Flowing maps to minimal surfaces (Q2816474)

The authors show how a flow can find this branched minimal immersion in one go by simultaneously flowing both the map and the domain in a coupled way, following the gradient flow of the harmonic map energy. Intuitively, one would mainly like to evolve the domain by deforming its conformal structure, and the authors achieve that by flowing a Riemannian metric on the domain surface within the class of constant curvature metrics (with fixed area in the genus 1 case).NEWLINENEWLINE When the domain is a sphere, there is only one conformal structure available (modulo diffeomorphisms) and their flow will be seen to correspond to the classical harmonic map flow of \textit{J. Eells jun.} and \textit{J. H. Sampson} [Am. J. Math. 86, 109--160 (1964; Zbl 0122.40102)], which evolves a map by the \(L^2\)-gradient flow of the harmonic map energy.NEWLINENEWLINE In the special case when the domain is a torus, the authors show that the flow agrees exactly with a flow introduced by \textit{W. Ding} et al. [Invent. Math. 165, No. 2, 225--242 (2006; Zbl 1109.53066)], which those authors had previously presented as a technical modification of an alternative flow.NEWLINENEWLINE In the higher-genus case, the flow is new, and is a little harder to deal with because of the more complicated structure of the Teichmüller space. But when combined with an existence theorem by the first author [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, No. 2, 349--368 (2014; Zbl 1301.53008)], the authors show that incompressible maps flow for all time without the domain metric degenerating, and subconverge to a conformal harmonic map.