Lecture Minimal Surfaces WiSe 2022/23

Dr. Matthias Kemper

Entry in the course catalogue

Tuesday and Friday, 14:15–15:45 in M6, Einsteinstr. 64, starting on October 11.

Program

  • Motivation from physics: soap films, bubbles
  • Minimal surfaces and their relatives: surfaces of prescribed mean curvature, almost minimizers, horizons in general relativity. All of that in manifolds of arbitrary dimension.
  • The isoperimetric problem: enclosing volume with least area
  • Geometric measure theory: Existence and regularity for area-minimizing hypersurfaces
  • Hypersurface singularities and how to cope with them

Mode

  • two lectures a week, including quizzes, and a few optional exercises every other week

Target group

  • You should be familiar with the basic notions of differential geometry (manifolds, Riemannian metrics, geodesics, curvature) and not be afraid of PDEs.
  • Master students can use this lecture …
    • as a type II lecture in the specializations Differential Geometry or Geometric Structures,
    • as Specialization Supplement (Ergänzungsmodul), or
    • just to learn more about minimal surfaces 😎
  • Physicists, advanced bachelor students, and PhD students are welcome, too!

Exercises

Optional excercise will appear from time to time in lecture.

Sheet 1 (for discussion on 21 25 October)

Literature

Introduction

Geometry of Submanifolds

  • Lee, John: Riemannian Manifolds. Springer 1997.
    Gentle standard introduction, including basics of submanifold geometry.
  • Chavel, Isaac: Riemannian Geometry. Cambridge 1994, 2006.
    This and the following are more broad, complete introductory texts
  • Petersen, Peter: Riemannian Geometry. Springer 1998, 2006, 2016.
  • Spivak, Michael: A Comprehensive Introduction to Differential Geometry, Volume Four. Publish or Perish 1970, 1979, 1999.
    Contains derivations of the first and second variation formula.

Caccioppoli Sets

  • Giusti, Enrico: Minimal surfaces and functions of bounded variation. Birkhäuser 1984.
    Our primary source.
  • Maggi, Francesco: Sets of finite perimeter and geometric variational problems. Cambridge 2012.
    Newer account with more details. Caccioppoli sets in Part II.
  • Giaquinta, Mariano; Modica, Giuseppe; Souček, Jiří: Cartesian Currents in the Calculus of Variations I. Springer 1998.
    Large reference. Relation between currents and Caccioppoli sets in Chapter 4.
  • De Philippis, G.; Paolini, E.: A short proof of the minimality of Simons cone. Rend. Semin. Mat. Univ. Padova 121, 233–241 (2009). DOI:10.4171/RSMUP/121-14.

Currents

Until January 17, you should get familiar with differential forms. Important topics: exterior derivative, integration, Stokes’ theorem, de Rham cohomology.
They are covered in the following books:

  • Jänich, Klaus: Vector analysis. Springer 2001.
    Concise presentation of all the basic results; you may skip the chapters on manifolds. There is also a German version.
  • Bott, Raoul; Tu, Loring: Differential Forms in Algebraic Topology. Springer 1982.
    This books presents lots of topics from a standard algebraic topology course, up to spectral sequences and characteristic classes, in the language of differential forms. For our purposes, the Introduction and §§ 1, 3, 4 are more than enough.

Literature on currents:

  • Morgan, Frank: Geometric Measure Theory. Academic Press 1988, 1995, 2000, 2009, 2016.
    Nice exposition with lots of pictures, originated as an accessible companion to Federer’s Geometric Measure Theory.
  • Federer, Herbert: Geometric Measure Theory. Springer 1969.
    This book has every detail in the most general case you could possibly wish for.
  • Simon, Leon: Lectures on Geometric Measure Theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, 1984.
    Friendly presentation of not only currents, but also Caccioppoli sets and varifolds, written for analysts, with just the right amount of details.
A rotating double soap bubble.
© Matthias Kemper