# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## A sharp Sobolev trace inequality involving the mean curvature on Riemannian manifoldsHTML articles powered by AMS MathViewer

by Tianling Jin and Jingang Xiong
Trans. Amer. Math. Soc. 367 (2015), 6751-6770 Request permission

## Abstract:

Let $(M,g)$ be a smooth compact $n$-dimensional Riemannian manifold with smooth boundary $\partial M$ for $n\ge 5$. We prove a trace inequality, that is $\|u\|^2_{L^q(\partial M)}\leq S\left (\int _{M}|\nabla _g u|^2 \mathrm {d} v_g+ \frac {n-2}{2}\int _{\partial M}h_g u^2 \mathrm {d} s_g\right )+A\|u\|^2_{L^r(\partial M)}$ for all $u\in H^1(M)$, where $S=\frac {2}{n-2}\omega _n^{-1/(n-1)}$ with $\omega _n$ the volume of the unit sphere in $\mathbb {R}^n$, $q=\frac {2(n-1)}{n-2}$, $r=\frac {2(n-1)}{n}$, $h_g$ is the mean curvature of $\partial M$, $\mathrm {d} v_g$ is the volume form of $(M,g)$, $\mathrm {d} s_g$ is the induced volume form on $\partial M$, and $A$ is a positive constant depending only on $(M, g)$. This inequality is sharp in the sense that $S$ cannot be replaced by any smaller constant, $h$ in general cannot be replaced by any smooth function which is smaller than $h$ at some point on $\partial M$, and $r$ cannot be replaced by any smaller number.
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• Tianling Jin
• Affiliation: Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
• Email: tj@math.uchicago.edu
• Jingang Xiong
• Affiliation: Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China
• Email: jxiong@math.pku.edu.cn
• Received by editor(s): December 30, 2013
• Received by editor(s) in revised form: March 3, 2014
• Published electronically: November 12, 2014
• Additional Notes: The second author was supported in part by the First Class Postdoctoral Science Foundation of China (No. 2012M520002).