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Journal ArticleDOI

$\alpha$-Dirac-harmonic maps from closed surfaces

TL;DR: In this article, the existence of nontrivial perturbed Dirac-harmonic maps with non-positive curvature was proved. And the regularity theorem was shown that they are actually smooth.
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 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.

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Citations
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TL;DR: In this article, the authors studied the general existence theory of Dirac-harmonic maps from closed surfaces via the heat flow and blow-up analysis, and they showed that any initial map along which the Dirac operator has nontrivial minimal kernel in the homotopy class of the given initial map can be chosen to restart the flow.
Abstract: 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 ($\alpha\geq1$) 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.

7 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to a nonpositive curved manifold via the scheme of Sacks and Uhlenbeck was studied.
Abstract: We study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to a nonpositive 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.

4 citations

Journal ArticleDOI
TL;DR: In this article, the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to non-positive curved manifold via the scheme of Sacks and Uhlenbeck was studied.
Abstract: 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.

3 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the energy identity and necklessness hold during the interior blow-up process for a sequence of Dirac-harmonic maps from a Riemann surface M to a compact manifold N with uniformly bounded energy.
Abstract: Let $$(\phi _\alpha , \psi _\alpha )$$ be a sequence of $$\alpha $$ -Dirac-harmonic maps from a Riemann surface M to a compact Riemannian manifold N with uniformly bounded energy. If the target N is a sphere $$S^{K-1}$$ , we show that the energy identity and necklessness hold during the interior blow-up process as $$\alpha \searrow 1$$ for such a sequence .

3 citations

References
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Book
01 Jul 1986
TL;DR: The mountain pass theorem and its application in Hamiltonian systems can be found in this paper, where the saddle point theorem is extended to the case of symmetric functionals with symmetries and index theorems.
Abstract: An overview The mountain pass theorem and some applications Some variants of the mountain pass theorem The saddle point theorem Some generalizations of the mountain pass theorem Applications to Hamiltonian systems Functionals with symmetries and index theorems Multiple critical points of symmetric functionals: problems with constraints Multiple critical points of symmetric functionals: the unconstrained case Pertubations from symmetry Variational methods in bifurcation theory.

3,685 citations

Book
01 Jan 1995
TL;DR: A very readable introduction to Riemannian geometry and geometric analysis can be found in this paper, where the author focuses on using analytic methods in the study of some fundamental theorems in Riemmannian geometry, e.g., the Hodge theorem, the Rauch comparison theorem, Lyusternik and Fet theorem and the existence of harmonic mappings.
Abstract: * Established textbook * Continues to lead its readers to some of the hottest topics of contemporary mathematical research This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research.This new edition introduces and explains the ideas of the parabolic methods that have recently found such a spectacular success in the work of Perelman at the examples of closed geodesics and harmonic forms. It also discusses further examples of geometric variational problems from quantum field theory, another source of profound new ideas and methods in geometry. From the reviews: "This book provides a very readable introduction to Riemannian geometry and geometric analysis. The author focuses on using analytic methods in the study of some fundamental theorems in Riemannian geometry, e.g., the Hodge theorem, the Rauch comparison theorem, the Lyusternik and Fet theorem and the existence of harmonic mappings. With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome. [..] The book is made more interesting by the perspectives in various sections." Mathematical Reviews

1,959 citations

Book
01 Jan 1990
TL;DR: Variational problems are part of our classical cultural heritage as discussed by the authors, and variational methods have been extensively studied in the literature, including lower semi-continuity results, the compensated compactness method, the concentration compactness methods, Ekeland's variational principle, and duality methods or minimax methods including the mountain pass theorems, index theory, perturbation theory, linking and extensions of these techniques to non-differentiable functionals and functionals defined on convex sets.
Abstract: Variational problems are part of our classical cultural heritage. The book gives an introduction to variational methods and presents on overview of areas of current research in this field. Particular topics included are the direct methods including lower semi-continuity results, the compensated compactness method, the concentration compactness method, Ekeland's variational principle, and duality methods or minimax methods, including the mountain pass theorems, index theory, perturbation theory, linking and extensions of these techniques to non-differentiable functionals and functionals defined on convex sets - and limit cases. All results are illustrated by specific examples, involving Hamiltonian systems, non-linear elliptic equations and systems, and non-linear evolution problems. These examples often represent the current state of the art in their fields and open perspective for further research. Special emphasis is laid on limit cases of the Palais-Smale condition.

1,794 citations