Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds

We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant nonnegative 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 manifolds of nonnegative sectional curvature. Using a concept of “duality” for strictly convex hypersurfaces, we also obtain a new type of inequality, so-called “pseudo”-Harnack inequality, for expanding flows in the sphere and in the hyperbolic space.

Differential Harnack Estimates for a Semilinear Parabolic System

In this paper, we prove differential Harnack inequalities for positive solutions of a semilinear parabolic system on hyperbolic space. We use the inequalities to construct classical Harnack estimates by integrating along space-time.

Singularities of null mean curvature flow of null hypersurfaces in Lorentzian manifolds

We classify two main singularities, as type I and type II, associated with null mean curvature flow of screen conformal null hypersurfaces in Lorentzian manifolds. We prove that the flow at a type I singularity is asymptotically self-similar, whereas at a type II singularity there is a blow-up solution which is an eternal solution. For further analysis of the above two singularities, we define null translating solitons and use them to prove some Harnack estimates for null mean curvature flow under certain geometric conditions.