Abstract
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.
Highlights
Harmonic maps from closed Riemann surfaces and their variants are important both in mathematics as tools to probe the geometry of a Riemannian manifold and in physics as ground states of the nonlinear sigma model of quantum field theory
In [19], Sacks–Uhlenbeck introduced the notion of α-harmonic maps which for α > 1 makes the problem subcritical for the Palais– Smale condition
For the one-dimensional case, Takeshi Isobe [8] proves the existence of nontrivial nonlinear Dirac-geodesics on flat tori, which are critical point of
Summary
Harmonic maps from closed Riemann surfaces and their variants are important both in mathematics as tools to probe the geometry of a Riemannian manifold and in physics as ground states of the nonlinear sigma model of quantum field theory. We shall first prove the Palais–Smale condition for the action functional Lα of perturbed α-Dirac-harmonic maps from closed. Since the proof of Theorem 1.3 depends on the oscillation of φ, which may not be uniformly controlled for φk, we want to prove another estimate which is called the ε-regularity estimate This kind of estimate was introduced by Sacks and Uhlenbeck for the α-harmonic maps in [19]. For the α0 given in Theorem 1.4 and each α ∈ (1, α0), if the sequence of perturbed α-Dirac-harmonic maps {(φk , ψk)} satisfies the uniform bounded energy condition (1.9), there exists a coupled α-Dirac-harmonic map from M to N with φ in the given homotopy class θ. We prove Theorem 1.4 and Theorem 1.5
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