Pluriharmonic Maps into Compact Lie Groups and Factorization into Unitons

Abstract

Grassmann manifold. The interesting problems are the removability or resolution of the singularity in the factorization for a pluriharmonic map and the explicit construction of pluriharmonic maps from a specific complex manifold into U(N).We shall introduce the notion of meromorphically pluriharmonic maps, which is the smallest class of maps invariant under the addition of meromorphic unitons, and prove a factorization theorem for meromorphically pluriharmonic maps by the energy-reduction process with Harder-Narasimhan filtration.Moreover, by the methods of [33,34,25] and [18,22,10], we also can make the action of the loop algebra and loop group on the space of pluriharmonic maps into a compact Lie group.The authors wish to thank Professor J. Eells for his kind suggestions. The firstnamed author wishes to thank Dr S. Udagawa and Dr M. Bergvelt for some useful communications. He was supported by the Max-Planck-Institut fur Mathematik in Bonn and partially by the London Mathematical Society during the preparation of this paper. He wishes to thank the Max-Planck-Institut and the Universities of Leeds, Warwick and Durham for their hospitality. The second-named author wishes to thank the University of Catania and the International Centre for Theoretical Physics in Trieste for their hospitality and financial support. Pluriharmonic mapsLet M be a connected complex manifold and N be a connected Riemannian manifold with a Riemannian metric g N . Let cp: Af -»N be a smooth map from M to N. The differential dcp: TM-*cp~1TN extends by complex linearity to dcp: TM c ->cp~1TN c . Relative to the complex structure / of M we have a decomposition TM C = rM ( 1 0 ) © rM (01) . By restricting d<p to each factor we define the bundle maps dcp: TM {h0)^> cp^TN* 2 and dcp: TM (0A)^> cp~xTN c . Using the induced connection V v and the 3-operator of TM (X ' Q) , we define the (0, l)-exterior derivative of dcp by (z>& d<p)(Z) = for each Z, W e C°°(rM (1>0) ). Then cp is called pluriharmonic if cp satisfies D" dcp = 0. We immediately see the following. PROPOSITION 1.1. A smooth map cp from a complex manifold M to a Riemannian manifold N is pluriharmonic if and only if for any holomorphic curve i: C-+M, the composite cp°i is always harmonic.Note that a pluriharmonic map cp: M-*N is harmonic with respect to any Kahler metric on M (we can always give a Kahler metric in a small neighborhood of M).Assume that M is a Kahler manifold. Denote by g M and V M the Kahler metric of M and its Riemannian connection. The second fundamental form V dcp of the map cp is defined byfor each X, Y e C°°(TM C ). Since V M is a Riemannian connection of a Kahler metric g M , the (0, l)-part of V M is the 3-operator of TM (1 -0) . Hence the (1, l)-part of the second fundamental form V dcp coincides with D" dcp. 548 YOSHIHIRO OHNITA AND GIORGIO VALLI LEMMA 1.2. Let <p: M->N be a pluriharmonic map from a complex manifold M to a Riemannian manifold N. Then we have R N (d<p(Z), d<p(V)) dq>(W) = 0 for each Z,V,We T X M^X' O) and each xeM, where R N denotes the curvatu...

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