# Colloquium Series Seminar: Area-minimizing integral currents: singularities and structure

Camillo De Lellis (Institute for Advanced Study)

Area-minimizing integral currents are natural generalization of area-minimizing oriented surfaces, which allow to get existence of minimizers even in those cases in which they cannot be smooth.

Almgren's famous Big Regularity Paper proves that the interior singular set of any $m$-dimensional area-minimizing integral current in any smooth Riemannian manifold $\mathcal{M}$ has (Hausdorff) dimension at most $m-2$. Except for the case $m=2$, when it was proved that interior singularities are isolated, little is known about the structure of the singular set. Moreover a recent theorem by Liu proves that we cannot expect it to be a $C^1$ $m-2$-dimensional submanifold (unless the ambient $\mathcal{M}$ is real-analytic) as in fact it can be a fractal set of any Hausdorff dimension $\alpha \leq m-2$. On the other hand it seems likely that it is an $(m-2)$-rectifiable set, i.e. that it can be covered by countably many $C^1$ submanifolds.

In this talk I will explain why the problem is very challenging and how it can be broken down into easier pieces following a recent joint work with Anna Skorobogatova.

Almgren's famous Big Regularity Paper proves that the interior singular set of any $m$-dimensional area-minimizing integral current in any smooth Riemannian manifold $\mathcal{M}$ has (Hausdorff) dimension at most $m-2$. Except for the case $m=2$, when it was proved that interior singularities are isolated, little is known about the structure of the singular set. Moreover a recent theorem by Liu proves that we cannot expect it to be a $C^1$ $m-2$-dimensional submanifold (unless the ambient $\mathcal{M}$ is real-analytic) as in fact it can be a fractal set of any Hausdorff dimension $\alpha \leq m-2$. On the other hand it seems likely that it is an $(m-2)$-rectifiable set, i.e. that it can be covered by countably many $C^1$ submanifolds.

In this talk I will explain why the problem is very challenging and how it can be broken down into easier pieces following a recent joint work with Anna Skorobogatova.

Building: | East Hall |
---|---|

Event Type: | Workshop / Seminar |

Tags: | Mathematics |

Source: | Happening @ Michigan from Department of Mathematics, Colloquium Series - Department of Mathematics |