Wednesday

Singularities along the Lagrangian mean curvature flow.

We study singularity formation along the Lagrangian mean curvature flow of surfaces. On the one hand we show that if a tangent flow at a singularity is the special Lagrangian union of two transverse planes, then the flow undergoes a "neck pinch", and can be continued past the flow. This can be related to the Thomas-Yau conjecture on stability conditions along the Lagrangian mean curvature flow. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Felix Schulze.

Singularities and diffeomorphisms

  • Speaker: Tobias Colding
  • Time: 10:10-11:10
  • Abstract: Comparing and recognizing metrics can be extraordinarily difficult because of the group of diffeomorphisms. Two metrics, that could even be the same, could look completely different in different coordinates. This is the gauge problem. The general gauge problem is extremely subtle, especially for non-compact spaces. Instead of dealing with it one uses some additional structure of the particular situation. However, in many problems there is no additional structure. Instead, we solve the gauge problem directly in great generality. The techniques and ideas apply to many problems. We use them to solve a well-known open problem in Ricci flow. We solve the gauge problem by solving a nonlinear system of PDEs. The PDE produces a diffeomorphism that fixes an appropriate gauge in the spirit of the slice theorem for group actions. We then show optimal bounds for the displacement function of the diffeomorphism.

Revisiting a proof of formality of Kahler manifolds, with a view towards complex manifolds and special metrics (d-dc + 3)

  • Speaker: Scott Wilson
  • Time: 11:20-12:20
  • Abstract: From one such proof of formality we obtain a new long exact sequence, defined for all complex manifolds, that relates de Rham cohomology to two other complex analytic groups. These groups relate in a transparent way to other well-studied notions, such as the Bott-Chern and Aeppeli cohomologies, and pure Hodge structures, while yielding new numerical (in)equalities involving Betti numbers as well. A mild weakening of the so-called d-dc condition, that we call (d-dc + 3), is equivalent to the vanishing of the connecting homomorphism in this sequence. Moreover, it implies E1-degeneration and is stable under many blowups. I’ll discuss numerous examples: all compact complex surfaces, higher Hopf surfaces (but not all products of odd spheres), and complex manifolds which admit locally conformal Kahler metrics satisfying a certain parallel-ness condition (i.e., Vaisman metrics). This gives a new complex analytic obstruction to the existence of certain compatible metrics and highlights a metric condition that implies E1-degeneration and much more (namely, d-dc + 3). This is joint work with Jonas Stelzig.

Stable minimal hypersurfaces in R^4

  • Speaker: Chao Li
  • Time: 2:00-3:00
  • Abstract: In this talk, I will discuss the Bernstein problem for minimal surfaces, and the solution to the stable Bernstein problem for minimal hypersurfaces in R^4: a complete, two-sided, stable minimal hypersurface in R^4 is flat.

Spectral shape optimization and new behaviors for free boundary minimal surface

Though the study of isoperimetric problems for Laplace eigenvalues dates back to the 19th century, the subject has undergone a renaissance in recent decades, due in part to the discovery of connections with harmonic maps and minimal surfaces. By the combined work of several authors, we now know that unit-area metrics maximizing the first nonzero Laplacian eigenvalue exist on any closed surface, and are induced by (branched) minimal immersions into round spheres. At the same time, work of Fraser-Schoen, Matthiesen-Petrides and others yields analogous results for the first eigenvalue of the Dirichlet-to-Neumann map on surfaces with boundary, with maximizing metrics induced by free boundary minimal immersions into Euclidean balls. After surveying these developments, I'll describe a series of recent results characterizing the (perhaps surprising) asymptotic behavior of these free boundary minimal immersions (and associated Steklov-maximizing metrics) as the number of boundary components becomes large. (Joint work with Mikhail Karpukhin.)

Lojasiewicz Inequalities for almost harmonic maps

For (almost) critical points of geometric variational problems one often only has weak, rather than strong, compactness results. As a consequence there are many situations where the seminal results of Simon on Lojasiewicz inequalities are not applicable since sequences of almost critical points can form singularities or converge to a limit with a different topology. In this talk we consider this problem for almost harmonic maps from surfaces into analytic manifolds. We present a method, which is based on joint work with A. Malchiodi and B. Sharp, that allows us to prove Lojasiewicz inequalities for sequences that converge to a simple bubble tree, and as a result obtain new conclusions about the energy spectrum and the convergence of harmonic map flow for low energy maps from surfaces of positive genus. As we shall discuss, our method is not restricted to integrable settings, but allows us to lift general Lojasiewicz-Simon inequalities from the regular setting to the singular setting of simple bubble trees whenever the bubble is attached at a non-branched point.