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Total mean curvatures of Riemannian hypersurfaces

Total mean curvatures of Riemannian hypersurfaces 1IntroductionTotal mean curvatures of a hypersurface Γ\Gamma in a Riemannian nn-manifold MMare integrals of symmetric functions of its principal curvatures. These quantities are known as quermassintegrals or mixed volumes when Γ\Gamma is convex and MMis 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 rthr{\rm{th}}mean curvatures of a pair of nested hypersurfaces Γ\Gamma and γ\gamma in a Riemannian manifold MMin terms of the sectional curvatures of MMand the principal curvatures of a family of hypersurfaces that fibrate the region between Γ\Gamma and γ\gamma . This formula simplifies when r=1r=1, Γ\Gamma and γ\gamma are parallel, or MMhas 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\Gamma (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.2Newton operatorsThroughout this work, MMdenotes an nn-dimensional Riemannian manifold with metric ⟨⋅,⋅⟩\langle \cdot ,\cdot \rangle and covariant derivative ∇\nabla . Furthermore, uuis a C1,1{{\mathcal{C}}}^{1,1}function on MM. In particular, uuis twice differentiable at almost every point ppof MM, and the computations below take place at such a point. The gradient of uuis the tangent vector ∇u∈TpM\nabla u\in {T}_{p}Mgiven by ⟨∇u(p),X⟩≔∇Xu\langle \nabla u\left(p),X\rangle := {\nabla }_{X}ufor all X∈TpMX\in {T}_{p}M. The Hessian operator ∇2u:TpM→TpM{\nabla }^{2}u:{T}_{p}M\to {T}_{p}Mis the self-adjoint linear map given by ∇2u(X)≔∇X(∇u).{\nabla }^{2}u\left(X):= {\nabla }_{X}\left(\nabla u).The symmetric elementary functions σr:Rk→R{\sigma }_{r}:{{\bf{R}}}^{k}\to {\bf{R}}, for 1≤r≤k1\le r\le k, and x=(x1,…,xk)x=\left({x}_{1},\ldots ,{x}_{k})are defined by σr(x)≔∑i1<⋯<irxi1…xir.{\sigma }_{r}\left(x):= \sum _{{i}_{1}\hspace{0.33em}\lt \cdots \lt {i}_{r}}{x}_{{i}_{1}}\ldots {x}_{{i}_{r}}.We set σ0≔1{\sigma }_{0}:= 1and σr≔0{\sigma }_{r}:= 0for r≥k+1r\ge k+1by convention. Let λ(∇2u)≔(λ1,…,λn)\lambda \left({\nabla }^{2}u):= \left({\lambda }_{1},\ldots ,{\lambda }_{n})denote the eigenvalues of ∇2u{\nabla }^{2}u. Then, we set σr(∇2u)≔σr(λ(∇2u)).{\sigma }_{r}\left({\nabla }^{2}u):= {\sigma }_{r}(\lambda \left({\nabla }^{2}u)).These functions form the coefficients of the characteristic polynomial P(λ)≔det(λI−∇2u)=∑i=0n(−1)iσi(∇2u)λn−i.P\left(\lambda ):= \det \left(\lambda I-{\nabla }^{2}u)=\mathop{\sum }\limits_{i=0}^{n}{\left(-1)}^{i}{\sigma }_{i}\left({\nabla }^{2}u){\lambda }^{n-i}.Let δj1…jmi1…im{\delta }_{{j}_{1}\ldots {j}_{m}}^{{i}_{1}\ldots {i}_{m}}be the generalized Kronecker tensor, which is equal to 1 (−1-1) if i1,…,im{i}_{1},\ldots ,{i}_{m}are distinct and (j1,…,jm)\left({j}_{1},\ldots ,{j}_{m})is an even (odd) permutation of (i1,…,im)\left({i}_{1},\ldots ,{i}_{m}); otherwise, it is equal to 0. Then [18, Prop. 1.2(a)], (1)σr(∇2u)=1r!δj1…jri1…irui1j1⋯uirjr,{\sigma }_{r}\left({\nabla }^{2}u)=\frac{1}{r\&#x0021;}{\delta }_{{j}_{1}\ldots {j}_{r}}^{{i}_{1}\ldots {i}_{r}}{u}_{{i}_{1}{j}_{1}}\cdots {u}_{{i}_{r}{j}_{r}},where uij≔∇iju{u}_{ij}:= {\nabla }_{ij}udenote the second partial derivatives of uuwith respect to an orthonormal frame Ei∈TpM{E}_{i}\in {T}_{p}M, which we extend to an open neighborhood of ppby parallel translation along geodesics. So ∇EiEj=0{\nabla }_{{E}_{i}}{E}_{j}=0at pp. We call Ei{E}_{i}a local parallel frame centered at ppand set ∇i≔∇Ei{\nabla }_{i}:= {\nabla }_{{E}_{i}}, ∇ij≔∇i∇j{\nabla }_{ij}:= {\nabla }_{i}{\nabla }_{j}. Each of the indices in (1) ranges from 1 to nn, and we employ Einstein’s convention by summing over repeated indices throughout the article. The Newton operators Tru:TpM→TpM{{\mathcal{T}}}_{r}^{u}:{T}_{p}M\to {T}_{p}M[17,18] are defined recursively by setting T0u≔I{{\mathcal{T}}}_{0}^{u}:= I, the identity map, and for r≥1r\ge 1, (2)Tru≔σr(∇2u)I−Tr−1u∘∇2u=∑i=0r(−1)iσi(∇2u)(∇2u)r−i.{{\mathcal{T}}}_{r}^{u}:= {\sigma }_{r}\left({\nabla }^{2}u)I-{{\mathcal{T}}}_{r-1}^{u}\circ {\nabla }^{2}u=\mathop{\sum }\limits_{i=0}^{r}{\left(-1)}^{i}{\sigma }_{i}\left({\nabla }^{2}u){\left({\nabla }^{2}u)}^{r-i}.Thus, Tru{{\mathcal{T}}}_{r}^{u}is the truncation of the polynomial P(∇2u)P\left({\nabla }^{2}u)obtained by removing the terms of order higher than rr. In particular, Tnu=P(∇2u){{\mathcal{T}}}_{n}^{u}=P\left({\nabla }^{2}u). So, by the Cayley-Hamilton theorem, Tnu=0{{\mathcal{T}}}_{n}^{u}=0. Consequently, when ∇2u{\nabla }^{2}uis nondegenerate, (2) yields that (3)Tn−1u=σn(∇2u)(∇2u)−1=det(∇2u)(∇2u)−1=Tu,{{\mathcal{T}}}_{n-1}^{u}={\sigma }_{n}\left({\nabla }^{2}u){\left({\nabla }^{2}u)}^{-1}=\det \left({\nabla }^{2}u){\left({\nabla }^{2}u)}^{-1}={{\mathcal{T}}}^{u},where Tu{{\mathcal{T}}}^{u}is the Hessian cofactor operator discussed in [13, Sec. 4]. See [18, Prop. 1.2] for other basic identities that relate σ\sigma and T{\mathcal{T}}. In particular, by [18, Prop. 1.2(c)], we have Trace(Tru⋅∇2u)=(r+1)σr+1(∇2u)\hspace{0.1em}\text{Trace}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}\cdot {\nabla }^{2}u)=\left(r+1){\sigma }_{r+1}\left({\nabla }^{2}u). So, by Euler’s identity for homogeneous polynomials, (4)(Tru)ijuij=Trace(Tru∘∇2u)=(r+1)σr+1(∇2u)=∂σr+1(∇2u)∂uijuij.{\left({{\mathcal{T}}}_{r}^{u})}_{ij}{u}_{ij}=\hspace{0.1em}\text{Trace}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}\circ {\nabla }^{2}u)=\left(r+1){\sigma }_{r+1}\left({\nabla }^{2}u)=\frac{\partial {\sigma }_{r+1}\left({\nabla }^{2}u)}{\partial {u}_{ij}}{u}_{ij}.Thus, it follows from (1) that (5)(Tru)ij=∂σr+1(∇2u)∂uij=1r!δjj1…jrii1…irui1j1⋯uirjr.{({{\mathcal{T}}}_{r}^{u})}_{ij}=\frac{\partial {\sigma }_{r+1}\left({\nabla }^{2}u)}{\partial {u}_{ij}}=\frac{1}{r\&#x0021;}{\delta }_{j{j}_{1}\ldots {j}_{r}}^{i{i}_{1}\ldots {i}_{r}}{u}_{{i}_{1}{j}_{1}}\cdots {u}_{{i}_{r}{j}_{r}}.Furthermore, by [17, Prop. 1(11)] (note that the sign of the Riemann tensor RRin [17] is opposite to the one in this article), we have (6)(div(Tru))j=1(r−1)!δjj1…jrii1…irui1j1⋯uir−1jr−1Rijrirkuk,(\hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}){)}_{j}=\frac{1}{\left(r-1)\&#x0021;}{\delta }_{j{j}_{1}\ldots {j}_{r}}^{i{i}_{1}\ldots {i}_{r}}{u}_{{i}_{1}{j}_{1}}\cdots {u}_{{i}_{r-1}{j}_{r-1}}{R}_{i{j}_{r}{i}_{r}k}{u}_{k},where Rijkl≔R(Ei,Ej,Ek,Eℓ)=⟨∇i∇jEk−∇j∇iEk,Eℓ⟩{R}_{ijkl}:= R\left({E}_{i},{E}_{j},{E}_{k},{E}_{\ell })=\langle {\nabla }_{i}{\nabla }_{j}{E}_{k}-{\nabla }_{j}{\nabla }_{i}{E}_{k},{E}_{\ell }\rangle . Another useful identity [17, p. 462] is (7)div(Tru(∇u))=⟨Tru,∇2u⟩+⟨div(Tru),∇u⟩,\hspace{0.1em}\text{div}\hspace{0.1em}({{\mathcal{T}}}_{r}^{u}\left(\nabla u))=\langle {{\mathcal{T}}}_{r}^{u},{\nabla }^{2}u\rangle +\langle \hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}),\nabla u\rangle ,where ⟨⋅,⋅⟩\langle \cdot ,\cdot \rangle here indicates the Frobenius inner product (i.e., ⟨A,B⟩≔AijBij\langle A,B\rangle := {A}_{ij}{B}_{ij}for any pair of matrices of the same dimension). The divergence of Tru{{\mathcal{T}}}_{r}^{u}may be defined by virtually the same argument used for Tu{{\mathcal{T}}}^{u}in [13, Sec. 4] to yield the following generalization of [13, (14)]: (8)(div(Tru))j=∇i(Tru)ij.(\hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}){)}_{j}={\nabla }_{i}{\left({{\mathcal{T}}}_{r}^{u})}_{ij}.Recall that Tu=Tn−1u{{\mathcal{T}}}^{u}={{\mathcal{T}}}_{n-1}^{u}by (3). Furthermore, Tnu=0{{\mathcal{T}}}_{n}^{u}=0as we mentioned earlier. Thus, the following observation generalizes [13, Lem. 4.2].Lemma 2.1divTr−1u∇u∣∇u∣r=div(Tr−1u),∇u∣∇u∣r+r⟨Tru(∇u),∇u⟩∣∇u∣r+2.{\rm{div}}\left({{\mathcal{T}}}_{r-1}^{u}\left(\frac{\nabla u}{| \nabla u{| }^{r}}\right)\right)=\left\langle \hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r-1}^{u}),\frac{\nabla u}{| \nabla u{| }^{r}}\right\rangle +r\frac{\langle {{\mathcal{T}}}_{r}^{u}\left(\nabla u),\nabla u\rangle }{| \nabla u{| }^{r+2}}.ProofBy Leibniz rule and (8), we have divTr−1u∇u∣∇u∣r=∇i(Tr−1u)ijuj∣∇u∣r=div(Tr−1u),∇u∣∇u∣r+(Tr−1u)ijuij∣∇u∣r−rujuℓuℓi∣∇u∣r+2,\hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r-1}^{u}\left(\frac{\nabla u}{| \nabla u{| }^{r}}\right)\right)={\nabla }_{i}\left({\left({{\mathcal{T}}}_{r-1}^{u})}_{ij}\frac{{u}_{j}}{| \nabla u{| }^{r}}\right)=\left\langle \hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r-1}^{u}),\frac{\nabla u}{| \nabla u{| }^{r}}\right\rangle +{\left({{\mathcal{T}}}_{r-1}^{u})}_{ij}\left(\frac{{u}_{ij}}{| \nabla u{| }^{r}}-r\frac{{u}_{j}{u}_{\ell }{u}_{\ell i}}{| \nabla u{| }^{r+2}}\right),where the computation to obtain the second term on the right is identical to the one performed earlier in [13, Lem. 4.2]. To develop this term further, note that by (2) (Tr−1u)ijuℓi=σr(∇2u)δℓj−(Tru)ℓj,{\left({{\mathcal{T}}}_{r-1}^{u})}_{ij}{u}_{\ell i}={\sigma }_{r}\left({\nabla }^{2}u){\delta }_{\ell j}-{\left({{\mathcal{T}}}_{r}^{u})}_{\ell j},which in turn yields (Tr−1u)ijuℓiujuℓ∣∇u∣2=σr(∇2u)−(Tru)ijuiuj∣∇u∣2.