Simon Morgan
Rice University

Blowing up high codimension sets to give them boundary data for harmonic maps.

Abstract:
If we wish to take a disc in R^3 and map it to the identity on its boundary and map the center to an arbitrary point in R^3, in general there will be non existence of a harmonic map. A sequence of decreasing energy maps can be found whose limit is not continuous; the point separates from the rest of the disc. This is true for assigning boundary data to sets of codimension greater than one.

If on the other hand we map a cylinder (a circle cross the unit interval) in R^3 the same way, that is the identity on the circle cross zero and the circle cross 1 is mapped to the arbitrary point in R^3, we will have existence of a harmonic map. Such a cylinder can be seen as a blow up of the disc because the metric on the cylinder can be pulled back to the disc.

We examine results of different kinds of blow up on certain boundary data problems. Also we show how this technique has applications in determining energy functionals on topologically singular domains, such as real varieties, to give continuous energy minimizing maps.