A comparison theorem with applications to sharp geometric inequalities for submanifolds

TL;DR

This paper derives a new comparison theorem for submanifolds using an explicit Jacobian formula, leading to geometric inequalities related to curvature and volume growth.

Contribution

It introduces an explicit Jacobian determinant expression for the normal exponential map, establishing a new comparison theorem and deriving geometric inequalities for submanifolds.

Findings

Derived an explicit Jacobian formula for the normal exponential map.

Established a new comparison theorem related to ambient geometry.

Obtained geometric inequalities for submanifolds in noncompact manifolds.

Abstract

Inspired by the work of Cordero-Erausquin, McCann and Schmuckenschl\"ager [{\it Invent. Math.,} 2001], we derive an explicit expression for the Jacobian determinant of the normal exponential map on a submanifold, establishing a relationship with its ambient counterpart. This formula leads to a new comparison theorem which is closely related to the comparison theorem of Heintze-Karcher [{\it Ann. Sci. \'Ecole Norm. Sup.,} 1978] and the esitimate of Brendle [{\it Comm. Pure Appl. Math.,} 2023]. As applications, inspired by Wang [{\it Ann. Fac. Sci. Toulouse Math.,} 2023] (and hence also by Heintze-Karcher), we obtain a Fenchel-Borsuk-Chern-Lashof-type inequality and a Willmore-Chen-type inequality on closed submanifolds in complete noncompact manifolds with nonnegative curvature and Euclidean volume growth.