Volume estimates for tubes around submanifolds using integral curvature bounds

TL;DR

This paper extends classical volume inequalities for tubes around submanifolds by incorporating integral bounds on $k$-Ricci curvature, leading to broader geometric estimates and comparison theorems.

Contribution

It generalizes existing inequalities by replacing sectional curvature bounds with integral $k$-Ricci bounds and introduces a Hessian comparison theorem for $k$-Ricci curvature.

Findings

Derived volume estimates for tubes using integral $k$-Ricci bounds.

Established a Hessian comparison theorem for $k$-Ricci curvature.

Generalized classical inequalities and comparison results in Riemannian geometry.

Abstract

We generalize an inequality of E. Heintze and H. Karcher [8] for the volume of tubes around minimal submanifolds to an inequality based on integral bounds for $k$-Ricci curvature. Even in the case of a pointwise bound, this generalizes the classical inequality by replacing a sectional curvature bound with a $k$-Ricci bound. This work is motivated by the estimates of Petersen-Shteingold-Wei for the volume of tubes around a geodesic [12] and generalizes their result. Using similar ideas we also prove a Hessian comparison theorem for $k$-Ricci curvature which generalizes the usual Hessian and Laplacian comparison for distance functions from a point and give several applications.