Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds

Abstract

We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant non-negative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifold of non-negative sectional curvature. Using a concept of "duality" for strictly convex hypersurfaces, we also obtain a new type of inequalities, so-called "pseudo"-Harnack inequalities, for expanding flows in the sphere and in the hyperbolic space.

References in corpus (5)

• The Cross Curvature Flow of 3-manifolds with Negative Sectional Curvature
• Isotropic functions revisited
• Harnack inequalities for evolving hypersurfaces on the sphere
• Mean curvature flow of pinched submanifolds of ${\mathbb{CP}}^n$
• Dual flows in hyperbolic space and de Sitter space

Cited by in corpus (6)

• Inverse curvature flows in Riemannian warped products
• The Minkowski inequality in de Sitter space
• Parabolic approaches to curvature equations
• Blaschke-Santaló type inequalities and quermassintegral inequalities in space forms
• Constant rank theorems for curvature problems via a viscosity approach
• Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings In Minkowski Space And The Cross Curvature Flow