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

chikako mess首席研究员提议研究几何变分问题，特别强调奇异空间中的调和映射。在几何和代数问题的激励下，奇异目标的调和映射的研究，特别是曲率从上面有界的度量空间，吸引了许多数学家的注意。特别是Gromov-Schoen、korevar - schoen和Jost等人取得了许多显著的结果。PI建议发展最小曲面理论，作为谐波映射理论研究的延续。π证明了曲率度规空间中的极小曲面推广了经典极小曲面的几个重要性质。我们感兴趣的是发展一种高维的类似于目前所提出的理论。PI还将研究曲面之间的谐波映射，当目标被给定一个曲率度规时。当目标度规具有与曲率集中相关的奇点时，我们特别感兴趣的是理解调和映射的行为。此外，我们希望获得谐波映射的行为的理解，当我们改变目标指标在某一类。这反过来将用于研究Teichmuller空间。谐波图的数学研究具有自然和物理意义。这是因为物理学表明，大多数自然行为都是以最小化某些数量的方式发生的。例如，众所周知，光在空间中的路径受到重力的影响，这种路径可以通过一条相对于某一度量最小化弧长的曲线来实现。薄膜（如在封闭的金属丝框架上形成的肥皂膜）将形成一种使表面积最小化的结构。这两种构型都可以用能量最小化图来表示。谐波映射作为能量泛函的临界点，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.