Comparison formulas for total mean curvatures of Riemannian hypersurfaces

1 Introduction

The total rth mean curvature of an oriented C 1,1 hypersurface Γ in a Riemannian n-manifold M, for 0 ≤ r ≤ n − 1, is given by

where κ ≔ (κ ₁, …, κ _n−1) denotes the principal curvatures of Γ, with respect to the choice of orientation, and σ _r : R ⁿ⁻¹ → R is the rth symmetric function; so

We set σ ₀ ≔ 1, and σ _r ≔ 0 for r ≥ n by convention. Thus M 0 ( Γ ) is the (n − 1)-dimensional volume, M 1 ( Γ ) is the total mean curvature, and M n − 1 ( Γ ) is the total Gauss-Kronecker curvature of Γ. Up to multiplicative constants, these quantities form the coefficients of Steiner’s polynomial, and are known as quermassintegrals when Γ is a convex hypersurface in Euclidean space. The following result was established in [1, Thm. 3.1] generalizing earlier work in [2, Thm. 4.7]:

In [1], the above theorem was established via Reilly’s identities [3]. Here we present a somewhat shorter and conceptually simpler proof using differential forms which we construct after Chern [4], as Borbély [5], [6] had also done earlier. More specifically, we devise a differential (n − 1)-form Φ _r on M so that M r ( Γ i ) correspond to integration of Φ _r on Γ _i . Then computing the exterior derivative dΦ _r yields (1) via Stokes theorem. Various applications of Theorem 1.1 are developed in [2], [7], including total curvature bounds, and rigidity results in Riemannian geometry. See also [1] for more results of this type.

2 Basic formulas

As in the statement of Theorem 1.1, we let M be a compact orientable Riemannian n-manifold with boundary ∂M = Γ₁ ∪ Γ₀. Furthermore, ⟨⋅, ⋅⟩ denotes the metric on M, with induced norm |⋅| ≔ ⟨⋅,⋅⟩^1/2, connection ∇, and curvature operator

for vector fields X, Y, Z on M. The sectional curvature of M with respect to a pair of orthonormal vectors x, y in the tangent space T _p M may be defined as

where X, Y are local extensions of x, y. With u as in the statement of Theorem 1.1, and for 0 ≤ t ≤ 1, let Γ _t ≔ u ⁻¹(t) be the level hypersurface of u at height t. Since u is C 1,1 , Γ _t is twice differentiable almost everywhere by Rademacher’s theorem. At every such point p of Γ _t , let e _i , i = 1, …, n, be the orthonormal frame mentioned above, i.e.,

and e ₁, …, e _n−1 form a set of orthonormal principal directions of Γ _t at p. Furthermore we assume that e _i is positively oriented, i.e.,

where dvol _M denotes the volume form of M. We call e _i a principal frame associated to (level sets of) u. Let θ ⁱ be the corresponding dual one forms on T _p M given by

where δ j i is the Kronecker function. Note that e _i may be extended to a C 1 orthonormal frame e ̄ i in a neighborhood of p in M so that e ̄ n = e n and thus e ̄ 1 , … , e ̄ n − 1 remain tangent to Γ _t (though they may no longer be principal directions). The corresponding connection 1-forms on T _p M are then given by

for 1 ≤ i, j ≤ n. Since e _i , i = 1, …, n − 1 are principal directions, and e ̄ n = e n is the normal of Γ _t ,

where κ _i are the principal curvatures of Γ _t with respect to e _n . We also record that,

where | ∇ u | i = ∇ e i | ∇ u | , and the second equality is due to the symmetry of the Hessian of u. Next, we compute ω i j for i, j ≠ n. We may assume that e ̄ 1 , … , e ̄ n − 1 are parallel translations of e ₁, …, e _n−1 on Γ _t , i.e., ∇ ̄ e i e ̄ j = 0 , for 1 ≤ i, j ≤ n − 1 where ∇ ̄ ≔ ∇ ⊤ is the induced connection on Γ _t . Then ω i j ( e k ) = ⟨ ∇ ̄ e k e ̄ j , e i ⟩ = 0 , for 1 ≤ i, j, k ≤ n − 1. Furthermore, we may assume that e ̄ 1 , … , e ̄ n − 1 are parallel translated along integral curves of e _n . Then ∇ e n e ̄ i = 0 for 1 ≤ i ≤ n − 1, which yields ω i j ( e n ) = 0 , for 1 ≤ i, j ≤ n − 1. So we record that

Cartan’s structure equations state that

where Ω j i are the curvature 2-forms given by

Note that R _ℓkij = −R _kℓij . We also set

Finally we record some basic formulas from exterior algebra which will be used in the next section. If λ is a k-form, and ϕ is an ℓ-form, then

where the sum ranges over 1 ≤ i ₁, …, i _k+ℓ ≤ k + ℓ, with i ₁ < … < i _k , and i _k+1 < … < i _k+ℓ ; furthermore, ɛ(i ₁ … i _n ) ≔ 1, or −1 depending on whether i ₁ … i _n is an even or odd permutation of 1 … n respectively. Note that

since ɛ(i ₁ … i _n−1) = ɛ(i ₁ … i _n−1 n). The following identities will also be useful

where the sums range over 1 ≤ i ₁, …, i _k ≤ k with i ₂ < … < i _k in the first sum, and i ₁ < … < i _k−1 in the second sum.

