Volume preserving mean curvature flow in the Hyperbolic space

By Esther Cabezas-Rivas, V. Miquel. 2006

64 citations recorded by the source corpus.

We prove: "If $M$ is a compact hypersurface of the hyperbolic space, convex by horospheres and evolving by the volume preserving mean curvature flow, then it flows for all time, convexity by horospheres is preserved and the flow converges, exponentially, to a geodesic sphere". In addition, we show that the same conclusions about long time existence and convergence hold if $M$ is not convex by horospheres but it is close enough to a geodesic sphere.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e68e578e4ef1b547750398729bf22c2a2d08af24

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space (2013, background): “[9], Theorem 4), which implies that u is also bounded above.”
Volume-preserving flow by powers of the mth mean curvature (2009, methodology): “Recently, the first author and Miquel [9] considered volume-preserving mean curvature flow in hyperbolic space and proved convergence to geodesic spheres.”
Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space (2018, methodology): “On the other hand, since each Ωt is h-convex, the inner radius and outer radius of Ωt satisfy ρ+(t) ≤ c(ρ−(t) + ρ−(t)) for some uniform positive constant c (see [14, 25]).”
Global solutions to the volume-preserving mean-curvature flow (2015, background): “Several solutions to volume-preserving mean-curvature flow have been proposed in the literature: existence and uniqueness of a global in time smooth solution and its convergence to a sphere is shown in [13, 18] for smooth convex initial data and in [11, 20] for initial data close to a sphere (for fu”
Quermassintegral preserving curvature flow in Hyperbolic space (2017, methodology): “In previous work, the convergence of solutions as t → ∞ was deduced using either the monotonicity of curvature pinching ratios [20, 24, 25, 26, 27, 13, 14, 17, 28] or of isoperimetric ratios [3, 29, 11, 12, 8].”