# Analysis Seminar

#### Nonclassical area minimizing oriented surfaces

**Speaker:**
Camillo DE LELLIS, IAS

**Location:**
Warren Weaver Hall 1302

**Date:**
Thursday, October 31, 2019, 11 a.m.

**Synopsis:**

Consider a smooth closed simple curve $\Gamma$ in a given Riemannian

manifold. Following the classical work of Douglas and Rado it can be

shown that, given any natural number $g$, there is an oriented surface

which bounds $\Gamma$ and has least area among all surfaces with genus

at most $g$. Obviously as we increase $g$ the area of the corresponding

minimizer can only decrease. If the ambient manifold has dimension $3$

and the curve is sufficiently regular ($C^{^2}$ suffices), works of De

Giorgi and Hardt and Simon guarantee that such number stabilizes, in

other words the absolute (oriented) minimizer has finite topology. In a

joint work with Guido De Philippis and Jonas Hirsch we show that the

latter property might fail in higher codimension even if the curve is

$C^\infty$. Some results point instead to its validity for analytic

curves (and analytic ambient metrics), confirming a conjecture of Brian

White.