  Motivated by questions arising in the study of harmonic maps and Yang Mills
theory, we study new techniques for producing optimal monotonicity relations
for geometric partial differential equations. We apply these results to sharpen
epsilon regularity results. As a sample application, we analyze energy
minimizing maps from compact manifolds to the space of Hermitian matrices,
where the energy of the map includes the usual kinetic term and a singular
potential designed to force the image of the map to lie in a set homotopic to a
Grassmannian.
