Global existence and convergence of solutions to gradient systems and applications to Yang-Mills gradient flow

Abstract

In this monograph, we develop results on global existence and convergence of solutions to abstract gradient flows on Banach spaces for a potential function that obeys the Lojasiewicz-Simon gradient inequality. We prove a Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional over closed, smooth Riemannian manifolds of arbitrary dimension and apply the resulting framework to prove new results for the gradient flow equation for the Yang-Mills energy functional on a principal bundle, with compact Lie structure group, over a closed, smooth Riemannian manifolds, including the following. If the initial connection is close enough to a local minimum of the Yang-Mills energy functional, in a norm sense when the base manifold has arbitrary dimension or in an energy sense when the base manifold has dimension four, then the Yang-Mills gradient flow exists for all time and converges to a Yang-Mills connection. If the initial connection is allowed to have arbitrary energy but we restrict to the setting of a Hermitian vector bundle over a compact, complex, Hermitian (but not necessarily Kaehler) surface and the initial connection has curvature of type (1,1), then the Yang-Mills gradient flow exists for all time, though bubble singularities may (and in certain cases must) occur in the limit as time tends to infinity.

xx + 475 pages

References in corpus (1)

• Ricci Flow and the Poincare Conjecture

Cited by in corpus (21)

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• Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces
• Energy gap for Yang-Mills connections, I: Four-dimensional closed Riemannian manifolds
• Some recent progress in singular stochastic PDEs
• A lower bound on the solutions of Kapustin-Witten equations
• Discreteness for energies of Yang-Mills connections over four-dimensional manifolds
• Optimal Lojasiewicz-Simon inequalities and Morse-Bott Yang-Mills energy functions
• An Energy Gap for Complex Yang-Mills Equations
• Stochastic quantization of an Abelian gauge theory
• Asymptotic convergence of evolving hypersurfaces
• Morse theory for the Yang-Mills energy function near flat connections
• A state space for 3D Euclidean Yang-Mills theories
• Gluing in geometric analysis via maps of Banach manifolds with corners and applications to gauge theory
• Lojasiewicz--Simon gradient inequalities for the harmonic map energy function
• Non-existence of Higgs fields on Calabi-Yau Manifolds
• Kurdyka-Lojasiewicz-Simon inequality for gradient flows in metric spaces
• Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds
• Heat flow of Yang-Mills-Higgs functionals in dimension two
• On a topology property for the moduli space of Kapustin-Witten equations
• Equivariant gluing theory on regular instanton moduli spaces
• Stabilities of homothetically shrinking Yang-Mills solitons