Convexity Properties of Energy on Teichmüller Space
Let M be a closed surface of genus at least two, N a manifold of non-positive Hermitian curvature (the Siu-Sampson condition) and fix a homotopy class of maps from M to N (or a representation of the fundamental group of M in the group of isometries of N). For each complex structure J on M there is a harmonic map f:M->N (or an equivariant harmonic map of the universal covers). In situations where this map is unique it depends smoothly on J and its energy E defines a smooth function on the Teichmüller space of M. We prove that this function is plurisubharmonic, and study conditions when it is strictly plurisubharmonic.
This result was suggested by Gromov as an alternative way of developing and strengthening the Siu-Sampson rigidity theory. Indications of these applications will be given as time permits.