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Uniqueness of closed self-similar solutions to the Gauss curvature flow
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abstract
We show the uniqueness of strictly convex closed smooth self-similar solutions to the $\alpha$-Gauss curvature flow with $(1/n) < \alpha < 1+(1/n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $\alpha$-Gauss curvature flow with $(1/n) < \alpha < 1+(1/n)$ shrinks a strictly convex closed smooth hypersurface to a round sphere.
Forward citations
Cited by 3 Pith papers
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Free boundary flows by powers of the Gauss curvature in the unit ball
Strictly convex free-boundary hypersurfaces in the unit ball under α-Gauss curvature flow extinct at a boundary point, and for α>1/(n+2) the volume-normalized Cayley images converge smoothly to the unit hemisphere.
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The capillary Gauss curvature flow
A new curvature flow for convex hypersurfaces with capillary boundary shrinks to a point and, after rescaling, converges to a soliton equation.
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Uniqueness of self-similar solutions to flows by quotient curvatures
Closed strictly convex self-similar hypersurfaces for quotient curvature flows (σ_k/σ_l)^α = ⟨X,ν⟩ are spheres whenever α > 1/(k-l).
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