Yamabe Memorial: Past symposia
2022
Friday, Sept. 30 to Sunday, Oct. 2, 2022
Pengfei Guan, McGill University
Curvature flows and monotonicity of associated geometric functionals
We discuss relationships of specific geometric functionals and curvature flows. In one direction, geometric functionals play essential roles for the convergence of curvature flows. Of special interest is Gauss curvature flow where entropy functional is crucial for non-collapsing estimates and convergence. In recent works, the arguments have been further implemented to deal with convergence of non-homogeneous Gauss curvature flow and Gauss curvature flow in space forms. The key is monotonicity or almost monotonicity of the entropy along the flows. In the reverse direction, curvature flows can be designed to solve isoperimetric type problems for geometric functionals. They serve as paths to optimal solutions for isoperimetric problem with proper monotonicity. This approach has been carried out successfully for classical quermassintegrals of non-convex domains and their variations in space forms. We will discuss some recent progresses and related open problems.
Sumio Yamada, Gakushuin University
Harmonic maps in general relativity
H. Weyl in 1916 described the Schwarzschild metric; the first exact solution to the Einstein equation by a single harmonic function with a pole. Since then, the Einstein equation has been regarded as an elliptic variational problem, and I will report on the recent progress in this direction, in particular a series of collaborative work with Gilbert Weinstein and Marcus Khuri. We will introduce spactimes of dimension four and five, and discuss the difference in 4 and 5, and the geometric consequences.
Song Sun, Berkeley
Complete Ricci-flat Kahler metrics asymptotic to cones
Complete Ricci-flat metrics with Euclidean volume growth provide singularity models for non-collapsed limits of Ricci-flat metrics. It is known that such a metric is always asymptotic to cones in a weak sense; when the metric had quadratic curvature decay, Colding-Minicozzi proved it is always asymptotic to a unique cone at a logarithmic rate. I will explain that when the metric is Kahler, the asymptotic rate is always polynomial. This result hinges on the global completeness of the metric, and is in sharp contrast to the case of local singularities where the logarithmic rate is optimal. This talk is based on joint work with Junsheng Zhang.
Ailana Fraser, University of British Columbia
When are stable minimal surfaces holomorphic?
The question of whether stable minimal surfaces are holomorphic under suitable geometric conditions has been much studied, beginning with work of Simons in complex projective space, and the proof of the Frankel conjecture by Siu-Yau. A theorem of Micallef shows that a stable minimal immersion of a complete oriented parabolic surface into Euclidean 4-space is holomorphic with respect to an orthogonal complex structure, and the same result in all dimensions if additionally the minimal surface has finite total curvature and genus zero. However, Arezzo, Micallef and Pirola gave a counterexample in general. It is therefore necessary to strengthen the stability condition in the general question. We will discuss some recent progress on this question. This is joint work with R. Schoen.
Antoine Song, University of California Berkeley
Groups homology classes, and Plateau's problem
Consider a countable group G, and an integer homology class h of G. The class h can be naturally represented by a geometric cycle in a Hilbert spherical quotient manifold via the regular representation of G. I will talk about the corresponding volume minimization problem and some of its properties.
