Fenchel-Willmore and Sobolev-type inequalities for submanifolds in non-negatively curved manifolds

Abstract

In this paper, we uncover a novel connection between the Fenchel-Willmore inequality and a new logarithmic Sobolev inequality for mean-convex submanifolds immersed in non-negatively curved manifolds with Euclidean volume growth. Building on this connection, we establish extensions of the Fenchel-Willmore inequality to submanifolds with boundary and to complete non-compact submanifolds. In addition, we derive a sharp Sobolev-type inequality for submanifolds in the same setting. These Sobolev-type inequalities admit a number of applications, including topological consequences in the surface case.