{\left({{\mathcal{T}}}_{r-1}^{u})}_{ij}{u}_{\ell i}\frac{{u}_{j}{u}_{\ell }}{| \nabla u{| }^{2}}={\sigma }_{r}\left({\nabla }^{2}u)-{\left({{\mathcal{T}}}_{r}^{u})}_{ij}\frac{{u}_{i}{u}_{j}}{| \nabla u{| }^{2}}.Hence, (Tr−1u)ijuij∣∇u∣r−rujuℓuℓi∣∇u∣r+2=rσr(∇2u)∣∇u∣r−r∣∇u∣rσr(∇2u)−(Tru)ijuiuj∣∇u∣2=r(Tru)ijuiuj∣∇u∣r+2,{\left({{\mathcal{T}}}_{r-1}^{u})}_{ij}\left(\frac{{u}_{ij}}{| \nabla u{| }^{r}}-r\frac{{u}_{j}{u}_{\ell }{u}_{\ell i}}{| \nabla u{| }^{r+2}}\right)=\frac{r{\sigma }_{r}\left({\nabla }^{2}u)}{| \nabla u{| }^{r}}-\frac{r}{| \nabla u{| }^{r}}\left({\sigma }_{r}\left({\nabla }^{2}u)-{\left({{\mathcal{T}}}_{r}^{u})}_{ij}\frac{{u}_{i}{u}_{j}}{| \nabla u{| }^{2}}\right)=r{\left({{\mathcal{T}}}_{r}^{u})}_{ij}\frac{{u}_{i}{u}_{j}}{| \nabla u{| }^{r+2}},which completes the proof.□Below we assume, as was the case in [13, Sec. 4], that all local computations take place with respect to a principal curvature frame Ei∈TpM{E}_{i}\in {T}_{p}Mof uu, which is defined as follows. Assuming ∣∇u(p)∣≠0| \nabla u\left(p)| \ne 0, we set En≔∇u(p)/∣∇u(p)∣{E}_{n}:= \nabla u\left(p)\hspace{0.1em}\text{/}\hspace{0.1em}| \nabla u\left(p)| , and let E1,…,En−1{E}_{1},\ldots ,{E}_{n-1}be the principal directions of the level set of uupassing through pp. Then, we extend Ei{E}_{i}to a local parallel frame near pp. The first partial derivatives of uuwith respect to Ei{E}_{i}, ui≔∇iu{u}_{i}:= {\nabla }_{i}u, satisfy (9)ui=0;fori≠n,andun=∣∇u∣.{u}_{i}=0;\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}i\ne n,\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{u}_{n}=| \nabla u| .Furthermore, for the second partial derivatives, uij=∇iju{u}_{ij}={\nabla }_{ij}u, we have (10)uij=0,fori≠j≤n−1,anduii∣∇u∣≕κiu,fori≠n,{u}_{ij}=0,\hspace{2.22144pt}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{2.22144pt}i\ne j\le n-1,\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\frac{{u}_{ii}}{| \nabla u| }=: {\kappa }_{i}^{u},\hspace{2.22144pt}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{2.22144pt}i\ne n,where κ1u,…,κn−1u{\kappa }_{1}^{u},\ldots ,{\kappa }_{n-1}^{u}are the principal curvatures of level sets of uuwith respect to En{E}_{n}, i.e., they are eigenvalues corresponding to E1,…,En−1{E}_{1},\ldots ,{E}_{n-1}of the shape operator X↦∇XνX\mapsto {\nabla }_{X}\nu on the tangent space of level sets of uu, where ν≔∇u/∣∇u∣\nu := \nabla u\hspace{0.1em}\text{/}\hspace{0.1em}| \nabla u| . We set κu≔(κ1u,…,κn−1u){\kappa }_{u}:= \left({\kappa }_{1}^{u},\ldots ,{\kappa }_{n-1}^{u}). So σr(κu){\sigma }_{r}\left({\kappa }^{u})is the rthr{\rm{th}}mean curvature of the level set of uuat pp. In particular, σn−1(κu){\sigma }_{n-1}\left({\kappa }^{u})is the Gauss-Kronecker curvature of the level sets. The next observation generalizes [13, Lem. 4.1].Lemma 2.2σr(κu)=⟨Tru(∇u),∇u⟩∣∇u∣r+2.{\sigma }_{r}\left({\kappa }^{u})=\frac{\langle {{\mathcal{T}}}_{r}^{u}\left(\nabla u),\nabla u\rangle }{| \nabla u{| }^{r+2}}.Proof(5) together with (9) and (10) yields that (Tru)ijuiuj∣∇u∣r+2=1r!δjj1⋯jrii1⋯irui1j1⋯uirjruiuj∣∇u∣r+2=1r!δnj1⋯jrni1⋯irui1j1⋯uirjr∣∇u∣r⋅un2∣∇u∣2=1r!δni1⋯irni1⋯irκi1u…κiru.□\begin{array}{rcl}\hspace{13.8em}{\left({{\mathcal{T}}}_{r}^{u})}_{ij}\frac{{u}_{i}{u}_{j}}{| \nabla u{| }^{r+2}}& =& \frac{1}{r\&#x0021;}{\delta }_{j{j}_{1}\cdots {j}_{r}}^{i{i}_{1}\cdots {i}_{r}}{u}_{{i}_{1}{j}_{1}}\cdots {u}_{{i}_{r}{j}_{r}}\frac{{u}_{i}{u}_{j}}{| \nabla u{| }^{r+2}}\\ & =& \frac{1}{r\&#x0021;}{\delta }_{n{j}_{1}\cdots {j}_{r}}^{n{i}_{1}\cdots {i}_{r}}\frac{{u}_{{i}_{1}{j}_{1}}\cdots {u}_{{i}_{r}{j}_{r}}}{| \nabla u{| }^{r}}\cdot \frac{{u}_{n}^{2}}{| \nabla u{| }^{2}}\\ & =& \frac{1}{r\&#x0021;}{\delta }_{n{i}_{1}\cdots {i}_{r}}^{n{i}_{1}\cdots {i}_{r}}{\kappa }_{{i}_{1}}^{u}\ldots {\kappa }_{{i}_{r}}^{u}.\hspace{17em}\square \end{array}3Comparison formulaHere, we establish the main result of this work. For a C1,1{{\mathcal{C}}}^{1,1}hypersurface Γ\Gamma in a Riemannian nn-manifold MM, oriented by a choice of normal vector field ν\nu , and 0≤r≤n−10\le r\le n-1, we let ℳr(Γ)≔∫Γσr(κ){{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma ):= \mathop{\int }\limits_{\Gamma }{\sigma }_{r}\left(\kappa )be the total rthr{\rm{th}}mean curvature of Γ\Gamma , where κ≔(κ1,…,κn−1)\kappa := \left({\kappa }_{1},\ldots ,{\kappa }_{n-1})denotes principal curvatures of Γ\Gamma with respect to ν\nu . Note that ℳ0(Γ)=∣Γ∣{{\mathcal{ {\mathcal M} }}}_{0}\left(\Gamma )=| \Gamma | , the volume of Γ\Gamma , since σ0=1{\sigma }_{0}=1, and ℳn−1(Γ){{\mathcal{ {\mathcal M} }}}_{n-1}\left(\Gamma )is the total Gauss-Kronecker curvature of Γ\Gamma (denoted by G(Γ){\mathcal{G}}\left(\Gamma )in [13]). A domain Ω⊂M\Omega \subset Mis an open set with a compact closure cl(Ω){\rm{cl}}\left(\Omega ). If Γ\Gamma bounds a domain Ω\Omega , then by convention we set ℳ−1(Γ)≔∣Ω∣{{\mathcal{ {\mathcal M} }}}_{-1}\left(\Gamma ):= | \Omega | , the volume of Ω\Omega . The following theorem generalizes [13, Thm. 4.7], where this result had been established for r=n−1r=n-1. It also uses less regularity than was required in [13, Thm. 4.7].Theorem 3.1Let Γ\Gamma and γ\gamma be closed C1,1{{\mathcal{C}}}^{1,1}hypersurfaces in a Riemannian n-manifold M bounding domains Ω\Omega and D, respectively, with cl(D)⊂Ω{\rm{cl}}\left(D)\subset \Omega . Suppose there exists a C1,1{{\mathcal{C}}}^{1,1}function uuon cl(Ω⧹D){\rm{cl}}\left(\Omega \setminus D)with ∇u≠0\nabla u\ne 0, which is constant on Γ\Gamma and γ\gamma . Let κu≔(κ1u,…,κn−1u){\kappa }^{u}:= \left({\kappa }_{1}^{u},\ldots ,{\kappa }_{n-1}^{u})be the principal curvatures of level sets of u with respect to En≔∇u/∣∇u∣{E}_{n}:= \nabla u\hspace{0.1em}\text{/}\hspace{0.1em}| \nabla u| , and let E1,…,En−1{E}_{1},\ldots ,{E}_{n-1}be the corresponding principal directions. Then, for 0≤r≤n−10\le r\le n-1, ℳr(Γ)−ℳr(γ)=(r+1)∫Ω⧹Dσr+1(κu)+∫Ω⧹D−∑κi1u…κir−1uKirn+1∣∇u∣∑κi1u…κir−2u∣∇u∣ir−1Ririr−1irn,{{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )-{{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )=\left(r+1)\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{r+1}\left({\kappa }^{u})+\mathop{\int }\limits_{\Omega \setminus D}\left(-\sum {\kappa }_{{i}_{1}}^{u}\ldots {\kappa }_{{i}_{r-1}}^{u}{K}_{{i}_{r}n}+\frac{1}{| \nabla u| }\sum {\kappa }_{{i}_{1}}^{u}\ldots {\kappa }_{{i}_{r-2}}^{u}| \nabla u{| }_{{i}_{r-1}}{R}_{{i}_{r}{i}_{r-1}{i}_{r}n}\right),where ∣∇u∣i≔∇Ei∣∇u∣| \nabla u{| }_{i}:= {\nabla }_{{E}_{i}}| \nabla u| , Rijkl=R(Ei,Ej,Ek,El){R}_{ijkl}=R\left({E}_{i},{E}_{j},{E}_{k},{E}_{l})are components of the Riemann curvature tensor of MM, Kij=Rijij{K}_{ij}={R}_{ijij}is the sectional curvature, and the summations take place over distinct values of 1≤i1,…,ir≤n−11\le {i}_{1},\ldots ,{i}_{r}\le n-1, with i1<⋯<ir−1{i}_{1}\hspace{0.33em}\lt \cdots \lt {i}_{r-1}in the first sum and i1<⋯<ir−2{i}_{1}\hspace{0.33em}\lt \cdots \lt {i}_{r-2}in the second sum.ProofBy Lemmas 2.1 and 2.2, (11)divTru∇u∣∇u∣r+1=(r+1)σr+1(κu)+div(Tru),∇u∣∇u∣r+1.\hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}\left(\frac{\nabla u}{| \nabla u{| }^{r+1}}\right)\right)=\left(r+1){\sigma }_{r+1}\left({\kappa }^{u})+\left\langle \hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}),\frac{\nabla u}{| \nabla u{| }^{r+1}}\right\rangle .By Stokes’ theorem and Lemma 2.2, ∫Ω⧹DdivTru∇u∣∇u∣r+1=∫Γ∪γTru∇u∣∇u∣r+1,∇u∣∇u∣=ℳr(Γ)−ℳr(γ).\mathop{\int }\limits_{\Omega \setminus D}\hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}\left(\frac{\nabla u}{| \nabla u{| }^{r+1}}\right)\right)=\mathop{\int }\limits_{\Gamma \cup \gamma }\left\langle {{\mathcal{T}}}_{r}^{u}\left(\frac{\nabla u}{| \nabla u{| }^{r+1}}\right),\frac{\nabla u}{| \nabla u| }\right\rangle ={{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )-{{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma ).So integrating both sides of (11) yields ℳr(Γ)−ℳr(γ)=(r+1)∫Ω⧹Dσr+1(κu)+∫Ω⧹Ddiv(Tru),∇u∣∇u∣r+1.{{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )-{{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )=\left(r+1)\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{r+1}\left({\kappa }^{u})+\mathop{\int }\limits_{\Omega \setminus D}\left\langle \hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}),\frac{\nabla u}{| \nabla u{| }^{r+1}}\right\rangle .Using (6) and (9), we have div(Tru),∇u∣∇u∣r+1=1(r−1)!δjj1…jrii1…irui1j1⋯uir−1jr−1Rijrirkukuj∣∇u∣r+1=1(r−1)!δnj1…jrii1…irui1j1∣∇u∣⋯uir−1jr−1∣∇u∣Rijrirn.\begin{array}{rcl}\left\langle \hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}),\frac{\nabla u}{| \nabla u{| }^{r+1}}\right\rangle & =& \frac{1}{\left(r-1)\&#x0021;}{\delta }_{j{j}_{1}\ldots {j}_{r}}^{i{i}_{1}\ldots {i}_{r}}{u}_{{i}_{1}{j}_{1}}\cdots {u}_{{i}_{r-1}{j}_{r-1}}{R}_{i{j}_{r}{i}_{r}k}\frac{{u}_{k}{u}_{j}}{| \nabla u{| }^{r+1}}\\ & =& \frac{1}{\left(r-1)\&#x0021;}{\delta }_{n{j}_{1}\ldots {j}_{r}}^{i{i}_{1}\ldots {i}_{r}}\frac{{u}_{{i}_{1}{j}_{1}}}{| \nabla u| }\cdots \frac{{u}_{{i}_{r-1}{j}_{r-1}}}{| \nabla u| }{R}_{i{j}_{r}{i}_{r}n}.\end{array}The last expression may be written as the sum of two components, AAand BB, which consist of terms with i=ni=nand i≠ni\ne n, respectively. Note that we may assume j1,…,jr≠n{j}_{1},\ldots ,{j}_{r}\ne n, for otherwise δnj1…jrii1…ir=0{\delta }_{n{j}_{1}\ldots {j}_{r}}^{i{i}_{1}\ldots {i}_{r}}=0. To compute AA, note that if i=ni=n, then for δnj1…jrii1…ir{\delta }_{n{j}_{1}\ldots {j}_{r}}^{i\hspace{0.33em}{i}_{1}\ldots {i}_{r}}not to vanish, we must have i1,…,ir≠n{i}_{1},\ldots ,{i}_{r}\ne n. Then, by (10), uikjk=0{u}_{{i}_{k}{j}_{k}}=0unless ik=jk{i}_{k}={j}_{k}, which yields that A=1(r−1)!