (for example, see [11]).

Definition 2.5

The anisotropic Gauss map ν F : M → ∂ W from an oriented hypersurface M in R n to the Wulff shape ∂ W is defined by

ν F : M → ∂ W , X ↦ D F ( ν ( X ) ) = F ( ν ( X ) ) ν ( X ) + ∇ S n − 1 F ( ν ( X ) ) ,

where ν is the unit normal vector of M .

Remark 23

The vector ν F is also called the anisotropic unit normal of the hypersurface.

Let A F be the 2-tensor on S n − 1 defined by

A F ( ξ ) = ( ∇ S n − 1 ) 2 F ( ξ ) + F ( ξ ) σ , ξ ∈ S n − 1 ,

where σ is the standard metric on S n − 1 .

Definition 2.6

The anisotropic principal curvatures κ 1 F , … , κ n − 1 F of a smooth oriented hypersurface M in R n are defined as the eigenvalues of the tangent map

d ν F : T X M → T ν F ( X ) ∂ W ≅ T X M .

The anisotropic mean curvature is defined as follows:

H F ≔ tr ( d ν F ) = ∑ i κ i F = ∑ i , j , k ( A F ) i j ( ν ( X ) ) g i k ( X ) h k j ( X ) ,

where g and h are the first and second fundamental forms of the hypersurface respectively. We call M anisotropically mean convex or F -mean convex if H F > 0 on M .

Now we are ready to state the result concerning the inverse anisotropic mean curvature flow in the Euclidean space. In [28], following the (isotropic) works [7,25], Chao Xia considered the inverse anisotropic mean curvature flow for a star-shaped F -mean convex hypersurface, i.e., the parabolic evolution equation

(2.4) ∂ t X = 1 H F ν F ,

where X : N × [ 0 , T ) → R n is a smooth family of embeddings from a closed manifold N to R n . He proved the following result on this flow.

Theorem 24

[28] Let M be a smooth compact star-shaped and F-mean convex hypersurface without boundary in R n ( n ≥ 3 ). Then the inverse anisotropic mean curvature flow starting from M exists for all time and converges smoothly and exponentially to an expanded Wulff shape determined by the initial hypersurface M.

3 Upper bounds in H n via the inverse mean curvature flow

Consider the inverse mean curvature flow

(3.1) ∂ t X = 1 σ 1 ν

in the hyperbolic space H n with the initial hypersurface ∂ K and the resulting hypersurfaces M t , where σ 1 is the mean curvature of M t . Then we obtain (see Section 3 in [18] for a similar derivation)

(3.2) Cap p ( K ) ≤ ∫ 0 ∞ T p − 1 ⁄ ( p − 1 ) ( t ) d t 1 − p , p > 1

and

T p ( t ) = ∫ M t σ 1 p − 1 d μ t .

Remark 25

Recall that the 1-capacity Cap 1 ( K ) enjoys the characterization (see, e.g., [20, page 149])

(3.3) Cap 1 ( K ) = inf ∣ ∂ Ω ∣ ,

where the infimum is taken over all bounded open sets Ω ⊂ H n with smooth boundary containing K . So wherever applicable, the limit as p → 1 + of an upper bound derived from the inequality (3.2) will obey this characterization.

In the following, we consider three cases separately.

3.1 The case n ≥ 2 and p = 2

For this case, we need to estimate ∫ M t σ 1 d μ t .

Recall the quermassintegrals

W 0 ( K ) ≔ ∣ K ∣ , W 1 ( K ) ≔ 1 n ∣ ∂ K ∣ , W 2 ( K ) ≔ 1 n ( n − 1 ) ∫ ∂ K σ 1 d μ − 1 n ∣ K ∣ .

Under the inverse mean curvature flow, by using the variational formula (3.5) in [26], we obtain (for n = 2 , the right-hand side of the first line below is understood to be 0; see the simplified argument at the end of this part)

d d t W 2 ( K t ) = n − 2 n ∫ ∂ K t σ 2 C n − 1 2 1 σ 1 d μ t ≤ n − 2 n ( n − 1 ) 2 ∫ ∂ K t σ 1 d μ t = n − 2 n − 1 1 n ( n − 1 ) ∫ ∂ K t σ 1 d μ t − 1 n ∣ K t ∣ + n − 2 n ( n − 1 ) ∣ K t ∣ = n − 2 n − 1 W 2 ( K t ) + n − 2 n ( n − 1 ) ∣ K t ∣ .

Here, K t is the compact set enclosed by M t , so that ∂ K t = M t and K 0 = K . Let I be the isoperimetric function on H n , i.e., ∣ B r ∣ = I ( ∣ ∂ B r ∣ ) . Then

(3.4) ∣ K t ∣ ≤ I ( ∣ ∂ K t ∣ ) = I ( e t ∣ ∂ K 0 ∣ ) ,

where we used ∣ ∂ K t ∣ = e t ∣ ∂ K 0 ∣ .

So we obtain

W 2 ( K t ) ≤ e n − 2 n − 1 t W 2 ( K 0 ) + n − 2 n ( n − 1 ) ∫ 0 t e − n − 2 n − 1 τ I ( e τ ∣ ∂ K 0 ∣ ) d τ .

