Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces

Abstract

We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasiewicz proved by Simon (1983). We also prove that the optimal exponent of the Lojasiewicz-Simon gradient inequality is obtained when the functional is Morse-Bott, improving on similar results due to Chill (2003, 2006), Haraux and Jendoubi (2007), and Simon (1996). In our article arXiv:1903.01953, we apply our abstract Lojasiewicz-Simon gradient inequalities to prove a Lojasiewicz-Simon gradient inequalities for the harmonic map energy functional using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Those inequalities for the harmonic map energy functional generalize those of Kwon (2002), Liu and Yang (2010), Simon (1983, 1985), and Topping (1997). In our monograph arXiv:1510.03815, we prove Lojasiewicz--Simon gradient inequalities for coupled Yang--Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. Those inequalities generalize that of the pure Yang--Mills energy function due to the first author (Theorems 23.1 and 23.17 in arXiv:1409.1525) for base manifolds of arbitrary dimension and due to Rade (1992) for dimensions two and three.

31 pages, incorporating final galley proof corrections. To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal)

References in corpus (7)

• Stability and instability of Ricci solitons
• Energy gap for Yang-Mills connections, II: Arbitrary closed Riemannian manifolds
• On the uniqueness of asymptotic limits of the Ricci flow
• Discreteness for energies of Yang-Mills connections over four-dimensional manifolds
• Optimal Lojasiewicz-Simon inequalities and Morse-Bott Yang-Mills energy functions
• Relative energy gap for harmonic maps of Riemann surfaces into real analytic Riemannian manifolds
• Lojasiewicz--Simon gradient inequalities for the harmonic map energy function

Cited by in corpus (11)

• Global existence and convergence of a flow to Kazdan-Warner equation with non-negative prescribed function
• On the Lojasiewicz-Simon gradient inequality on submanifolds
• Resolution of singularities and geometric proofs of the Lojasiewicz inequalities
• Optimal Lojasiewicz-Simon inequalities and Morse-Bott Yang-Mills energy functions
• Existence and convergence of the length-preserving elastic flow of clamped curves
• Morse theory for the Yang-Mills energy function near flat connections
• Convergence of solutions for the fractional Cahn-Hilliard system
• Lojasiewicz--Simon gradient inequalities for the harmonic map energy function
• Geometry of Lagrangian self-shrinking tori and applications to the Piecewise Lagrangian Mean Curvature Flow
• The Prescribed $Q$-Curvature Flow for Arbitrary Even Dimension in a Critical Case
• Non-Uniqueness of Bubbling for Wave Maps