Finite-time blow-up for the heat flow of pseudoharmonic maps
Tong ChangShu-Cheng Chang
Primary 32V0532V20Secondary 53C56.CR Paneitz operatorEnergy densityHeisenberg groupMonotonicity inequalityMoser's Harnack inequalityPseudoharmonic mapPseudoharmonic map heat flowPseudohermitian manifoldPseudohermitian Ricci tensorsPseudohermitian torsionSub-Laplacian
In this paper, we consider the heat flow for pseudoharmonic maps from a closed pseudohermitian manifold $(M^{2n+1},J,\theta)$ into a compact Riemannian manifold $(N^m,g)$. In our pervious work, we proved global existence of the solution for the pseudoharmonic map heat flow, provided that the sectional curvature of the target manifold $N$ is nonpositive. In this present paper, we show that the solution of the pseudoharmonic map heat flow blows up in finite time if the initial map belongs to a nontrivial homotopy class and its initial energy is sufficiently small. As a consequence, we obtain global existence for the pseudoharmonic map heat flow without the curvature assumption on the target manifold.
Indiana University Mathematics Journal
2015
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10.1512/iumj.2015.64.5499
10.1512/iumj.2015.64.5499
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Indiana Univ. Math. J. 64 (2015) 441 - 470
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