Short-time existence of the α-Dirac-harmonic map ﬂow and applications
Jürgen Jost and Jingyong Zhu
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Submission date: 24. Dec. 2019
MSC-Numbers: 53C43, 58E20
Keywords and phrases: Dirac-harmonic map, $\alpha$-Dirac-harmonic map flow, existence, minimal kernel
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In this paper, we discuss the general existence theory of Dirac-harmonic maps from closed surfaces via the heat ﬂow for α-Dirac-harmonic maps and blow-up analysis. More precisely, given any initial map along which the Dirac operator has nontrivial minimal kernel, we ﬁrst prove the short time existence of the heat ﬂow for α-Dirac-harmonic maps. The obstacle to the global existence is the singular time when the kernel of the Dirac operator no longer stays minimal along the ﬂow. In this case, the kernel may not be continuous even if the map is smooth with respect to time. To overcome this issue, we use the analyticity of the target manifold to obtain the density of the maps along which the Dirac operator has minimal kernel in the homotopy class of the given initial map. Then, when we arrive at the singular time, this density allows us to pick another map which has lower energy to restart the ﬂow. Thus, we get a ﬂow which may not be continuous at a set of isolated points. Furthermore, with the help of small energy regularity and blow-up analysis, we ﬁnally get the existence of nontrivial α-Dirac-harmonic maps (α ≥ 1) from closed surfaces. Moreover, if the target manifold does not admit any nontrivial harmonic sphere, then the map part stays in the same homotopy class as the given initial map.