Quantization for a nonlinear Dirac equation
HTML articles powered by AMS MathViewer
- by Miaomiao Zhu PDF
- Proc. Amer. Math. Soc. 144 (2016), 4533-4544 Request permission
Abstract:
We study solutions of certain nonlinear Dirac-type equations on Riemann spin surfaces. We first improve an energy identity theorem for a sequence of such solutions with uniformly bounded energy in the case of a fixed domain. Then, we prove the corresponding energy identity in the case that the equations have constant coefficients and the domains possibly degenerate to a spin surface with only Neveu-Schwarz type nodes.References
- Bernd Ammann, The smallest Dirac eigenvalue in a spin-conformal class and cmc immersions, Comm. Anal. Geom. 17 (2009), no. 3, 429–479. MR 2550205, DOI 10.4310/CAG.2009.v17.n3.a2
- Bernd Ammann and Christian Bär, Dirac eigenvalue estimates on surfaces, Math. Z. 240 (2002), no. 2, 423–449. MR 1900319, DOI 10.1007/s002090100392
- Bernd Ammann and Emmanuel Humbert, The first conformal Dirac eigenvalue on 2-dimensional tori, J. Geom. Phys. 56 (2006), no. 4, 623–642. MR 2199284, DOI 10.1016/j.geomphys.2005.04.007
- Christian Bär, The Dirac operator on hyperbolic manifolds of finite volume, J. Differential Geom. 54 (2000), no. 3, 439–488. MR 1823312
- Qun Chen, Jürgen Jost, Jiayu Li, and Guofang Wang, Regularity theorems and energy identities for Dirac-harmonic maps, Math. Z. 251 (2005), no. 1, 61–84. MR 2176464, DOI 10.1007/s00209-005-0788-7
- Qun Chen, Jürgen Jost, Jiayu Li, and Guofang Wang, Dirac-harmonic maps, Math. Z. 254 (2006), no. 2, 409–432. MR 2262709, DOI 10.1007/s00209-006-0961-7
- Q. Chen, J. Jost, and G. Wang, Liouville theorems for Dirac-harmonic maps, J. Math. Phys. 48 (2007), no. 11, 113517, 13. MR 2370260, DOI 10.1063/1.2809266
- Qun Chen, Jürgen Jost, and Guofang Wang, Nonlinear Dirac equations on Riemann surfaces, Ann. Global Anal. Geom. 33 (2008), no. 3, 253–270. MR 2390834, DOI 10.1007/s10455-007-9084-6
- Weiyue Ding and Gang Tian, Energy identity for a class of approximate harmonic maps from surfaces, Comm. Anal. Geom. 3 (1995), no. 3-4, 543–554. MR 1371209, DOI 10.4310/CAG.1995.v3.n4.a1
- Jürgen Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. A Wiley-Interscience Publication. MR 1100926
- Tyler J. Jarvis, Takashi Kimura, and Arkady Vaintrob, Moduli spaces of higher spin curves and integrable hierarchies, Compositio Math. 126 (2001), no. 2, 157–212. MR 1827643, DOI 10.1023/A:1017528003622
- H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
- John Lott, $\hat A$-genus and collapsing, J. Geom. Anal. 10 (2000), no. 3, 529–543. MR 1794576, DOI 10.1007/BF02921948
- Thomas H. Parker, Bubble tree convergence for harmonic maps, J. Differential Geom. 44 (1996), no. 3, 595–633. MR 1431008
- Thomas H. Parker and Jon G. Wolfson, Pseudo-holomorphic maps and bubble trees, J. Geom. Anal. 3 (1993), no. 1, 63–98. MR 1197017, DOI 10.1007/BF02921330
- I. A. Taĭmanov, The two-dimensional Dirac operator and the theory of surfaces, Uspekhi Mat. Nauk 61 (2006), no. 1(367), 85–164 (Russian, with Russian summary); English transl., Russian Math. Surveys 61 (2006), no. 1, 79–159. MR 2239773, DOI 10.1070/RM2006v061n01ABEH004299
- Changyou Wang, A remark on nonlinear Dirac equations, Proc. Amer. Math. Soc. 138 (2010), no. 10, 3753–3758. MR 2661574, DOI 10.1090/S0002-9939-10-10438-9
- Rugang Ye, Gromov’s compactness theorem for pseudo holomorphic curves, Trans. Amer. Math. Soc. 342 (1994), no. 2, 671–694. MR 1176088, DOI 10.1090/S0002-9947-1994-1176088-1
- Liang Zhao, Energy identities for Dirac-harmonic maps, Calc. Var. Partial Differential Equations 28 (2007), no. 1, 121–138. MR 2267756, DOI 10.1007/s00526-006-0035-z
- Miaomiao Zhu, Harmonic maps from degenerating Riemann surfaces, Math. Z. 264 (2010), no. 1, 63–85. MR 2564932, DOI 10.1007/s00209-008-0452-0
- Miaomiao Zhu, Dirac-harmonic maps from degenerating spin surfaces. I. The Neveu-Schwarz case, Calc. Var. Partial Differential Equations 35 (2009), no. 2, 169–189. MR 2481821, DOI 10.1007/s00526-008-0201-6
Additional Information
- Miaomiao Zhu
- Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany
- Address at time of publication: Department of Mathematics, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai 200240, People’s Republic of China
- MR Author ID: 863941
- Email: mizhu@sjtu.edu.cn
- Received by editor(s): July 9, 2015
- Received by editor(s) in revised form: November 20, 2015
- Published electronically: March 17, 2016
- Communicated by: Guofang Wei
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4533-4544
- MSC (2010): Primary 58J05, 53C27
- DOI: https://doi.org/10.1090/proc/13041
- MathSciNet review: 3531200