Then we obtain an upper bound for ∫ ∂ K t σ 1 d μ t as follows:

1 n ( n − 1 ) ∫ ∂ K t σ 1 d μ t = W 2 ( K t ) + 1 n ∣ K t ∣ ≤ e n − 2 n − 1 t W 2 ( K 0 ) + n − 2 n ( n − 1 ) ∫ 0 t e − n − 2 n − 1 τ I ( e τ ∣ ∂ K 0 ∣ ) d τ + 1 n I ( e t ∣ ∂ K 0 ∣ ) ,

where again we used ∣ ∂ K t ∣ = e t ∣ ∂ K 0 ∣ .

Then we obtain

Cap p ( K ) ≤ n ( n − 1 ) ∫ 0 ∞ e n − 2 n − 1 t W 2 ( K 0 ) + n − 2 n ( n − 1 ) ∫ 0 t e − n − 2 n − 1 τ I ( e τ ∣ ∂ K 0 ∣ ) d τ + 1 n I ( e t ∣ ∂ K 0 ∣ ) − 1 d t − 1 .

In particular, when n = 2 , W 2 ( K t ) is a constant π by the Gauss-Bonnet theorem. So

∫ ∂ K t σ 1 d μ t = 2 π + ∣ K t ∣ ≤ 4 π 2 + e 2 t ∣ ∂ K 0 ∣ 2 ,

since by ∣ ∂ B r ∣ 2 = 4 π ∣ B r ∣ + ∣ B r ∣ 2 , we know

(3.5) I ( s ) = 4 π 2 + s 2 − 2 π .

Then we obtain

Cap 2 ( K ) ≤ ∫ 0 ∞ ( 4 π 2 + e 2 t ∣ ∂ K 0 ∣ 2 ) − 1 ⁄ 2 d t − 1 = 2 π arsinh ( 2 π ⁄ ∣ ∂ K 0 ∣ ) .

Finally, assume the equality holds. Then the equality case of the isoperimetric inequality used in the aforementioned argument implies that the boundary ∂ K 0 must be a geodesic sphere. So we finish the proof of Theorem 3.

3.2 The case n = 2 and p ≥ 3

In this case, we obtain

d d t T p ( t ) = d d t ∫ M t σ 1 p − 1 d μ t = ∫ M t ( p − 1 ) σ 1 p − 2 − Δ 1 σ 1 − σ 1 + 1 σ 1 + σ 1 p − 1 d μ t = − ( p − 1 ) ( p − 2 ) ∫ M t σ 1 p − 5 ∣ ∇ σ 1 ∣ 2 d μ t − ( p − 2 ) ∫ M t σ 1 p − 1 d μ t + ( p − 1 ) ∫ M t σ 1 p − 3 d μ t .

Noting p ≥ 3 , we see

d d t T p ( t ) ≤ − ( p − 2 ) ∫ M t σ 1 p − 1 d μ t + ( p − 1 ) ∫ M t σ 1 p − 3 d μ t .

When p = 3 , noting ∣ M t ∣ = e t ∣ M ∣ , we can obtain

∫ M t σ 1 2 d μ t ≤ e − t ∫ M σ 1 2 d μ t + 2 sinh t ∣ M ∣ ,

∣ M t ∣ ∫ M t σ 1 2 d μ t − ∣ M t ∣ 2 ≤ ∣ M ∣ ∫ M σ 1 2 d μ − ∣ M ∣ 2 .

Next consider p > 3 . Then by the Hölder inequality, we obtain

∫ M t σ 1 p − 3 d μ t ≤ ∫ M t σ 1 ( p − 3 ) p − 1 p − 3 d μ t p − 3 p − 1 ∣ M t ∣ 2 p − 1 = ∫ M t σ 1 p − 1 d μ t p − 3 p − 1 ∣ M t ∣ 2 p − 1 .

So, we obtain

d d t T p ( t ) ≤ − ( p − 2 ) T p ( t ) + ( p − 1 ) ( T p ( t ) ) p − 3 p − 1 ∣ M t ∣ 2 p − 1 .

Solving it yields

( ∣ M t ∣ p − 2 T p ( t ) ) 2 p − 1 − ∣ M t ∣ 2 ≤ ( ∣ M ∣ p − 2 T p ( 0 ) ) 2 p − 1 − ∣ M ∣ 2 .

Note that this result also covers the case p = 3 .

In summary, for p ≥ 3 , we obtain

( T p ( t ) ) 1 p − 1 ≤ ∣ M t ∣ − p − 2 p − 1 ∣ M t ∣ 2 + ( ∣ M ∣ p − 2 T p ( 0 ) ) 2 p − 1 − ∣ M ∣ 2 1 2 .

So, we obtain

Cap p ( K ) ≤ ∫ 0 ∞ ∣ M t ∣ p − 2 p − 1 ∣ M t ∣ 2 + ( ∣ M ∣ p − 2 T p ( 0 ) ) 2 p − 1 − ∣ M ∣ 2 − 1 2 d t 1 − p .

Finally, assume the equality holds. Then, by checking the aforementioned argument, we see that σ 1 is constant on the flow curve M t . Thus, M t is a geodesic circle and so is ∂ K = M 0 . We finish the proof of Theorem 4.

3.3 The case n = 3 and 1 < p ≤ 3

Let us first estimate ∫ M t σ 1 2 d μ t using the inverse mean curvature flow.

We compute

d d t ∫ M t σ 1 2 d μ t = ∫ M t 2 σ 1 − Δ 1 σ 1 − 1 σ 1 ∣ h ∣ 2 + 2 σ 1 + σ 1 2 d μ t ≤ ∫ M t − 2 ∣ h ∣ 2 + 4 + σ 1 2 d μ t .