δni1…irni1…irui1i1∣∇u∣⋯uir−1ir−1∣∇u∣Rnirirn=−∑κi1u…κir−1uKirn,A=\frac{1}{\left(r-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\delta }_{n{i}_{1}\ldots {i}_{r}}^{n{i}_{1}\ldots {i}_{r}}\frac{{u}_{{i}_{1}{i}_{1}}}{| \nabla u| }\cdots \frac{{u}_{{i}_{r-1}{i}_{r-1}}}{| \nabla u| }{R}_{n{i}_{r}{i}_{r}n}=-\sum {\kappa }_{{i}_{1}}^{u}\ldots {\kappa }_{{i}_{r-1}}^{u}{K}_{{i}_{r}n},where the sum ranges over all distinct values of 1≤i1,…,ir≤n−11\le {i}_{1},\ldots ,{i}_{r}\le n-1, with i1<⋯<ir−1{i}_{1}\hspace{0.33em}\lt \cdots \lt {i}_{r-1}as desired. To find BBnote that if i≠ni\ne n, then for δnj1…jrii1…ir{\delta }_{n{j}_{1}\ldots {j}_{r}}^{i\hspace{0.33em}{i}_{1}\ldots {i}_{r}}not to vanish, we must have ik=n{i}_{k}=nfor some 1≤k≤r1\le k\le r. If k=rk=r, then Rijrirn=Rijknn=0{R}_{i{j}_{r}{i}_{r}n}={R}_{i{j}_{k}nn}=0. In particular, B=0B=0when r=1r=1. Now assume that r≥2r\ge 2. Then, we may assume that k≠rk\ne r, or ir≠n{i}_{r}\ne n. Then, by (10), uirjr=0{u}_{{i}_{r}{j}_{r}}=0unless ir=jr{i}_{r}={j}_{r}. So, we may assume that ir=jr{i}_{r}={j}_{r}for r≠kr\ne k, which in turn implies that jk=i{j}_{k}=i. Thus, B=∑k=1r−1BkB={\sum }_{k=1}^{r-1}{B}_{k}, where Bk=1(r−1)!δni1…ik−1iik+1…irii1…ik−1nik+1…irui1i1∣∇u∣…uik−1ik−1∣∇u∣uni∣∇u∣uik+1ik+1∣∇u∣…uir−1ir−1∣∇u∣Riirirn=−1(r−1)∑κi1u…κik−1u∣∇u∣i∣∇u∣κik+1u…κir−1uRiirirn,\begin{array}{rcl}{B}_{k}& =& \frac{1}{\left(r-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\delta }_{n{i}_{1}\ldots {i}_{k-1}i\hspace{0.33em}{i}_{k+1}\ldots {i}_{r}}^{i\hspace{0.33em}{i}_{1}\ldots {i}_{k-1}n{i}_{k+1}\ldots {i}_{r}}\frac{{u}_{{i}_{1}{i}_{1}}}{| \nabla u| }\ldots \frac{{u}_{{i}_{k-1}{i}_{k-1}}}{| \nabla u| }\frac{{u}_{ni}}{| \nabla u| }\frac{{u}_{{i}_{k+1}{i}_{k+1}}}{| \nabla u| }\ldots \frac{{u}_{{i}_{r-1}{i}_{r-1}}}{| \nabla u| }{R}_{i{i}_{r}{i}_{r}n}\\ & =& \frac{-1}{\left(r-1)}\displaystyle \sum {\kappa }_{{i}_{1}}^{u}\ldots {\kappa }_{{i}_{k-1}}^{u}\frac{| \nabla u{| }_{i}}{| \nabla u| }{\kappa }_{{i}_{k+1}}^{u}\ldots {\kappa }_{{i}_{r-1}}^{u}{R}_{i{i}_{r}{i}_{r}n},\end{array}since un=∣∇u∣{u}_{n}=| \nabla u| . Here, the sum ranges over all distinct indices 1≤i,i1,…,ik−1,ik,…,ir≤n−11\le i,{i}_{1},\ldots ,{i}_{k-1},{i}_{k},\ldots ,{i}_{r}\le n-1, with i1<⋯<ik−1<ik+1<⋯<ir−1{i}_{1}\hspace{0.33em}\lt \cdots \lt {i}_{k-1}\lt {i}_{k+1}\hspace{0.33em}\lt \cdots \lt {i}_{r-1}. Note that B1=⋯=Br−1{B}_{1}=\cdots ={B}_{r-1}. Thus, B=(r−1)Br−1=1∣∇u∣∑κi1u…κir−2u∣∇u∣iRiriirn,B=\left(r-1){B}_{r-1}=\frac{1}{| \nabla u| }\sum {\kappa }_{{i}_{1}}^{u}\ldots {\kappa }_{{i}_{r-2}}^{u}| \nabla u{| }_{i}{R}_{{i}_{r}i{i}_{r}n},which completes the proof (after renaming iito ir−1{i}_{r-1}).□4ApplicationsHere, we develop some consequences of Theorem 3.1. A subset of a Cartan-Hadamard manifold MMis convex if it contains the (unique) geodesic segment connecting every pair of its points. A convex hypersurface Γ⊂M\Gamma \subset Mis the boundary of a compact convex set with interior points. If Γ\Gamma is of class C1,1{{\mathcal{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\Gamma \subset Mare all nonnegative, then Γ\Gamma 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\Gamma if it lies in the convex domain bounded by Γ\Gamma .Corollary 4.1Let Γ\Gamma and γ\gamma be C1,1{{\mathcal{C}}}^{1,1}convex hypersurfaces in a Cartan-Hadamard nn-manifold. Suppose that γ\gamma is nested inside Γ\Gamma . Then, ℳ1(Γ)≥ℳ1(γ){{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )\ge {{\mathcal{ {\mathcal M} }}}_{1}\left(\gamma ).ProofSetting r=1r=1in the comparison formula of Theorem 3.1 yields (12)ℳ1(Γ)−ℳ1(γ)=2∫Ω⧹Dσ2(κu)−∫Ω⧹DRic∇u∣∇u∣,{{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )-{{\mathcal{ {\mathcal M} }}}_{1}\left(\gamma )=2\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{2}\left({\kappa }^{u})-\mathop{\int }\limits_{\Omega \setminus D}\hspace{0.1em}\text{Ric}\hspace{0.1em}\left(\frac{\nabla u}{| \nabla u| }\right),where Ric stands for Ricci curvature; more explicitly, in a principal curvature frame where En≔∇u/∣∇u∣{E}_{n}:= \nabla u\hspace{0.1em}\text{/}\hspace{0.1em}| \nabla u| , Ric(En)\hspace{0.1em}\text{Ric}\hspace{0.1em}\left({E}_{n})is the sum of sectional curvatures Kin{K}_{{\rm{in}}}, for 1≤i≤n−11\le i\le n-1. So Ric(En)≤0\hspace{0.1em}\text{Ric}\hspace{0.1em}\left({E}_{n})\le 0. If Γ\Gamma and γ\gamma are smooth (C∞{{\mathcal{C}}}^{\infty }) and strictly convex, we may let uuin Theorem 3.1 be a function with convex level sets [4, Lem. 1]. Then, σ2(κu)≥0{\sigma }_{2}\left({\kappa }^{u})\ge 0, which yields ℳ1(Γ)≥ℳ1(γ){{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )\ge {{\mathcal{ {\mathcal M} }}}_{1}\left(\gamma )as desired. This completes the proof since we may approximate Γ\Gamma and γ\gamma by smooth strictly convex hypersurfaces, e.g., by applying the Greene-Wu convolution to their distance functions, see [12, Lem. 3.3]; furthermore, total mean curvatures will converge here since they constitute “valuations” in the sense of integral geometry, see [13, Note 3.7] or [3, Prop. 3.8].□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.Lemma 4.2Let Sρ{S}_{\rho }be a geodesic sphere of radius ρ\rho centered at a point in a Riemannian manifold. As ρ→0\rho \to 0, ℳr(Sρ){{\mathcal{ {\mathcal M} }}}_{r}\left({S}_{\rho })converges to 0 for r≤n−2r\le n-2and to ∣Sn−1∣| {{\bf{S}}}^{n-1}| for r=n−1r=n-1.ProofA power series expansion [6, Thm. 3.1] of the second fundamental form of Sρ{S}_{\rho }in normal coordinates shows that the principal curvatures of Sρ{S}_{\rho }are given by κiρ=(1+O(ρ2))/ρ{\kappa }_{i}^{\rho }=\left(1+O\left({\rho }^{2}))\hspace{0.1em}\text{/}\hspace{0.1em}\rho . So σr(κρ)=n−1r1ρr(1+O(ρ2)).{\sigma }_{r}\left({\kappa }^{\rho })=\left(\begin{array}{c}n-1\\ r\end{array}\right)\frac{1}{{\rho }^{r}}\left(1+O\left({\rho }^{2})).Another power series expansion [15, Thm. 3.1] yields ∣Sρ∣=∣Sn−1∣ρn−1(1+O(ρ2)).| {S}_{\rho }| =| {{\bf{S}}}^{n-1}| {\rho }^{n-1}\left(1+O\left({\rho }^{2})).So, it follows that ℳr(Sρ)=n−1r∣Sn−1∣ρn−1−r(1+O(ρ2)),{{\mathcal{ {\mathcal M} }}}_{r}\left({S}_{\rho })=\left(\begin{array}{c}n-1\\ r\end{array}\right)| {{\bf{S}}}^{n-1}| {\rho }^{n-1-r}(1+O\left({\rho }^{2})),which completes the proof.□Gallego and Solanes showed [10, Cor. 3.2] that if Γ\Gamma is a convex hypersurface bounding a domain Ω\Omega in a hyperbolic nn-space of constant curvature a<0a\lt 0, then ℳ1(Γ)>−(n−1)2a∣Ω∣.{{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )\gt -{\left(n-1)}^{2}a| \Omega | .When comparing formulas, note that in [10], mean curvature is defined as the average of κi{\kappa }_{i}, as opposed to the sum of κi{\kappa }_{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:Corollary 4.3Let Γ\Gamma be a C1,1{{\mathcal{C}}}^{1,1}convex hypersurface in a Cartan-Hadamard n-manifold M bounding a domain Ω\Omega . Suppose that curvature of M is bounded above by a≤0a\le 0. Then, ℳ1(Γ)>−(n−1)a∣Ω∣.{{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )\gt -\left(n-1)a| \Omega | .Furthermore, if n=3n=3, thenℳ1(Γ)>−4a∣Ω∣.{{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )\gt -4a| \Omega | .ProofLet γ=γρ\gamma ={\gamma }_{\rho }in (12) be a geodesic sphere of radius ρ\rho . By Lemma 4.2, ℳ1(γρ)→0{{\mathcal{ {\mathcal M} }}}_{1}\left({\gamma }_{\rho })\to 0as ρ→0\rho \to 0, which yields ℳ1(Γ)=2∫Ωσ2(κu)−∫ΩRic∇u∣∇u∣>−(n−1)a∣Ω∣,{{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )=2\mathop{\int }\limits_{\Omega }{\sigma }_{2}\left({\kappa }^{u})-\mathop{\int }\limits_{\Omega }\hspace{0.1em}\text{Ric}\hspace{0.1em}\left(\frac{\nabla u}{| \nabla u| }\right)\gt -\left(n-1)a| \Omega | ,as desired. When n=3n=3, Gauss’ equation states that σ2(κu)=Ku−KMu,{\sigma }_{2}\left({\kappa }^{u})={K}^{u}-{K}_{M}^{u},where Ku{K}^{u}is the sectional curvature of level sets of uuand KMu{K}_{M}^{u}is the sectional curvature of MMwith respect to tangent planes to level sets of uu. Thus, ℳ1(Γ)=2∫ΩKu−2∫ΩKMu−∫ΩRic∇u∣∇u∣>−4a∣Ω∣,{{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )=2\mathop{\int }\limits_{\Omega }{K}^{u}-2\mathop{\int }\limits_{\Omega }{K}_{M}^{u}-\mathop{\int }\limits_{\Omega }\hspace{0.1em}\text{Ric}\hspace{0.1em}\left(\frac{\nabla u}{| \nabla u| }\right)\gt -4a| \Omega | ,which completes the proof.