Total mean curvatures of Riemannian hypersurfaces

1 Introduction

Total mean curvatures of a hypersurface Γ in a Riemannian n -manifold M are integrals of symmetric functions of its principal curvatures. These quantities are known as quermassintegrals or mixed volumes when Γ is convex and M is the Euclidean space. They are fundamental in geometric variational problems, as they feature in Steiner’s polynomial, Brunn-Minkowski theory, and Alexandrov-Fenchel inequalities [11,14,19,20], which were all originally developed in Euclidean space. Extending these notions to Riemannian manifolds has been a major topic of investigation. In particular, total mean curvatures have been studied extensively in hyperbolic space in recent years [2,22, 23,24]. Here, we study these integrals in the broader setting of Cartan-Hadamard spaces, i.e., complete simply connected manifolds of nonpositive curvature and generalize a number of inequalities that had been established in Euclidean or hyperbolic space.

The main result of this article, Theorem 3.1, expresses the difference between the total r th mean curvatures of a pair of nested hypersurfaces Γ and γ in a Riemannian manifold M in terms of the sectional curvatures of M and the principal curvatures of a family of hypersurfaces that fibrate the region between Γ and γ . This formula simplifies when r = 1 , Γ and γ are parallel, or M has constant curvature, leading to a number of applications. In particular, we establish the monotonicity property of the total first mean curvature for nested convex hypersurfaces in Cartan-Hadamard manifolds (Corollary 4.1). This leads to a sharp lower bound in dimension 3 for the total first mean curvature in terms of the volume bounded by Γ (Corollary 4.3), which generalizes a result of Gallego-Solanes in hyperbolic 3-space [10, Cor. 3.2]. We also extend to all mean curvatures some monotonicity results of Schroeder-Strake [21] and Borbely [4] for total Gauss-Kronecker curvature (Corollaries 4.4 and 4.5). Finally, we include a characterization of hyperbolic balls as minimizers of total mean curvatures among balls of equal radii in Cartan-Hadamard manifolds (Corollary 4.7).

Theorem 3.1 is a generalization of the comparison result we had obtained earlier in [13] for the Gauss-Kronecker curvature, motivated by Kleiner’s approach to the Cartan-Hadamard conjecture on the isoperimetric inequality [16]. Similar to [13], our starting point here, in Section 2, will be an identity (Lemma 2.1) for the divergence of Newton operators, which were developed by Reilly [18,17] to study the invariants of Hessians of functions on Riemannian manifolds. This formula, together with Stokes’ theorem, leads to the proof of Theorem 3.1 in Section 3. Then, in Section 4, we develop the applications of that result.

2 Newton operators

Throughout this work, M denotes an n -dimensional Riemannian manifold with metric ⟨ ⋅ , ⋅ ⟩ and covariant derivative ∇ . Furthermore, u is a C 1 , 1 function on M . In particular, u is twice differentiable at almost every point p of M , and the computations below take place at such a point. The gradient of u is the tangent vector ∇ u ∈ T p M given by ⟨ ∇ u ( p ) , X ⟩ ≔ ∇ X u for all X ∈ T p M . The Hessian operator ∇ 2 u : T p M → T p M is the self-adjoint linear map given by ∇ 2 u ( X ) ≔ ∇ X ( ∇ u ) . The symmetric elementary functions σ r : R k → R , for 1 ≤ r ≤ k , and x = ( x 1 , … , x k ) are defined by

We set σ 0 ≔ 1 and σ r ≔ 0 for r ≥ k + 1 by convention. Let λ ( ∇ 2 u ) ≔ ( λ 1 , … , λ n ) denote the eigenvalues of ∇ 2 u . Then, we set

