# 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.