□We say Γ\Gamma is an outer parallel hypersurface of a convex hypersurface γ\gamma if all points of Γ\Gamma are at a constant distance λ≥0\lambda \ge 0from the convex domain bounded by γ\gamma . Since the distance function of a convex set in a Cartan-Hadamard manifold is convex [5, Prop. 2.4], Γ\Gamma is convex. Furthermore, Γ\Gamma is C1,1{{\mathcal{C}}}^{1,1}for λ>0\lambda \gt 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].Corollary 4.4Let M be a Cartan-Hadamard n-manifold, and Γ\Gamma and γ\gamma be C1,1{{\mathcal{C}}}^{1,1}convex hypersurfaces in M. Suppose that Γ\Gamma is an outer parallel hypersurface of γ\gamma . Then, ℳr(Γ)≥ℳr(γ){{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )\ge {{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )for 1≤r≤n−11\le r\le n-1.ProofWe may let uuin Theorem 3.1 be the distance function of the convex domain bounded by Γ\Gamma . Then, ∣∇u∣| \nabla u| is constant on level sets of uu. So, ∣∇u∣i=0| \nabla u{| }_{i}=0for 1≤i≤n−11\le i\le n-1, which yields ℳr(Γ)−ℳr(γ)≥(r+1)∫Ω⧹Dσr+1(κu)−a(n−r)∫Ω⧹Dσr−1(κu),{{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )-{{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )\ge \left(r+1)\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{r+1}\left({\kappa }^{u})-a\left(n-r)\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{r-1}\left({\kappa }^{u}),where a≤0a\le 0is the upper bound for sectional curvatures of MM. Since uuis convex, σr(κu)≥0{\sigma }_{r}\left({\kappa }^{u})\ge 0, which completes the proof.□The next result generalizes [13, Cor. 5.2] and observation of Borbely [4, Thm. 1] for Gauss-Kronecker curvature.Corollary 4.5Let M be a Cartan-Hadamard n-manifold with constant curvature, and Γ\Gamma and γ\gamma be C1,1{{\mathcal{C}}}^{1,1}convex hypersurfaces in M, with γ\gamma nested inside Γ\Gamma . Then, ℳr(Γ)≥ℳr(γ){{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )\ge {{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma ), for 1≤r≤n−11\le r\le n-1.ProofAgain we may assume that the function uuin Theorem 3.1 is convex [4, Lem. 1]. If MMhas constant curvature aa, then Rijkℓ=a(δikδjℓ−δiℓδjk){R}_{ijk\ell }=a\left({\delta }_{ik}{\delta }_{j\ell }-{\delta }_{i\ell }{\delta }_{jk}). Thus, Theorem 3.1 yields (13)ℳr(Γ)−ℳr(γ)=(r+1)∫Ω⧹Dσr+1(κu)−a(n−r)∫Ω⧹Dσr−1(κu).{{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )-{{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )=\left(r+1)\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{r+1}\left({\kappa }^{u})-a\left(n-r)\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{r-1}\left({\kappa }^{u}).By assumption a≤0a\le 0, and since uuis convex, σr(κu)≥0{\sigma }_{r}\left({\kappa }^{u})\ge 0, which completes the proof.□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\Gamma is a convex hypersurface in a Cartan-Hadamard manifold, then its outer parallel hypersurface at distance ε\varepsilon , denoted by Γε{\Gamma }^{\varepsilon }, is C1,1{{\mathcal{C}}}^{1,1}for all ε>0\varepsilon \gt 0[13, Lem. 2.6]. So ℳr(Γε){{\mathcal{ {\mathcal M} }}}_{r}\left({\Gamma }^{\varepsilon })is well defined. By Corollary 4.4, ℳr(Γε){{\mathcal{ {\mathcal M} }}}_{r}\left({\Gamma }^{\varepsilon })is decreasing in ε\varepsilon . Hence, its limit as ε→0\varepsilon \to 0exists, and we may set ℳr(Γ)≔limε→0ℳr(Γε).{{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma ):= {\mathrm{lim}}_{\varepsilon \to 0}{{\mathcal{ {\mathcal M} }}}_{r}\left({\Gamma }^{\varepsilon }).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!!k\hspace{0.1em}\text{&#x0021;&#x0021;}\hspace{0.1em}, when kkis a positive integer, stands for the product of all positive odd (even) integers up to kk, when kkis odd (even). For k≤0k\le 0, we set k!!=1k\hspace{0.1em}\text{&#x0021;&#x0021;}\hspace{0.1em}=1.Corollary 4.6Let Γ\Gamma be a closed C1,1{{\mathcal{C}}}^{1,1}hypersurface in an n-manifold M bounding a domain Ω\Omega . Suppose that M has constant curvature a, and cl(Ω){\rm{cl}}\left(\Omega )is diffeomorphic to a ball. Then, ℳn−1(Γ)=∣Sn−1∣−∑i=1n−(nmod2)2(2i−1)!!(n−2i−2)!!(n−2)!!aiℳn−2i−1(Γ).{{\mathcal{ {\mathcal M} }}}_{n-1}\left(\Gamma )=| {{\bf{S}}}^{n-1}| -\mathop{\sum }\limits_{i=1}^{\frac{n-\left(n\hspace{0.33em}\hspace{0.1em}\text{mod}\hspace{0.1em}\hspace{0.33em}2)}{2}}\frac{\left(2i-1)\hspace{0.1em}\text{&#x0021;&#x0021;}\hspace{0.1em}\left(n-2i-2)\hspace{0.1em}\text{&#x0021;&#x0021;}\hspace{0.1em}}{\left(n-2)\hspace{0.1em}\text{&#x0021;&#x0021;}\hspace{0.1em}}{a}^{i}{{\mathcal{ {\mathcal M} }}}_{n-2i-1}\left(\Gamma ).ProofLet ϕ:cl(Ω)→Bn\phi :{\rm{cl}}\left(\Omega )\to {B}^{n}be a diffeomorphism to the unit ball in Rn{{\bf{R}}}^{n}and set u(x)≔∣ϕ(x)∣2u\left(x):= | \phi \left(x){| }^{2}. All regular level sets γ\gamma of uusatisfy (13). Furthermore, these level sets are convex near the minimum point x0{x}_{0}of uu, since uuhas positive definite Hessian at x0{x}_{0}. So by Corollary 4.5, for these small level sets, ℳr(S)≤ℳr(γ)≤ℳr(S′),{{\mathcal{ {\mathcal M} }}}_{r}\left(S)\le {{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )\le {{\mathcal{ {\mathcal M} }}}_{r}\left(S^{\prime} ),where SSand S′S^{\prime} are geodesic spheres centered at x0{x}_{0}such that SSis nested inside γ\gamma and γ\gamma is nested inside S′S^{\prime} . Consequently, by Lemma 4.2, as γ\gamma shrinks to x0{x}_{0}, ℳn−1(γ){{\mathcal{ {\mathcal M} }}}_{n-1}\left(\gamma )converges to ∣Sn−1∣| {{\bf{S}}}^{n-1}| , while ℳr(γ){{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )vanishes for r≤n−2r\le n-2. Thus, since σn(κu)=0{\sigma }_{n}\left({\kappa }^{u})=0, (13) yields ℳn−1(Γ)=∣Sn−1∣−a∫Ωσn−2(κu){{\mathcal{ {\mathcal M} }}}_{n-1}\left(\Gamma )=| {{\bf{S}}}^{n-1}| -a\mathop{\int }\limits_{\Omega }{\sigma }_{n-2}\left({\kappa }^{u})and ∫Ωσr(κu)=1rℳr−1(Γ)+a(n−r+1)r∫Ωσr−2(κu)\mathop{\int }\limits_{\Omega }{\sigma }_{r}\left({\kappa }^{u})=\frac{1}{r}{{\mathcal{ {\mathcal M} }}}_{r-1}\left(\Gamma )+\frac{a\left(n-r+1)}{r}\mathop{\int }\limits_{\Omega }{\sigma }_{r-2}\left({\kappa }^{u})for r≤n−2r\le n-2. Using these expressions iteratively completes the proof.□Finally, we include a characterization for hyperbolic balls, which extends to all mean curvatures a previous result of the authors on Gauss-Kronecker curvature [13, Cor. 5.5].Corollary 4.7Let M be a Cartan-Hadamard n-manifold with curvature ≤a≤0\le a\le 0, and Bρ{B}_{\rho }be a ball of radius ρ\rho in MM. Then, for 1≤r≤n−11\le r\le n-1, ℳr(∂Bρ)≥ℳr(∂Bρa),{{\mathcal{ {\mathcal M} }}}_{r}\left(\partial {B}_{\rho })\ge {{\mathcal{ {\mathcal M} }}}_{r}\left(\partial {B}_{\rho }^{a}),where Bρa{B}_{\rho }^{a}denotes a ball of radius ρ\rho in a manifold of constant curvature a. Equality holds only if Bρ{B}_{\rho }is isometric to Bρa{B}_{\rho }^{a}.ProofFor r=n−1r=n-1, the desired inequality has already been established [13, Cor. 5.5]. Suppose then that r≤n−2r\le n-2. We will show that (14)ℳr(∂Bρ)≥(r+1)∫Bρσr+1(κu)−a(n−r)∫Bρσr−1(κu)≥ℳr(∂Bρa).{{\mathcal{ {\mathcal M} }}}_{r}\left(\partial {B}_{\rho })\ge \left(r+1)\mathop{\int }\limits_{{B}_{\rho }}{\sigma }_{r+1}\left({\kappa }^{u})-a\left(n-r)\mathop{\int }\limits_{{B}_{\rho }}{\sigma }_{r-1}\left({\kappa }^{u})\ge {{\mathcal{ {\mathcal M} }}}_{r}\left(\partial {B}_{\rho }^{a}).Letting uube the distance squared function from the center ooof Bρ{B}_{\rho }, and γ\gamma shrink to ooin Theorem 3.1, yields the first inequality in (14) via Lemma 4.2. The principal curvatures of ∂Bρ\partial {B}_{\rho }are bounded below by −acoth(−aρ)\sqrt{-a}\coth \left(\sqrt{-a}\rho )[16, p. 184], which are the principal curvatures of ∂Bρa\partial {B}_{\rho }^{a}. Hence, the mean curvatures of ∂Bρ\partial {B}_{\rho }satisfy σr(κu)≥n−1r(−acoth(−aρ))r=σra(κu),{\sigma }_{r}\left({\kappa }^{u})\ge \left(\begin{array}{c}n-1\\ r\end{array}\right)(\sqrt{-a}\coth \left(\sqrt{-a}\rho ){)}^{r}={\sigma }_{r}^{a}\left({\kappa }^{u}),where σra(κu){\sigma }_{r}^{a}\left({\kappa }^{u})are the mean curvatures of ∂Bρa\partial {B}_{\rho }^{a}. Furthermore, if A(ρ,θ)dθA\left(\rho ,\theta ){\rm{d}}\theta denotes the volume element of ∂Bρ\partial {B}_{\rho }in geodesic spherical coordinates, then by [16, (1.5.4)], A(ρ,θ)≥sinh(−aρ)−an−1=Aa(ρ,θ),A\left(\rho ,\theta )\ge {\left(\frac{\sinh \left(\sqrt{-a}\rho )}{\sqrt{-a}}\right)}^{n-1}={A}^{a}\left(\rho ,\theta ),where Aa(ρ,θ)dθ{A}^{a}\left(\rho ,\theta )d\theta is the volume element of ∂Bρa\partial {B}_{\rho }^{a}; see [13, Cor. 5.5]. Thus, ∫Bρσr(κu)≥∫0ρ∫Sn−1σra(κu)Aa(t,θ)dθdt=∫Bρaσra(κu),\begin{array}{r}\mathop{\displaystyle \int }\limits_{{B}_{\rho }}{\sigma }_{r}\left({\kappa }^{u})\ge \underset{0}{\overset{\rho }{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{{\bf{S}}}^{n-1}}{\sigma }_{r}^{a}\left({\kappa }^{u}){A}^{a}\left(t,\theta ){\rm{d}}\theta {\rm{d}}t=\mathop{\displaystyle \int }\limits_{{B}_{\rho }^{a}}{\sigma }_{r}^{a}\left({\kappa }^{u}),\end{array}which yields the second inequality in (14). If ℳr(∂Bρ)=ℳr(∂Bρa){{\mathcal{ {\mathcal M} }}}_{r}\left(\partial {B}_{\rho })={{\mathcal{ {\mathcal M} }}}_{r}\left(\partial {B}_{\rho }^{a}), then equality holds in the first inequality of (14). So Krn=a{K}_{rn}=a, i.e., the radial sectional curvatures of Bρ{B}_{\rho }are constant, which forces Bρ{B}_{\rho }to have constant curvature aa[13, Lem. 5.4]. Hence, Bρ{B}_{\rho }is isometric to Bρa{B}_{\rho }^{a}.□ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advanced Nonlinear Studies de Gruyter

Total mean curvatures of Riemannian hypersurfaces

Advanced Nonlinear Studies , Volume 23 (1): 1 – Jan 1, 2023

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de Gruyter
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© 2023 the author(s), published by De Gruyter
