Title:
Stability of Llarull's Theorem and scalar curvature in
dimension three
Abstract: As a consequence of Llarull's work, the unit n-sphere
has the following characterization: it is the only Riemannian metric
on the sphere Sn whose scalar curvature and distance
function are both at least as large as the unit sphere's. Following a
developing program of Sormani and Gromov, this talk considers the
associated "stability" problem which probes the flexibility of
Llarull's rigidity statement. In joint work with Allen and Bryden, we
provide a solution in dimension 3. The main result states that a
sequence of Riemannian 3-spheres becoming closer and closer to
satisfying the hypothesis of Llarull's Theorem must converge to the
unit 3-sphere in the Sormani-Wenger intrinsic flat sense, so long as
the sequence satisfies a uniform Cheeger constant bound. The argument
relies on a new proof of Llarull's Theorem in dimension 3 which was
inspired by the Positive Mass Theorem. We will also discuss new
'drawstring' examples with Kai Xu which probe the stability of other
scalar curvature rigidity theorems.