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.
On a theorem of Fenchel-Borsuk-Willmore-Chern-Lashof
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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.