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Surfaces expanding by non-concave curvature functions (1609.00570v2)

Published 2 Sep 2016 in math.DG and math.AP

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.

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