Yanyan Li: A Geometric Problem and the Hopf Lemma
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.