Open accessJournal Article

# Short-time existence of the α-Dirac-harmonic map flow and applications

04 Mar 2021-Communications in Partial Differential Equations (Informa UK Limited)-Vol. 46, Iss: 3, pp 442-469
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 initia...

Topics: Dirac (software) (61%), Flow (mathematics) (60%), Harmonic map (54%)
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Open accessJournal Article
Jürgen Jost1, Jingyong Zhu1Institutions (1)
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.

Topics: Harmonic map (50%)

4 Citations

Open accessJournal Article
Jürgen Jost1, Jingyong Zhu1Institutions (1)
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.

Topics: Harmonic map (58%), , Manifold (50%)

1 Citations

Open accessJournal Article
Jürgen Jost1, Jingyong Zhu1Institutions (1)
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.

1 Citations

Open accessJournal Article
Jürgen Jost1, Jingyong Zhu1Institutions (1)
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.

Topics: Harmonic map (58%), , Manifold (50%)

Journal Article
Lei Liu1, Miaomiao Zhu2Institutions (2)
Abstract: Dirac-harmonic maps are critical points of an action functional that is motivated from the nonlinear σ-model of quantum field theory. They couple a harmonic map like field with a nonlinear spinor field. In this article, we shall discuss the latest progress on heat flow approaches for the existence of Dirac-harmonic maps under appropriate boundary conditions. Also, we discuss the refined blow-up analysis for two types of approximating Dirac-harmonic maps arising from those heat flow approaches.

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01 Jan 1966-
Abstract: "The monograph by T Kato is an excellent textbook in the theory of linear operators in Banach and Hilbert spaces It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4) Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8) The fundamentals of semigroup theory are given in chapter 9 The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10 The first edition is now 30 years old The revised edition is 20 years old Nevertheless it is a standard textbook for the theory of linear operators It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field Zentralblatt MATH, 836

18,840 Citations

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01 Jan 1995-
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

1,823 Citations

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06 Nov 1996-
Abstract: Maximum principles introduction to the theory of weak solutions Holder estimates existence, uniqueness and regularity of solutions further theory of weak solutions strong solutions fixed point theorems and their applications comparison and maximum principles boundary gradient estimates global and local gradient bounds Holder gradient estimates and existence theorems the oblique derivative problem for quasilinear parabolic equations fully nonlinear equations I - introduction fully nonlinear equations II - Monge-Ampere and Hessian equations.

1,540 Citations

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