Topping’s conjecture on continuity of the harmonic-map-flow body map

Determine whether real-analyticity of the target Riemannian manifold is sufficient to rule out essential discontinuities in the body map at every finite-time singularity of harmonic map flow from a two-dimensional domain.

Background

For harmonic map flow from a two-dimensional domain into a compact target, the body map at a finite singular time is obtained as the weak limiting map away from the finite singular set. At a singular point, bubbles—nonconstant harmonic maps from the two-sphere—may form, and the body map can potentially fail to extend continuously across that point.

Peter Topping’s conjecture asserts that real-analyticity of the target prevents such essential discontinuities. The paper proves this conjecture only in the case where the relevant singularity produces isolated bubbles, so the general assertion remains unresolved beyond the isolated-bubble setting addressed by the main theorem.