Geometric inequalities and the Alexandrov-Bakelman-Pucci technique
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912971915001856 |
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| author | Brendle, S. |
| author_facet | Brendle, S. |
| contents | In this expository paper, we discuss a unified framework for proving various geometric inequalities, based on the so-called Alexandrov-Bakelman-Pucci technique. Examples include Cabré's proof of the classical isoperimetric inequality in Euclidean space; the Fenchel-Willmore-Chen inequality for the mean curvature of a submanifold; the sharp version of the Michael-Simon Sobolev inequality for submanifolds; the sharp version of Ecker's logarithmic Sobolev inequality for submanifolds; and the Sobolev inequality for complete manifolds with nonnegative Ricci curvature and Euclidean volume growth. Finally, we discuss a connection to the work of Heintze and Karcher on the volume of a tubular neighborhood of a hypersurface in a manifold with nonnegative Ricci curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12025 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric inequalities and the Alexandrov-Bakelman-Pucci technique Brendle, S. Differential Geometry In this expository paper, we discuss a unified framework for proving various geometric inequalities, based on the so-called Alexandrov-Bakelman-Pucci technique. Examples include Cabré's proof of the classical isoperimetric inequality in Euclidean space; the Fenchel-Willmore-Chen inequality for the mean curvature of a submanifold; the sharp version of the Michael-Simon Sobolev inequality for submanifolds; the sharp version of Ecker's logarithmic Sobolev inequality for submanifolds; and the Sobolev inequality for complete manifolds with nonnegative Ricci curvature and Euclidean volume growth. Finally, we discuss a connection to the work of Heintze and Karcher on the volume of a tubular neighborhood of a hypersurface in a manifold with nonnegative Ricci curvature. |
| title | Geometric inequalities and the Alexandrov-Bakelman-Pucci technique |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2603.12025 |