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This paper concerns closed hypersurfaces of dimension $n\geq 2$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature $\kappa$ evolving in direction of its normal vector, where the speed equals a power $\beta \geq1$ of the mean curvature. The main result is that if the initial closed, weakly $h$-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to $1$, depending only on $n$ and $\beta$, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in ${\mathbb{H}}_{\kappa}^{n+1}$ in a maximal finite time, and when rescaling appropriately, the evolving hypersurfaces exponential convergence to a unit geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$.
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참고문헌 (24)
참고문헌 신청
B. H. Andrews / 1994 / Contraction of convex hypersurfaces in Euclidean space / Calc. Var. Partial Differential Equations 2(2):151~171
B. H. Andrews / 1994 / Contraction of convex hypersurfaces in Riemannian spaces / J. Differential Geom 39(2):407~431
B. H. Andrews / 1999 / Gauss curvature flow: the fate of the rolling stones / Invent. Math 138(1):151~161
B. H. Andrews / 2000 / Motion of hypersurfaces by Gauss curvature / Pacific J. Math 195(1):1~34
B. H. Andrews / 2004 / Moving surfaces by non-concave curvature functions / arXiv:math.DG/0402273
B. H. Andrews / 1994 / Contraction of convex hypersurfaces in Riemannian spaces / J. Differential Geom 39(2):407~431
B. H. Andrews / 1999 / Gauss curvature flow: the fate of the rolling stones / Invent. Math 138(1):151~161
B. H. Andrews / 2000 / Motion of hypersurfaces by Gauss curvature / Pacific J. Math 195(1):1~34
B. H. Andrews / 2004 / Moving surfaces by non-concave curvature functions / arXiv:math.DG/0402273
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