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.
Sharp geometric inequali- ties for closed hypersurfaces in manifolds with nonnegative Ricci curvature.Invent
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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.