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Brouwer degree and genus for the isotropic harmonic closed surfaces on CP^n
by
Xiaohuan Mo
School of Mathematics - Peking Uniiversity
One of the moet fundamental topological invariant for closed Riemann surface M is the genus. If \phi is a map from M to a manifold N such that H2(N, Z)=Z, the Brouwer degree deg(\phi) of \phi is the image of the generator of H2(N, Z) in H2(M, Z)=Z by the homomorphism induced by \phi. If N is an almost Keahler manifold, then deg(\phi) is a smooth homotopy invariant. This talk will present some basic relations between genus, Brouwer degree and main geometric invariants, (such as area with respect to the induced metric and ramification index of the associate curve), for an isotropic har- monic compact surface on a complex projective space. Their Applica- tions are: 1). Topolocigal obstacle of a totally unramified pseudo-holomorphic curve in CPn; 2). Necessary condition of isotropic harmonic map to be Lagrangian; 3). Area value-distribution of harmonic twe-sphere with few higher order singularities; 4). A quantization result of the Gauss curvature for Lagrangian minimal surface in CPn.
Date received: May 26, 1998
Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabo-10.