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Abstract Suppose that $$\Sigma ^{n}\subset \mathbb{S}^{n+1}$$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue $$\lambda _{1}$$ of the induced Laplace–Beltrami operator on $$\Sigma $$ satisfies $$\lambda _{1} \geq \frac{n}{2}+ a_{n}(\Lambda ^{6} + b_{n})^{-1}$$, where $$a_{n}$$ and $$b_{n}$$ are explicit dimensional constants and $$\Lambda $$ is an upper bound for the length of the second fundamental form of $$\Sigma $$. This provides the first explicitly computable improvement on Choi and Wang’s lower bound $$\lambda _{1} \geq \frac{n}{2}$$ without any further assumptions on $$\Sigma $$.
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We show that a compact Riemannian -manifold with strictly convex simply connected boundary and sectional curvature is isometric to a convex domain in a complete simply connected space of constant curvature , provided that on planes tangent to the boundary of . This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard -manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for .more » « less
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Using harmonic mean curvature flow, we establish a sharp Minkowski-type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard $$3$$-manifolds. This inequality also improves the known estimates for total mean curvature in hyperbolic $$3$$-space. As an application, we obtain a Bonnesen-style isoperimetric inequality for surfaces with convex distance function in nonpositively curved $$3$$-spaces, via monotonicity results for total mean curvature. This connection between the Minkowski and isoperimetric inequalities is extended to Cartan–Hadamard manifolds of any dimension.more » « less
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Abstract We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ \Gamma in a Cartan-Hadamard manifold M M . In particular, we show that the first mean curvature integral of a convex hypersurface γ \gamma nested inside Γ \Gamma cannot exceed that of Γ \Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ \Gamma in terms of the volume it bounds in M M in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ \gamma is parallel to Γ \Gamma , or M M has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.more » « less
