A class of inverse curvature flows in \(\mathbb{R}^{n+1}\). II. (English) Zbl 1456.53076
Summary: : We consider closed, star-shaped, admissible hypersurfaces in \(\mathbb{R}^{n+1}\) expanding along the flow \(\dot{X}=|X|^{\alpha-1}F^{-\beta}\), \(\alpha\leq 1\), \(\beta> 0\), and prove that for the case \(\alpha\leq 1,\beta> 0, \alpha+\beta\leq 2\), this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a round sphere after rescaling. Besides, for the case \(\alpha\leq 1, \alpha+\beta> 2\), 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.
MSC:
| 53E10 | Flows related to mean curvature |
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
| 35K96 | Parabolic Monge-Ampère equations |
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