3 Proof of Theorem 1.1

Let θ ⁱ be the dual 1-forms, and ω j i be the connection forms corresponding to the principal frame e _i of u discussed in the last section. For 0 ≤ r ≤ n − 1, we define the (n − 1)-forms

where the sum ranges over 1 ≤ i ₁, …, i _n−1 ≤ n − 1 with i ₁ < … < i _r , and i _r+1 < … < i _n−1. For r = n − 1, this form appears in Chern [4], and later in Borbély [6] (where it is denoted as “Φ₀” and “Φ” respectively). The form Φ₁ has also been used by Borbély in [5]. One quickly checks, using (3), (4), and (9), that

which is the main feature of these forms. Recall that Γ _t ≔ u ⁻¹(t) is the level hypersurface of u at height t, for 0 ≤ t ≤ 1. Let Φ r | Γ t denote the pull back of Φ _r via the inclusion map Γ _t → M. Since Φ r | Γ t is an (n − 1)-form on Γ _t , it is a multiple of the volume form of Γ _t , which is given by

Note that here we have used the assumption (2) that e _i is positively oriented. So it follows from (12) and (13) that

This shows that Φ _r depends only on e _n , not the choice of e ₁, …, e _n−1 (which also follows from transformation rules for ω n i and θ ⁱ under a change of frame e i → e i ′ with e n = e n ′ ; see [5, p. 269]). In addition, (14) shows that

Consequently, by Stokes theorem, for the left hand side of (1) we have

Here we have used the assumption that u | Γ 1 > u | Γ 0 , which ensures that e _n points outward on Γ₁ and inward on Γ₀ with respect to M. Furthermore, since Φ _r depends only on e _n and u is C 1,1 , it follows that Φ _r is Lipschitz (in local coordinates). Hence dΦ _r is integrable, and the use of Stokes theorem here is justified.

Next we compute dΦ _r . Since ω n i 1 ∧ … ∧ ω n i r is an r-form, the product rule for exterior differentiation yields that

where the sums still range over i ₁ < … < i _r and i _r+1 < … < i _n−1. By (11), the structure equations (7), and (6), the first term in (16) reduces to

where the sums now range over i ₁ < … < i _r , and i _r+2 < … < i _n−1. The factor (r + 1) appears in the last line because definition of Φ _r+1 requires that i ₁ < … < i _r+1. Applying (11) and (7) also to the second term in (16), we obtain

where the sum ranges over i ₁ < … < i _r−1, and i _r+1 < … < i _n−1. For r = 1, this formula had been computed earlier by Borbély [5, (6)].

By (15), it remains to show that (−1) ⁿ⁻¹ ∫ _M dΦ _r yields the right hand side of (1). To see this first note that, by (9) and (12),

Thus the first term on the right hand side of (17) quickly yields the first integral on the right hand side of (1). To obtain the second integral there, we evaluate the sum in (17) at e _i , which yields

where the sums range over 1 ≤ j ₁ … j _n ≤ n with j _r < j _r+1 by (9), and the range for 1 ≤ i ₁, …, i _n−1 ≤ n − 1 remains as in (17), i.e., i ₁ < … < i _r−1, and i _r+1 < … < i _n−1. The last sum may be partitioned into A + B, where A consists of terms with j _r+1 = n, and B of terms with j _r+1 ≠ n. If j _r+1 = n, then j ₁, …, j _r−1 ≠ n, which yields i _k = j _k+1 for k = 1, …, r − 2 by (4). This in turn forces j _r = i _r , as they are the only remaining indices. So by (10) and (8),

where we still have i ₁ < … < i _r−1. This yields the first term in the second integral in (1), after multiplication by the sign factors (−1) ^r−1 from (17) and (−1) ⁿ⁻¹ from (15), which ensures the desired sign −1. Next, to compute B, note that if j _r+1 ≠ n, then j _r ≠ n either, since j _r < j _r+1, which forces j _k = n, for some 1 ≤ k ≤ r − 1. We may assume k = r − 1 after reindexing. Then j ₁, …, j _r−2 ≠ n, which yields i _k = j _k for k = 1, …, r − 2 by (4). So by (5)

where the second equality holds because {j _r , j _r+1} = {i _r−1, i _r }, since these are the only remaining indices. We may assume then that j _r = i _r−1, and j _r+1 = i _r , since switching j _r and j _r+1 does not change the sign of the right hand side of the first equality for B. The sign (−1) ^n−r in the third equality is due to (10) and switching two indices in the Riemann tensor coefficient. Finally note that the restriction on the range of indices in the last sum is now i ₁ < … < i _r−2, since i _r−1 corresponds to j _r−1, and we set r − 1 = k during the reindexing above. So B yields the second term in the second integral in (1), after multiplication by (−1) ^r−1 and (−1) ⁿ⁻¹, as was the case for A, which ensures the desired sign +1. This concludes the proof of Theorem 1.1.