A comparison theorem with applications to sharp geometric inequalities for submanifolds

In this paper, 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.

As applications, 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.