A new conformal heat flow of harmonic maps (Q6057175)

In the paper under review, the author introduces and studies a conformal heat flow of harmonic maps defined by an evolution equation for a pair consisting of a map and a conformal factor of metric on the two-dimensional domain. Section 2 contains some preliminaries, including volume formula and its asymptotic limit if the map \(f\) is steady solution, i.e., harmonic. In Sections 3, 4, certain Hilbert spaces \(X,Y,Z\) and closed subsets of them \(B\) and \(B'\) are considered. An operator \(S: B\times B'\to B\times B'\) is defined in Section 5. It is a contraction for short time. Different types of singularity are investigated. The flow introduced in this paper is designed to postpone finite time singularity but does not get rid of possibility of bubble forming. The author shows that a Struwe-type global weak solution exists, which is smooth except at most finitely many points. Some remarks about finite time singularity are provided.