Fenchel-Willmore inequality for submanifolds in manifolds with non-negative $k$-Ricci curvature

TL;DR

This paper proves a sharp inequality relating curvature and volume growth for submanifolds in manifolds with non-negative intermediate Ricci curvature, extending classical and recent geometric inequalities.

Contribution

It generalizes the Fenchel-Willmore inequality to submanifolds in manifolds with non-negative intermediate Ricci curvature and characterizes the equality case.

Findings

Established a sharp Fenchel-Willmore inequality for submanifolds

Extended classical inequalities to broader curvature conditions

Characterized the equality case in the inequality

Abstract

We establish a sharp Fenchel-Willmore inequality for closed submanifolds of arbitrary dimension and codimension immersed in a complete Riemannian manifold with non-negative intermediate Ricci curvature and Euclidean volume growth. In the hypersurface case, this reduces to non-negative Ricci curvature. We also characterize the equality case. This generalizes the recent work of Agostiniani, Fogagnolo, and Mazzieri \cite{Agostiniani-Fogagnolo-Mazzieri}, as well as classical results by Chen, Fenchel, Willmore, and others.