Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 102, 18 pages      arXiv:2508.01420      https://doi.org/10.3842/SIGMA.2025.102

The Space of Dirac-Minimal Metrics is Connected in Dimensions 2 and 4

Bernd Ammann a and Mattias Dahl b
a) Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
b) Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden

Received August 11, 2025, in final form November 25, 2025; Published online December 06, 2025

Abstract
Let $M$ be a closed connected spin manifold. Index theory provides a topological lower bound on the dimension of the kernel of the Dirac operator which depends on the choice of Riemannian metric. Riemannian metrics for which this bound is attained are called Dirac-minimal. We show that the space of Dirac-minimal metrics on $M$ is connected if $M$ is of dimension 2 or 4.

Key words: Dirac operator; Atiyah-Singer index theorem; generic Riemannian metrics; minimal kernel.

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References

  1. Abraham R., Marsden J.E., Ratiu T., Manifolds, tensor analysis, and applications, 2nd ed., Appl. Math. Sci., Vol. 75, Springer, New York, 1988.
  2. Ammann B., Dahl M., Humbert E., Surgery and harmonic spinors, Adv. Math. 220 (2009), 523-539, arXiv:math.DG/0606224.
  3. Ammann B., Humbert E., The spinorial $\tau$-invariant and 0-dimensional surgery, J. Reine Angew. Math. 624 (2008), 27-50, arXiv:math.DG/0607716.
  4. Ammann B., Weiss H., Witt F., A spinorial energy functional: critical points and gradient flow, Math. Ann. 365 (2016), 1559-1602, arXiv:1207.3529.
  5. Ammann B., Weiss H., Witt F., The spinorial energy functional on surfaces, Math. Z. 282 (2016), 177-202, arXiv:1407.2590.
  6. Bamler R.H., Kleiner B., Ricci flow and contractibility of spaces of metrics, arXiv:1909.08710.
  7. Bär C., Gauduchon P., Moroianu A., Generalized cylinders in semi-Riemannian and Spin geometry, Math. Z. 249 (2005), 545-580, arXiv:math.DG/0303095.
  8. Bär C., Schmutz P., Harmonic spinors on Riemann surfaces, Ann. Global Anal. Geom. 10 (1992), 263-273.
  9. Bourguignon J.-P., Gauduchon P., Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys. 144 (1992), 581-599.
  10. Dahl M., On the space of metrics with invertible Dirac operator, Comment. Math. Helv. 83 (2008), 451-469, arXiv:math.DG/0603018.
  11. Farkas H.M., Kra I., Riemann surfaces, 2nd ed., Grad. Texts in Math., Vol. 71, Springer, New York, 1992.
  12. Friedrich T., Dirac operators in Riemannian geometry, Grad. Stud. Math., Vol. 25, American Mathematical Society, Providence, RI, 2000.
  13. Glöckner H., Fundamentals of submersions and immersions between infinite-dimensional manifolds, arXiv:1502.05795.
  14. Große N., Pederzani N., Homotopy equivalence of spaces of metrics with invertible Dirac operator, arXiv:1910.09167.
  15. Harvey F.R., Spinors and calibrations, Perspect. Math., Vol. 9, Academic Press, Inc., Boston, MA, 1990.
  16. Hitchin N., Harmonic spinors, Adv. Math. 14 (1974), 1-55.
  17. Lawson Jr. H.B., Michelsohn M.-L., Spin geometry, Princeton Math. Ser., Vol. 38, Princeton University Press, Princeton, NJ, 1989.
  18. Maier S., Generic metrics and connections on Spin- and Spinc-manifolds, Comm. Math. Phys. 188 (1997), 407-437.
  19. Marques F.C., Deforming three-manifolds with positive scalar curvature, Ann. of Math. 176 (2012), 815-863, arXiv:0907.2444.
  20. Müller O., Nowaczyk N., A universal spinor bundle and the Einstein-Dirac-Maxwell equation as a variational theory, Lett. Math. Phys. 107 (2017), 933-961, arXiv:1504.01034.
  21. Neeb K.-H., Wagemann F., Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds, Geom. Dedicata 134 (2008), 17-60, arXiv:math.DG/0703460.
  22. Nowaczyk N., Dirac eigenvalues of higher multiplicity, Ph.D. Thesis, University of Regensburg, 2015, https://doi.org/10.5283/epub.31209, arXiv:1501.04045.
  23. Nowaczyk N., Continuity of Dirac spectra, Ann. Global Anal. Geom. 44 (2013), 541-563, arXiv:1303.6561.
  24. Pederzani N., On the spin cobordism invariance of the homotopy type of the space $\mathcal{R}^{\mathrm{inv}}(M)$, Ph.D. Thesis, University of Leipzig, 2018, https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-213917.
  25. Ruberman D., Positive scalar curvature, diffeomorphisms and the Seiberg-Witten invariants, Geom. Topol. 5 (2001), 895-924, arXiv:math.DG/0105027.
  26. Swift S.T., Natural bundles. II. Spin and the diffeomorphism group, J. Math. Phys. 34 (1993), 3825-3840.

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