Calculus of Variations and Geometric Measure Theory

Almgren minimals sets, minimal cones, unions and products

Xiangyu Liang

created by scharrer on 05 Jul 2024

17 jul 2024 -- 10:00   [open in google calendar]

Agenda: Get-together (30 min), presentation Xiangyu Liang (60 min), questions and discussions (30 min).

Registration required for new participants. Please go to our seminar website (allow one work day for processing).


The notion of Almgren minimal sets is a way to try to solve Plateau’s problem in the setting of sets. To study local structures for these sets, one does blow-ups at each point, and the blow-up limits turn out to be minimal cones. People then would like to know the list of all minimal cones.

The list of 1 or 2-dimensional minimal cones in $\mathbb R^3$ are known for over a century. For other dimensions and codimensions, much less is known. Up to now there is no general way to classify all possible minimal cones. One typical way is to test unions and products of known minimal cones.

In this talk, we will first introduce basic notions and facts on Almgren minimal sets and minimal cones. Then we will discuss the minimality of unions and products of two minimal cones.