A comparison theorem with applications to sharp geometric inequalities for submanifolds
arXiv:2605.06074
The authors derive an explicit formula for the Jacobian determinant of the normal exponential map on a submanifold, use it to prove a new comparison theorem related to Heintze‑Karcher, and apply the result to obtain sharp Fenchel‑Borsuk‑Chern‑Lashof and Willmore‑Chen type inequalities for closed submanifolds in complete noncompact manifolds with nonnegative curvature and Euclidean volume growth.
Abstract
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.
42 pages, comments are welcome!