# Lecture Minimal Surfaces WiSe 2022/23

Dr. Matthias Kemper

**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

- 3D-XplorMath Minimal Surface Gallery

*Lots of 2d solutions of the minimal surface equation in 3d Euclidean space.* - Klarreich, Erica: ‘Monumental’ Math Proof Solves Triple Bubble Problem and More.
*Quanta Magazine*(2022-10-06).

*Popular science article on recent progress on multi-bubble conjectures.* - Blåsjö, Viktor: The isoperimetric problem.
*Amer. Math. Monthly***112**, 526–566 (2005). DOI:10.2307/30037526.

*Historical overview and several proofs in the plane.* - 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.*

#### 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.*