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Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions

  • Paul M. N. Feehan ORCID logo EMAIL logo
Published/Copyright: January 12, 2021
Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 15 Issue 4

Abstract

For any compact Lie group 𝐺 and closed, smooth Riemannian manifold ( X , g ) of dimension d ≥ 2 , we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal 𝐺-bundle over 𝑋 supporting a connection with L p -small curvature, when p > d / 2 , to the case of a connection with L d / 2 -small curvature. We prove an optimal Łojasiewicz–Simon gradient inequality for abstract Morse–Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis (2019), principally by removing the hypothesis that the Hessian operator be Fredholm with index zero. We apply this result to prove the optimal Łojasiewicz–Simon gradient inequality for the self-dual Yang–Mills energy function near regular anti-self-dual connections over closed Riemannian four-manifolds and for the full Yang–Mills energy function over closed Riemannian manifolds of dimension d ≥ 2 , when known to be Morse–Bott at a given Yang–Mills connection. We also prove the optimal Łojasiewicz–Simon gradient inequality by direct analysis near a given flat connection that is a regular point of the curvature map.

Keywords: Flat connections; gauge theory; infinite-dimensional Morse theory; Łojasiewicz inequalities; Łojasiewicz–Simon inequalities; Morse–Bott functions; Yang–Mills connections
MSC 2010: 58E15; 57R57; 37D15; 58D27; 70S15

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1510064

Funding statement: The author was partially supported by National Science Foundation grant DMS-1510064, the Simons Center for Geometry and Physics, Stony Brook, the Dublin Institute for Advanced Studies, and the Institut des Hautes Études Scientifiques, Bures-sur-Yvette, during the preparation of this article.

Acknowledgements

I am very grateful to the National Science Foundation for their support and to the Simons Center for Geometry and Physics, Stony Brook, the Dublin Institute for Advanced Studies, and the Institut des Hautes Études Scientifiques, Bures-sur-Yvette, for their hospitality and support during the preparation of this article. I thank Manousos Maridakis for many helpful conversations regarding Łojasiewicz–Simon gradient inequalities, Yasha Berchenko-Kogan for useful communications regarding Yang–Mills gauge theory, Changyou Wang for useful conversations regarding geometric analysis, and George Daskalopoulos, Richard Wentworth, and Graeme Wilkin for helpful correspondence regarding the Yang–Mills equations over Riemann surfaces. Lastly, I am most grateful to the anonymous referee for a thoughtful review of our article and for alerting me to the fact that Theorem 8 should hold and suggesting a proof.

  1. Communicated by: Frank Duzaar

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Received: 2020-04-13
Revised: 2020-11-16
Accepted: 2020-11-23
Published Online: 2021-01-12
Published in Print: 2022-10-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Borderline regularity for fully nonlinear equations in Dini domains
  3. Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions
  4. Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝ N
  5. A note on Kazdan–Warner equation on networks
  6. Integral representation for energies in linear elasticity with surface discontinuities
  7. Minkowski inequalities and constrained inverse curvature flows in warped spaces
  8. Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian
  9. Pairings between bounded divergence-measure vector fields and BV functions
  10. Equivalence between distributional and viscosity solutions for the double-phase equation
  11. Vanishing John–Nirenberg spaces
  12. Extremals in nonlinear potential theory
  13. On the structure of divergence-free measures on ℝ2
  14. Solutions of the (free boundary) Reifenberg Plateau problem
  15. Equilibrium measure for a nonlocal dislocation energy with physical confinement
https://doi.org/10.1515/acv-2020-0034
Keywords for this article
Flat connections; gauge theory; infinite-dimensional Morse theory; Łojasiewicz inequalities; Łojasiewicz–Simon inequalities; Morse–Bott functions; Yang–Mills connections

Articles in the same Issue

  1. Frontmatter
  2. Borderline regularity for fully nonlinear equations in Dini domains
  3. Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions
  4. Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝ N
  5. A note on Kazdan–Warner equation on networks
  6. Integral representation for energies in linear elasticity with surface discontinuities
  7. Minkowski inequalities and constrained inverse curvature flows in warped spaces
  8. Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian
  9. Pairings between bounded divergence-measure vector fields and BV functions
  10. Equivalence between distributional and viscosity solutions for the double-phase equation
  11. Vanishing John–Nirenberg spaces
  12. Extremals in nonlinear potential theory
  13. On the structure of divergence-free measures on ℝ2
  14. Solutions of the (free boundary) Reifenberg Plateau problem
  15. Equilibrium measure for a nonlocal dislocation energy with physical confinement
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