(in Polish) Geometryczna teoria miary i zagadnienia wariacyjne

mathematics

elective monographs

Measure Theory 1000-135TM

Differential geometry 1000-135GR

In addition to the material of the compulsory courses at the undergraduate level, the student should familiarize himself with the material of the lecture "Measure Theory". In particular, the Besicowitch covering theorem, theory of differentiation of measures, Riesz's representation theorem, the concept of weak convergence of measures will be useful. One should also recall from the course in Mathematical Analysis the basic facts about differential manifolds embedded in Rⁿ, differential forms, and Stokes' theorem, and from the course in Topology, among others, Tikhonov's theorem on the compactness of the Cartesian product of any family of compact spaces.

Familiarity with the definitions of tensor product and exterior power (by the universal property) and a general openness to a somewhat algebraic and categorical way of thinking may be helpful.

Classroom

Geometric measure theory is the study of geometric objects using the methods of measure theory. A manifold embedded in Rⁿ can be associated a Hausdorff measure restricted to a given manifold or to a tangent bundle of that manifold. Considering a sequence of such measures and passing to the weak limit we get more general objects, e.g., varifolds or currents. We study functionals defined on such objects and their critical points, i.e. stationary varifolds (generalization of minimal surfaces). The lecture aims to present the current knowledge in this field to the extent that allows independent research. We will start with the classic area and coareavformulas and introduce the concept of rectifiability. Next, we shall discuss the basic theory of varifolds. Finally, we will focus on the pivotal concept of ellipticity of functionals, which has not been sufficiently explored so far.

The lecture is a natural continuation of the course "Measure Theory", the material of which will be the entry point for further considerations. The end point is intended to be the current state of knowledge of geometric variational problems and knowledge of the main open problems in this field. In particular, we will focus on the hitherto poorly understood, but crucial, notion of ellipticity.

Inevitably, a lot of material will be presented in an illustrative way, without giving detailed proofs, although always with reference to specific scientific papers. The aim is to present the current state of knowledge at a level sufficient to undertake independent research.

Lecture

1. Crash-course of multilinear algebra [1, §1] (1 lecture)
1. Tensor product of linear spaces
2. Tensor algebra and exterior algebra
3. The isomorphism: Hom(A,B) ≃ A* ⊗ B
4. Norm and scalar product on the exterior power
5. Wedge product and contraction operations for multilinear forms and multivectors
6. Oriented and non-oriented Grassmannian
2. Area and coarea [1, §3.2.1-12] (2 lectures)
1. Approximate Jacobian
2. Area formula, i.e., a generalisation of the change of variable formula
3. The coarea formula
3. Rectifiable sets and measures [1, §3.2.14-22] (2 lectures)
1. Formulation of the Whitney extension theorem and the Rademacher theorem on differentiability of Lipschitz functions
2. Definition and examples of rectifiable and purely unrectifiable sets
3. Briefly on area and coarea formulas for maps between rectifiable sets
4. Hausdorff measure truncated to a manifold as the surface area measure
5. Several characterisations of rectifiable sets and measures without proofs, e.g. Preiss (1987) and Azzam and Tolsa (2015)
6. Rectifiable measures as weak limits of sequences of smooth manifolds
4. Varifolds [10] (4 lectures)
1. General, rectifiable, and integral varifolds
2. Flat norm, convergence, and compactness of families of varifolds (by Tikhonov's theorem)
3. Tangent measures and varifolds
4. Push-forward of a varifold by a Lipschitz map
5. Second fundamental form and mean curvature of embedded smooth manifolds
6. First variation of the varifold with respect to an anisotropic integrand and the generalised mean curvature
7. Proof of monotonicity of density quotients and some illustrative remarks on the consequences
8. Weak maximum principle based on [8].
5. Ellipticity (4 lectures)
1. Definitions of Almgren ellipticity (AE) [11] and the Atomic Condition (AC) [4].
2. Dependence of the notion of ellipticity on the choice of competitors.
3. Overview of the consequences of ellipticity:
   • existence of minima in the class of rectifiable varifolds [12];
   • rectifiability of critical points [4];
   • partial regularity of minima [11].
4. condition BC and relations between AE and AC [6].
5. Geometrical characterisation of AC.
6. Scalar atomic condition (SAC) and regularity of critical graphs [7].
7. Constructions of k-dimensional translation invariant measures in Rⁿ and the ellipticity problem [5].

