Abstract
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 () 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.
Highlights
Motivated by the supersymmetric nonlinear sigma model from quantum field theory, see [8], Dirac-harmonic maps from spin Riemann surfaces into Riemannian manifolds were introduced in [3]
When we analogously introduce α-Diracharmonic maps, the Palais-Smale condition fails due to the following existence result for uncoupled α-Dirac-harmonic maps, which directly follows from the proof of Theorem 4.1
We are only able to prove it for perturbed α-Dirac-harmonic maps, and approximate the α-Dirac-harmonic map by a sequence of perturbed α-Dirac-harmonic maps. In this approach, it is not easy to control the energies of the perturbed α-Dirac-harmonic maps, which are constructed by a Min-Max method over increasingly large domains in the configuration space
Summary
Motivated by the supersymmetric nonlinear sigma model from quantum field theory, see [8], Dirac-harmonic maps from spin Riemann surfaces into Riemannian manifolds were introduced in [3]. In this approach, it is not easy to control the energies of the perturbed α-Dirac-harmonic maps, which are constructed by a Min-Max method over increasingly large domains in the configuration space Due to these two problems, in this paper, we would like to use the heat flow method to get the existence of Dirac-harmonic maps from closed surfaces to general manifolds where the harmonic map type equation is parabolized and the first order Dirac equation is carried along as an elliptic side constraint [5]. In the Appendix, we solve the constraint equation and prove Lipschitz continuity of the solution with respect to the map
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