Abstract
We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with large coupling constant.
Highlights
Introduction and resultsThe proof of Proposition 1.3 starts, where we show that the evolution of the underlying conformal structure, described by the horizontal curve g0(t), is well controlled and in particular that the injectivity radius of g0(t) is a priori bounded away from zero on any given time interval of finite length
Introduction and resultsIn this article, we study Harmonic Ricci Flow, which was introduced by the first author in [20, 22], with some special cases previously studied in [17, 18]
We study Harmonic Ricci Flow
Summary
The proof of Proposition 1.3 starts, where we show that the evolution of the underlying conformal structure, described by the horizontal curve g0(t), is well controlled and in particular that the injectivity radius of g0(t) is a priori bounded away from zero on any given time interval of finite length. This corresponds to saying that g0(t) does not degenerate in. We will explain how the two-dimensional Harmonic Ricci Flow (or any flow of Riemannian metrics on a closed surface) can be split into evolutions in conformal, horizontal and Lie-derivative directions.
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