On stability of capillary surfaces in a ball
The authors extend earlier results by \textit{A. Ros} and \textit{E. Vergasta} [Geom. Dedicata 56, 19-33 (1995; Zbl 0912.53009)] on stable partitioning of convex sets. In the earlier paper, the natural boundary condition of orthogonality appears. In the present work, the authors prescribe the ``capillary contact angle'' \(\gamma\) on the contact manifold, corresponding to the physical problem of determining the free surface interface when a container of homogeneous material is partially filled with a prescribed volume of fluid. The present results are less definitive than the earlier ones but they do contain significant new information, and are suggestive for further investigation. The authors prove that if \(\gamma\in(0,\pi)\), a stable stationary surface of genus zero for the energy functional in an Euclidean ball is necessarily a flat disk or a spherical cap; thus, for example, tubular constant mean curvature surfaces, such as a cylinder or a catenoid, are excluded as unstable. This contrasts with the configuration of two parallel plates, for which tubular stable surfaces meeting the plates at prescribed angles do exist under reasonable conditions. In the event the partitioning surface is minimal, the authors find that it can only be a flat disk or a surface of genus one with boundary having at most three connected components. Stronger conclusions are obtained when \(\gamma\) is close to 0 or \(\pi\), and analogous properties are shown to hold for particular metrizations of the ball. Stability in this paper is understood as positivity of the second variation, and is nominally a weaker condition than that of being a (weak or strong) relative minimum. As a useful bonus, the Appendix offers a derivation of a second variation formula, apparently first given by Wente and which was later used also by others, but for which no published version seems to have been available.
scientific article; zbMATH DE number 1054057