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Register a new userThe qualitative behavior at the free boundary for approximate harmonic maps from surfaces
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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.
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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.
$alpha$-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.
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.
Our main result in this paper is the following: Given $H^m, H^n$ hyperbolic spaces of dimensional $m$ and $n$ corresponding, and given a Holder function $f=(s^1,...,f^{n-1}):partial H^mto partial H^n$ between geometric boundaries of $H^m$ and $H^n$.
Then for each $epsilon >0$ there exists a harmonic map $u:H^mto H^n$ which is continuous up to the boundary (in the sense of Euclidean) and $u|_{partial H^m}=(f^1,...,f^{n-1},epsilon)$.
We adapt the viscosity method introduced by Rivi`ere to the free boundary case. Namely, given a compact oriented surface $Sigma$, possibly with boundary, a closed ambient Riemannian manifold $(mathcal{M}^m,g)$ and a closed embedded submanifold $mathcal{N}^nsubsetmathcal{M}$, we study the asymptotic behavior of (almost) critical maps $Phi$ for the functional begin{align*} &E_sigma(Phi):=operatorname{area}(Phi)+sigmaoperatorname{length}(Phi|_{partialSigma})+sigma^4int_Sigma|{mathrm {I!I}}^Phi|^4,operatorname{vol}_Phi end{align*} on immersions $Phi:Sigmatomathcal{M}$ with the constraint $Phi(partialSigma)subseteqmathcal{N}$, as $sigmato 0$, assuming an upper bound for the area and a suitable entropy condition. As a consequence, given any collection $mathcal{F}$ of compact subsets of the space of smooth immersions $(Sigma,partialSigma)to(mathcal{M},mathcal{N})$, assuming $mathcal{F}$ to be stable under isotopies of this space we show that the min-max value begin{align*} &beta:=inf_{Ainmathcal{F}}max_{Phiin A}operatorname{area}(Phi) end{align*} is the sum of the areas of finitely many branched minimal immersions $Phi_{(i)}:Sigma_{(i)}tomathcal{M}$ with $partial_ uPhi_{(i)}perp Tmathcal{N}$ along $partialSigma_{(i)}$, whose (connected) domains $Sigma_{(i)}$ can be different from $Sigma$ but cannot have a more complicated topology. We adopt a point of view which exploits extensively the diffeomorphism invariance of $E_sigma$ and, along the way, we simplify several arguments from the original work. Some parts generalize to closed higher-dimensional domains, for which we get a rectifiable stationary varifold in the limit.
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