A comparison theorem with applications to sharp geometric inequalities for submanifolds

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.