A logarithmic Gauss curvature flow and the Minkowski problem |
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Authors: | Kai-Seng Chou Xu-Jia Wang |
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Institution: | a Department of Mathematic, The Chinese University of Hong Kong, Shatin, Hong Kong;b School of Mathematical Sciences, Australian National University, Canberra, ACT0200, Australia |
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Abstract: | Let X0 be a smooth uniformly convex hypersurface and f a postive smooth function in Sn. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX0 along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ*>0 such that if θ<θ*, they shrink to a point in finite time and, if θ>θ*, they expand to an asymptotic sphere. Finally, when θ=θ*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x). |
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Keywords: | Parabolic Monge– Ampè re equation Gauss curvature Minkowski problem Asymptotic behavior |
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