These functions form the coefficients of the characteristic polynomial

Let δ j 1 … j m i 1 … i m be the generalized Kronecker tensor, which is equal to 1 ( − 1 ) if i 1 , … , i m are distinct and ( j 1 , … , j m ) is an even (odd) permutation of ( i 1 , … , i m ) ; otherwise, it is equal to 0. Then [18, Prop. 1.2(a)],

where u i j ≔ ∇ i j u denote the second partial derivatives of u with respect to an orthonormal frame E i ∈ T p M , which we extend to an open neighborhood of p by parallel translation along geodesics. So ∇ E i E j = 0 at p . We call E i a local parallel frame centered at p and set ∇ i ≔ ∇ E i , ∇ i j ≔ ∇ i ∇ j . Each of the indices in (1) ranges from 1 to n , and we employ Einstein’s convention by summing over repeated indices throughout the article. The Newton operators T r u : T p M → T p M [17,18] are defined recursively by setting T 0 u ≔ I , the identity map, and for r ≥ 1 ,

Thus, T r u is the truncation of the polynomial P ( ∇ 2 u ) obtained by removing the terms of order higher than r . In particular, T n u = P ( ∇ 2 u ) . So, by the Cayley-Hamilton theorem, T n u = 0 . Consequently, when ∇ 2 u is nondegenerate, (2) yields that

where T u is the Hessian cofactor operator discussed in [13, Sec. 4]. See [18, Prop. 1.2] for other basic identities that relate σ and T . In particular, by [18, Prop. 1.2(c)], we have Trace ( T r u ⋅ ∇ 2 u ) = ( r + 1 ) σ r + 1 ( ∇ 2 u ) . So, by Euler’s identity for homogeneous polynomials,

Thus, it follows from (1) that

Furthermore, by [17, Prop. 1(11)] (note that the sign of the Riemann tensor R in [17] is opposite to the one in this article), we have

where R i j k l ≔ R ( E i , E j , E k , E ℓ ) = ⟨ ∇ i ∇ j E k − ∇ j ∇ i E k , E ℓ ⟩ . Another useful identity [17, p. 462] is

where ⟨ ⋅ , ⋅ ⟩ here indicates the Frobenius inner product (i.e., ⟨ A , B ⟩ ≔ A i j B i j for any pair of matrices of the same dimension). The divergence of T r u may be defined by virtually the same argument used for T u in [13, Sec. 4] to yield the following generalization of [13, (14)]:

Recall that T u = T n − 1 u by (3). Furthermore, T n u = 0 as we mentioned earlier. Thus, the following observation generalizes [13, Lem. 4.2].

Below we assume, as was the case in [13, Sec. 4], that all local computations take place with respect to a principal curvature frame E i ∈ T p M of u , which is defined as follows. Assuming ∣ ∇ u ( p ) ∣ ≠ 0 , we set E n ≔ ∇ u ( p ) / ∣ ∇ u ( p ) ∣ , and let E 1 , … , E n − 1 be the principal directions of the level set of u passing through p . Then, we extend E i to a local parallel frame near p . The first partial derivatives of u with respect to E i , u i ≔ ∇ i u , satisfy

Furthermore, for the second partial derivatives, u i j = ∇ i j u , we have

where κ 1 u , … , κ n − 1 u are the principal curvatures of level sets of u with respect to E n , i.e., they are eigenvalues corresponding to E 1 , … , E n − 1 of the shape operator X ↦ ∇ X ν on the tangent space of level sets of u , where ν ≔ ∇ u / ∣ ∇ u ∣ . We set κ u ≔ ( κ 1 u , … , κ n − 1 u ) . So σ r ( κ u ) is the r th mean curvature of the level set of u at p . In particular, σ n − 1 ( κ u ) is the Gauss-Kronecker curvature of the level sets. The next observation generalizes [13, Lem. 4.1].

