Yanyan Li: A Geometric Problem and the Hopf Lemma

Published: 23 November 2016
on channel: HarveyMuddCollegeEDU

616

2008 HMC Mathematics Conference on Nonlinear Functional Analysis

Yanyan Li (Rutgers University)

A classical result of A.D. Alexandrov states that a connected compact smooth $n$-dimensional manifold without boundary, embedded in $R^{n+1}$, and such that its mean curvature is constant, is a sphere.

Here we study the problem of symmetry of $M$ in a hyperplane $X_{n+1} = \mathrm{constant}$ in case $M$ satisfies: for any two points $(X', X_{n+1}), (X', \widehat X_{n+1})$ on $M$, with $X_{n+1}$ < $\widehat X_{n+1}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, we establish it under some additional conditions. Some variations of the Hopf Lemma will also be presented. Several open problems will be described. These are joint works with Louis Nirenberg.

Watch video Yanyan Li: A Geometric Problem and the Hopf Lemma online, duration hours minute second in high quality that is uploaded to the channel HarveyMuddCollegeEDU 23 November 2016. Share the link to the video on social media so that your subscribers and friends will also watch this video. This video clip has been viewed 616 times and liked it 8 visitors.

41,486

340