Capillary surfaces: stability, index and curvature estimates

Abstract: In this paper we investigate the connection between the index and the geometry and topology of capillary surfaces. We prove an index estimate for compact capillary surfaces immersed in general 3-manifolds with boundary. We also study noncompact capillary surfaces with finite index and show that, under suitable curvature assumptions, such surface is conformally equivalent to a compact Riemann surface with boundary, punctured at finitely many points. We then prove that a weakly stable capillary surface immersed in a half-space of $\mathbb{R}^3$ which is minimal or has a contact angle less than or equal to $\pi/2$ must be a half-plane. Using this uniqueness result we obtain curvature estimates for strongly stable capillary surfaces immersed in a 3-manifold with bounded geometry.