Deforming pinched hypersurfaces of the hyperbolic space by powers of the mean curvature into spheres
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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 ${\mathbb{H}}_{\kappa}^{n+1}$.
Abstract
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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