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Willmore-type inequalities for closed hypersurfaces in weighted manifolds
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In this paper, we prove some Willmore-type inequalities for closed hypersurfaces in weighted manifolds with nonnegative Bakry-\'Emery Ricci curvature. In particular, we give a sharp Willmore type inequality in steady gradient Ricci solitons. We also prove a sharp Willmore-like inequality in shrinking gradient Ricci solitons. Moreover, we characterize the equality cases of Willmore-type inequalities. These results can be regarded as weighted versions of Agostiniani-Fogagnolo-Mazzieri's Willmore-type inequality. As applications, we derive some sharp isoperimetric type inequalities in weighted manifolds under the existence assumption of a critical set of weighted isoperimetric functional.
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A Willmore-type inequality for hypersurfaces with asymptotic or integral Ricci curvature bounds
For bounded domains in complete non-compact manifolds, Willmore-type inequalities hold under asymptotic or Lp Ricci curvature bounds, recovering the pointwise Jin-Yin theorem as a limit.
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