AI 中文总结
该研究探讨单位球中正交于支撑球面的严格凸超曲面在α-高斯曲率流下的演化，证明其有限时间消失且α>1/(n+2)时归一化后收敛至单位半球面，结合多种边界分析方法完成证明。
AI 中文摘要
我们研究单位球中光滑紧致严格凸超曲面，这些超曲面正交于支撑球面，且按α-高斯曲率流∂ₜX=-K^αν演化，其中α>0。我们证明该解保持严格凸性，在有限时间内消失并收缩至支撑球面上的单点。若α>1/(n+2)，我们应用一类凯莱型共形映射，将消失点映射至欧氏半空间的原点，随后对包围的半空间体积进行归一化处理。所得归一化超曲面光滑收敛至单位半球面。该证明结合了球面自由边界的边界恒等式、适配边界的Tso估计、半空间熵的几乎单调性公式以及归一化流的一致曲率估计。
英文摘要
We study smooth compact strictly convex hypersurfaces in the unit ball that meet the support sphere orthogonally and evolve by the $α$-Gauss curvature flow $\partial_tX=-K^αν$, $α>0$. We prove that the solution remains strictly convex, becomes extinct in finite time and contracts to a single point on the support sphere. If $α>\frac{1}{n+2}$, we apply a Cayley-type conformal map that sends the extinction point to the origin of a Euclidean half-space and then normalize the enclosed half-space volume. The resulting normalized hypersurfaces converge smoothly to the unit hemisphere. The proof combines boundary identities for the spherical free boundary, a boundary-adapted Tso estimate, an almost-monotonicity formula for a half-space entropy, and uniform curvature estimates for the normalized flow.
Comments52 pages
