In this talk we will consider min-max minimal surfaces in
three-manifolds and describe some rigidity results.
For instance, we will discuss the proof that any metric on a 3-sphere
which has scalar curvature greater than or equal to 6 and is not round
must have an embedded minimal sphere of area strictly smaller than
$4\pi$ and index at most one.
We will also mention some sharp estimates for the width in the case of
positive Ricci curvature. The proofs use Ricci flow.
This is joint work with Andre Neves.