Now use the Gauss equation 2 K M = σ 1 2 − ∣ h ∣ 2 − 2 to replace ∣ h ∣ 2 . Here, K M is the Gauss curvature for M t so that ∫ M t K M d μ t = 4 π . Therefore,

d d t ∫ M t σ 1 2 d μ t ≤ ∫ M t ( 4 K M − 2 σ 1 2 + 4 ) + 4 + σ 1 2 d μ t = − ∫ M t σ 1 2 d μ t + 8 e t ∣ M ∣ + 16 π ,

which implies

d d t e t ∫ M t σ 1 2 d μ t ≤ e t ( 8 e t ∣ M ∣ + 16 π ) .

Here, we may rewrite the aforementioned inequality as follows:

d d t ∣ M t ∣ ∫ M t σ 1 2 d μ t − 4 ∣ M t ∣ 2 − 16 π ∣ M t ∣ ≤ 0 .

So we can define the modified Hawking mass in the hyperbolic space by

m H ( M t ) ≔ ∣ M t ∣ 16 π + 4 ∣ M t ∣ − ∫ M t σ 1 2 d μ t ,

which is a nondecreasing quantity in t .

Remark 26

The modified Hawking mass in the hyperbolic space was introduced by Hung and Wang in [15], and its monotonicity along the inverse mean curvature flow was also proved there. Here, we include the proof of the monotonicity for the convenience of readers.

Remark 27

By using the Gauss equation and the Gauss-Bonnet theorem, we see m H ( M t ) ≤ 0 and m H ( M t ) = 0 if and only if the surface M t is a geodesic sphere.

So we have

∫ M t σ 1 2 d μ t ≤ ∫ M σ 1 2 d μ − 4 ∣ M ∣ − 16 π e − t + 4 ∣ M ∣ e t + 16 π .

Now for 1 < p ≤ 3 , we can derive

T p ( t ) = ∫ M t σ 1 p − 1 d μ t ≤ ∫ M t σ 1 2 d μ t p − 1 2 ∣ M t ∣ 3 − p 2 .

T p ( t ) ≤ ∫ M σ 1 2 d μ − 4 ∣ M ∣ − 16 π e − t + 4 ∣ M ∣ e t + 16 π p − 1 2 e 3 − p 2 t ∣ M ∣ 3 − p 2 .

Then

Cap p ( K ) ≤ ∫ 0 ∞ T p − 1 ⁄ ( p − 1 ) ( t ) d t 1 − p ≤ ∫ 0 ∞ ∫ M σ 1 2 d μ − 4 ∣ M ∣ − 16 π e − t + 4 ∣ M ∣ e t + 16 π − 1 2 e − 3 − p 2 ( p − 1 ) t d t 1 − p ∣ M ∣ 3 − p 2 .

Finally, assume the equality holds. Then we can check that the modified Hawking mass m H ( M t ) is constant along the inverse mean curvature flow, which implies that the flow hypersurfaces M t are geodesic spheres. So ∂ K = M 0 is a geodesic sphere, and we finish the proof of Theorem 5.

Remark 28

In particular, for p = 2 , we obtain

Cap 2 ( K ) ≤ ∫ 0 ∞ ( a e − t + b e t + c ) − 1 2 e − 1 2 t d t − 1 ∣ M ∣ 1 2 ,

where

a = ∫ M σ 1 2 d μ − 4 ∣ M ∣ − 16 π , b = 4 ∣ M ∣ , c = 16 π .

If 4 a b > c 2 , we obtain

∫ 0 ∞ ( a e − t + b e t + c ) − 1 2 e − 1 2 t d t = − 1 a arsinh 2 a e − t + c 4 a b − c 2 0 ∞ = 1 a arsinh 2 a + c 4 a b − c 2 − 1 a arsinh c 4 a b − c 2 .

If 4 a b < c 2 , we obtain

∫ 0 ∞ ( a e − t + b e t + c ) − 1 2 e − 1 2 t d t = − 1 a arcosh 2 a e − t + c c 2 − 4 a b 0 ∞ = 1 a arcosh 2 a + c c 2 − 4 a b − 1 a arcosh c c 2 − 4 a b .

If 4 a b = c 2 , we obtain

∫ 0 ∞ ( a e − t + b e t + c ) − 1 2 e − 1 2 t d t = − 1 a log ( 2 a e − t + c ) 0 ∞ = 1 a log ( 2 a + c ) − 1 a log c .

4 The upper bound in H n via the unit-speed normal flow

In this section, we consider a convex hypersurface M 0 and use the unit-speed normal flow

(4.1) ∂ t X = ν ,

where X : N × [ 0 , T ) → H n is a smooth family of embeddings from a closed manifold N to H n .

Here, we use the hyperboloid model in the Minkowski space R n , 1 for H n . More precisely, the Minkowski space R n , 1 is the linear space R n + 1 equipped with the Lorentz metric

d s 2 = d x 1 2 + d x 2 2 + … + d x n 2 − d x n + 1 2 .

Then H n is viewed as the set

H n = { x ∈ R n , 1 ∣ x n + 1 = 1 + x 1 2 + x 2 2 + … + x n 2 }

with the induced metric.

So under the unit-speed normal flow (4.1), we have

X ( q , t ) = cosh t ⋅ X ( q , 0 ) + sinh t ⋅ ν ( q , 0 ) , ( q , t ) ∈ N × [ 0 , T ) .