Exercises

1. Operations on multi-vectors and multi-linear forms.
2. Various characterizations of embed manifolds beyond the material of Analysis II (applications of the constant rank theorem).
3. The Grassmannian as an embedded manifold.
4. The Hausdorff measure in a normed space expressed by the integral of some integrand with respect to the Euclidean Hausdorff measure.
5. An example showing that a Hausdorff measure is not a product of lower dimensional Hausdorff measures.
6. Derivation of the formula for the anisotropic perimeter of a set.
7. Holmes-Thompson measure.
8. Relationship between the weak topology and the product topology.
9. Relationship between the flat norm and week convergence of measures.
10. Calculation of derivatives of the Jacobian and other functions whose domain is a set of linear maps.

[1] Herbert Federer

Geometric measure theory, 1969

[2] Luigi Ambrosio, Nicola Fusco, Diego Pallara

Functions of bounded variation and free discontinuity problems

Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.

ISBN: 0-19-850245-1

[3] Pertti Mattila

Geometry of sets and measures in Euclidean spaces, 1995

[4] Philippis, Guido De / Rosa, Antonio De / Ghiraldin, Francesco

Rectifiability of Varifolds with Locally Bounded First Variation with Respect to Anisotropic Surface Energies

Communications on Pure and Applied Mathematics , Vol. 71, No. 6, 2018

[5] J. C. Álvarez Paiva, A. C. Thompson

Volumes on normed and Finsler spaces

A sampler of Riemann-Finsler geometry, Vol. 50, 2004

[6] Antonio De Rosa, Sławomir Kolasiński

Equivalence of the ellipticity conditions for geometric variational problems

Communications on Pure and Applied Mathematics , Vol. 73, No. 11, 2020

[7] Antonio De Rosa, Riccardo Tione

Regularity for graphs with bounded anisotropic mean curvature

Inventiones mathematicae , Vol. 230 p. 463 - 507, 2020

[8] Brian White

The maximum principle for minimal varieties of arbitrary codimension

Communications in Analysis and Geometry, Vol. 18, No. 3, p. 421 - 432, 2010

[9] Nicolas Bourbaki

Topological vector spaces. Chapters 1-5.

Elements of Mathematics (Berlin).

Springer-Verlag, Berlin, 1987.

[10] William K. Allard

On the first variation of a varifold.

Ann. of Math. (2) 95, 1972

[11] Frederick J., Jr. Almgren

Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure.

Ann. of Math. (2) 87, 1968

[12] Yangqin Fang, Sławomir Kolasiński

Existence of solutions to a general geometric elliptic variational problem.

Calc. Var. Partial Differential Equations 57, 2018

Familiarity with the current state of knowledge on geometric variational problems at a level sufficient to undertake independent research. In particular:

• Understanding of what geometric variational problems are and the problems encountered in their study
• Knowledge of previous achievements in the field of regularity of critical points and minima of functionals defined on subsets of Rⁿ
• Understanding the difficulties arising from the lack of proof of monotonicity of density quotients in the anisotropic case
• Knowledge of the construction of classical Busemann-Hausdorff and Holmes-Thompson integrands
• Knowledge of various notions of ellipticity and their properties
• Knowledge of the literature on the subject
• Ability to apply area and coarea formulas

Oral exam

The following activities can have a positive impact on the final grade:

• performing calculations at the blackboard
• presenting some topics (filling gaps from the lecture)
• presenting homework assignments
• writing down lecture notes in LaTeX