Global well-posedness in the critical Besov space of the skew mean curvature flow in $\mathbb{R}^d: d\ge 5$

Abstract

In this paper we prove small-data global well-posedness for the skew mean curvature flow of codimension-two submanifolds of \(\mathbb R^{d+2}\) (\(d\ge5\)) in the critical Besov space. With harmonic coordinates and Coulomb gauge, the flow is formulated as a quasilinear Schr\"odinger equation for the complex mean curvature coupled to an elliptic system for the geometric and gauge variables. The main difficulty is to control the frequency interactions at critical regularity, where no derivative margin is available. Our argument combines two complementary spacetime estimates derived from the mass and momentum balance laws: a new div-curl lemma introduced by the fourth author yields a bilinear estimate with a half-derivative gain, providing the key control of low-high interactions; while a quasilinear interaction Morawetz estimate provides critical spacetime bounds for comparable and high-high frequency interactions. These estimates coupled with the Gauss-Codazzi structure of the curvature equations yield the unique global solutions to the gauge-reduced system in the critical Besov space, and improves the previous small-data global regularity results.