3 Comparison formula

Here, we establish the main result of this work. For a C 1 , 1 hypersurface Γ in a Riemannian n -manifold M , oriented by a choice of normal vector field ν , and 0 ≤ r ≤ n − 1 , we let

be the total r th mean curvature of Γ , where κ ≔ ( κ 1 , … , κ n − 1 ) denotes principal curvatures of Γ with respect to ν . Note that ℳ 0 ( Γ ) = ∣ Γ ∣ , the volume of Γ , since σ 0 = 1 , and ℳ n − 1 ( Γ ) is the total Gauss-Kronecker curvature of Γ (denoted by G ( Γ ) in [13]). A domain Ω ⊂ M is an open set with a compact closure cl ( Ω ) . If Γ bounds a domain Ω , then by convention we set ℳ − 1 ( Γ ) ≔ ∣ Ω ∣ , the volume of Ω . The following theorem generalizes [13, Thm. 4.7], where this result had been established for r = n − 1 . It also uses less regularity than was required in [13, Thm. 4.7].

4 Applications

Here, we develop some consequences of Theorem 3.1. A subset of a Cartan-Hadamard manifold M is convex if it contains the (unique) geodesic segment connecting every pair of its points. A convex hypersurface Γ ⊂ M is the boundary of a compact convex set with interior points. If Γ is of class C 1 , 1 , then its principal curvatures are nonnegative at all twice differentiable points with respect to the outward normal. Conversely, if the principal curvatures of a closed hypersurface Γ ⊂ M are all nonnegative, then Γ is convex [1]. See [13, Sec. 2 and 3] for the basic properties of convex sets in Cartan-Hadamard manifolds. A set is nested inside Γ if it lies in the convex domain bounded by Γ .

Dekster [9] constructed examples of nested convex hypersurfaces in Cartan-Hadamard manifolds where the monotonicity property in the last result does not hold for Gauss-Kronecker curvature. So the aforementioned corollary cannot be extended to all mean curvatures without further assumptions, which we will discuss below. First, we need to record the following observation.

Gallego and Solanes showed [10, Cor. 3.2] that if Γ is a convex hypersurface bounding a domain Ω in a hyperbolic n -space of constant curvature a < 0 , then

When comparing formulas, note that in [10], mean curvature is defined as the average of κ i , as opposed to the sum of κ i , which is our convention. Large balls show that the above inequality is sharp. Here, we extend this inequality to Cartan-Hadamard 3-manifolds as follows:

We say Γ is an outer parallel hypersurface of a convex hypersurface γ if all points of Γ are at a constant distance λ ≥ 0 from the convex domain bounded by γ . Since the distance function of a convex set in a Cartan-Hadamard manifold is convex [5, Prop. 2.4], Γ is convex. Furthermore, Γ is C 1 , 1 for λ > 0 [13, Lem. 2.6]. The following corollary generalizes [13, Cor. 5.3] and a theorem of Schroeder-Strake [21, Thm. 3], where this result was established for Gauss-Kronecker curvature; see also [13, Note 6.9].

The next result generalizes [13, Cor. 5.2] and observation of Borbely [4, Thm. 1] for Gauss-Kronecker curvature.

The above result had been observed earlier by Solanes [22, Cor. 9]. It is due to the integral formula for quermassintegrals [22, Def. 2.1], which immediately yields that quermassintegrals of convex domains are increasing with respect to inclusion. Monotonicity of total mean curvatures follows due to a formula [22, Prop. 7] relating quermassintegrals to total mean curvatures. As an application of the last corollary, one may extend the definition of total mean curvatures to non-regular convex hypersurfaces as follows. If Γ is a convex hypersurface in a Cartan-Hadamard manifold, then its outer parallel hypersurface at distance ε , denoted by Γ ε , is C 1 , 1 for all ε > 0 [13, Lem. 2.6]. So ℳ r ( Γ ε ) is well defined. By Corollary 4.4, ℳ r ( Γ ε ) is decreasing in ε . Hence, its limit as ε → 0 exists, and we may set ℳ r ( Γ ) ≔ lim ε → 0 ℳ r ( Γ ε ) .

Next, we derive a formula that appears in Solanes [22, (1) and (2)] and follows from Gauss-Bonnet-Chern theorems [8,7]; see also [22, Cor. 8]. Here k !! , when k is a positive integer, stands for the product of all positive odd (even) integers up to k , when k is odd (even). For k ≤ 0 , we set k !! = 1 .