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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.
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참고문헌 (21)
참고문헌 신청
S. Brendle / 2016 / A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold / Comm. Pure Appl. Math 69(1):124~144
L. Caarelli / 1985 / The Dirichlet problem for nonlinear second order elliptic equations, III; Functions of the eigenvalue of the Hessian / Acta. Math. 155:261~301
H. Chen / The Lp-dual Minkowski problem and related parabolic ows
L. Chen / 2018 / Non-parametric inverse curvature ows in the AdS-Schwarzschild manifold / J. Geom. Anal 28(2):921~949
L. Chen / 2020 / Asymptotic convergence for a class of inverse mean curvature ows in Rn+1 / Proc. Amer. Math. Soc 148(1):379~392
L. Caarelli / 1985 / The Dirichlet problem for nonlinear second order elliptic equations, III; Functions of the eigenvalue of the Hessian / Acta. Math. 155:261~301
H. Chen / The Lp-dual Minkowski problem and related parabolic ows
L. Chen / 2018 / Non-parametric inverse curvature ows in the AdS-Schwarzschild manifold / J. Geom. Anal 28(2):921~949
L. Chen / 2020 / Asymptotic convergence for a class of inverse mean curvature ows in Rn+1 / Proc. Amer. Math. Soc 148(1):379~392
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