论文标题
dirac-harmonic用微不足道的指数
Dirac-harmonic maps with trivial index
论文作者
论文摘要
对于封闭的Riemannian歧管$ M $和一般歧管$ n $之间的同型类$ [U] $ [U] $，我们希望在给定同型类中找到带有地图组件的Dirac-Harmonic地图。大多数已知的结果要求该指数是非平凡的。当指数是微不足道的时，几个已知结果都是建设性的，并且产生了未偶联的溶液。在本文中，我们定义了一个新数量。作为证明该新数量的同型不变性的副产品，我们发现了一个新的简单证明，证明了表面中所有狄拉克谐波球的均未耦合。更重要的是，通过使用该新数量的同型不变性，我们证明了来自琐事索引中的歧管的狄拉克谐波图。特别是，当域是封闭的黎曼表面时，我们证明了$α$ -Dirac-Harmonic Map在琐碎的索引情况下的短期存在。加上最小内核的密度，我们从封闭的riemann表面到Kähler歧管的狄拉克谐波地图的存在结果，这扩展了第一和第三作者的先前结果。这在琐碎指数的背景下建立了狄拉克谐波地图的一般存在理论。
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 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 the $α$-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 Kähler 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 trivial index.
