Harmonic Maps and Minimal Surfaces into Spaces of Curvature Bounded from Above

DMS-0072483 Chikako Mese首席研究员建议研究几何变分问题，特别是具有奇点的空间中的调和映射。在几何和代数问题的推动下，到奇异目标的调和映射的研究，特别是从上面有界的曲率度量空间，引起了许多数学家的注意。特别是格罗莫夫-斯库恩、科尔瓦尔-斯库恩和约斯特取得了许多显著的成果。PI建议发展极小曲面理论，作为调和映射理论研究的继续。PI证明了曲率度量空间中的极小曲面推广了经典极小曲面的几个重要性质。我们对目前为止提出的理论的高维类比的发展很感兴趣。PI还将研究曲面之间的调和映射，当目标被给予从上方有界的曲率度量时。我们特别感兴趣的是，当目标度规具有与曲率浓度相关的奇点时，调和映射的行为。此外，我们还希望了解调和映射在某一类目标度量变化时的行为。这反过来将被用来研究TeichMuller空间。调和映射的数学研究具有自然和物理意义。这是因为物理学规定，大多数自然行为都是以最小化某些量的方式发生的。例如，众所周知，光在太空中的路径受到重力的影响，这一路径可以通过一条曲线来实现，该曲线相对于某一度量使弧长最小。薄膜(如形成在闭合线框上的肥皂膜)将形成使表面积最小的结构。这两种配置也可以用能量最小化地图来表示。作为能量泛函的临界点，调和映射是自然现象的数学模型，提供了一种有趣的数学研究。

DMS-0072483Chikako MeseThe principal investigator proposes to study geometric variational problems, with particular emphasis on harmonic maps in spaces with singularities. Motivated by questions in geometry and algebra, the study of harmonic maps to singular targets, particularly metric spaces of curvature bounded from above, has attracted the attention of many mathematicians. In particular, many remarkable results have been obtained by Gromov-Schoen, Korevaar-Schoen, and Jost. The PI proposes to develop the minimal surface theory as a continuation of the study of the harmonic map theory. The PI has shown that minimalsurfaces in metric spaces of curvature bounded from above generalize several important properties of the classical minimal surfaces. We are interested in the development of a higherdimensional analogue of the theory advanced thus far. The PI will also investigate harmonic maps between surfaces when thetarget is given a metric of curvature bounded from above. We are particularly interested in understanding the behavior of harmonic maps when the target metric has singularities associated with curvature concentrations. Furthermore, we hope to gain an understanding of the behavior of harmonic maps when we vary the target metrics in a certain class. This in turn will be usedto study Teichmuller spaces.The mathematical study of harmonic maps is natural and physically significant. This is because physics dictates that most natural actions occur in a way to minimize certain quantities. For instance, it is well known that the pathof light in space is affected by gravity, and this path can be realized by a curve which minimizes arclength with respect to a certain metric. A thin film (such as a soap film formed on a closed wire frame) will come to a configuration which minimizes the surface area. Both these configurations can alsobe represented by energy minimizing maps. As critical points of the energy functional, harmonic maps are mathematical models of natural phenomena and provide an interesting mathematical study.