Symmetric Rearrangement and Geometric Inequalities on Riemannian Manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929602194046976 |
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| author | Stone, Richard |
| author_facet | Stone, Richard |
| contents | This paper starts by introducing results from geometric measure theory to prove symmetric decreasing rearrangement inequalities on $\mathbb{R}^n$, which give multiple proofs of the isoperimetric and Pólya-Szegő inequalities. Then we consider smooth oriented Riemannian manifolds of the form $M^n = (0,\infty)\times Σ^{n-1}$, and test what results carry over from the $\mathbb{R}^n$ setting or what assumptions about $M^n$ need to be added. Of particular interest was proving the smooth co-area formula in the Riemannian manifolds setting and re-formulating particular geometric inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15412 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetric Rearrangement and Geometric Inequalities on Riemannian Manifolds Stone, Richard Differential Geometry 53C21 (Primary) 28A75 (Secondary) This paper starts by introducing results from geometric measure theory to prove symmetric decreasing rearrangement inequalities on $\mathbb{R}^n$, which give multiple proofs of the isoperimetric and Pólya-Szegő inequalities. Then we consider smooth oriented Riemannian manifolds of the form $M^n = (0,\infty)\times Σ^{n-1}$, and test what results carry over from the $\mathbb{R}^n$ setting or what assumptions about $M^n$ need to be added. Of particular interest was proving the smooth co-area formula in the Riemannian manifolds setting and re-formulating particular geometric inequalities. |
| title | Symmetric Rearrangement and Geometric Inequalities on Riemannian Manifolds |
| topic | Differential Geometry 53C21 (Primary) 28A75 (Secondary) |
| url | https://arxiv.org/abs/2411.15412 |