Dirac-harmonic maps with the trivial index  

作　　者：Jürgen Jost Linlin Sun Jingyong Zhu 

机构地区：[1]Max Planck Institute for Mathematics in the Sciences,Leipzig 04103,Germany [2]School of Mathematics and Statistics,Guangxi Normal University,Guilin 541004,China [3]School of Mathematics,Sichuan University,Chengdu 610065,China

出　　处：《Science China Mathematics》2025年第4期917-938,共22页中国科学(数学英文版)

基　　金：supported by National Natural Science Foundation of China(Grant No.12201440);the Fundamental Research Funds for the Central Universities。

摘　　要：For a homotopy class[u]of maps between a closed Riemannian manifold M and a general manifold N,we want to find a Dirac-harmonic map with the map component in the given homotopy class.Most known results require the index to be nontrivial.When the index is trivial,the few known results are all constructive and produce uncoupled solutions.In this paper,we define a new quantity.As a byproduct of proving the homotopy invariance of this new quantity,we find a new simple proof for the fact that all the Dirac-harmonic spheres in surfaces are uncoupled.More importantly,by using the homotopy invariance of this new quantity,we prove the existence of Dirac-harmonic maps from manifolds in the trivial index case.In particular,when the domain is a closed Riemann surface,we prove the short-time existence of theα-Dirac-harmonic map flow in the trivial index case.Together with the density of the minimal kernel,we get an existence result for Dirac-harmonic maps from closed Riemann surfaces to K?hler manifolds,which extends the previous result of the first and third authors.This establishes a general existence theory for Dirac-harmonic maps in the context of the trivial index.

关 键 词：Dirac-harmonic map Q-Dirac-harmonic map flow minimal kernel EXISTENCE Kahler manifolds 

分 类 号：O186.12[理学—数学]