Topics in Geometric Measure Theory (I)
2020/9/18-2021/1/15 Every Friday, 14:20-17:20
Rm 509, Chee-Chun Leung Cosmology Hall, National Taiwan University
Speaker:
Ulrich Menne 孟悟理 (National Taiwan Normal University)
Organizer:
Chun Chu Lin 林俊吉 (National Taiwan Normal University)
Background & Purpose
Amongst sets of finite H^m measure, the (H^m, m) rectifiable subsets of R^n arguably constitute the broadest possible generalisation of the notion of m dimensional submanifold of class 1 in R^n. The study of existence of solutions to geometric variational problems involving the area integrand requires compact classes of objects on which the integrand is lower-semicontinuous (for minimisers) or continuous (for stationary points). This led to the development of integral currents and integral varifolds, respectively. Both classes are functional- analytically defined augmentations of the concept of (H^m, m) rectifiable sets. After solving the existence problem for the afore-mentioned variational problems in these broad classes, the aim of the subsequent regularity theory is to study to which degree these solutions resemble the original differential geometric object (i.e., an m dimensional submanifolds of class 1 of R^n). In this process, varifolds play a central role since each (H^m, m) rectifiable subset and each integral current possess an associated integral varifold.
Outline
This course provides a thorough introduction of the classical parts of varifold theory including the fundamental compactness theorems and Allard’s regularity theorem. Most of the necessary background from locally convex spaces, distribution theory, Grassmann manifolds, and curvature of submanifolds shall be developed in the course. However, we doassume knowledge of real analysis—in particular, concerning the representation of linear functionals by measures and differentiation theory of measures (e.g., covering theorems and densities). The main topics also employ basic properties of Hausdorff measures, the concepts of (H^m, m) rectifiability of subsets of R^n, and some Grassmann algebra (centred around m vectors and alternating forms). Lecture notes on Geometric Measure Theory (see [Men20]) are made available upon request; they include all afore-mentioned prerequisites. Participants meeting some but not all of these prerequisites are likely to find a suitable topic amongst the preparatory and supplementary ones listed below.
Topics are presented by the participants and are assigned in consultation with the teacher taking into account the prior knowledge of the individual participants. Remote participants may employ a virtual whiteboard for their presentation. Each participant usually presents two sessions and all topics should take two sessions unless indicated otherwise.