The geometry of locally symmetric spaces via natural maps

封闭曲面到非正曲率无欧氏因子的局部对称流形M的调和映射是理解这类流形的重要工具。在一个固定的（足够复杂的）同伦类中，这种映射的能量是表面的teichmller空间上的光滑函数。该项目的目标是在以下情况下通过能量剖面对歧管进行几何表征。1)流形为3维、非紧化、拟樱氏或双简并。2)流形是封闭的，维数为3，双曲，由最小heegard分解给出。接枝是在双曲曲面上关联一个新的度量，而这个度量不再是双曲曲面。4)通过接枝构造，可以将表面群表示的Hitchin分量中的Fuchsian点的显式变形构造为PSL（n，R）。能量分布图给出了特征变化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.