Pseudoharmonic maps and vector fields on CR manifolds

Abstract

Building on the work by J. Jost and C.-J. Xu (32), and E. Barletta et al. (3), we study smooth pseudoharmonic maps from a compact strictly pseudoconvex CR manifold and their generalizations e.g. pseudoharmonic unit tangent vector fields. The purpose of this paper is to study several analogs to differential geometric objects appearing in Riemannian geometry and admitting a treatment based on elliptic theory e.g. the Laplace-Beltrami operator (cf. (40)), harmonic maps among Riemannian manifolds (cf. (49)), and harmonic vector fields (regarded as smooth maps of a Riemannian manifold into the total space of the tangent bundle endowed with the Sasaki metric, cf. (51 )a nd (52)). We obtain the following results. Boundary values of Bergman-harmonic maps � :� ! S from a smoothly bounded strictly pseudoconvex domain � � C n into a Riemannian manifoldS are shown to be pseudoharmonic maps, provided their normal derivatives vanish. We prove that @b-pluriharmonic maps are pseudoharmonic maps. A pseudoharmonic map� : M ! Sfrom a compact strictly pseudoconvex CR manifold into a sphere is shown either to link or to meet any codimension 2 totally geodesic sphere in S � . Also we prove that a smooth vector field X : M ! T ðMÞ from a strictly pseudoconvex CR manifold M is a pseudoharmonic map if and only if X is parallel (with respect to the Tanaka-Webster connection) along the maximally complex, or Levi, distribution. We start a theory of pseudoharmonic vector fields i.e. unit vector fields X 2 U ðM;� Þ which are critical points of the energy functional EðX Þ¼ 1 R M traceG� ð� HXS� Þ� ^ð d� Þ n relative to variations through unit vector fields. Any such critical point X is shown to satisfy the nonlinear subelliptic systembX þk r H Xk 2 X ¼ 0 .A lsoinfX2U ðM;� ÞEðX Þ¼ n VolðM;� Þ yet E is unbounded from above. We establish first and second variation formulae for E : U ðM;� Þ!½ 0; þ1Þ and give applications.