A class of inverse curvature flows in $\mathbb{R}^{n+1}$, II
J. Korean Math. Soc. 2020 Vol. 57, No. 5, 1299-1322
Published online September 1, 2020
Jin-Hua Hu, Jing Mao, Qiang Tu, Di Wu
Hubei University; University of Lisbon; Hubei University; 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
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).
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