Lojasiewicz inequalities for maps of the 2-sphere

TL;DR

The paper proves a Lojasiewicz-Simon inequality for maps $u:S^2\to S^2$, quantifying how close the energy $E(u)$ is to the quantized value $4\pi n$ in terms of the tension $\|\mathcal{T}(u)\|$, with exponent $\alpha=1$ generally and $\alpha>1$ when the body map is nonconstant. By combining Topping’s repulsion estimates with a bubble-tree analysis, it establishes lower bounds on energy densities in neck regions and propagates them through ghost bubbles, yielding a global and a localized Lojasiewicz inequality for almost-harmonic sequences. This framework immediately yields polynomial convergence of weak harmonic-map-flow solutions on compact domains away from the bubbling set, provided the body map is nonconstant, and implies uniqueness of subsequential limits in that regime. The work builds on and extends prior results on energy quantization, bubble-tree convergence, and gradient-flow convergence, offering a robust toolkit for analyzing long-time behavior and singularity formation in harmonic map flows. It also develops new neck-energy control mechanisms that may apply to other geometric flows with bubbling phenomena.

Abstract

We prove a Lojasiewicz-Simon inequality $$ \left| E(u) - 4πn \right| \leq C \| \mathcal{T}(u) \|^α$$ for maps $u \in W^{2,2}\left( S^2, S^2 \right).$ The inequality holds with $α= 1$ in general and with $α> 1$ unless $u$ is nearly constant on an open set. We obtain polynomial convergence of weak solutions of harmonic map flow $u(t) : S^2 \to S^2$ as $t \to \infty$ on compact domains away from the singular set, assuming that the body map is nonconstant. The proof uses Topping's repulsion estimates together with polynomial lower bounds on the energy density coming from a bubble-tree induction argument.

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Remark 2.1
• Lemma 3.1
• proof
• Lemma 3.2