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A structure theorem of Dirac-harmonic maps between spheres
TL;DR: For an arbitrary Dirac-harmonic map between compact oriented Riemannian surfaces, the authors studied the order of the zeros of the Dirac map and the genus of the genus.
Abstract: For an arbitrary Dirac-harmonic map $(\phi,\psi)$ between compact oriented Riemannian surfaces, we shall study the zeros of $|\psi|$. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of $|\psi|$ and the genus of $M$ and $N$. On the basis, we could clarify all of nontrivial Dirac-harmonic maps from $S^2$ to $S^2$.
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09 Sep 2022
TL;DR: In this paper , the existence of Dirac-harmonic maps from Riemannian surfaces to K-ahler manifolds was shown in the trivial index case, where the domain is a closed Rieman surface.
Abstract: . For a homotopy class [ u ] of maps between a closed Riemannian manifold M and a general manifold N , we want to find a Dirac-harmonic map with map component in the given homotopy class. Most known results require the index to be nontrivial. When the index is trivial, the few known results are all constructive and produce uncoupled solutions. In this paper, we define a new quantity. As a byproduct of proving the homotopy invariance of this new quantity, we find a new simple proof for the fact that all Dirac-harmonic spheres in surfaces are uncoupled. More importantly, by using the homotopy invariance of this new quantity, we prove the existence of Dirac-harmonic maps from manifolds in the trivial index case. In particular, when the domain is a closed Riemann surface, we prove the short-time existence of the α -Dirac-harmonic map flow in the trivial index case. Together with the density of the minimal kernel, we get an existence result for Dirac-harmonic maps from closed Riemann surfaces to K¨ahler manifolds, which extends the previous result of the first and third authors. This establishes a general existence theory for Dirac-harmonic maps in the context of trivial index.
1 citations
07 Sep 2022
TL;DR: In this article , it was shown that under some minimality assumption Dirac-harmonic maps defined on a closed domain are uncoupled, and they are defined as critical points of the super-symmetric energy functional.
Abstract: . Dirac-harmonic maps ( f,ϕ ) consist of a map f ∶ M → N and a twisted spinor ϕ ∈ Γ ( Σ M ⊗ f ∗ TN ) and they are defined as critical points of the super-symmetric energy functional. A Dirac-harmonic map is called uncoupled , if f is a harmonic map. We show that under some minimality assumption Dirac-harmonic maps defined on a closed domain are uncoupled.
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15 Aug 2000TL;DR: In this paper, the authors present an analysis of the Dirac operator and twistor spinors for the Clifford algebras and spin representation, including principal bundles and connections.
Abstract: Clifford algebras and spin representation Spin structures Dirac operators Analytical properties of Dirac operators Eigenvalue estimates for the Dirac operator and twistor spinors Seiberg-Witten invariants Principal bundles and connections Bibliography Index.
561 citations
"A structure theorem of Dirac-harmon..." refers background in this paper
...We refer to [5] and [4] for more background material on spin structures and Dirac operators....
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Book•
01 Jul 1997
TL;DR: Schoen as discussed by the authors used the Fells/Sampson theorem to reprove the rigidity theorem of Masfow and superrigidity of Marqulis in harmonic maps.
Abstract: A presentation of research on harmonic maps, based on lectures given at the University of California, San Diego. Schoen has worked to use the Fells/Sampson theorem to reprove the rigidity theorem of Masfow and superrigidity theorem of Marqulis. Many of these developments are recorded here.
287 citations
"A structure theorem of Dirac-harmon..." refers background or methods in this paper
...18) on M , in conjunction with divergence theorem and Gauss-Bonnet formula, we can proceed as [6]pp....
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...From it several interesting formulas easily follow, which tell us the relationship between the order of the zeros of log |∂u| and log |∂̄u| and the genus of M and N ; and moreover we can obtain some uniqueness theorems and non-existence theorems (see [6] Chapter I)....
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Book•
30 Apr 1996TL;DR: In this article, the authors examine a fundamental mathematical concept connected to differential geometry - stochastic processes, and propose a monograph that examines the relation between the two concepts and differential geometry.
Abstract: This monograph examines a fundamental mathematical concept connected to differential geometry - stochastic processes.
181 citations
TL;DR: In this article, it was shown that for a sequence of Dirac-harmonic maps from a compact Riemannian surface to a n-dimensional compact manifold with uniformly bounded energy, the energy identities hold during the blow-up process.
Abstract: We prove that for a sequence of Dirac-harmonic maps from a compact Riemannian surface to a n dimensional compact Riemannian manifold N with uniformly bounded energy, the energy identities hold during the blow-up process.
50 citations