Surfaces expanding by non-concave curvature functions

Abstract

In this paper, we first investigate the flow of convex surfaces in the space form ${\mathbb{R}}^3(\kappa )(\kappa =0,1,-1)$ expanding by $F^{-\alpha }$, where $F$ is a smooth, symmetric, increasing and homogeneous of degree one function of the principal curvatures of the surfaces and the power $\alpha \in (0,1]$ for $\kappa =0,-1$ and $\alpha =1$ for $\kappa =1$. By deriving that the pinching ratio of the flow surface $M_t$ is no greater than that of the initial surface $M_0$, we prove the long time existence and the convergence of the flow. No concavity assumption of $F$ is required. We also show that for the flow in ${\mathbb{H}}^3$ with $\alpha \in (0,1)$, the limit shape may not be necessarily round after rescaling.

36 pages, accepted version for Annals of Global Analysis and Geometry

References in corpus (6)

• Fully nonlinear parabolic equations in two space variables
• New pinching estimates for Inverse curvature flows in space forms
• Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature
• Inverse curvature flows in Riemannian warped products
• Inverse mean curvature flows in warped product manifolds
• Inverse mean curvature flow in complex hyperbolic space

Cited by in corpus (8)

• New pinching estimates for Inverse curvature flows in space forms
• Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature
• Inverse curvature flows in Riemannian warped products
• Shifted inverse curvature flows in hyperbolic space
• Contraction of surfaces in hyperbolic space and in sphere
• A Class Of Curvature Flows Expanded By Support Function And Curvature Function In The Euclidean Space And Hyperbolic Space
• Inverse Gauss curvature flows with free boundaries in a cone
• Contracting axially symmetric hypersurfaces by powers of the ${\sigma }_k$-curvature