15 research outputs found
Spacelike capillary surfaces in the Lorentz-Minkowski space
For a compact spacelike constant mean curvature surface with nonempty
boundary in the three-dimensional Lorentz-Minkowski space, we introduce a
rotation index of the lines of curvature at the boundary umbilic point, which
was developed by Choe \cite{Choe}. Using the concept of the rotation index at
the interior and boundary umbilic points and applying the Poincar\'{e}-Hopf
index formula, we prove that a compact immersed spacelike disk type capillary
surface with less than vertices in a domain of bounded by
(spacelike or timelike) totally umbilic surfaces is part of a (spacelike) plane
or a hyperbolic plane. Moreover we prove that the only immersed spacelike disk
type capillary surface inside de Sitter surface in is part of
(spacelike) plane or a hyperbolic plane
Rigidity results for mean curvature flow graphical translators moving in non-graphical direction
In this paper, we study the rigidity results of complete graphical
translating hypersurfaces when the translating direction is not in the
graphical direction. We proved that any entire graphical translating surface in
the translating direction not parallel to the graphical one is flat if either
the translating surface is mean convex or the entropy of the translating
surface is smaller than . For higher dimensional case, we show that the same
conclusion holds if the graphical translating hypersurface satisfies certain
growth condition