Anisotropic Gauss curvature flows and their associated Dual Orlicz-Minkowski problems

2021 ◽  
pp. 1-15
Author(s):  
Li Chen
Keyword(s):  
Type Equation ◽  
Gauss Curvature ◽  
Curvature Flow ◽  
Strictly Convex ◽  
Convex Solution ◽  
Orlicz Minkowski Problem ◽  
Closed Hypersurfaces ◽  
Smooth Measures ◽  
Gauss Curvature Flow ◽  
Monge Ampere

In this paper we study a normalized anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Ampère type equation and we obtain a new existence result of solutions to the Dual Orlicz-Minkowski problem for smooth measures, especially for even smooth measures.

scholarly journals The evolution of complete non-compact graphs by powers of Gauss curvature

2019 ◽  
Vol 2019 (757) ◽  
pp. 131-158
Author(s):  
Kyeongsu Choi ◽  
Panagiota Daskalopoulos ◽  
Lami Kim ◽  
Ki-Ahm Lee
Keyword(s):  
Gauss Curvature ◽  
Curvature Flow ◽  
Positive Power ◽  
Strictly Convex ◽  
Gauss Curvature Flow ◽  
Time Existence

AbstractWe prove the all-time existence of non-compact, complete, strictly convex solutions to the α-Gauss curvature flow for any positive power α.

scholarly journals Exterior Dirichlet Problem for Translating Solutions of Gauss Curvature Flow in Minkowski Space

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Hongjie Ju
Keyword(s):  
Dirichlet Problem ◽  
Minkowski Space ◽  
Existence Of Solutions ◽  
Gauss Curvature ◽  
Curvature Flow ◽  
Exterior Domains ◽  
Gauss Curvature Flow ◽  
Monge Ampere

We prove the existence of solutions to a class of Monge-Ampère equations on exterior domains inℝn(n≥2)and the solutions are close to a cone. This problem comes from the study of the flow by powers of Gauss curvature in Minkowski space.

scholarly journals Complete Self-Shrinking Solutions for Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Ruiwei Xu ◽  
Linfen Cao
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Indefinite Metric ◽  
Bernstein Theorem ◽  
Strictly Convex ◽  
Convex Solution

Letf(x)be a smooth strictly convex solution ofdet(∂2f/∂xi∂xj)=exp(1/2)∑i=1nxi(∂f/∂xi)-fdefined on a domainΩ⊂Rn; then the graphM∇fof∇fis a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean spaceRn2nwith the indefinite metric∑dxidyi. In this paper, we prove a Bernstein theorem for complete self-shrinkers. As a corollary, we obtain if the Lagrangian graphM∇fis complete inRn2nand passes through the origin then it is flat.

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