Harmonic map flow for almost-holomorphic maps

TL;DR

This paper studies the harmonic map flow from a surface to a Kähler manifold, showing that solutions starting from almost-holomorphic maps extend continuously at singularities without forming necks, under certain curvature conditions.

Contribution

It establishes the continuous extension of harmonic map flow solutions at singular times for almost-holomorphic initial maps into Kähler manifolds with nonnegative holomorphic bisectional curvature.

Findings

Solutions extend continuously over bubble points.

No necks appear during the flow.

Results depend on the curvature condition of the target manifold.

Abstract

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