Existence of (Dirac-)harmonic Maps from Degenerating (Spin) Surfaces
Jürgen Jost,Jingyong Zhu +1 more
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TLDR
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.read more
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Asymptotic analysis for Dirac-harmonic maps from degenerating spin surfaces and with bounded index
Jürgen Jost,Lei Liu,Miaomiao Zhu +2 more
TL;DR: In this article, the authors studied the refined blow-up behavior of a sequence of Dirac-harmonic maps from degenerating spin surfaces with uniformly bounded energy in the case that the domain surfaces converge to a spin surface with only Neveu-Schwarz type nodes, and showed that the limit of the map part of each neck is a geodesic in the target manifold.
On triviality of dirac-harmonic maps
TL;DR: In this article , it was shown that all Dirac-harmonic maps with compact domain are R -trivial and that this heat is just an extension of the classical heat flow for harmonic maps, for which short-time existence and uniqueness was proven by Chen, Jost, Sun and Zhu.
Are all Dirac-harmonic maps uncoupled?
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.
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Uniqueness of Dirac-harmonic maps from a compact surface with boundary
Jürgen Jost,Jingyong Zhu +1 more
TL;DR: In this article , the authors studied the uniqueness problem of Dirac-harmonic maps from a compact surface with boundary and proved the energy convexity of weakly Dirac maps from the unit disk with small energy.
References
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Book
Riemannian geometry and geometric analysis
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.
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The existence of minimal immersions of $2$-spheres
Jonathan Sacks,Karen Uhlenbeck +1 more
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Gromov’s compactness theorem for pseudo holomorphic curves
TL;DR: In this paper, a complete proof for Gromov's compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary has been given for both closed and non-closed curves.