The harmonic map heat flow was introduced by Eels and Sampson in to order to find harmonic maps. If the flow exists for all time with derivative bounds, then the limit is a harmonic map. This program runs into difficulties due to the non-linearities of the flow, and various results in the 1990’s demonstrated that finite time blow-up may occur. Similar behaviour occurs for the Yang-Mills flow and this connection was illuminated by Grotowski-Shatah who phrased equivariant Yang-Mills flow as equivariant harmonic map heat flow into a warped product metric with conical singularity. They showed that in the critical dimension (where the problems are conformally invariant), the phenomena of blow is independent of the conical singularity and is fact related to whether the linearisation admits singular solutions or not. I will describe the general construction covering a number of related equations.
This is joint work with Joe Grotowski and Julie Clutterbuck.