Curvature flows in the sphere

2013/08/07 by Gerhardt, Claus · 2 citations
#35J60 #53C21 #53C44 #53C50 #58J05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1308.1607

Abstract

We consider contracting and expanding curvature flows in \Ss. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gauß map. The contracting hypersurfaces shrink to a point x₀ while the expanding hypersurfaces converge to the equator of the hemisphere \mc H(-x₀). After rescaling, by the same scale factor, the rescaled hypersurfaces converge to the unit spheres with centers x₀ \resp -x₀ exponentially fast in C^\un(\Ss[n]).

Cited by

• Contraction Maps Generated by Inverse Mean Curvature Flow
• Spectral Obstructions to Contracting Transport Maps on Curved Spaces

Related

• Flow by the power of the Gauss curvature
• Flow by powers of the Gauss curvature in space forms
• Dual flows in hyperbolic space and de Sitter space
• Surfaces expanding by the inverse Gauss curvature flow
• Contracting convex hypersurfaces by curvature