Let { e i } i = 1 n − 1 be orthonormal principal directions in a neighborhood of a point X ( q , 0 ) on M 0 . Then the vectors

X ( ⋅ , t ) * ( e i ) = ( cosh t + sinh t ⋅ κ i ) e i

form an orthogonal frame in a neighborhood of the point X ( q , t ) . So

d μ t = ∏ i = 1 n − 1 ( cosh t + sinh t ⋅ κ i ) d μ = ∑ i = 0 n − 1 cosh n − 1 − i t ⋅ sinh i t ⋅ σ i d μ .

Therefore, noting that the level-set function ψ satisfies D ψ = ν , we obtain

T p ( t ) = ∫ M t ∣ D ψ ∣ p − 1 d μ t = ∣ M t ∣ = ∫ M ∑ i = 0 n − 1 cosh n − 1 − i t ⋅ sinh i t ⋅ σ i d μ ,

and then

Cap p ( K ) ≤ ∫ 0 ∞ T p − 1 ⁄ ( p − 1 ) ( t ) d t 1 − p = ∫ 0 ∞ ∫ M ∑ i = 0 n − 1 cosh n − 1 − i t ⋅ sinh i t ⋅ σ i d μ − 1 p − 1 d t 1 − p .

Finally, a direct computation shows that for a geodesic ball K = B r ¯ , the aforementioned inequality becomes an equality. So we finish the proof of Theorem 7.

Remark 29

Consider the special case n = 2 and p = 2 . Then we can compute to obtain a more transparent inequality

Cap 2 ( K ) ≤ ∫ M κ d μ 2 − ∣ M ∣ 2 log ∫ M κ d μ + ∫ M κ d μ 2 − ∣ M ∣ 2 ∣ M ∣ − 1 ,

provided ∫ M κ d μ ≥ ∣ M ∣ . For comparison, if K = B r ¯ , we have the explicit expression

Cap 2 ( B r ) = 2 π ∫ r ∞ sinh − 1 t d t − 1 = 4 π log cosh r + 1 cosh r − 1 − 1 ,

which is equal to the right-hand side of the aforementioned inequality.

On the other hand, if ∫ M κ d μ < ∣ M ∣ , we obtain

Cap 2 ( K ) ≤ 1 2 ∣ M ∣ 2 − ∫ M κ d μ 2 arctan ∣ M ∣ − ∫ M κ d μ ∣ M ∣ + ∫ M κ d μ − 1 .

5 The upper bound in R 3 via the weak inverse mean curvature flow

Proof of Theorem 9

In the setting of Theorem 9, we choose the test function f ( x ) = f ¯ ( ϕ ( x ) ) for some C 1 function f ¯ : [ 0 , ∞ ) → R satisfying f ¯ ( 0 ) = 1 and f ¯ ( ∞ ) = 0 to be determined. It is a classical fact that this kind of functions can be admissible test functions; see [4, Definition 1] and the remark below [23, Definition 1.2]. Therefore,

Cap p ( K ) ≤ ∫ U ∣ ∇ f ∣ p d x = ∫ U ( f ¯ ′ ( ϕ ( x ) ) ) p ∣ ∇ ϕ ∣ p d x .

By using the co-area formula, we obtain

∫ U ( f ¯ ′ ( ϕ ( x ) ) ) p ∣ ∇ ϕ ∣ p d x = ∫ 0 ∞ ( f ¯ ′ ( t ) ) p ∫ Σ t ∣ ∇ ϕ ∣ p − 1 d μ t d t .

Note that

∫ Σ t ∣ ∇ ϕ ∣ p − 1 d μ t = ∫ Σ t H p − 1 d μ t ≤ ∫ Σ t H 2 d μ t ( p − 1 ) ⁄ 2 ∣ Σ t ∣ ( 3 − p ) ⁄ 2 ≤ 16 π − e − t 16 π − ∫ Σ ′ H ′ 2 d μ ( p − 1 ) ⁄ 2 e 3 − p 2 t ∣ Σ ′ ∣ ( 3 − p ) ⁄ 2 ,

where we used the Hölder inequality and the monotonicity of the modified Hawking mass. Here, H ′ denotes the mean curvature of the surface Σ ′ . Thus, we obtain

∫ U ∣ ∇ f ∣ p d x ≤ ∫ 0 ∞ ( f ¯ ′ ( t ) ) p 16 π + e − t ∫ Σ ′ H ′ 2 d μ − 16 π ( p − 1 ) ⁄ 2 e 3 − p 2 t d t × ∣ Σ ′ ∣ ( 3 − p ) ⁄ 2 .

Meanwhile, note that by the Hölder inequality, we have

1 = ( f ¯ ( 0 ) ) p = − ∫ 0 ∞ f ¯ ′ ( t ) d t p ≤ ∫ 0 ∞ ( f ¯ ′ ( t ) ) p 16 π + e − t ∫ Σ ′ H ′ 2 d μ − 16 π ( p − 1 ) ⁄ 2 e 3 − p 2 t d t × ∫ 0 ∞ 16 π + e − t ∫ Σ ′ H ′ 2 d μ − 16 π − 1 ⁄ 2 e − 3 − p 2 ( p − 1 ) t d t p − 1 ,

with the equality when

f ¯ ′ ( t ) = c 16 π + e − t ∫ Σ ′ H ′ 2 d μ − 16 π − 1 ⁄ 2 e − 3 − p 2 ( p − 1 ) t , c ∈ R .

