# Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds

## TL;DR

This paper establishes Harnack inequalities for various curvature flows in Riemannian and Lorentzian manifolds, including new estimates and duality-based inequalities for convex hypersurfaces in different geometric settings.

## Contribution

It introduces new Harnack and pseudo-Harnack inequalities for curvature flows in diverse curved spaces, extending previous results and employing duality concepts.

## Key findings

- Harnack estimates for curvature flows in constant curvature manifolds.
- A Harnack estimate with a bonus term for mean curvature flow in Einstein manifolds.
- Pseudo-Harnack inequalities for expanding flows in spheres and hyperbolic spaces.

## 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.