ISSN
1536-1365
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2169-0375
DOI
10.1515/ans-2022-0029
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Abstract

1IntroductionTotal mean curvatures of a hypersurface Γ\Gamma in a Riemannian nn-manifold MMare integrals of symmetric functions of its principal curvatures. These quantities are known as quermassintegrals or mixed volumes when Γ\Gamma is convex and MMis 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 rthr{\rm{th}}mean curvatures of a pair of nested hypersurfaces Γ\Gamma and γ\gamma in a Riemannian manifold MMin terms of the sectional curvatures of MMand the principal curvatures of a family of hypersurfaces that fibrate the region between Γ\Gamma and γ\gamma . This formula simplifies when r=1r=1, Γ\Gamma and γ\gamma are parallel, or MMhas 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\Gamma (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.2Newton operatorsThroughout this work, MMdenotes an nn-dimensional Riemannian manifold with metric ⟨⋅,⋅⟩\langle \cdot ,\cdot \rangle and covariant derivative ∇\nabla . Furthermore, uuis a C1,1{{\mathcal{C}}}^{1,1}function on MM. In particular, uuis twice differentiable at almost every point ppof MM, and the computations below take place at such a point. The gradient of uuis the tangent vector ∇u∈TpM\nabla u\in {T}_{p}Mgiven by ⟨∇u(p),X⟩≔∇Xu\langle \nabla u\left(p),X\rangle := {\nabla }_{X}ufor all X∈TpMX\in {T}_{p}M. The Hessian operator ∇2u:TpM→TpM{\nabla }^{2}u:{T}_{p}M\to {T}_{p}Mis the self-adjoint linear map given by ∇2u(X)≔∇X(∇u).{\nabla }^{2}u\left(X):= {\nabla }_{X}\left(\nabla u).The symmetric elementary functions σr:Rk→R{\sigma }_{r}:{{\bf{R}}}^{k}\to {\bf{R}}, for 1≤r≤k1\le r\le k, and x=(x1,…,xk)x=\left({x}_{1},\ldots ,{x}_{k})are defined by σr(x)≔∑i1<⋯<irxi1…xir.{\sigma }_{r}\left(x):= \sum _{{i}_{1}\hspace{0.33em}\lt \cdots \lt {i}_{r}}{x}_{{i}_{1}}\ldots {x}_{{i}_{r}}.We set σ0≔1{\sigma }_{0}:= 1and σr≔0{\sigma }_{r}:= 0for r≥k+1r\ge k+1by convention. Let λ(∇2u)≔(λ1,…,λn)\lambda \left({\nabla }^{2}u):= \left({\lambda }_{1},\ldots ,{\lambda }_{n})denote the eigenvalues of ∇2u{\nabla }^{2}u. Then, we set σr(∇2u)≔σr(λ(∇2u)).{\sigma }_{r}\left({\nabla }^{2}u):= {\sigma }_{r}(\lambda \left({\nabla }^{2}u)).These functions form the coefficients of the characteristic polynomial P(λ)≔det(λI−∇2u)=∑i=0n(−1)iσi(∇2u)λn−i.P\left(\lambda ):= \det \left(\lambda I-{\nabla }^{2}u)=\mathop{\sum }\limits_{i=0}^{n}{\left(-1)}^{i}{\sigma }_{i}\left({\nabla }^{2}u){\lambda }^{n-i}.Let δj1…jmi1…im{\delta }_{{j}_{1}\ldots {j}_{m}}^{{i}_{1}\ldots {i}_{m}}be the generalized Kronecker tensor, which is equal to 1 (−1-1) if i1,…,im{i}_{1},\ldots ,{i}_{m}are distinct and (j1,…,jm)\left({j}_{1},\ldots ,{j}_{m})is an even (odd) permutation of (i1,…,im)\left({i}_{1},\ldots ,{i}_{m}); otherwise, it is equal to 0. Then [18, Prop. 1.2(a)], (1)σr(∇2u)=1r!δj1…jri1…irui1j1⋯uirjr,{\sigma }_{r}\left({\nabla }^{2}u)=\frac{1}{r\&#x0021;}{\delta }_{{j}_{1}\ldots {j}_{r}}^{{i}_{1}\ldots {i}_{r}}{u}_{{i}_{1}{j}_{1}}\cdots {u}_{{i}_{r}{j}_{r}},where uij≔∇iju{u}_{ij}:= {\nabla }_{ij}udenote the second partial derivatives of uuwith respect to an orthonormal frame Ei∈TpM{E}_{i}\in {T}_{p}M, which we extend to an open neighborhood of ppby parallel translation along geodesics. So ∇EiEj=0{\nabla }_{{E}_{i}}{E}_{j}=0at pp. We call Ei{E}_{i}a local parallel frame centered at ppand set ∇i≔∇Ei{\nabla }_{i}:= {\nabla }_{{E}_{i}}, ∇ij≔∇i∇j{\nabla }_{ij}:= {\nabla }_{i}{\nabla }_{j}. Each of the indices in (1) ranges from 1 to nn, and we employ Einstein’s convention by summing over repeated indices throughout the article. The Newton operators Tru:TpM→TpM{{\mathcal{T}}}_{r}^{u}:{T}_{p}M\to {T}_{p}M[17,18] are defined recursively by setting T0u≔I{{\mathcal{T}}}_{0}^{u}:= I, the identity map, and for r≥1r\ge 1, (2)Tru≔σr(∇2u)I−Tr−1u∘∇2u=∑i=0r(−1)iσi(∇2u)(∇2u)r−i.{{\mathcal{T}}}_{r}^{u}:= {\sigma }_{r}\left({\nabla }^{2}u)I-{{\mathcal{T}}}_{r-1}^{u}\circ {\nabla }^{2}u=\mathop{\sum }\limits_{i=0}^{r}{\left(-1)}^{i}{\sigma }_{i}\left({\nabla }^{2}u){\left({\nabla }^{2}u)}^{r-i}.Thus, Tru{{\mathcal{T}}}_{r}^{u}is the truncation of the polynomial P(∇2u)P\left({\nabla }^{2}u)obtained by removing the terms of order higher than rr. In particular, Tnu=P(∇2u){{\mathcal{T}}}_{n}^{u}=P\left({\nabla }^{2}u). So, by the Cayley-Hamilton theorem, Tnu=0{{\mathcal{T}}}_{n}^{u}=0. Consequently, when ∇2u{\nabla }^{2}uis nondegenerate, (2) yields that (3)Tn−1u=σn(∇2u)(∇2u)−1=det(∇2u)(∇2u)−1=Tu,{{\mathcal{T}}}_{n-1}^{u}={\sigma }_{n}\left({\nabla }^{2}u){\left({\nabla }^{2}u)}^{-1}=\det \left({\nabla }^{2}u){\left({\nabla }^{2}u)}^{-1}={{\mathcal{T}}}^{u},where Tu{{\mathcal{T}}}^{u}is the Hessian cofactor operator discussed in [13, Sec. 4]. See [18, Prop. 1.2] for other basic identities that relate σ\sigma and T{\mathcal{T}}. In particular, by [18, Prop. 1.2(c)], we have Trace(Tru⋅∇2u)=(r+1)σr+1(∇2u)\hspace{0.1em}\text{Trace}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}\cdot {\nabla }^{2}u)=\left(r+1){\sigma }_{r+1}\left({\nabla }^{2}u). So, by Euler’s identity for homogeneous polynomials, (4)(Tru)ijuij=Trace(Tru∘∇2u)=(r+1)σr+1(∇2u)=∂σr+1(∇2u)∂uijuij.{\left({{\mathcal{T}}}_{r}^{u})}_{ij}{u}_{ij}=\hspace{0.1em}\text{Trace}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}\circ {\nabla }^{2}u)=\left(r+1){\sigma }_{r+1}\left({\nabla }^{2}u)=\frac{\partial {\sigma }_{r+1}\left({\nabla }^{2}u)}{\partial {u}_{ij}}{u}_{ij}.Thus, it follows from (1) that (5)(Tru)ij=∂σr+1(∇2u)∂uij=1r!δjj1…jrii1…irui1j1⋯uirjr.{({{\mathcal{T}}}_{r}^{u})}_{ij}=\frac{\partial {\sigma }_{r+1}\left({\nabla }^{2}u)}{\partial {u}_{ij}}=\frac{1}{r\&#x0021;}{\delta }_{j{j}_{1}\ldots {j}_{r}}^{i{i}_{1}\ldots {i}_{r}}{u}_{{i}_{1}{j}_{1}}\cdots {u}_{{i}_{r}{j}_{r}}.Furthermore, by [17, Prop. 1(11)] (note that the sign of the Riemann tensor RRin [17] is opposite to the one in this article), we have (6)(div(Tru))j=1(r−1)!δjj1…jrii1…irui1j1⋯uir−1jr−1Rijrirkuk,(\hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}){)}_{j}=\frac{1}{\left(r-1)\&#x0021;}{\delta }_{j{j}_{1}\ldots {j}_{r}}^{i{i}_{1}\ldots {i}_{r}}{u}_{{i}_{1}{j}_{1}}\cdots {u}_{{i}_{r-1}{j}_{r-1}}{R}_{i{j}_{r}{i}_{r}k}{u}_{k},where Rijkl≔R(Ei,Ej,Ek,Eℓ)=⟨∇i∇jEk−∇j∇iEk,Eℓ⟩{R}_{ijkl}:= R\left({E}_{i},{E}_{j},{E}_{k},{E}_{\ell })=\langle {\nabla }_{i}{\nabla }_{j}{E}_{k}-{\nabla }_{j}{\nabla }_{i}{E}_{k},{E}_{\ell }\rangle . Another useful identity [17, p. 462] is (7)div(Tru(∇u))=⟨Tru,∇2u⟩+⟨div(Tru),∇u⟩,\hspace{0.1em}\text{div}\hspace{0.1em}({{\mathcal{T}}}_{r}^{u}\left(\nabla u))=\langle {{\mathcal{T}}}_{r}^{u},{\nabla }^{2}u\rangle +\langle \hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}),\nabla u\rangle ,where ⟨⋅,⋅⟩\langle \cdot ,\cdot \rangle here indicates the Frobenius inner product (i.e., ⟨A,B⟩≔AijBij\langle A,B\rangle := {A}_{ij}{B}_{ij}for any pair of matrices of the same dimension). The divergence of Tru{{\mathcal{T}}}_{r}^{u}may be defined by virtually the same argument used for Tu{{\mathcal{T}}}^{u}in [13, Sec. 4] to yield the following generalization of [13, (14)]: (8)(div(Tru))j=∇i(Tru)ij.(\hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}){)}_{j}={\nabla }_{i}{\left({{\mathcal{T}}}_{r}^{u})}_{ij}.Recall that Tu=Tn−1u{{\mathcal{T}}}^{u}={{\mathcal{T}}}_{n-1}^{u}by (3). Furthermore, Tnu=0{{\mathcal{T}}}_{n}^{u}=0as we mentioned earlier. Thus, the following observation generalizes [13, Lem. 4.2].Lemma 2.1divTr−1u∇u∣∇u∣r=div(Tr−1u),∇u∣∇u∣r+r⟨Tru(∇u),∇u⟩∣∇u∣r+2.{\rm{div}}\left({{\mathcal{T}}}_{r-1}^{u}\left(\frac{\nabla u}{| \nabla u{| }^{r}}\right)\right)=\left\langle \hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r-1}^{u}),\frac{\nabla u}{| \nabla u{| }^{r}}\right\rangle +r\frac{\langle {{\mathcal{T}}}_{r}^{u}\left(\nabla u),\nabla u\rangle }{| \nabla u{| }^{r+2}}.ProofBy Leibniz rule and (8), we have divTr−1u∇u∣∇u∣r=∇i(Tr−1u)ijuj∣∇u∣r=div(Tr−1u),∇u∣∇u∣r+(Tr−1u)ijuij∣∇u∣r−rujuℓuℓi∣∇u∣r+2,\hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r-1}^{u}\left(\frac{\nabla u}{| \nabla u{| }^{r}}\right)\right)={\nabla }_{i}\left({\left({{\mathcal{T}}}_{r-1}^{u})}_{ij}\frac{{u}_{j}}{| \nabla u{| }^{r}}\right)=\left\langle \hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r-1}^{u}),\frac{\nabla u}{| \nabla u{| }^{r}}\right\rangle +{\left({{\mathcal{T}}}_{r-1}^{u})}_{ij}\left(\frac{{u}_{ij}}{| \nabla u{| }^{r}}-r\frac{{u}_{j}{u}_{\ell }{u}_{\ell i}}{| \nabla u{| }^{r+2}}\right),where the computation to obtain the second term on the right is identical to the one performed earlier in [13, Lem. 4.2]. To develop this term further, note that by (2) (Tr−1u)ijuℓi=σr(∇2u)δℓj−(Tru)ℓj,{\left({{\mathcal{T}}}_{r-1}^{u})}_{ij}{u}_{\ell i}={\sigma }_{r}\left({\nabla }^{2}u){\delta }_{\ell j}-{\left({{\mathcal{T}}}_{r}^{u})}_{\ell j},which in turn yields (Tr−1u)ijuℓiujuℓ∣∇u∣2=σr(∇2u)−(Tru)ijuiuj∣∇u∣2.