Abstract
We study Dirac-harmonic maps from degenerating spin surfaces with uniformly bounded energy and show the so-called generalized energy identity in the case that the domain converges to a spin surface with only Neveu–Schwarz type nodes. We find condition that is both necessary and sufficient for the W 1,2 × L 4 modulo bubbles compactness of a sequence of such maps.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ammann B., Bär C.: Dirac eigenvalue estimates on surfaces. Math. Z. 240, 423–449 (2002)
Atiyah M.F.: Riemann surfaces and spin structures. Ann. Sci. École Norm. Sup. 4(4), 47–62 (1971)
Bär C.: The Dirac operator on hyperbolic manifolds of finite volume. J. Diff. Geom. 54, 439–488 (2000)
Bourguignon J.-P., Gauduchon P.: Spineurs, operateurs de Dirac et variations. Comm. Math. Phys. 144, 581–599 (1992)
Chen Q., Jost J., Li J.Y., Wang G.: Regularity and energy identities for Dirac-harmonic maps. Math. Z. 251, 61–84 (2005)
Chen Q., Jost J., Li J.Y., Wang G.: Dirac-harmonic maps. Math. Z. 254, 409–432 (2006)
Cornalba, M.: Moduli of curves and theta-characteristics. Lectures on Riemann surfaces (Trieste, 1987), pp. 560–589. World Sci. Publ., Teaneck, NJ (1989)
Deligne, P. et al. (eds.): Quantum Fields and Strings: A Course for Mathematicians, vols.1, 2. AMS & Inst. Adv. Study (1999)
Friedrich, T.: Dirac operators in Riemannian geometry. Graduate Studies in Mathematics, vol. 25. American Mathematical Society, Providence, RI (2000), xvi+195 pp.
Gilkey, P.: Invariance theory, the heat equation and the Atiyah-Singer index theorem. Mathematics Lecture Series, vol. 11. Publish or Perish, Inc., Wilmington, DE, 1984, viii+349 pp
Jost, J.: Riemannian Geometry and Geometric Analysis, 4th edn. Universitext. Springer-Verlag, Berlin (2005). xiv+566
Jarvis T.J., Kimura T., Vaintrob A.: Moduli spaces of higher spin curves and integrable hierarchies. Compositio Math. 126(2), 157–212 (2001)
Lawson H.B., Michelsohn M.-L.: Spin Geometry. Princeton University Press, Princeton (1989)
Lott J.: \({\widehat{A}}\) -genus and collapsing. J. Geom. Anal. 10, 529–543 (2000)
Maier S.: Generic metrics and connections on spin- and spinc-manifolds. Comm. Math. Phys. 188, 407–437 (1997)
Mumford D.: Theta characteristics of an algebraic curve. Ann. Sci. École Norm. Sup. 4(4), 181–192 (1971)
Sacks J., Uhlenbeck K.: The existence of minimal immersions of 2-spheres. Ann. Math. 113, 1–24 (1981)
Sacks J., Uhlenbeck K.: The minimal immersions of closed Riemann surfaces. Trans. Am. Math. Soc. 271, 639–652 (1982)
Ye R.G.: Gromov’s compactness theorem for pseudo-holomorphic curves. Trans. Am. Math. Soc. 342(2), 671–694 (1994)
Zhao L.: Energy identities for Dirac-harmonic maps. Calc. Var. Partial Differ. Equ. 28, 121–138 (2006)
Zhu, M.: Harmonic maps and Dirac-harmonic maps from degenerating surfaces. Ph.D. thesis, University of Leipzig (2007, submitted)
Zhu, M.: Harmonic maps from degenerating Riemann surfaces (2008, preprint)
Acknowledgments
This paper is part of the author’s Ph.D. thesis [21]. He is grateful to his advisor, Prof. Jürgen Jost, for guidance and encouragement. He would also like to thank Prof. Guofang Wang, Prof. Xiaohuan Mo and Guy Buss for helpful discussions.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by IMPRS “Mathematics in the Sciences” and the Klaus Tschira Foundation.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Zhu, M. Dirac-harmonic maps from degenerating spin surfaces I: the Neveu–Schwarz case. Calc. Var. 35, 169–189 (2009). https://doi.org/10.1007/s00526-008-0201-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-008-0201-6