Heat Flows with Prescribed Singularities from 3-dimensional Manifold

TL;DR

This paper investigates the existence, regularity, and convergence of singular heat flows from 3D manifolds into hyperbolic space, focusing on flows with prescribed singularities along closed curves.

Contribution

It establishes the existence and regularity of singular heat flows with prescribed singularities and proves their exponential convergence to singular harmonic maps.

Findings

Existence of singular heat flows with prescribed singularities.

Regularity results for these flows.

Exponential convergence to singular harmonic maps.

Abstract

In this paper, we study singular heat flows from a 3-dimensional complete bounded Riemannian manifold without boundary into the hyperbolic space with prescribe singularity along a closed curve. We prove the existence and regularity of the singular heat flows. Furthermore, we prove that the singular heat flows converge to a singular harmonic map at an exponential rate.