{\left({{\mathcal{T}}}_{r-1}^{u})}_{ij}{u}_{\ell i}\frac{{u}_{j}{u}_{\ell }}{| \nabla u{| }^{2}}={\sigma }_{r}\left({\nabla }^{2}u)-{\left({{\mathcal{T}}}_{r}^{u})}_{ij}\frac{{u}_{i}{u}_{j}}{| \nabla u{| }^{2}}.Hence, (Tr−1u)ijuij∣∇u∣r−rujuℓuℓi∣∇u∣r+2=rσr(∇2u)∣∇u∣r−r∣∇u∣rσr(∇2u)−(Tru)ijuiuj∣∇u∣2=r(Tru)ijuiuj∣∇u∣r+2,{\left({{\mathcal{T}}}_{r-1}^{u})}_{ij}\left(\frac{{u}_{ij}}{| \nabla u{| }^{r}}-r\frac{{u}_{j}{u}_{\ell }{u}_{\ell i}}{| \nabla u{| }^{r+2}}\right)=\frac{r{\sigma }_{r}\left({\nabla }^{2}u)}{| \nabla u{| }^{r}}-\frac{r}{| \nabla u{| }^{r}}\left({\sigma }_{r}\left({\nabla }^{2}u)-{\left({{\mathcal{T}}}_{r}^{u})}_{ij}\frac{{u}_{i}{u}_{j}}{| \nabla u{| }^{2}}\right)=r{\left({{\mathcal{T}}}_{r}^{u})}_{ij}\frac{{u}_{i}{u}_{j}}{| \nabla u{| }^{r+2}},which completes the proof.□Below we assume, as was the case in [13, Sec. 4], that all local computations take place with respect to a principal curvature frame Ei∈TpM{E}_{i}\in {T}_{p}Mof uu, which is defined as follows. Assuming ∣∇u(p)∣≠0| \nabla u\left(p)| \ne 0, we set En≔∇u(p)/∣∇u(p)∣{E}_{n}:= \nabla u\left(p)\hspace{0.1em}\text{/}\hspace{0.1em}| \nabla u\left(p)| , and let E1,…,En−1{E}_{1},\ldots ,{E}_{n-1}be the principal directions of the level set of uupassing through pp. Then, we extend Ei{E}_{i}to a local parallel frame near pp. The first partial derivatives of uuwith respect to Ei{E}_{i}, ui≔∇iu{u}_{i}:= {\nabla }_{i}u, satisfy (9)ui=0;fori≠n,andun=∣∇u∣.{u}_{i}=0;\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}i\ne n,\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{u}_{n}=| \nabla u| .Furthermore, for the second partial derivatives, uij=∇iju{u}_{ij}={\nabla }_{ij}u, we have (10)uij=0,fori≠j≤n−1,anduii∣∇u∣≕κiu,fori≠n,{u}_{ij}=0,\hspace{2.22144pt}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{2.22144pt}i\ne j\le n-1,\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\frac{{u}_{ii}}{| \nabla u| }=: {\kappa }_{i}^{u},\hspace{2.22144pt}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{2.22144pt}i\ne n,where κ1u,…,κn−1u{\kappa }_{1}^{u},\ldots ,{\kappa }_{n-1}^{u}are the principal curvatures of level sets of uuwith respect to En{E}_{n}, i.e., they are eigenvalues corresponding to E1,…,En−1{E}_{1},\ldots ,{E}_{n-1}of the shape operator X↦∇XνX\mapsto {\nabla }_{X}\nu on the tangent space of level sets of uu, where ν≔∇u/∣∇u∣\nu := \nabla u\hspace{0.1em}\text{/}\hspace{0.1em}| \nabla u| . We set κu≔(κ1u,…,κn−1u){\kappa }_{u}:= \left({\kappa }_{1}^{u},\ldots ,{\kappa }_{n-1}^{u}). So σr(κu){\sigma }_{r}\left({\kappa }^{u})is the rthr{\rm{th}}mean curvature of the level set of uuat pp. In particular, σn−1(κu){\sigma }_{n-1}\left({\kappa }^{u})is the Gauss-Kronecker curvature of the level sets. The next observation generalizes [13, Lem. 4.1].Lemma 2.2σr(κu)=⟨Tru(∇u),∇u⟩∣∇u∣r+2.{\sigma }_{r}\left({\kappa }^{u})=\frac{\langle {{\mathcal{T}}}_{r}^{u}\left(\nabla u),\nabla u\rangle }{| \nabla u{| }^{r+2}}.Proof(5) together with (9) and (10) yields that (Tru)ijuiuj∣∇u∣r+2=1r!δjj1⋯jrii1⋯irui1j1⋯uirjruiuj∣∇u∣r+2=1r!δnj1⋯jrni1⋯irui1j1⋯uirjr∣∇u∣r⋅un2∣∇u∣2=1r!δni1⋯irni1⋯irκi1u…κiru.□\begin{array}{rcl}\hspace{13.8em}{\left({{\mathcal{T}}}_{r}^{u})}_{ij}\frac{{u}_{i}{u}_{j}}{| \nabla u{| }^{r+2}}& =& \frac{1}{r\&#x0021;}{\delta }_{j{j}_{1}\cdots {j}_{r}}^{i{i}_{1}\cdots {i}_{r}}{u}_{{i}_{1}{j}_{1}}\cdots {u}_{{i}_{r}{j}_{r}}\frac{{u}_{i}{u}_{j}}{| \nabla u{| }^{r+2}}\\ & =& \frac{1}{r\&#x0021;}{\delta }_{n{j}_{1}\cdots {j}_{r}}^{n{i}_{1}\cdots {i}_{r}}\frac{{u}_{{i}_{1}{j}_{1}}\cdots {u}_{{i}_{r}{j}_{r}}}{| \nabla u{| }^{r}}\cdot \frac{{u}_{n}^{2}}{| \nabla u{| }^{2}}\\ & =& \frac{1}{r\&#x0021;}{\delta }_{n{i}_{1}\cdots {i}_{r}}^{n{i}_{1}\cdots {i}_{r}}{\kappa }_{{i}_{1}}^{u}\ldots {\kappa }_{{i}_{r}}^{u}.\hspace{17em}\square \end{array}3Comparison formulaHere, we establish the main result of this work. For a C1,1{{\mathcal{C}}}^{1,1}hypersurface Γ\Gamma in a Riemannian nn-manifold MM, oriented by a choice of normal vector field ν\nu , and 0≤r≤n−10\le r\le n-1, we let ℳr(Γ)≔∫Γσr(κ){{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma ):= \mathop{\int }\limits_{\Gamma }{\sigma }_{r}\left(\kappa )be the total rthr{\rm{th}}mean curvature of Γ\Gamma , where κ≔(κ1,…,κn−1)\kappa := \left({\kappa }_{1},\ldots ,{\kappa }_{n-1})denotes principal curvatures of Γ\Gamma with respect to ν\nu . Note that ℳ0(Γ)=∣Γ∣{{\mathcal{ {\mathcal M} }}}_{0}\left(\Gamma )=| \Gamma | , the volume of Γ\Gamma , since σ0=1{\sigma }_{0}=1, and ℳn−1(Γ){{\mathcal{ {\mathcal M} }}}_{n-1}\left(\Gamma )is the total Gauss-Kronecker curvature of Γ\Gamma (denoted by G(Γ){\mathcal{G}}\left(\Gamma )in [13]). A domain Ω⊂M\Omega \subset Mis an open set with a compact closure cl(Ω){\rm{cl}}\left(\Omega ). If Γ\Gamma bounds a domain Ω\Omega , then by convention we set ℳ−1(Γ)≔∣Ω∣{{\mathcal{ {\mathcal M} }}}_{-1}\left(\Gamma ):= | \Omega | , the volume of Ω\Omega . The following theorem generalizes [13, Thm. 4.7], where this result had been established for r=n−1r=n-1. It also uses less regularity than was required in [13, Thm. 4.7].Theorem 3.1Let Γ\Gamma and γ\gamma be closed C1,1{{\mathcal{C}}}^{1,1}hypersurfaces in a Riemannian n-manifold M bounding domains Ω\Omega and D, respectively, with cl(D)⊂Ω{\rm{cl}}\left(D)\subset \Omega . Suppose there exists a C1,1{{\mathcal{C}}}^{1,1}function uuon cl(Ω⧹D){\rm{cl}}\left(\Omega \setminus D)with ∇u≠0\nabla u\ne 0, which is constant on Γ\Gamma and γ\gamma . Let κu≔(κ1u,…,κn−1u){\kappa }^{u}:= \left({\kappa }_{1}^{u},\ldots ,{\kappa }_{n-1}^{u})be the principal curvatures of level sets of u with respect to En≔∇u/∣∇u∣{E}_{n}:= \nabla u\hspace{0.1em}\text{/}\hspace{0.1em}| \nabla u| , and let E1,…,En−1{E}_{1},\ldots ,{E}_{n-1}be the corresponding principal directions. Then, for 0≤r≤n−10\le r\le n-1, ℳr(Γ)−ℳr(γ)=(r+1)∫Ω⧹Dσr+1(κu)+∫Ω⧹D−∑κi1u…κir−1uKirn+1∣∇u∣∑κi1u…κir−2u∣∇u∣ir−1Ririr−1irn,{{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )-{{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )=\left(r+1)\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{r+1}\left({\kappa }^{u})+\mathop{\int }\limits_{\Omega \setminus D}\left(-\sum {\kappa }_{{i}_{1}}^{u}\ldots {\kappa }_{{i}_{r-1}}^{u}{K}_{{i}_{r}n}+\frac{1}{| \nabla u| }\sum {\kappa }_{{i}_{1}}^{u}\ldots {\kappa }_{{i}_{r-2}}^{u}| \nabla u{| }_{{i}_{r-1}}{R}_{{i}_{r}{i}_{r-1}{i}_{r}n}\right),where ∣∇u∣i≔∇Ei∣∇u∣| \nabla u{| }_{i}:= {\nabla }_{{E}_{i}}| \nabla u| , Rijkl=R(Ei,Ej,Ek,El){R}_{ijkl}=R\left({E}_{i},{E}_{j},{E}_{k},{E}_{l})are components of the Riemann curvature tensor of MM, Kij=Rijij{K}_{ij}={R}_{ijij}is the sectional curvature, and the summations take place over distinct values of 1≤i1,…,ir≤n−11\le {i}_{1},\ldots ,{i}_{r}\le n-1, with i1<⋯<ir−1{i}_{1}\hspace{0.33em}\lt \cdots \lt {i}_{r-1}in the first sum and i1<⋯<ir−2{i}_{1}\hspace{0.33em}\lt \cdots \lt {i}_{r-2}in the second sum.ProofBy Lemmas 2.1 and 2.2, (11)divTru∇u∣∇u∣r+1=(r+1)σr+1(κu)+div(Tru),∇u∣∇u∣r+1.\hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}\left(\frac{\nabla u}{| \nabla u{| }^{r+1}}\right)\right)=\left(r+1){\sigma }_{r+1}\left({\kappa }^{u})+\left\langle \hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}),\frac{\nabla u}{| \nabla u{| }^{r+1}}\right\rangle .By Stokes’ theorem and Lemma 2.2, ∫Ω⧹DdivTru∇u∣∇u∣r+1=∫Γ∪γTru∇u∣∇u∣r+1,∇u∣∇u∣=ℳr(Γ)−ℳr(γ).\mathop{\int }\limits_{\Omega \setminus D}\hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}\left(\frac{\nabla u}{| \nabla u{| }^{r+1}}\right)\right)=\mathop{\int }\limits_{\Gamma \cup \gamma }\left\langle {{\mathcal{T}}}_{r}^{u}\left(\frac{\nabla u}{| \nabla u{| }^{r+1}}\right),\frac{\nabla u}{| \nabla u| }\right\rangle ={{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )-{{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma ).So integrating both sides of (11) yields ℳr(Γ)−ℳr(γ)=(r+1)∫Ω⧹Dσr+1(κu)+∫Ω⧹Ddiv(Tru),∇u∣∇u∣r+1.{{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )-{{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )=\left(r+1)\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{r+1}\left({\kappa }^{u})+\mathop{\int }\limits_{\Omega \setminus D}\left\langle \hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}),\frac{\nabla u}{| \nabla u{| }^{r+1}}\right\rangle .Using (6) and (9), we have div(Tru),∇u∣∇u∣r+1=1(r−1)!δjj1…jrii1…irui1j1⋯uir−1jr−1Rijrirkukuj∣∇u∣r+1=1(r−1)!δnj1…jrii1…irui1j1∣∇u∣⋯uir−1jr−1∣∇u∣Rijrirn.\begin{array}{rcl}\left\langle \hspace{0.1em}\text{div}\hspace{0.1em}\left({{\mathcal{T}}}_{r}^{u}),\frac{\nabla u}{| \nabla u{| }^{r+1}}\right\rangle & =& \frac{1}{\left(r-1)\&#x0021;}{\delta }_{j{j}_{1}\ldots {j}_{r}}^{i{i}_{1}\ldots {i}_{r}}{u}_{{i}_{1}{j}_{1}}\cdots {u}_{{i}_{r-1}{j}_{r-1}}{R}_{i{j}_{r}{i}_{r}k}\frac{{u}_{k}{u}_{j}}{| \nabla u{| }^{r+1}}\\ & =& \frac{1}{\left(r-1)\&#x0021;}{\delta }_{n{j}_{1}\ldots {j}_{r}}^{i{i}_{1}\ldots {i}_{r}}\frac{{u}_{{i}_{1}{j}_{1}}}{| \nabla u| }\cdots \frac{{u}_{{i}_{r-1}{j}_{r-1}}}{| \nabla u| }{R}_{i{j}_{r}{i}_{r}n}.