Global existence of the harmonic map heat flow into Lorentzian manifolds
Xiaoli Han, Jürgen Jost, Lei Liu, and Liang Zhao
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Submission date: 11. Oct. 2016
published in: Journal de mathématiques pures et appliquées, 130 (2019), p. 130-156
DOI number (of the published article): 10.1016/j.matpur.2019.01.011
Keywords and phrases: heat flow, harmonic map, Lorentzian manifold, warped product, blow up
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We investigate a parabolic-elliptic system for maps from a compact Riemann surface M into a Lorentzian manifold N × ℝ with a warped product metric. We prove a global existence result of the parabolic-elliptic system by assuming either some geometric conditions on the target Lorentzian manifold or small energy of the initial maps. The result implies the existence of a harmonic map in a given homotopy class with fixed boundary data.