Taiwan Mathematics School: Topics in Geometric Measure Theory I

14:20-17:20, every Friday,
September 18, 2020 - January 15, 2021 R509, Cosmology Building, NTU

Speaker:
Ulrich Menne (National Taiwan Normal University)

Organizers:
Chun-Chi Lin (National Taiwan Normal University)

1. Background:

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 regularitytheory 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.

2. Purpose and prerequisites:

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, curvature of submanifolds, and elliptic partial differential equations 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.

3. Course 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.

4. Detailed Time and Other information of the Course:

The following seven topics are of preparatory nature.

(1)

Locally convex spaces induced by a nonempty family of real valued seminorms,

see [Men16c, §2]; one session.

(2)

Locally convex spaces of continuous functions with and without compact support, see [Fed69, 2.5.19] and [Men16b, 2.10–2.12, 2.23].

(3)

Decomposition of Daniell integrals and weak convergence of linear functionals on spaces of continuous functions with compact support, see [Fed69, 2.5.20] and [AFP00, 1.61–1.63]; one session.

(4)

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].

(5)

The Grassmann manifolds, Jacobians, and curvatures of submanifolds, see [Fed69, 3.2.18(1)(2)(4)], [All72, 2.3–2.5], and [Men18, §6].

(6)

Some structure theory, see [Fed69, 3.3.1, 3.3.5, 3.3.6, 3.3.17] and [Men18, §13].

(7)

Regularity of solutions of certain partial differential equations: L2 and Hölder conditions, strongly elliptic systems, Sobolev’s inequality, and strongly elliptic systems, see [Fed69, 5.2.1–5.2.6]; four sessions.

The following six main topics are devoted to foundational results on varifolds.

(8)

Basic properties of varifolds, see [All72, 3.1–3.5], [MS18, 4.1], and [Men18,§15].

(9)

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].

(10)

Radial deformations and the rectifiability theorem, see [All72, 2.6(3), §5], [Kol15, §3], [Men16b, §4], and [Men18, §17].

(11)

The compactness theorem for integral varifolds, see [All72, §6] and [HM86].

(12)

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].

(13)

The regularity theorem, see [All72, §8]; four sessions.

Finally, the following three supplementary topics are independent of the main line of development and serve to complete the picture.

(14)

Curves of finite length, see [Fed69, 2.9.21–2.9.23, 2.10.10–20.10.14, 3.2.6].

(15)

Almgren’s bi-Lipschitzian embedding of the space Q_{Q}(R^{n}), consisting of all unordered Q tuples in R^{n}, into a suitable Euclidean space, see [Alm00, 1.1(1)–(4), 1.2].

Topics are assigned on a first-come-first-served basis. Please contact the instructor by email. Participants are recommended to use the time before the start of the lecture period to prepare the topic chosen to present.

5. References

[AFP00]

Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000.

[All72]

William K. Allard. On the first variation of a varifold. Ann. of Math. (2), 95:417–491, 1972. URL: https://doi.org/10.2307/1970868.

[Alm00]

Frederick J. Almgren, Jr. Almgren’s big regularity paper, volume 1 of World Scientific Monograph Series in Mathematics. World Scientific Publishing Co. Inc., River Edge, NJ, 2000. Q-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2, With a preface by Jean E. Taylor and Vladimir Scheffer. URL: https://doi.org/10.1142/9789812813299.

[Fed69]

Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969. URL: https://doi.org/10.1007/978-3-642-62010-2.

[HM86]

John E. Hutchinson and Michael Meier. A remark on the nonuniqueness of tangent cones. Proc. Amer. Math. Soc., 97(1):184–185, 1986. URL: https://doi.org/10.2307/2046103.

[Kol15]

Jan Kolář. Non-unique conical and non-conical tangents to rectifiable stationary varifolds in R^4. Calc. Var. Partial Differential Equations, 54(2):1875–1909, 2015. URL: https://doi.org/10.1007/ s00526-015-0847-9.

[Men09]

Ulrich Menne. Some applications of the isoperimetric inequality for integral varifolds. Adv. Calc. Var., 2(3):247–269, 2009. URL: https://doi.org/10.1515/ACV.2009.010.

[Men16a]

Ulrich Menne. Topologies on D(U, Y ), 2016. Max Planck Institute for Gravitational Physics and University of Potsdam, 3 pages, unpublished.

Ulrich Menne. Geometric variational problems, 2018. Lecture notes, University of Leipzig.

[Men20]

Ulrich Menne. Geometric measure theory, 2020. Lecture notes, National Taiwan Normal University.

[MS18]

Ulrich Menne and Christian Scharrer. An isoperimetric inequality for diffused surfaces. Kodai Math. J., 41(1):70–85, 2018. URL: https://doi.org/10.2996/kmj/1521424824.

[Sim83]

Leon Simon. Lectures on geometric measure theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University. Australian National University Centre for Mathematical Analysis, Canberra, 1983.