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arXiv · 2411.15412

Symmetric Rearrangement and Geometric Inequalities on Riemannian Manifolds

Richard Stone
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Abstract

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 P\'{o}lya-Szeg\H{o} inequalities. Then we consider smooth oriented Riemannian manifolds of the form $M^n = (0,\infty)\times \Sigma^{n-1}$, 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.

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Richard Stone. 2024-11-23. Symmetric Rearrangement and Geometric Inequalities on Riemannian Manifolds. https://arxiv.org/abs/2411.15412

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