Surveys in Differential Geometry

Volume 21 (2016)

The size of the singular set of area-minimizing currents

Pages: 1 – 83



Camillo De Lellis (Mathematik Institut der Universität Zürich, Switzerland)


A well-known monograph of Almgren proves that the singular set of a general $n$-dimensional area-minimizing integral currents has dimension at most $n-2$, which is an optimal bound when the dimension of the ambient manifold is larger than $n+1$. Almgren’s original (typewritten) manuscript was more than 1700 pages long. In a recent series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren’s theory, building upon large portions of his program but also bringing some new ideas from partial differential equations, metric analysis and metric geometry.


area-minimizing currents, singular set, multiple valued functions, center manifold

2010 Mathematics Subject Classification

Primary 49Q15. Secondary 53C42, 58A20.

Published 13 January 2022