Research in Geometric Analysis
On the existence of a rotational, embedded lambda-hypersurface.
For n > 1, we show the existence of a closed, embedded lambda-hypersurface embedded in R^2n. The hypersurface is diffeomorphic to (S^(n-1) x S^(n-1) x S^1) and exhibits SO(n) x SO(n) symmetry. Our approach uses a "shooting method" similar to the approaches used by Angenent and McGrath in the discovery of their self-shrinkers.
The hyperplane is the only stable, smooth solution to the isoperimetric problem in Gaussian space.
Joint with M. McGonagle. We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean space, by studying hypersurfaces that are second-order stable critical points of a weighted area minimizing function for compact variations that preserve weighted volume. Our first main result is that for non-planar hypersurfaces, bounds on the index of the Jacobi operator L for the second variation split off a linear space. We also show that, for 2-dimensional hypersurfaces in a large ball B_R, there is a gradient decay estimate that recovers the hyperplane result as R goes to infinity.
Stability and rigidity results results for lambda-hypersurfaces
Ph.D. Thesis. We prove several stability and rigidity results regarding lambda-hypersurfaces. First, we give a full treatment to the first and second variation formulas, classifying lambda-hypersurfaces as those surfaces whose first variation of Gaussian-weighted area is stationary under all compact variations that preserve Gaussian-weighted volume. Next, we examine the second variation formula, and prove that the hyperplane is the only stable lambda-hypersurface. Finally, we prove some rigidity results about lambda-hypersurfaces.