Some convexity and unique continuation properties of harmonic mappings

In this thesis we first study the energy functional defined on the Teichmuller space, where the energy is the Dirichlet energy of a harmonic map from a fixed Riemannian manifold to a closed hyperbolic surface. It is a smooth functional defined on the Teichmuller space since under certain nondegeneracy conditions, the harmonic map is uniquely determined by the hyperbolic metric of the image surface. We focus in particular on the variation of the energy functional along a path of metrics in the Teichmuller space. We choose the path to be a geodesic with respect to the Weil-Petersson metric on the Teichmuller space, and we show that the energy functional is strictly convex along each such path parametrized by arc length.; Secondly we study the local behavior of a smooth harmonic map, specifically the Hausdorff dimension of the set of points where the map has low rank. The map can be either degenerate, so that the image of the harmonic map is a point or a geodesic, or else we show that the set of points where the differential of the map has rank zero is of codimension two, and the set of points where the differential has rank one is of codimension one. The map being smooth and harmonic is a much weaker assumption than the map being analytic, yet these results indicate that the behavior of a smooth harmonic map resembles that of an analytic map to some extent. In order to exploit the resemblance, the unique continuation property of elliptic systems is used.; In the last part of this thesis, we combine the previous two subjects to look into further possibilities for understanding the geometry of harmonic maps in terms of variations of the image metric as well as variations of the map itself.