Journal of the
Korean Mathematical Society JKMS

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

MSC numbers: Primary 53C44; Secondary 35K96

Supported by: This research was supported in part by the National Natural Science Foundation of China (Grant Nos. 11401131 and 11801496), China Scholarship Council, the Fok Ying-Tung Education Foundation (China), and Hubei Key Laboratory of Applied Mathematics (Hubei University).