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$$\alpha $$ α -Dirac-harmonic maps from closed surfaces
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-05-11 , DOI: 10.1007/s00526-021-01955-1
Jürgen Jost , Jingyong Zhu

\(\alpha \)-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to \(\alpha \)-harmonic maps that were introduced by Sacks–Uhlenbeck to attack the existence problem for harmonic maps from closed surfaces. For \(\alpha >1\), the latter are known to satisfy a Palais–Smale condition, and so, the technique of Sacks–Uhlenbeck consists in constructing \(\alpha \)-harmonic maps for \(\alpha >1\) and then letting \(\alpha \rightarrow 1\). The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed \(\alpha \)-Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth if the perturbation function is smooth. By \(\varepsilon \)-regularity and suitable perturbations, we can then show that such a sequence of perturbed \(\alpha \)-Dirac-harmonic maps converges to a smooth coupled \(\alpha \)-Dirac-harmonic map.



中文翻译:

$$ \ alpha $$α-封闭曲面上的狄拉克调和图

\(\阿尔法\) -Dirac谐波地图是狄拉克调和映射的变型中,类似于\(\阿尔法\)由袋-乌伦贝克引入攻击的存在问题为从闭曲面调和映射该K谐波地图。对于\(\ alpha> 1 \),已知后者满足Palais–Smale条件,因此,Sacks–Uhlenbeck的技术包括为\(\ alpha> 1构造\(\ alpha \)-谐波映射。\),然后让\(\ alpha \ rightarrow 1 \)。将该方案扩展到狄拉克-调和地图遇到了许多困难,在本文中,我们开始对这些问题进行攻击。我们首先证明非平凡摄动\(\ alpha \)的存在当目标流形具有非正曲率时,-狄拉克谐波映射。然后,规则性定理表明,如果扰动函数是平滑的,则它们实际上是平滑的。通过\(\ varepsilon \) -正则性和适当的扰动,我们可以证明扰动\(\ alpha \)-狄拉克-调和图的序列收敛到平滑耦合\(\ alpha \)-狄拉克-调和图。

更新日期:2021-05-11
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