Harmonic map flow for almost-holomorphic maps

TL;DR

This work analyzes the harmonic map flow from a compact surface into a compact Kähler manifold with nonnegative holomorphic bisectional curvature, under the almost-holomorphic-energy premise that either $E_{p}(u(0))$ or $E_{ar\bp}(u(0))$ is small. By leveraging an ε-regularity framework compatible with the split Bochner formula, the authors obtain a uniform bound on $|ar{\bp}u|$ and translate a uniform stress-energy bound into a decay mechanism for the energy scale $oldsymbol{\lambda}(t)$, improving the blow-up rate from the usual type-II bound. A neck-analysis via a carefully constructed supersolution yields strong decay in the angular energy, which, combined with energy-scale control, leads to Hölder continuity of the body map and a bubble-tree convergence with no necks. Consequently, at every finite singular time, the limit map extends continuously over bubble points and the full bubble-tree decomposition satisfies the energy identity with connected bubble images, yielding a robust, no-neck bubble-tree theory for almost-holomorphic maps under positive curvature constraints. The results advance the understanding of singularity formation in two-dimensional harmonic map flow under Kähler targets and provide precise quantitative control over neck formation and energy dissipation in the almost-holomorphic regime.

Abstract

Let $Σ$ be a compact oriented surface and $N$ a compact Kähler manifold with nonnegative holomorphic bisectional curvature. For a solution of harmonic map flow starting from an almost-holomorphic map $Σ\to N$ (in the energy sense), the limit at each singular time extends continuously over the bubble points and no necks appear.

Theorems & Definitions (58)

• Theorem 1.1
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Remark 2.4
• Lemma 2.5: Liu and Yang LiuYang2010
• Lemma 3.1: Struwe struwevariationalmethods, Lemma III.6.7
• proof
• Proposition 3.2: Cf. Waldron2019, Proposition 3.4