Stability of non-uniform liquid bridges in all dimensions

Static equilibrium configurations of continuum supported by surface tension are given by constant-mean-curvature (CMC) surfaces which are the critical points of a variational problem to extremize the area with keeping the volume fixed. The geometry of CMC surfaces and their properties such as stability are of special importance in differential geometry and a variety of physical science. In this paper, we examine the stability of CMC hypersurfaces in general dimensions possibly having boundaries on two parallel hyperplanes, by investigating the second variation of area functional. We reveal the stability of non-uniform liquid bridges or unduloids for the first time in all dimensions and all parameter (the ratio of the neck radius to bulge radius) regimes. The analysis is assisted by numerical computations.Comment: 36 pages, 3 figures, 3 table

Heinz-type mean curvature estimates in Lorentz-Minkowski space

We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, we give a unified vanishing theorem of mean curvature for these entire graphs of constant mean curvature.Comment: 11 pages, no figure, minor revisio

R^3ナイ ノ ゴクショウ キョクメン ノ アンテイセイ ニ ツイテ

Koiso, Miyuki; On the stability of minimal surfaces in R^3. Journal of the Mathematical Society of Japan. 1984. 36(3), p.523-541