VVZ API is not affiliated with ETH Zurich. Data might be outdated or incorrect. Please view the official ETHZ Vorlesungsverzeichnis for binding information.
Regularity for Minimal Surfaces
Last Updated: 2026-06-01 11:30:59
Abstract
Minimal surfaces are critical points of the area functional, and in general are not smooth submanifolds of the ambient space. Nevertheless, one can hope to understand the size and structure of singularities in certain situations, and behavior of the surface nearby.
Objective
The aim of this course is to provide an introduction to the regularity of minimal hypersurfaces using the weak framework of stationary integral varifolds, and develop an intuition for various regularity results and the conditions under which they hold.
Content
- Background on Hausdorff measure, dimension and countably rectifiable sets - Integral varifolds: definitions, first and second variation - Monotonicity for mass ratios, tangent cones - Allard-De Giorgi Regularity Theorem: proof for Lipschitz graphs (ideas of Lipschitz approximation step will be in notes) - Stratification for singular set, Federer's dimension reduction - Simons' Theorem for low-dimensional stable minimal hypercones, dimension estimate on singular set for area-minimizing hypersurfaces (using the framework of currents) - Bernstein Theorems and curvature estimates in low dimensions - Existence of singular area-minimizing cones and failure of Bernstein Theorems in high dimensions TIME PERMITTING: Hardt-Simon foliation, ideas and difficulties in higher codimension
Resources
Lecture Notes
Notes will be uploaded weekly on my webpage:https://sites.google.com/view/askorobogatova/teaching
Literature
See "Learning Materials" tab
Learning Materials (Links)
- Literature
- Geometry of sets and measures in Euclidean spaces - P. Mattila
- Introduction to Geometric Measure Theory - L. Simon
- Sets of finite perimeter and geometric variational problems - F. Maggi
- Additional links
- A Course in Minimal Surfaces - T. H. Colding & W. P. Minicozzi
- Lectures on Minimal Surface Theory - B. White
General Information
- Language
- English
- Levels
- DR , MSC
Examination
- Type
- session examination
- Mode
- oral 20 minutes
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture | Regularity for Minimal Surfaces |
|
2 h weekly |
Offered In
-
-
Wahlfächer (Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 14 KP der erforderlichen 26 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen.)
-
-
-
Doktorat Mathematik (Mehr Informationen unter: )
-
Vertiefung Fachwissen (Die Liste der Lehrveranstaltungen für Doktoratsstudentinnen und Doktoratsstudenten wird jedes Semester im Newsletter der ZGSM veröffentlicht.)
-
Graduate School (Offizielle Website der Zurich Graduate School in Mathematics: )
-
-