A class of inverse curvature flows in $\mathbb{R}^{n+1}$, II

J. Korean Math. Soc. Published online March 23, 2020

Jinhua Hu, Jing Mao, Qiang Tu, and Di Wu
Hubei University

Abstract : We consider closed, star-shaped, admissible hypersurfaces in $\mathbb{R}^{n+1}$ expanding along the flow $\dot{X}=|X|^{\alpha-1}F^{-\beta}$, $\alpha\leq1$, $\beta>0$, and prove that for the case $\alpha\leq1$, $\beta>0$, $\alpha+\beta\leq2$, this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a round sphere after rescaling. Besides, for the case $\alpha\leq1$, $\alpha+\beta>2$, if furthermore the initial closed hypersurface is strictly convex, then the strict convexity is preserved during the evolution process and the flow blows up at finite time.

Keywords : Inverse curvature flows, star-shaped, principal curvatures.