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Register a new userExistence of (Dirac-)harmonic Maps from Degenerating (Spin) Surfaces
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We study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to non-positive curved manifold via the scheme of Sacks and Uhlenbeck. By choosing a suitable sequence of $alpha$-(Dirac-)harmonic maps from a sequence of suitable closed surfaces degenerating to a hyperbolic surface, we get the convergence and a cleaner energy identity under the uniformly bounded energy assumption. In this energy identity, there is no energy loss near the punctures. As an application, we obtain an existence result about (Dirac-)harmonic maps from degenerating (spin) surfaces. If the energies of the map parts also stay away from zero, which is a necessary condition, both the limiting harmonic map and Dirac-harmonic map are nontrivial.
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$alpha$-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 surfaces. For $alpha >1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in constructing $alpha$-harmonic maps for $alpha >1$ and then letting $alpha to 1$. 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. By $varepsilon$-regularity and suitable perturbations, we can then show that such a sequence of perturbed $alpha$-Dirac-harmonic maps converges to a smooth nontrivial $alpha$-Dirac-harmonic map.
For a sequence of coupled fields ${(phi_n,psi_n)}$ from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold with uniformly bounded energy and satisfying the Dirac-harmonic system up to some uniformly controlled error terms, we show that the energy identity holds during a blow-up process near the boundary. As an application to the heat flow of Dirac-harmonic maps from surfaces with boundary, when such a flow blows up at infinite time, we obtain an energy identity.
Let ${u_n}$ be a sequence of maps from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold $N$ with free boundary on a smooth submanifold $Ksubset N$ satisfying [ sup_n left(| abla u_n|_{L^2(M)}+|tau(u_n)|_{L^2(M)}right)leq Lambda, ] where $tau(u_n)$ is the tension field of the map $u_n$. We show that the energy identity and the no neck property hold during a blow-up process. The assumptions are such that this result also applies to the harmonic map heat flow with free boundary, to prove the energy identity at finite singular time as well as at infinity time. Also, the no neck property holds at infinity time.
In this paper, we discuss the general existence theory of Dirac-harmonic maps from closed surfaces via the heat flow for $alpha$-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 $alpha$-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 $alpha$-Dirac-harmonic maps ($alphageq1$) 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.
In this paper, we prove the Lipschitz regularity of continuous harmonic maps from an finite dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in cite{lin97}. The proof extends the argument of Huang-Wang cite {hua-w10}.
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