Volume Estimates for Tubes Around Submanifolds Using Integral Curvature Bounds

Abstract

We generalize an inequality of Heintze and Karcher (Ann Sci Ecole Norm Sup 11(4):451–470, 1978) 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 (Geom Funct Anal 7(6):1011–1030, 1997) for the volume of tubes around a geodesic 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.