September 8 at 3:30 pm
Ricci Flow and the Sphere Theorem
In 1926, Hopf showed that every compact,
simply connected manifold with constant curvature 1 is isometric
to the standard round sphere. Motivated by this result,
Hopf posed the question whether a compact, simply connected
manifold with sufficiently pinched curvatured must be a
sphere topologically. This question has been studied by
many authors during the past decades, a milestone being
the topological sphere theorem of Berger and Klingenberg.
I will discuss the history of this problem and sketch the
proof of the Differentiable Sphere Theorem. The proof relies
on the Ricci flow
method pioneered by Richard Hamilton."
September 9 at 3:30 pm
Minimal tori in $S^3$ and the Lawson Conjecture
In 1966, Almgren showed that any immersed
minimal surface in $S^3$ of genus $0$ is totally geodesic,
hence congruent to the equator. In 1970, Blaine Lawson constructed
many examples of minimal surfaces in $S^3$ of higher genus;
he also constructed numerous examples of immersed minimal
tori. Motivated by these results, Lawson conjectured that
any embedded minimal surface in $S^3$ of genus $1$ must
be congruent to the Clifford torus.
In this lecture, I will describe a proof
of Lawson's conjecture. The proof involves an application
of the maximum principle to a function that depends on a
pair of points on the surface."
September 10 at 3:30 pm
New Estimates for Mean Curvature Flow
We describe a sharp noncollapsing estimate
for mean curvature flow, which improves earlier work of
White and Andrews. This estimate holds for any solution
which has positive mean curvature and is
free of self-intersections. The estimate is particularly
useful for two-dimensional surfaces in $\mathbb{R}^3$; in
this case, it provides a substitute for the cylindrical
estimates established by Huisken and Sinestrari in the higher-dimensional
case.
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