Almgren's multiple-valued functions and geometric measure theory

项目编号：dms -0905347首席研究员：朱伟（音译）。该奖项由2009年美国复苏与再投资法案（公法111-5）资助。这些研究项目涉及几何测度理论与偏微分方程的相互作用。具体项目将研究平稳调和多值函数、多值函数的频率和分支集、积分变量的正则性。在这些问题上的工作将解决几何分析中一些长期存在的问题，如一般协维中面积最小化积分流的边界正则性，以及在平均曲率或第二基本形式的自然假设下积分变分的奇异性估计。几何学中一些最有用和最重要的子流形被描述为自然能量或面积函数的临界点。例如，肥皂泡形成的表面是平均曲率泛函的临界点，而测地线（最短路径）是长度泛函的临界点。几何测量理论通过将允许对象的类别从曲面扩展到广义曲面，为确定临界对象的存在提供了有力的工具。这些方法往往留下了解决方案的规律性或平滑性的开放性问题，这些问题在这些项目中得到了强调。

AbstractAward: DMS-0905347Principal Investigator: Wei ZhuThis award is funded under the American Recovery and Reinvestment Act of 2009(Public Law 111-5).These research projects concern the interaction of geometricmeasure theory and partial differential equations. Particularprojects will study stationary-harmonic multiple-valuedfunctions, frequency and branch sets of multiple-valuedfunctions, and regularity of integral varifolds. Work on theseproblems will address some longstanding questions in geometricanalysis such as boundary regularity of area minimizing integralcurrents in general codimensions and the singularity estimate ofintegral varifolds under natural assumptions on the mean curvatureor the second fundamental form.Some of most useful and significant classes of submanifolds ingeometry arise are characterized as critical points of naturalenergy or area functions. For example, surfaces formed by soapbubbles are critical points of the mean curvature functional,while geodesics (shortest paths) are critical points for thelength functional. Geometric measure theory offers powerfultools for establishing that critical objects exist by expandingthe class of allowable objects from surfaces to generalizedsurfaces. These methods often leave open questions ofregularity, or smoothness, of solutions, and those issues areemphasized in these projects.