\end{array}The last expression may be written as the sum of two components, AAand BB, which consist of terms with i=ni=nand i≠ni\ne n, respectively. Note that we may assume j1,…,jr≠n{j}_{1},\ldots ,{j}_{r}\ne n, for otherwise δnj1…jrii1…ir=0{\delta }_{n{j}_{1}\ldots {j}_{r}}^{i{i}_{1}\ldots {i}_{r}}=0. To compute AA, note that if i=ni=n, then for δnj1…jrii1…ir{\delta }_{n{j}_{1}\ldots {j}_{r}}^{i\hspace{0.33em}{i}_{1}\ldots {i}_{r}}not to vanish, we must have i1,…,ir≠n{i}_{1},\ldots ,{i}_{r}\ne n. Then, by (10), uikjk=0{u}_{{i}_{k}{j}_{k}}=0unless ik=jk{i}_{k}={j}_{k}, which yields that A=1(r−1)!δni1…irni1…irui1i1∣∇u∣⋯uir−1ir−1∣∇u∣Rnirirn=−∑κi1u…κir−1uKirn,A=\frac{1}{\left(r-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\delta }_{n{i}_{1}\ldots {i}_{r}}^{n{i}_{1}\ldots {i}_{r}}\frac{{u}_{{i}_{1}{i}_{1}}}{| \nabla u| }\cdots \frac{{u}_{{i}_{r-1}{i}_{r-1}}}{| \nabla u| }{R}_{n{i}_{r}{i}_{r}n}=-\sum {\kappa }_{{i}_{1}}^{u}\ldots {\kappa }_{{i}_{r-1}}^{u}{K}_{{i}_{r}n},where the sum ranges over all distinct values of 1≤i1,…,ir≤n−11\le {i}_{1},\ldots ,{i}_{r}\le n-1, with i1<⋯<ir−1{i}_{1}\hspace{0.33em}\lt \cdots \lt {i}_{r-1}as desired. To find BBnote that if i≠ni\ne n, then for δnj1…jrii1…ir{\delta }_{n{j}_{1}\ldots {j}_{r}}^{i\hspace{0.33em}{i}_{1}\ldots {i}_{r}}not to vanish, we must have ik=n{i}_{k}=nfor some 1≤k≤r1\le k\le r. If k=rk=r, then Rijrirn=Rijknn=0{R}_{i{j}_{r}{i}_{r}n}={R}_{i{j}_{k}nn}=0. In particular, B=0B=0when r=1r=1. Now assume that r≥2r\ge 2. Then, we may assume that k≠rk\ne r, or ir≠n{i}_{r}\ne n. Then, by (10), uirjr=0{u}_{{i}_{r}{j}_{r}}=0unless ir=jr{i}_{r}={j}_{r}. So, we may assume that ir=jr{i}_{r}={j}_{r}for r≠kr\ne k, which in turn implies that jk=i{j}_{k}=i. Thus, B=∑k=1r−1BkB={\sum }_{k=1}^{r-1}{B}_{k}, where Bk=1(r−1)!δni1…ik−1iik+1…irii1…ik−1nik+1…irui1i1∣∇u∣…uik−1ik−1∣∇u∣uni∣∇u∣uik+1ik+1∣∇u∣…uir−1ir−1∣∇u∣Riirirn=−1(r−1)∑κi1u…κik−1u∣∇u∣i∣∇u∣κik+1u…κir−1uRiirirn,\begin{array}{rcl}{B}_{k}& =& \frac{1}{\left(r-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\delta }_{n{i}_{1}\ldots {i}_{k-1}i\hspace{0.33em}{i}_{k+1}\ldots {i}_{r}}^{i\hspace{0.33em}{i}_{1}\ldots {i}_{k-1}n{i}_{k+1}\ldots {i}_{r}}\frac{{u}_{{i}_{1}{i}_{1}}}{| \nabla u| }\ldots \frac{{u}_{{i}_{k-1}{i}_{k-1}}}{| \nabla u| }\frac{{u}_{ni}}{| \nabla u| }\frac{{u}_{{i}_{k+1}{i}_{k+1}}}{| \nabla u| }\ldots \frac{{u}_{{i}_{r-1}{i}_{r-1}}}{| \nabla u| }{R}_{i{i}_{r}{i}_{r}n}\\ & =& \frac{-1}{\left(r-1)}\displaystyle \sum {\kappa }_{{i}_{1}}^{u}\ldots {\kappa }_{{i}_{k-1}}^{u}\frac{| \nabla u{| }_{i}}{| \nabla u| }{\kappa }_{{i}_{k+1}}^{u}\ldots {\kappa }_{{i}_{r-1}}^{u}{R}_{i{i}_{r}{i}_{r}n},\end{array}since un=∣∇u∣{u}_{n}=| \nabla u| . Here, the sum ranges over all distinct indices 1≤i,i1,…,ik−1,ik,…,ir≤n−11\le i,{i}_{1},\ldots ,{i}_{k-1},{i}_{k},\ldots ,{i}_{r}\le n-1, with i1<⋯<ik−1<ik+1<⋯<ir−1{i}_{1}\hspace{0.33em}\lt \cdots \lt {i}_{k-1}\lt {i}_{k+1}\hspace{0.33em}\lt \cdots \lt {i}_{r-1}. Note that B1=⋯=Br−1{B}_{1}=\cdots ={B}_{r-1}. Thus, B=(r−1)Br−1=1∣∇u∣∑κi1u…κir−2u∣∇u∣iRiriirn,B=\left(r-1){B}_{r-1}=\frac{1}{| \nabla u| }\sum {\kappa }_{{i}_{1}}^{u}\ldots {\kappa }_{{i}_{r-2}}^{u}| \nabla u{| }_{i}{R}_{{i}_{r}i{i}_{r}n},which completes the proof (after renaming iito ir−1{i}_{r-1}).□4ApplicationsHere, we develop some consequences of Theorem 3.1. A subset of a Cartan-Hadamard manifold MMis convex if it contains the (unique) geodesic segment connecting every pair of its points. A convex hypersurface Γ⊂M\Gamma \subset Mis the boundary of a compact convex set with interior points. If Γ\Gamma is of class C1,1{{\mathcal{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\Gamma \subset Mare all nonnegative, then Γ\Gamma 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\Gamma if it lies in the convex domain bounded by Γ\Gamma .Corollary 4.1Let Γ\Gamma and γ\gamma be C1,1{{\mathcal{C}}}^{1,1}convex hypersurfaces in a Cartan-Hadamard nn-manifold. Suppose that γ\gamma is nested inside Γ\Gamma . Then, ℳ1(Γ)≥ℳ1(γ){{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )\ge {{\mathcal{ {\mathcal M} }}}_{1}\left(\gamma ).ProofSetting r=1r=1in the comparison formula of Theorem 3.1 yields (12)ℳ1(Γ)−ℳ1(γ)=2∫Ω⧹Dσ2(κu)−∫Ω⧹DRic∇u∣∇u∣,{{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )-{{\mathcal{ {\mathcal M} }}}_{1}\left(\gamma )=2\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{2}\left({\kappa }^{u})-\mathop{\int }\limits_{\Omega \setminus D}\hspace{0.1em}\text{Ric}\hspace{0.1em}\left(\frac{\nabla u}{| \nabla u| }\right),where Ric stands for Ricci curvature; more explicitly, in a principal curvature frame where En≔∇u/∣∇u∣{E}_{n}:= \nabla u\hspace{0.1em}\text{/}\hspace{0.1em}| \nabla u| , Ric(En)\hspace{0.1em}\text{Ric}\hspace{0.1em}\left({E}_{n})is the sum of sectional curvatures Kin{K}_{{\rm{in}}}, for 1≤i≤n−11\le i\le n-1. So Ric(En)≤0\hspace{0.1em}\text{Ric}\hspace{0.1em}\left({E}_{n})\le 0. If Γ\Gamma and γ\gamma are smooth (C∞{{\mathcal{C}}}^{\infty }) and strictly convex, we may let uuin Theorem 3.1 be a function with convex level sets [4, Lem. 1]. Then, σ2(κu)≥0{\sigma }_{2}\left({\kappa }^{u})\ge 0, which yields ℳ1(Γ)≥ℳ1(γ){{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )\ge {{\mathcal{ {\mathcal M} }}}_{1}\left(\gamma )as desired. This completes the proof since we may approximate Γ\Gamma and γ\gamma by smooth strictly convex hypersurfaces, e.g., by applying the Greene-Wu convolution to their distance functions, see [12, Lem. 3.3]; furthermore, total mean curvatures will converge here since they constitute “valuations” in the sense of integral geometry, see [13, Note 3.7] or [3, Prop. 3.8].□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.Lemma 4.2Let Sρ{S}_{\rho }be a geodesic sphere of radius ρ\rho centered at a point in a Riemannian manifold. As ρ→0\rho \to 0, ℳr(Sρ){{\mathcal{ {\mathcal M} }}}_{r}\left({S}_{\rho })converges to 0 for r≤n−2r\le n-2and to ∣Sn−1∣| {{\bf{S}}}^{n-1}| for r=n−1r=n-1.ProofA power series expansion [6, Thm. 3.1] of the second fundamental form of Sρ{S}_{\rho }in normal coordinates shows that the principal curvatures of Sρ{S}_{\rho }are given by κiρ=(1+O(ρ2))/ρ{\kappa }_{i}^{\rho }=\left(1+O\left({\rho }^{2}))\hspace{0.1em}\text{/}\hspace{0.1em}\rho . So σr(κρ)=n−1r1ρr(1+O(ρ2)).{\sigma }_{r}\left({\kappa }^{\rho })=\left(\begin{array}{c}n-1\\ r\end{array}\right)\frac{1}{{\rho }^{r}}\left(1+O\left({\rho }^{2})).Another power series expansion [15, Thm. 3.1] yields ∣Sρ∣=∣Sn−1∣ρn−1(1+O(ρ2)).| {S}_{\rho }| =| {{\bf{S}}}^{n-1}| {\rho }^{n-1}\left(1+O\left({\rho }^{2})).So, it follows that ℳr(Sρ)=n−1r∣Sn−1∣ρn−1−r(1+O(ρ2)),{{\mathcal{ {\mathcal M} }}}_{r}\left({S}_{\rho })=\left(\begin{array}{c}n-1\\ r\end{array}\right)| {{\bf{S}}}^{n-1}| {\rho }^{n-1-r}(1+O\left({\rho }^{2})),which completes the proof.□Gallego and Solanes showed [10, Cor. 3.2] that if Γ\Gamma is a convex hypersurface bounding a domain Ω\Omega in a hyperbolic nn-space of constant curvature a<0a\lt 0, then ℳ1(Γ)>−(n−1)2a∣Ω∣.{{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )\gt -{\left(n-1)}^{2}a| \Omega | .When comparing formulas, note that in [10], mean curvature is defined as the average of κi{\kappa }_{i}, as opposed to the sum of κi{\kappa }_{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:Corollary 4.3Let Γ\Gamma be a C1,1{{\mathcal{C}}}^{1,1}convex hypersurface in a Cartan-Hadamard n-manifold M bounding a domain Ω\Omega . Suppose that curvature of M is bounded above by a≤0a\le 0. Then, ℳ1(Γ)>−(n−1)a∣Ω∣.{{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )\gt -\left(n-1)a| \Omega | .Furthermore, if n=3n=3, thenℳ1(Γ)>−4a∣Ω∣.{{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )\gt -4a| \Omega | .ProofLet γ=γρ\gamma ={\gamma }_{\rho }in (12) be a geodesic sphere of radius ρ\rho . By Lemma 4.2, ℳ1(γρ)→0{{\mathcal{ {\mathcal M} }}}_{1}\left({\gamma }_{\rho })\to 0as ρ→0\rho \to 0, which yields ℳ1(Γ)=2∫Ωσ2(κu)−∫ΩRic∇u∣∇u∣>−(n−1)a∣Ω∣,{{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )=2\mathop{\int }\limits_{\Omega }{\sigma }_{2}\left({\kappa }^{u})-\mathop{\int }\limits_{\Omega }\hspace{0.1em}\text{Ric}\hspace{0.1em}\left(\frac{\nabla u}{| \nabla u| }\right)\gt -\left(n-1)a| \Omega | ,as desired. When n=3n=3, Gauss’ equation states that σ2(κu)=Ku−KMu,{\sigma }_{2}\left({\kappa }^{u})={K}^{u}-{K}_{M}^{u},where Ku{K}^{u}is the sectional curvature of level sets of uuand KMu{K}_{M}^{u}is the sectional curvature of MMwith respect to tangent planes to level sets of uu. Thus, ℳ1(Γ)=2∫ΩKu−2∫ΩKMu−∫ΩRic∇u∣∇u∣>−4a∣Ω∣,{{\mathcal{ {\mathcal M} }}}_{1}\left(\Gamma )=2\mathop{\int }\limits_{\Omega }{K}^{u}-2\mathop{\int }\limits_{\Omega }{K}_{M}^{u}-\mathop{\int }\limits_{\Omega }\hspace{0.1em}\text{Ric}\hspace{0.1em}\left(\frac{\nabla u}{| \nabla u| }\right)\gt -4a| \Omega | ,which completes the proof.