Stanford University
Department of Mathematics

# Geometry Seminar Spring 2013

Organizers: Richard Bamler (rbamler@math.*) and Yi Wang (wangyi@math.*)

Time: Wednesdays at 4 PM

Location: 383N

(*=stanford.edu)

## Next Seminar

22 May

Speaker: Otis Chodosh (Stanford)

Title: Ricci solitons asymptotic to cones

Abstract: We'll discuss the relationship between expanding Ricci solitons and metric cones, in particular the works by Schulze--Simon and Cabezas-Rivas–Wilking constructing Ricci flows with certain cones as initial conditions. We will then discuss two recent uniqueness results for expanding solitons asymptotic to "round" cones (one of these results is joint work with Frederick Tsz-Ho Fong).

## Spring Quarter

3 April

Speaker: Clifford Taubes (Harvard)

Title: How many monopoles can fit in your refrigerator?

Abstract: Bolognese conjectured that a large number, N, of non-Abelian SU(2) monopoles on R3 (solutions to the time independent self-dual equations) can fit in a ball of radius R only if R is no smaller than N. I explain the sense in which Bolognese’s bound is true and describe a construction of N-monopole solutions that come close to realizing Bolognese’s minimal radius bound.

10 April

Speaker: Martin Li (UBC)

Title: Minimal surfaces in the unit ball

Abstract: Minimal surfaces in the unit ball arise as solutions to an extremal eigenvalue problem for surfaces with boundary, due to the work of A. Fraser and R. Schoen. In this talk, we will discuss some recent results concerning the geometry and the variational properties of these minimal surfaces. For the embedded ones, we give an eigenvalue estimate and a smooth compactness result. As an application, we construct new examples of embedded Willmore surfaces of arbitrary genus. (Part of this is joint work with A. Fraser.)

12 April, 2pm (NOTE: SPECIAL DAY AND TIME)

Speaker: Artem Pulemotov (Queensland)

Title: The Dirichlet problem for the prescribed Ricci curvature equation

Abstract: We will discuss the following question: is it possible to find a Riemannian metric whose Ricci curvature is equal to a given tensor on a manifold $M$? To answer this question, one must analyze a weakly elliptic second-order geometric PDE. In the first part of the talk, we will review the history of the subject and state several classical theorems. After that, our focus will be on new results concerning the case where $M$ has nonempty boundary.

17 April

Speaker: TBA

Title: TBA

Abstract: TBA

24 April

Speaker: Nick Edelen (Stanford)

Title: Constant mean curvature, flux conservation, and symmetry

Abstract: As first observed by Korevaar, Kusner and Solomon, constant mean curvature implies a homological conservation law for hypersurfaces in ambient spaces with Killing fields. I will discuss joint work with Bruce Solomon, in which we generalize this law by relaxing the topological restrictions assumed by KKS, and by allowing a weighted mean curvature functional. We also prove a partial converse. Roughly, it states: when flux is conserved along enough Killing fields, the hypersurface splits into a region with constant mean curvature, and a region fixed by the given Killing fields. I will demonstrate our theory by using it to derive a first integral for twizzlers, i.e. helicoidal surfaces of constant mean curvature in R3.

1 May

Speaker: TBA

Title: TBA

Abstract: TBA

8 May

Speaker: Qi Zhang (Riverside)

Title: Volume non-inflating under Ricci flow and a convergence result

Abstract: First we present a proof of the so called volume non-inflating property of Ricci flows, which can be regarded as the opposite statement of Perelman's volume non-collapsing property. This implies, combining with Perelman's result, that the Kaehler Ricci flow in the positive Chern class case, converges to a metric space in the Gromov-Hausdorff topology. Second we discuss a gradient bound for the heat equation which is independent of a lower bound of the Ricci curvature. Applying this together with Cheeger's result, one can show that the limiting metric space has certain regularity. The talk is based on a paper by the speaker and on a joint paper with Gang Tian.

14 May, 4pm-5pm (NOTE: SPECIAL DAY and SPECIAL ROOM GESB 131)

Title: Ancient Solutions to the Yamabe Flow

Abstract: We will discuss the existence of new type II ancient compact solutions to the Yamabe flow. These solutions are rotationally symmetric and converge, as $t \to -\infty$, to a tower of spheres. We will also discuss complete non-compact Yamabe shrinking solitons and the singularity formation of complete non-compact solutions to the conformally flat Yamabe flow whose conformal factors have cylindrical behavior at in?nity.

15 May

Speaker: Curtis McMullen (Harvard)

Title: The Gauss-Bonnet theorem for cone manifolds and moduli spaces

Abstract: TBA

22 May

Speaker: Otis Chodosh (Stanford)

Title: Ricci solitons asymptotic to cones

Abstract: We'll discuss the relationship between expanding Ricci solitons and metric cones, in particular the works by Schulze--Simon and Cabezas-Rivas–Wilking constructing Ricci flows with certain cones as initial conditions. We will then discuss two recent uniqueness results for expanding solitons asymptotic to "round" cones (one of these results is joint work with Frederick Tsz-Ho Fong).

29 May

Speaker: Karl-Theodor Sturm (Bonn)

Title: The space of mm spaces is an Alexandrov space

Abstract: The space X of all metric measure spaces (X,d,m) plays an important rôle in image analysis, in the investigation of limits of Riemannnian manifolds and metric graphs as well as in the study of geometric flows that develop singularities. We show that the space X – equipped with the L2-distortion distance ∆ – is a challenging object of geometric interest in its own. In particular, we show that it has nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on X are presented.

## Past Quarters

For the Winter 2013 Schedule go here

For the Fall 2012 Schedule go here

For the Spring 2012 Schedule go here

For the Winter 2012 Schedule go here

For the Fall 2011 Schedule go here

For the Spring 2011 Schedule go here

For the Winter 2011 Schedule go here

For the Fall 2010 Schedule go here

For the Spring 2010 Schedule go here

For the Winter 2010 Schedule go here

For the Fall 2009 Schedule go here