Set

s ′ ≔ ∫ Σ ′ H ′ 2 d μ 16 π − 1 .

So noticing f ¯ ( 0 ) = 1 and f ¯ ( ∞ ) = 0 , we may choose

f ¯ ( t ) = ∫ t ∞ ( 1 + s ′ e − τ ) − 1 ⁄ 2 e − 3 − p 2 ( p − 1 ) τ d τ ∫ 0 ∞ ( 1 + s ′ e − τ ) − 1 ⁄ 2 e − 3 − p 2 ( p − 1 ) τ d τ .

Then in this case, we obtain

∫ U ∣ ∇ f ∣ p d x ≤ ∫ 0 ∞ 16 π + e − t ∫ Σ ′ H ′ 2 d μ − 16 π − 1 ⁄ 2 e − 3 − p 2 ( p − 1 ) t d t 1 − p × ∣ Σ ′ ∣ ( 3 − p ) ⁄ 2 = ( 16 π ) p − 1 2 ∫ 0 ∞ ( 1 + s ′ e − t ) − 1 ⁄ 2 e − 3 − p 2 ( p − 1 ) t d t 1 − p ⋅ ∣ Σ ′ ∣ ( 3 − p ) ⁄ 2 .

As a result, we have

Cap p ( K ) ≤ ( 16 π ) p − 1 2 ∫ 0 ∞ ( 1 + s ′ e − t ) − 1 ⁄ 2 e − 3 − p 2 ( p − 1 ) t d t 1 − p ⋅ ∣ Σ ′ ∣ ( 3 − p ) ⁄ 2 .

Next we replace Σ ′ by Σ . First, recall that Σ ′ strictly minimizes area among all surfaces homologous to Σ . So ∣ Σ ′ ∣ ≤ ∣ Σ ∣ .

Second, because Σ is C 2 , the surface Σ ′ is C 1 , 1 , and moreover C ∞ where Σ ′ does not contact Σ . Besides, the mean curvature H ′ of Σ ′ satisfies

H ′ = 0 on Σ ′ \ Σ , and H ′ = H ≥ 0 a.e. on Σ ′ ∩ Σ .

Therefore, we see

∫ Σ ′ H ′ 2 d μ ≤ ∫ Σ H 2 d μ .

In conclusion, we derive

Cap p ( K ) ≤ ( 16 π ) p − 1 2 ∫ 0 ∞ ( 1 + s e − t ) − 1 ⁄ 2 e − 3 − p 2 ( p − 1 ) t d t 1 − p ⋅ ∣ Σ ∣ ( 3 − p ) ⁄ 2 ,

where

s ≔ ∫ Σ H 2 d μ 16 π − 1 .

If ∫ Σ H 2 d μ = 16 π , then Σ is a round sphere [5,22,27] with radius r 0 > 0 , and we can compute to obtain

Cap p ( K ) = 3 − p p − 1 p − 1 4 π r 0 3 − p .

If ∫ Σ H 2 d μ > 16 π , then s > 0 , and we take the change of variables

s e − t = r 2 ( p − 1 ) 3 − p .

Then we obtain

Cap p ( K ) ≤ 3 − p p − 1 p − 1 ( 4 π ) p − 1 2 ∣ Σ ∣ 3 − p 2 s 3 − p 2 θ 1 − p ,

where

θ ≔ ∫ 0 s 3 − p 2 ( p − 1 ) 1 + r 2 ( p − 1 ) 3 − p − 1 2 d r .

Next we claim that if ∫ Σ H 2 d μ > 16 π , then only the strict inequality can occur

Cap p ( K ) < 3 − p p − 1 p − 1 ( 4 π ) p − 1 2 ∣ Σ ∣ 3 − p 2 s 3 − p 2 θ 1 − p .

Otherwise assume the equality holds. Then checking the aforementioned proof, we see that

∣ Σ ′ ∣ = ∣ Σ ∣ , ∫ Σ ′ H ′ 2 d μ = ∫ Σ H 2 d μ , m ˜ H ( Σ t ) = m ˜ H ( Σ ′ ) , ∀ t > 0 ,

and f ( x ) = f ¯ ( ϕ ( x ) ) is a p -harmonic function on U with f ∣ Σ = 1 and f ( ∞ ) = 0 . So Σ is outer-minimizing and the modified Hawking mass m ˜ H ( Σ t ) is equal to m ˜ H ( Σ ) for all t . Moreover, since f ( x ) is p -harmonic in U , any level set of f ( x ) can not have nonempty interior by the strong maximum principle, and so the surfaces Σ t and Σ do not jump to Σ t ′ and Σ ′ , respectively, in the sense of [14] (meaning Σ t = Σ t ′ and Σ = Σ ′ ). Next fix any t > 0 and consider the exterior domain of Σ t in U . By using the fact f ( x ) is constant on Σ t and Σ t is at least C 1 , by the Hopf boundary lemma, we see that ∇ f never vanishes on Σ t . So ϕ = f ¯ − 1 ∘ f is a smooth function on U with ∇ ϕ ≠ 0 . It follows that the surfaces { Σ t } evolve smoothly by the inverse mean curvature flow. Then the equality case of m ˜ H ( Σ t ) ≥ m ˜ H ( Σ ) implies that H is constant on Σ t , and so Σ t is a round sphere. So Σ itself is a round sphere, which contradicts ∫ Σ H 2 d μ > 16 π . Therefore, when ∫ Σ H 2 d μ > 16 π , we have the strict inequality

Cap ( K ) < 3 − p p − 1 p − 1 ( 4 π ) p − 1 2 ∣ Σ ∣ 3 − p 2 s 3 − p 2 θ 1 − p .

