2026/07/29 by Tianci Luo, Yong Wei, Rong Zhou
Mathematics · #math.DG
52 pages
arxiv created 2026/07/29 · arxiv updated 2026/07/30
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 ∂tX=-Kαν, 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>(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.