Analysis Seminar

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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.