So the proof of Theorem 9 is complete.□

6 The upper bound in R 3 via the inverse anisotropic mean curvature flow

Proof of Theorem 13

For the proof, we use the inverse anisotropic mean curvature flow. Recall the estimate (3.1) in [18],

Cap F , p ( K ) ≤ ∫ 0 ∞ T p 1 1 − p ( t ) d t 1 − p ,

where (see Section 3.2 in [18])

(6.1) T p ( t ) = ∫ Σ t H F p − 1 d μ F ≤ ∫ Σ t H F 2 d μ F p − 1 2 ( ∣ Σ t ∣ F ) 3 − p 2 ,

after using the Hölder inequality.

Next we define the modified anisotropic Hawking mass

m H F ( Σ t ) ≔ ∣ Σ t ∣ F 4 ∣ ∂ W ∣ F 1 − ∫ Σ t H F 2 d μ F 4 ∣ ∂ W ∣ F .

Lemma 30

Let Σ be a compact star-shaped F-mean convex surface without boundary in R 3 . Along the inverse anisotropic mean curvature flow (2.4) starting from Σ , the modified anisotropic Hawking mass m H F ( Σ t ) is nondecreasing in t. Moreover, if ( d ⁄ d t ) m H F ( Σ t ) = 0 at some time t > 0 , then Σ is a translated scaled Wulff shape.

Proof

First recalling the computation in the formula (3.2) in [18], we obtain (let p = 3 there)

d d t ∫ Σ t H F 2 d μ F = ∫ Σ t − 2 ∣ ∇ ˆ H F ∣ g ˆ 2 H F 2 − 2 ∣ h ˆ ∣ g ˆ 2 + H F 2 d μ F .

Here, g ˆ is the anisotropic first fundamental form, h ˆ is the anisotropic second fundamental form, and ∇ ˆ is the Levi-Civita connection corresponding to g ˆ on the surface; see [18] for more details. By using H F 2 = 2 K F + ∣ h ˆ ∣ g ˆ 2 and Lemma 25 in [18], we obtain

d d t ∫ Σ t H F 2 d μ F = ∫ Σ t − 2 ∣ ∇ ˆ H F ∣ g ˆ 2 H F 2 − H F 2 + 4 K F d μ F ≤ 4 ∣ ∂ W ∣ F − ∫ Σ t H F 2 d μ F .

Therefore in view of ( d ⁄ d t ) ∣ Σ t ∣ F = ∣ Σ t ∣ F , we conclude

d d t ∣ Σ t ∣ F 4 ∣ ∂ W ∣ F − ∫ Σ t H F 2 d μ F ≥ ∣ Σ t ∣ F 4 ∣ ∂ W ∣ F − ∫ Σ t H F 2 d μ F − ∣ Σ t ∣ F 4 ∣ ∂ W ∣ F − ∫ Σ t H F 2 d μ F = 0 .

So the modified anisotropic Hawking mass is nondecreasing in t .

Now assume ( d ⁄ d t ) m H F ( Σ t ) = 0 at some time t > 0 . Then checking the aforementioned argument we see that H F is constant on Σ t , which implies that Σ t is a translated scaled Wulff shape [12]. So the initial surface Σ is a translated scaled Wulff shape. The proof is complete.□

Note that m H F ( Σ t ) is monotone nondecreasing in t . So

m H F ( Σ ) ≤ m H F ( Σ t ) = ∣ Σ t ∣ F 4 ∣ ∂ W ∣ F 1 − ∫ Σ t H F 2 d μ F 4 ∣ ∂ W ∣ F ,

which means

∫ Σ t H F 2 d μ F ≤ 4 ∣ ∂ W ∣ F 1 − 4 ∣ ∂ W ∣ F ∣ Σ t ∣ F m H F ( Σ ) .

Setting

(6.2) s = ∫ Σ H F 2 d μ F 4 ∣ ∂ W ∣ F − 1 ,

and noting ∣ Σ t ∣ F = ∣ Σ ∣ F e t , we obtain

∫ Σ t H F 2 d μ F ≤ 4 ∣ ∂ W ∣ F ( 1 + s e − t ) .

Consequently, from (6.1), we obtain

T p ( t ) ≤ ( 4 ∣ ∂ W ∣ F ) p − 1 2 ( 1 + s e − t ) p − 1 2 ( ∣ Σ t ∣ F ) 3 − p 2 = ( 4 ∣ ∂ W ∣ F ) p − 1 2 ∣ Σ ∣ F 3 − p 2 ( 1 + s e − t ) p − 1 2 e 3 − p 2 t ,

where we used ∣ Σ t ∣ F = ∣ Σ ∣ F e t .

So, we obtain

Cap F , p ( K ) ≤ ∫ 0 ∞ T p 1 1 − p ( t ) d t 1 − p ≤ ( 4 ∣ ∂ W ∣ F ) p − 1 2 ∣ Σ ∣ F 3 − p 2 ∫ 0 ∞ ( 1 + s e − t ) − 1 ⁄ 2 e 3 − p 2 ( 1 − p ) t d t 1 − p .

