非负曲率黎曼流形子流形上的Pólya--Szegö不等式及其应用

中文摘要 AI 辅助

我们证明了定义在完备非紧、截面曲率非负的黎曼流形的$n$维子流形$\Sigma$上的函数的Pólya--Szegö不等式。相应的重排是$\mathbb R^n$上的Schwarz重排，常数依赖于$\Sigma$的平均曲率的$L^n$范数以及由Brendle得到的等周量。作为应用，我们在总平均曲率较小的假设下，推导了任意余维子流形上的Sobolev、Log-Sobolev、Hardy和Gagliardo--Nirenberg不等式。在临界Sobolev情形下，我们获得了有限体积子流形上的Moser--Trudinger不等式以及无限体积子流形上的精确增长不等式。在适当假设下，Pólya--Szegö常数等于1；此时，不等式中的临界常数与尖锐的欧氏常数一致。

英文摘要

We prove a Pólya--Szegö inequality for functions defined on an $n$-dimensional submanifold $Σ$ of a complete noncompact Riemannian manifold with nonnegative sectional curvature. The associated rearrangement is a Schwarz rearrangement on $\mathbb R^n$, and the constant depends on the $L^n$-norm of the mean curvature of $Σ$ and an isoperimetric quantity obtained by Brendle. As applications, we derive Sobolev, Log-Sobolev, Hardy, and Gagliardo--Nirenberg inequalities on submanifolds of arbitrary codimension under a small total mean curvature assumption. In the critical Sobolev case, we obtain Moser--Trudinger inequalities on finite-volume submanifolds and exact growth inequalities on submanifolds with infinite volume. Under suitable assumptions, the Pólya--Szegö constant equals one; in this case, the critical constants in the inequalities coincide with the sharp Euclidean ones.

发表机构

• University of Connecticut（康涅狄格大学）
• Memorial University of Newfoundland（纽芬兰纪念大学）
• Federal University of ABC（ABC联邦大学）

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