The Geometry of Capillary and Constant Mean Curvature Surfaces

Constant mean curvature (CMC) surfaces are critical points of the area functional for variations that preserve the volume of the region enclosed by the surface. Capillary surfaces are defined in a similar way, but instead of the area functional, one considers a functional that is the sum of the surface area with a boundary term. Both of these types of surfaces arise in nature as the interface between a liquid and air. The index of a CMC or a capillary surface is an integer that measures how far the surface is from minimizing the functional. In this thesis, we explore the relationship between the index and the geometry of capillary and CMC surfaces. We begin by showing that the index together with the area bound the genus of compact CMC surfaces embedded in a compact 3-manifold. We also show that in the case where the surface is not minimal and the 3-manifold has finite fundamental group, the index and the mean curvature are sufficient to bound the genus. Then we move on to study capillary surfaces immersed in 3-manifolds. Amongst other results, we describe the conformal structure of noncompact capillary surfaces with finite index, one consequence of this description is that the only noncompact capillary surface immersed in a half-space with acute contact angle and zero index is the half-plane