If m H F ( Σ ) = 0 , then by Proposition 26 in [18], the surface Σ is a translated scaled Wulff shape r 0 ∂ W + x 0 ( r 0 > 0 , x 0 ∈ R 3 ), and we can compute directly to obtain

Cap F , p ( K ) = 3 − p p − 1 p − 1 ∣ ∂ W ∣ F r 0 3 − p .

If m H F ( Σ ) < 0 , then using the change of variables

s e − t = r 2 ( p − 1 ) 3 − p ,

we obtain by direct computation

Cap F , p ( K ) ≤ 3 − p p − 1 p − 1 ∣ ∂ W ∣ F p − 1 2 ∣ Σ ∣ F 3 − p 2 s 3 − p 2 θ 1 − p ,

where

θ ≔ ∫ 0 s 3 − p 2 ( p − 1 ) 1 + r 2 ( p − 1 ) 3 − p − 1 2 d r .

Should the equality hold, then (6.1) must be an equality. The Hölder inequality becomes an equality, which implies that H F is constant. So Σ t and then Σ are translated scaled Wulff shapes ([12]), which is impossible. Thus, we cannot have the equality in this case. The proof is now complete.□

Proof of Corollary 15

Let p = 2 in Theorem 13. If m H F ( Σ ) = 0 , then ∫ Σ H F 2 d μ F = 4 ∣ ∂ W ∣ F and the conclusion follows immediately.

If m H F ( Σ ) < 0 , then ∫ Σ H F 2 d μ F > 4 ∣ ∂ W ∣ F , and we obtain

Cap F , 2 ( K ) < ∣ ∂ W ∣ F ∣ Σ ∣ F s 1 ⁄ 2 ⋅ θ − 1 ,

where

θ = ∫ 0 s ( 1 + r 2 ) − 1 2 d r = arsinh s and s = ∫ Σ H F 2 d μ F 4 ∣ ∂ W ∣ F − 1 .

Thus, we obtain

Cap F , 2 ( K ) < ∣ ∂ W ∣ F ∣ Σ ∣ F s arsinh s .

So we finish the proof of Corollary 15.□

Acknowledgements

The authors would like to thank the anonymous referee for the careful reading of the paper and the valuable comments which improve the paper a lot.

Funding information: The first and second authors were partially supported by NSFC Grant no. 12471047. The third author was supported by National Key R and D Program of China 2021YFA1001800, NSFC Grant no. 12171334, and the funding (no. 1082204112549) from Sichuan University.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. All authors have made the equal contribution to the manuscript.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

References

[1] V. Agostiniani, M. Fogagnolo, and L. Mazzieri, Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature, Invent. Math. 222 (2020), no. 3, 1033–1101, https://doi.org/10.1007/s00222-020-00985-4. Suche in Google Scholar

[2] V. Agostiniani and L. Mazzieri, Riemannian aspects of potential theory, J. Math. Pures Appl. (9) 104 (2015), no. 3, 561–586, https://doi.org/10.1016/j.matpur.2015.03.008. Suche in Google Scholar

[3] V. Agostiniani and L. Mazzieri, Monotonicity formulas in potential theory, Calc. Var. Partial Differential Equations 59 (2020), no. 1, Paper No. 6, 32 pp, https://doi.org/10.1007/s00526-019-1665-2. Suche in Google Scholar

[4] H. Bray and P. Miao, On the capacity of surfaces in manifolds with nonnegative scalar curvature, Invent. Math. 172 (2008), no. 3, 459–475, https://doi.org/10.1007/s00222-007-0102-x. Suche in Google Scholar

[5] B.-Y. Chen, On the total curvature of immersed manifolds. I. An inequality of Fenchel–Borsuk–Willmore, Amer. J. Math. 93 (1971), 148–162, https://doi.org/10.2307/2373454. Suche in Google Scholar

[6] A. Freire and F. Schwartz, Mass-capacity inequalities for conformally flat manifolds with boundary, Comm. Partial Differential Equations 39 (2014), no. 1, 98–119, https://doi.org/10.1080/03605302.2013.851211. Suche in Google Scholar

[7] C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom. 32 (1990), no. 1, 299–314, https://doi.org/10.4310/jdg/1214445048. Suche in Google Scholar

[8] C. Gerhardt, Inverse curvature flows in hyperbolic space, J. Differential Geom. 89 (2011), no. 3, 487–527. DOI: https://doi.org/10.4310/jdg/1335207376.10.4310/jdg/1335207376Suche in Google Scholar

[9] A. Grigor’yan, Isoperimetric inequalities and capacities on Riemannian manifolds, The Mazaya anniversary collection, Vol. 1 (The Maz’ya anniversary collection, Vol. 1 (Rostock, 1998), 139–153, Oper. Theory Adv. Appl., vol. 109, Birkhäuser, Basel, 1999, https://doi.org/10.1007/978-3-0348-8675-8_9. Suche in Google Scholar

[10] A. Grigor’yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 2, 135–249. DOI: https://www.ams.org/journals/bull/1999-36-02/S0273-0979-99-00776-4/.10.1090/S0273-0979-99-00776-4Suche in Google Scholar

[11] Y. He and H. Li, Integral formula of Minkowski type and new characterization of the Wulff shape, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 4, 697–704, https://doi.org/10.1007/s10114-007-7116-6. Suche in Google Scholar

