Paper · Jul 29, 2026
Free boundary flows by powers of the Gauss curvature in the unit ball
As on arXiv
Abstract
We study smooth compact strictly convex hypersurfaces in the unit ball that meet the support sphere orthogonally and evolve by the -Gauss curvature flow , . We prove that the solution remains strictly convex, becomes extinct in finite time and contracts to a single point on the support sphere. If , 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.
Comments: 52 pages
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
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Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.
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