□We say Γ\Gamma is an outer parallel hypersurface of a convex hypersurface γ\gamma if all points of Γ\Gamma are at a constant distance λ≥0\lambda \ge 0from the convex domain bounded by γ\gamma . Since the distance function of a convex set in a Cartan-Hadamard manifold is convex [5, Prop. 2.4], Γ\Gamma is convex. Furthermore, Γ\Gamma is C1,1{{\mathcal{C}}}^{1,1}for λ>0\lambda \gt 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].Corollary 4.4Let M be a Cartan-Hadamard n-manifold, and Γ\Gamma and γ\gamma be C1,1{{\mathcal{C}}}^{1,1}convex hypersurfaces in M. Suppose that Γ\Gamma is an outer parallel hypersurface of γ\gamma . Then, ℳr(Γ)≥ℳr(γ){{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )\ge {{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )for 1≤r≤n−11\le r\le n-1.ProofWe may let uuin Theorem 3.1 be the distance function of the convex domain bounded by Γ\Gamma . Then, ∣∇u∣| \nabla u| is constant on level sets of uu. So, ∣∇u∣i=0| \nabla u{| }_{i}=0for 1≤i≤n−11\le i\le n-1, which yields ℳr(Γ)−ℳr(γ)≥(r+1)∫Ω⧹Dσr+1(κu)−a(n−r)∫Ω⧹Dσr−1(κu),{{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )-{{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )\ge \left(r+1)\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{r+1}\left({\kappa }^{u})-a\left(n-r)\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{r-1}\left({\kappa }^{u}),where a≤0a\le 0is the upper bound for sectional curvatures of MM. Since uuis convex, σr(κu)≥0{\sigma }_{r}\left({\kappa }^{u})\ge 0, which completes the proof.□The next result generalizes [13, Cor. 5.2] and observation of Borbely [4, Thm. 1] for Gauss-Kronecker curvature.Corollary 4.5Let M be a Cartan-Hadamard n-manifold with constant curvature, and Γ\Gamma and γ\gamma be C1,1{{\mathcal{C}}}^{1,1}convex hypersurfaces in M, with γ\gamma nested inside Γ\Gamma . Then, ℳr(Γ)≥ℳr(γ){{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )\ge {{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma ), for 1≤r≤n−11\le r\le n-1.ProofAgain we may assume that the function uuin Theorem 3.1 is convex [4, Lem. 1]. If MMhas constant curvature aa, then Rijkℓ=a(δikδjℓ−δiℓδjk){R}_{ijk\ell }=a\left({\delta }_{ik}{\delta }_{j\ell }-{\delta }_{i\ell }{\delta }_{jk}). Thus, Theorem 3.1 yields (13)ℳr(Γ)−ℳr(γ)=(r+1)∫Ω⧹Dσr+1(κu)−a(n−r)∫Ω⧹Dσr−1(κu).{{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma )-{{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )=\left(r+1)\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{r+1}\left({\kappa }^{u})-a\left(n-r)\mathop{\int }\limits_{\Omega \setminus D}{\sigma }_{r-1}\left({\kappa }^{u}).By assumption a≤0a\le 0, and since uuis convex, σr(κu)≥0{\sigma }_{r}\left({\kappa }^{u})\ge 0, which completes the proof.□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\Gamma is a convex hypersurface in a Cartan-Hadamard manifold, then its outer parallel hypersurface at distance ε\varepsilon , denoted by Γε{\Gamma }^{\varepsilon }, is C1,1{{\mathcal{C}}}^{1,1}for all ε>0\varepsilon \gt 0[13, Lem. 2.6]. So ℳr(Γε){{\mathcal{ {\mathcal M} }}}_{r}\left({\Gamma }^{\varepsilon })is well defined. By Corollary 4.4, ℳr(Γε){{\mathcal{ {\mathcal M} }}}_{r}\left({\Gamma }^{\varepsilon })is decreasing in ε\varepsilon . Hence, its limit as ε→0\varepsilon \to 0exists, and we may set ℳr(Γ)≔limε→0ℳr(Γε).{{\mathcal{ {\mathcal M} }}}_{r}\left(\Gamma ):= {\mathrm{lim}}_{\varepsilon \to 0}{{\mathcal{ {\mathcal M} }}}_{r}\left({\Gamma }^{\varepsilon }).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!!k\hspace{0.1em}\text{&#x0021;&#x0021;}\hspace{0.1em}, when kkis a positive integer, stands for the product of all positive odd (even) integers up to kk, when kkis odd (even). For k≤0k\le 0, we set k!!=1k\hspace{0.1em}\text{&#x0021;&#x0021;}\hspace{0.1em}=1.Corollary 4.6Let Γ\Gamma be a closed C1,1{{\mathcal{C}}}^{1,1}hypersurface in an n-manifold M bounding a domain Ω\Omega . Suppose that M has constant curvature a, and cl(Ω){\rm{cl}}\left(\Omega )is diffeomorphic to a ball. Then, ℳn−1(Γ)=∣Sn−1∣−∑i=1n−(nmod2)2(2i−1)!!(n−2i−2)!!(n−2)!!aiℳn−2i−1(Γ).{{\mathcal{ {\mathcal M} }}}_{n-1}\left(\Gamma )=| {{\bf{S}}}^{n-1}| -\mathop{\sum }\limits_{i=1}^{\frac{n-\left(n\hspace{0.33em}\hspace{0.1em}\text{mod}\hspace{0.1em}\hspace{0.33em}2)}{2}}\frac{\left(2i-1)\hspace{0.1em}\text{&#x0021;&#x0021;}\hspace{0.1em}\left(n-2i-2)\hspace{0.1em}\text{&#x0021;&#x0021;}\hspace{0.1em}}{\left(n-2)\hspace{0.1em}\text{&#x0021;&#x0021;}\hspace{0.1em}}{a}^{i}{{\mathcal{ {\mathcal M} }}}_{n-2i-1}\left(\Gamma ).ProofLet ϕ:cl(Ω)→Bn\phi :{\rm{cl}}\left(\Omega )\to {B}^{n}be a diffeomorphism to the unit ball in Rn{{\bf{R}}}^{n}and set u(x)≔∣ϕ(x)∣2u\left(x):= | \phi \left(x){| }^{2}. All regular level sets γ\gamma of uusatisfy (13). Furthermore, these level sets are convex near the minimum point x0{x}_{0}of uu, since uuhas positive definite Hessian at x0{x}_{0}. So by Corollary 4.5, for these small level sets, ℳr(S)≤ℳr(γ)≤ℳr(S′),{{\mathcal{ {\mathcal M} }}}_{r}\left(S)\le {{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )\le {{\mathcal{ {\mathcal M} }}}_{r}\left(S^{\prime} ),where SSand S′S^{\prime} are geodesic spheres centered at x0{x}_{0}such that SSis nested inside γ\gamma and γ\gamma is nested inside S′S^{\prime} . Consequently, by Lemma 4.2, as γ\gamma shrinks to x0{x}_{0}, ℳn−1(γ){{\mathcal{ {\mathcal M} }}}_{n-1}\left(\gamma )converges to ∣Sn−1∣| {{\bf{S}}}^{n-1}| , while ℳr(γ){{\mathcal{ {\mathcal M} }}}_{r}\left(\gamma )vanishes for r≤n−2r\le n-2. Thus, since σn(κu)=0{\sigma }_{n}\left({\kappa }^{u})=0, (13) yields ℳn−1(Γ)=∣Sn−1∣−a∫Ωσn−2(κu){{\mathcal{ {\mathcal M} }}}_{n-1}\left(\Gamma )=| {{\bf{S}}}^{n-1}| -a\mathop{\int }\limits_{\Omega }{\sigma }_{n-2}\left({\kappa }^{u})and ∫Ωσr(κu)=1rℳr−1(Γ)+a(n−r+1)r∫Ωσr−2(κu)\mathop{\int }\limits_{\Omega }{\sigma }_{r}\left({\kappa }^{u})=\frac{1}{r}{{\mathcal{ {\mathcal M} }}}_{r-1}\left(\Gamma )+\frac{a\left(n-r+1)}{r}\mathop{\int }\limits_{\Omega }{\sigma }_{r-2}\left({\kappa }^{u})for r≤n−2r\le n-2. Using these expressions iteratively completes the proof.□Finally, we include a characterization for hyperbolic balls, which extends to all mean curvatures a previous result of the authors on Gauss-Kronecker curvature [13, Cor. 5.5].Corollary 4.7Let M be a Cartan-Hadamard n-manifold with curvature ≤a≤0\le a\le 0, and Bρ{B}_{\rho }be a ball of radius ρ\rho in MM. Then, for 1≤r≤n−11\le r\le n-1, ℳr(∂Bρ)≥ℳr(∂Bρa),{{\mathcal{ {\mathcal M} }}}_{r}\left(\partial {B}_{\rho })\ge {{\mathcal{ {\mathcal M} }}}_{r}\left(\partial {B}_{\rho }^{a}),where Bρa{B}_{\rho }^{a}denotes a ball of radius ρ\rho in a manifold of constant curvature a. Equality holds only if Bρ{B}_{\rho }is isometric to Bρa{B}_{\rho }^{a}.ProofFor r=n−1r=n-1, the desired inequality has already been established [13, Cor. 5.5]. Suppose then that r≤n−2r\le n-2. We will show that (14)ℳr(∂Bρ)≥(r+1)∫Bρσr+1(κu)−a(n−r)∫Bρσr−1(κu)≥ℳr(∂Bρa).{{\mathcal{ {\mathcal M} }}}_{r}\left(\partial {B}_{\rho })\ge \left(r+1)\mathop{\int }\limits_{{B}_{\rho }}{\sigma }_{r+1}\left({\kappa }^{u})-a\left(n-r)\mathop{\int }\limits_{{B}_{\rho }}{\sigma }_{r-1}\left({\kappa }^{u})\ge {{\mathcal{ {\mathcal M} }}}_{r}\left(\partial {B}_{\rho }^{a}).Letting uube the distance squared function from the center ooof Bρ{B}_{\rho }, and γ\gamma shrink to ooin Theorem 3.1, yields the first inequality in (14) via Lemma 4.2. The principal curvatures of ∂Bρ\partial {B}_{\rho }are bounded below by −acoth(−aρ)\sqrt{-a}\coth \left(\sqrt{-a}\rho )[16, p. 184], which are the principal curvatures of ∂Bρa\partial {B}_{\rho }^{a}. Hence, the mean curvatures of ∂Bρ\partial {B}_{\rho }satisfy σr(κu)≥n−1r(−acoth(−aρ))r=σra(κu),{\sigma }_{r}\left({\kappa }^{u})\ge \left(\begin{array}{c}n-1\\ r\end{array}\right)(\sqrt{-a}\coth \left(\sqrt{-a}\rho ){)}^{r}={\sigma }_{r}^{a}\left({\kappa }^{u}),where σra(κu){\sigma }_{r}^{a}\left({\kappa }^{u})are the mean curvatures of ∂Bρa\partial {B}_{\rho }^{a}. Furthermore, if A(ρ,θ)dθA\left(\rho ,\theta ){\rm{d}}\theta denotes the volume element of ∂Bρ\partial {B}_{\rho }in geodesic spherical coordinates, then by [16, (1.5.4)], A(ρ,θ)≥sinh(−aρ)−an−1=Aa(ρ,θ),A\left(\rho ,\theta )\ge {\left(\frac{\sinh \left(\sqrt{-a}\rho )}{\sqrt{-a}}\right)}^{n-1}={A}^{a}\left(\rho ,\theta ),where Aa(ρ,θ)dθ{A}^{a}\left(\rho ,\theta )d\theta is the volume element of ∂Bρa\partial {B}_{\rho }^{a}; see [13, Cor. 5.5]. Thus, ∫Bρσr(κu)≥∫0ρ∫Sn−1σra(κu)Aa(t,θ)dθdt=∫Bρaσra(κu),\begin{array}{r}\mathop{\displaystyle \int }\limits_{{B}_{\rho }}{\sigma }_{r}\left({\kappa }^{u})\ge \underset{0}{\overset{\rho }{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{{\bf{S}}}^{n-1}}{\sigma }_{r}^{a}\left({\kappa }^{u}){A}^{a}\left(t,\theta ){\rm{d}}\theta {\rm{d}}t=\mathop{\displaystyle \int }\limits_{{B}_{\rho }^{a}}{\sigma }_{r}^{a}\left({\kappa }^{u}),\end{array}which yields the second inequality in (14). If ℳr(∂Bρ)=ℳr(∂Bρa){{\mathcal{ {\mathcal M} }}}_{r}\left(\partial {B}_{\rho })={{\mathcal{ {\mathcal M} }}}_{r}\left(\partial {B}_{\rho }^{a}), then equality holds in the first inequality of (14). So Krn=a{K}_{rn}=a, i.e., the radial sectional curvatures of Bρ{B}_{\rho }are constant, which forces Bρ{B}_{\rho }to have constant curvature aa[13, Lem. 5.4]. Hence, Bρ{B}_{\rho }is isometric to Bρa{B}_{\rho }^{a}.□

Journal

Advanced Nonlinear Studiesde Gruyter

Published: Jan 1, 2023

Keywords: Reilly’s formulas; quermassintegral; mixed volume; generalized mean curvature; hyperbolic space; Cartan-Hadamard manifold; Primary: 53C20; 58J05; Secondary: 52A38; 49Q15

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