A few Willmore-type inequalities in noncompact spaces

Abstract

In this work we obtain two Willmore-type inequalities for certain hypersurfaces in complete and noncompact Riemannian manifolds. The first is a sharp geometric inequality for closed hypersurfaces in Riemannian manifolds with asymptotically nonnegative Ricci curvature. The second concerns Riemannian manifolds with nonnegative Bakry-Émery Ricci curvature, whose sharpness is under current development. To accomplish this, we use standard comparison methods derived both from the Riccati and Jacobi equations, whose elements goes back to the work of E. Heintze and H. Karcher, specially in regards to volume growth of geodesic tubes around hypersurfaces. Moreover, these methods have been applied in a recent work by Wang to greatly simplify the proof of the Willmore-type inequality in complete noncompact Riemannian manifolds of nonnegative Ricci curvature, which was first proved by Agostiniani, Fagagnolo and Mazzieri.