Topics in Geometric Measure Theory (II)
2021/2/26-2021/6/25 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 Chi Lin 林俊吉 (National Taiwan Normal University)
Background & Purpose
Amongst sets of finite H m measure, the (H m, m) rectifiable subsets of Rn arguably constitute the broadest possible generalisation of the notion of m dimensional submanifold of class 1 in Rn. 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 Rn). In this process, varifolds play a central role since each (H m , m) rectifiable subset and each integral current possess an associated integral varifold.
Background & Purpose
Amongst sets of finite H m measure, the (H m, m) rectifiable subsets of Rn arguably constitute the broadest possible generalisation of the notion of m dimensional submanifold of class 1 in Rn. 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 Rn). 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
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.
The following two topics are of preparatory nature.
(1) Distributions, regularisation, and distributions representable by integration, see [Fed69, 4.1.1–4.1.5], [Men16b, 2.13–2.21, 2.24, 3.1], and [Men16a].
(2) Regularity of solutions of certain partial differential equations: Sobolev’s inequality, and strongly elliptic systems, see [Fed69, 5.2.4–5.2.6]. The following five main topics are devoted to foundational results on varifolds.
(3) The first variation (with respect to the area integrand) of a varifold, see [All72, 4.1–4.6, 4.8 (2) (3), 4.10 (1), 4.11–4.12] and [Men18, § 16].
(4) Radial deformations and the rectifiability theorem, see [All72, 2.6 (3), § 5], [Kol15, § 3], [Men16b, § 4], and [Men18, § 17].
(5) The compactness theorem for integral varifolds, see [All72, § 6] and [HM86].
(6) The isoperimetric inequality, see [MS18, § 3], [Men09, 2.5], [All72, 8.6], and [Men16b, 5.1, § 7]; with regard to [All72, 8.6], see also [Sim83, 17.8].
(7) The regularity theorem, see [All72, § 8]; four sessions.
Finally, the following supplementary topic is independent of the mainline of development and serves to complete the picture.
(8) Curves of finite length, see [Fed69, 2.9.21–2.9.23, 2.10.10–20.10.14, 3.2.6].
Topics are assigned on a first-come-first-served basis. Please contact the instructor by email (see below). Participants are recommended to use the time before the start of the lecture period to prepare the topic chosen to present.