My main research interest is geometric analysis with emphasis currently on the theory of minimal and constant mean curvature surfaces.
Minimal surfaces are defined as surfaces which are critical points for the area functional. The mean curvature of a minimal surface, the average of the two principal curvatures, is identically zero. Surfaces which are critical points for the area functional under a volume constraint are instead called constant mean curvature (CMC) surfaces and in fact the average of the two principal curvatures is constant. Not only are there plenty of mathematical examples for both of these surfaces (for instance planes, helicoids and catenoids are minimal surfaces while spheres, cylinders and Delaunay surfaces are non zero CMC surfaces), but they overtly present themselves in the real world. In nature, the shape of a soap film approximates with great accuracy that of a minimal surface while soap bubbles provide the analogous approximation for CMC surfaces. In other words, a soap bubble is the least-area surface that encloses the fixed volume inside.
The results in my papers and my recent interests can be divided into the following areas:
- Curvature estimates for nonzero CMC disks embedded in locally homogeneous manifolds.
- The asymptotic geometric structure of nonzero CMC surfaces properly embedded in locally homogeneous manifolds.
- The rigidity of CMC surfaces immersed in locally homogeneous manifolds.
- The geometry of surfaces embedded in Euclidean space with integral bounds for the mean curvature.
- Problems and results on the number, genus, shape and other geometric properties of compact CMC surfaces from the geometry of their boundary.