Curvature flows on four manifolds with boundary

TL;DR

This paper studies curvature flow equations on four-dimensional manifolds with boundary, proving global existence and convergence to metrics with prescribed curvature properties under conformally invariant conditions.

Contribution

It introduces new methods to analyze curvature flows with boundary conditions, establishing global existence and convergence results for $Q$- and $T$-curvature flows.

Findings

Proved global existence of curvature flows with boundary conditions.

Established convergence to metrics with prescribed curvature.

Applied integral methods under conformally invariant assumptions.

Abstract

Given a compact four dimensional smooth Riemannian manifold $(M,g)$ with smooth boundary, we consider the evolution equation by $Q$-curvature in the interior keeping the $T$-curvature and the mean curvature to be zero and the evolution equation by $T$-curvature at the boundary with the condition that the $Q$-curvature and the mean curvature vanish. Using integral method, we prove global existence and convergence for the $Q$-curvature flow (resp $T$-curvature flow) to smooth metric of prescribed $Q$-curvature (resp $T$-curvature) under conformally invariant assumptions.