A comparison theorem with applications to sharp geometric inequalities for submanifolds
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910197925019648 |
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| author | Pan, Shengliang Yi, Chengyang |
| author_facet | Pan, Shengliang Yi, Chengyang |
| contents | Inspired by the work of Cordero-Erausquin, McCann and Schmuckenschläger [{\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. École 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06074 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A comparison theorem with applications to sharp geometric inequalities for submanifolds Pan, Shengliang Yi, Chengyang Differential Geometry Inspired by the work of Cordero-Erausquin, McCann and Schmuckenschläger [{\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. École 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. |
| title | A comparison theorem with applications to sharp geometric inequalities for submanifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.06074 |