5:00 PM - 6:00 PM

Boundary regularity for area minimizing surfaces and a question of Almgren

Camillo de Lellis (IAS)

Consider an area minimizing oriented surface which has a smooth
boundary $\Gamma$ of multiplicity $1$
in some smooth Riemannian ambient manifold $\Sigma$. The celebrated work
of Federer and Fleming guarantees the existence of one such minimizer
in a suitable class of generalized oriented surfaces, called integral
currents.

In codimension $1$ a famous work of Hardt and Simon gives full
regularity of this object at the boundary, which is thus a classical
oriented hypersurface (with boundary) in a neighborhood of $\Gamma$. In
higher codimension the work of Allard can be used to conclude regularity
under some geometric assumptions, but for general smooth $\Gamma$ even
the existence of a single boundary regular point was open. In a recent
joint work with Guido De Philippis, Jonas Hirsch and Annalisa Massaccesi
we show that boundary regular points are always dense in $\Gamma$. This
has some interesting consequences on the structure of the minimizer and
in particular it allows us to answer positively to a question raised by
Almgren at the end of his `` Big regularity paper''.

Room: tba