Global existence of the harmonic map heat flow into Lorentzian manifolds
Abstract
We investigate a parabolic-elliptic system for maps $(u,v)$ from a compact Riemann surface $M$ into a Lorentzian manifold $N\times{\mathbb{R}}$ with a warped product metric. That system turns the harmonic map type equations into a parabolic system, but keeps the $v$-equation as a nonlinear second order constraint along the flow. 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 Lorentzian harmonic map in a given homotopy class with fixed boundary data.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2019
- DOI:
- 10.48550/arXiv.1901.00901
- arXiv:
- arXiv:1901.00901
- Bibcode:
- 2019arXiv190100901H
- Keywords:
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- Mathematics - Differential Geometry
- E-Print:
- to appear in J. Math. Pures Appl