The geometry of locally symmetric spaces via natural maps

闭曲面到不含欧氏因子的非正曲率局部对称流形M的调和映射是理解这类流形几何的重要工具。在一个固定的（足够复杂的）同伦类中，这样的映射的能量是曲面的Teichmüller空间上的光滑函数。 该项目的目标是在以下情况下，通过这种能量分布的流形的几何特征。1)该流形是3维的，非紧的，拟Fuchsian的或双退化的。2）该流形是3维闭的，双曲的，由极小Heegaard分解给出。3)嫁接是一种将一个新的度量与一个双曲曲面相关联的操作，该度量不再是双曲的。4)通过嫁接构造，可以将表面群表示的希钦分量中的Fuchsian点的显式变形构造为PSL（n，R）。能量分布给出了关于特征变量中的最小代表和退化的信息。

Harmonic maps of closed surfaces into locally symmetric manifolds M of non-positive curvature without Euclidean factors are an important tool for the geometric understanding of such manifolds. The energy of such a map in a fixed (sufficiently complicated) homotopy class is a smooth function on Teichmüller space of the surface. Goal of the project is a geometric characterization of the manifold via this energy profile in the following situations. 1) The manifold is of dimension 3, non-compact and quasi-Fuchsian or doubly degenerate.2) The manifold is closed, of dimension 3, hyperbolic and given by a minimal Heegaard decomposition. 3) Grafting is an operation which associates to a hyperbolic surface a new metric, which is not hyperbolic any more. 4) Via a grafting construction one can construct explicit deformations of Fuchsian points in the Hitchin component of surface group representations into PSL(n,R). The energy profile gives information about minimal representatives and degenerations in the character variety.