Dirac-harmonic maps with trivial index
J\"urgen Jost, Linlin Sun, Jingyong Zhu

TL;DR
This paper introduces a new invariant to establish the existence of Dirac-harmonic maps with trivial index, extending the theory to cases previously limited to constructive solutions and uncoupled solutions.
Contribution
It defines a new homotopy-invariant quantity and proves its invariance, enabling the existence of Dirac-harmonic maps in trivial index cases, including from Riemann surfaces to Kähler manifolds.
Findings
Proves homotopy invariance of the new quantity.
Establishes existence of Dirac-harmonic maps in trivial index cases.
Provides short-time existence of the $eta$-Dirac-harmonic map flow.
Abstract
For a homotopy class of maps between a closed Riemannian manifold and a general manifold , 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 -Dirac-harmonic map flow in the trivial index case. Together with…
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Homotopy and Cohomology in Algebraic Topology
