Second‐Order Parabolic Differential Equations

Nonlinear elliptic and parabolic equations of the second order

We consider the quermassintegral preserving flow of closed h-convex hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function f of the principal curvatures which is inverse concave and has dual f* approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is h-convex, then the solution of the flow becomes strictly h-convex for t > 0, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.

Quermassintegral preserving curvature flow in Hyperbolic space

We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the principal curvatures which is inverse concave and has dual $f_*$ approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is \emph{h-convex}, then the solution of the flow becomes strictly \emph{h-convex} for $t>0$, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.

We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarily homogeneous. For this class of speeds we prove the exponential convergence to a geodesic sphere. The proof is inspired by [9] and is based on the preserving of the convexity by horospheres that allows to bound the inner and outer radii and to give uniform bounds on the curvature by maximum principle arguments. In order to deduce the exponential trend, we study the behaviour of a suitable ratio associated to the hypersurface that converges exponentially in time to the value associated to a geodesic sphere.

Volume preserving non-homogeneous mean curvature flow in hyperbolic space

This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature $\kappa$ evolving in direction of its normal vector, where the speed is given by a power $\beta (\geq 1/m)$ of the $m$th mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the $\mbox{Gaus}$ curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvature is close enough to 1 everywhere, depending only on $n$, $m$, $\beta$ and $\kappa$, then under the flow this is maintained, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces exponentially converge to a geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$, enclosing the same volume as the initial hypersurface.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2017/0025/0002/CAG-2017-0025-0002-a003.pdf

Volume-Preserving flow by powers of the mth mean curvature in the hyperbolic space

This paper concerns the evolution of a closed hypersurface of the hyperbolic space, convex by horospheres, in direction of its inner unit normal vector, where the speed equals a positive powerof the positive mean curvature. It is shown that the flow exists on a finite maximal inter- val, convexity by horospheres is preserved and the hypersurfaces shrink down to a single point as the final time is approached.

/pdf/contraction-of-horosphere-convex-hypersurfaces-by-powers-of-3zdcf287hl.pdf

Contraction of horosphere-convex hypersurfaces by powers of the mean curvature in the hyperbolic space

Abstract. This paper concerns closed hypersurfaces of dimension n ≥ 2 in the hyperbolic space H κ of constant sectional curvature κ evolving in direction of its normal vector, where the speed equals a power β ≥ 1 of the mean curvature. The main result is that if the initial closed, weakly h-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to 1, depending only on n and β, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in H κ

Shunzi Guo

Papers

Volume-Preserving flow by powers of the mth mean curvature in the hyperbolic space

Contraction of horosphere-convex hypersurfaces by powers of the mean curvature in the hyperbolic space

Volume-Preserving flow by powers of the mth mean curvature in the hyperbolic space

Deforming pinched hypersurfaces of the hyperbolic space by powers of the mean curvature into spheres