The paper proves a Fenchel-Willmore-Chen inequality for submanifolds in smooth metric measure spaces with lower bounds on the 1-Bakry-Emery n-Ricci curvature, plus Sobolev and isoperimetric inequalities under nonnegative 1-weighted Ricci curvature.
Willmore-type inequality for closed hypersurfaces in complete manifolds with Ricci curvature bounded below
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abstract
In this paper, we establish a Willmore-type inequality for closed hypersurfaces in a complete Riemannian manifold of dimension $n+1$ with ${\rm Ric}\geq-ng$. It extends the classic result of Argostianiani, Fogagnolo, and Mazzieri in [1] to the Riemannian manifold of negative curvature. As an application, we construct a Willmore-type inequality for closed hypersurfaces in hyperbolic space and obtain the characterization of geodesic sphere.
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Fenchel-Willmore-Chen inequality under lower bounds on weighted intermediate Ricci curvature
The paper proves a Fenchel-Willmore-Chen inequality for submanifolds in smooth metric measure spaces with lower bounds on the 1-Bakry-Emery n-Ricci curvature, plus Sobolev and isoperimetric inequalities under nonnegative 1-weighted Ricci curvature.