# Minimal hypersurfaces and positive scalar curvature

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Thomas Schick

created by risa on 06 Apr 2018

11 apr 2018
-- 14:00

Aula di Consiglio, Dip.Matematica, Università "La Sapienza", Roma

**Abstract.**

There is a long history to find relations between the topology of a smooth
manifold and its (Riemannian) geometry. The first such is the Gauss-Bonnet
theorem which says that the Euler characterestic of a compact 2-dimensional
surface without boundary is (upto a positive constant) the scalar curvature
of that manifold. Particular conclusion: if the Euler characterestic is not
positive (i.e. if the surface is not a sphere or a real projective plane)
then there is no metric such that the scalar curvature is everywhere
positive. The use of the Dirac operator allows to obtain similar
obstructions to the existence of positive scalar curvature in higher
dimension; but only for spin manifolds (as otherwise this operator doesn't
exist). There is one further approach -invented by Schoen and Yau, which
does not rely on the spin condition, but rather uses minimal hypersurfaces.
We will present this approach and its main implications. There are two
crucial problems with this approach:

-in its initial incarnation, it requires regularity results on minimal
hypersurfaces which are available only in dimension less than 8,

- it needs a large integral first homology.

We will report on current work which aims to overcome part of these
problems, due to Schoen-Yau for the first problem, and developped in joint
work with Simone Cecchini for some aspects of the second problem.
Specifically, we will introduce and discuss the case of 'enlargeable
manfolds' (as introduced by Gromov and Lawson).