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On the existence of pseudoharmonic maps from pseudohermitian manifolds into Riemannian manifolds with nonpositive sectional curvature. (English) Zbl 1304.32024
In a seminal article [Am. J. Math. 86, 109–160 (1964; Zbl 0122.40102)], J. Eells jun. and J. H. Sampson proved the existence of harmonic maps from a closed Riemannian manifold to a Riemannian manifold of non-positive sectional curvature via the harmonic map heat flow. The aim of the paper under review is to develop an analogous theory for the pseudoharmonic map heat flow. This allows the authors to derive the existence of pseudoharmonic maps from a closed pseudohermitian manifold to a Riemannian manifold with non-positive sectional curvature.
After reviewing some facts about pseudohermitian geometry and the definition of pseudoharmonic maps as critical points of an energy functional, the authors derive a CR Bochner formula and the Euler Lagrange equations characterizing the pseudoharmonic maps. These are then used to derive an energy estimate for solutions of the pseudoharmonic heat flow, which in turn is used to prove that the pseudoharmonic heat flow converges to a pseudoharmonic map.
Reviewer: Andreas Cap (Wien)

MSC:
32V20 Analysis on CR manifolds
32V05 CR structures, CR operators, and generalizations
53C43 Differential geometric aspects of harmonic maps
53C56 Other complex differential geometry
58E20 Harmonic maps, etc.
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