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Transactions of the American Mathematical Society

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Spinor pairs and the concentration principle for Dirac operators


Author: Manousos Maridakis
Journal: Trans. Amer. Math. Soc. 369 (2017), 2231-2254
MSC (2010): Primary 53C27; Secondary 58J37
DOI: https://doi.org/10.1090/tran/6858
Published electronically: October 7, 2016
MathSciNet review: 3581233
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Abstract: We study perturbed Dirac operators of the form $ D_s= D + s\mathcal {A} :\Gamma (E)\rightarrow \Gamma (F)$ over a compact Riemannian manifold $ (X, g)$ with symbol $ c$ and special bundle maps $ \mathcal {A} : E\rightarrow F$ for $ s \gg 0$. Under a simple algebraic criterion on the pair $ (c, \mathcal {A})$, solutions of $ D_s\psi =0$ concentrate as $ s\to \infty $ around the singular set $ Z_{\mathcal {A}}\subset X$ of $ \mathcal {A}$. We give many examples, the most interesting ones arising from a general ``spinor pair'' construction.


References [Enhancements On Off] (What's this?)

  • [LM] H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
  • [PR] Igor Prokhorenkov and Ken Richardson, Perturbations of Dirac operators, J. Geom. Phys. 57 (2006), no. 1, 297-321. MR 2265473, https://doi.org/10.1016/j.geomphys.2006.03.004
  • [T1] Clifford Henry Taubes, Counting pseudo-holomorphic submanifolds in dimension $ 4$, J. Differential Geom. 44 (1996), no. 4, 818-893. MR 1438194
  • [T2] Clifford H. Taubes, $ {\rm SW}\Rightarrow {\rm Gr}$: from the Seiberg-Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996), no. 3, 845-918. MR 1362874, https://doi.org/10.1090/S0894-0347-96-00211-1
  • [W1] Edward Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), no. 4, 661-692 (1983). MR 683171
  • [W2] Edward Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), no. 6, 769-796. MR 1306021, https://doi.org/10.4310/MRL.1994.v1.n6.a13

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Additional Information

Manousos Maridakis
Affiliation: Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
Email: mmanos@math.rutgers.edu

DOI: https://doi.org/10.1090/tran/6858
Received by editor(s): June 19, 2015
Received by editor(s) in revised form: June 27, 2015, and October 16, 2015
Published electronically: October 7, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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