[12] Y. He, H. Li, H. Ma, and J. Ge, Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures, Indiana Univ. Math. J. 58 (2009), no. 2, 853–868, http://www.jstor.org/stable/24903217. 10.1512/iumj.2009.58.3515Suche in Google Scholar

[13] J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear potential theory of degenerate elliptic equations, Unabridged republication of the 1993 original, Dover Publications, Inc., Mineola, NY, 2006. DOI: https://store.doverpublications.com/products/9780486824253?srsltid=AfmBOopK4YsA-SnMOJVIYYdExDxvnvrOYt5HxuGH5yc1vfHyy68wQ0v3.Suche in Google Scholar

[14] G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353–437, https://doi.org/10.4310/jdg/1090349447. Suche in Google Scholar

[15] P.-K. Hung and M.-T. Wang, Inverse mean curvature flows in the hyperbolic 3-space revisited, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 119–126, https://doi.org/10.1007/s00526-014-0780-3. Suche in Google Scholar

[16] X. Jin, The relative volume function and the capacity of sphere on asymptotically hyperbolic manifolds, arXiv:2207.02012v3. DOI: https://arxiv.org/abs/2207.02012.Suche in Google Scholar

[17] H. Li, Y. Wei, and C. Xiong, A geometric inequality on hypersurface in hyperbolic space, Adv. Math. 253 (2014), 152–162, https://doi.org/10.1016/j.aim.2013.12.003. Suche in Google Scholar

[18] R. Li and C. Xiong, Sharp bounds for the anisotropic p-capacity of Euclidean compact sets, J. Differential Equations 317 (2022), 196–224, https://doi.org/10.1016/j.jde.2022.02.005. Suche in Google Scholar

[19] M. Ludwig, J. Xiao, and G. Zhang, Sharp convex Lorentz-Sobolev inequalities, Math. Ann. 350 (2011), no. 1, 169–197, https://doi.org/10.1007/s00208-010-0555-x. Suche in Google Scholar

[20] V. Maz’ya, Sobolev spaces with applications to elliptic partial differential equations, Second, revised and augmented edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011. DOI: https://link.springer.com/book/10.1007/978-3-642-15564-2.10.1007/978-3-642-15564-2Suche in Google Scholar

[21] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N.J., 1951, https://www.jstor.org/stable/j.ctt1b9rzzn. Suche in Google Scholar

[22] M. Ritoré and C. Sinestrari, Mean curvature flow and isoperimetric inequalities, Edited by Vicente Miquel and Joan Porti. Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2010. p. viii.113. DOI: https://link.springer.com/book/10.1007/978-3-0346-0213-6.10.1007/978-3-0346-0213-6Suche in Google Scholar

[23] M. Shubin, Capacity and its applications, https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.1404&rep=rep1&type=pdf. Suche in Google Scholar

[24] G. Szegö, Uber einige neue Extremaleigenschaften der Kugel, Math. Z. 33 (1931), no. 1, 419–425, https://doi.org/10.1007/BF01174362. Suche in Google Scholar

[25] J. I. E. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990), no. 3, 355–372, https://doi.org/10.1007/BF02571249. Suche in Google Scholar

[26] G. Wang and C. Xia, Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space, Adv. Math. 259 (2014), 532–556, https://doi.org/10.1016/j.aim.2014.01.024. Suche in Google Scholar

[27] T. J. Willmore, Mean curvature of immersed surfaces, An. Şti. Univ. “All. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 14 (1968), 99–103. DOI: https://mathscinet.ams.org/mathscinet/2006/mathscinet/search/publdoc.html?pg1=INDI&s1=183255&sort=Newest&vfpref=html&r=40&mx-pid=238220.Suche in Google Scholar

[28] C. Xia, Inverse anisotropic mean curvature flow and a Minkowski type inequality, Adv. Math. 315 (2017), 102–129, https://doi.org/10.1016/j.aim.2017.05.020. Suche in Google Scholar

[29] C. Xia and J. Yin, Anisotropic p-capacity and anisotropic Minkowski inequality, Sci. China Math. 65 (2022), no. 3, 559–582, https://doi.org/10.1007/s11425-021-1884-1. Suche in Google Scholar

[30] J. Xiao, The p-harmonic capacity of an asymptotically flat 3-manifold with non-negative scalar curvature, Ann. Henri Poincaré 17 (2016), no. 8, 2265–2283, https://doi.org/10.1007/s00023-016-0475-8. Suche in Google Scholar

[31] J. Xiao, P-capacity vs surface-area, Adv. Math. 308 (2017), 1318–1336, https://doi.org/10.1016/j.aim.2017.01.007. Suche in Google Scholar

[32] J. Xiao, A maximum problem of S.-T. Yau for variational p-capacity, Adv. Geom. 17 (2017), no. 4, 483–496, https://doi.org/10.1515/advgeom-2017-0035. Suche in Google Scholar

[33] J. Xiao and N. Zhang, Flux & radii within the subconformal capacity, Calc. Var. Partial Differential Equations 60 (2021), no. 3, 120, 30, https://doi.org/10.1007/s00526-021-01989-5. Suche in Google Scholar

[34] C. Xiong, Sharp bounds for the first two eigenvalues of an exterior Steklov eigenvalue problem, DOI: https://arxiv.org/abs/2304.11297.Suche in Google Scholar

Received: 2023-08-27

Revised: 2024-01-08

Accepted: 2025-01-24

Published Online: 2025-03-04

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hyperbolic space; capacity; inverse mean curvature flow; modified Hawking mass

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