Geometric measure theory -- varifolds 2017/2018

Lecture description (in Polish)

Technicalities

• Times:
  • lecture: Thursdays 8:30 - 10:00, room 5820;
  • exercise classes: Tuesdays 10:15 - 11:45, room 5840.
• Lecturer: Sławomir Kolasiński
• Contact: initial.lastname --at-- mimuw.edu.pl
• Office hours: room 5220 @ MIM UW, Fridays 10:15 - 11:45

Mini-course on the Allard regularity theorem

Downloads:

• Basic definitions and theorems (last update: November 6, 2017)
• Exercises (last update: January 9, 2018)
• Construction of the tensor product (last update: October 20, 2017)

Literature:

• H. Federer "Geometric measure theory"
• W. Allard "On the first variation of a varifold"

Auxiliary literature:

• L. Simon "Lectures on geometric measure theory"
• F. Almgren "Plateau's problem. An invitation to varifold geometry"
• F. Morgan "Geometric Measure Theory. A Beginner's Guide"
• F. Almgren "The theory of varifolds"
• U. Menne "The concept of varifold"
• U. Menne "Weakly differentiable functions on varifolds"
• U. Menne "Sobolev functions on varifolds"
• L. Evans and R. F. Gariepy "Measure theory and fine properties of functions"
• W. P. Ziemer "Weakly Differentiable Functions"
• E. M. Stein "Singular integrals and differentiability properties of functions"
• C. Scharrer Relating diameter and mean curvature for varifolds

Miscellaneous:

• Homepage of Kenneth A. Brakke creator of the Surface Evolver.

• Gallery of soap films and minimal surfaces by John M. Sullivan.

• De Philippis, De Rosa, and Ghiraldin found recently a sufficient and necessary condition on the integrand F for the following implication to hold: if V is F-stationary, then V is rectifiable.

Rules:

• There will be an oral exam during which I may ask for solutions of selected exercises.