# Yamabe Memorial Symposium 2022

## Speakers

## Schedule

#### Friday, September 30

**2-3:15pm** —** Registration***120 Vincent Hall Lounge*

**3:20-3:30pm **—** Welcome **

**3:30-4:30pm — Curvature flows and monotonicity of associated geometric functionals **(Pengfei Guan, McGill University)*16 Vincent Hall*

Abstract: 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.

**4:30-5pm **—** Coffee break***120 Vincent Hall Lounge*

**5-6pm **— **Harmonic maps in general relativity **(Sumio Yamada, Gakushuin University)*16 Vincent Hall*

Abstract: 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.

#### Saturday, October 1

**8-9am — Light breakfast***120 Vincent Hall Lounge*

**9-10am — Complete Ricci-flat Kahler metrics asymptotic to cones **(Song Sun, Berkeley)*16 Vincent Hall*

Abstract: 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.

**10:00-10:30am — Group photo and coffee break**

**10:30-11:30am — Geodesics and Minimal Surfaces I **(Andre Neves, University of Chicago)*16 Vincent Hall*

**11:30-1:30pm **—** Lunch break**

**1:30-2:30pm **—** Geodesics and Minimal Surfaces II **(Fernando Marques, Princeton University)*16 Vincent Hall*

**2:30-3pm **—** Coffee break***120 Vincent Hall Lounge*

**3-4pm **— **When are stable minimal surfaces holomorphic?** (Ailana Fraser, University of British Columbia)*16 Vincent Hall*

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.

**6pm — Banquet***Graduate Hotel*

#### Sunday, October 2

**8-9am **—** Light breakfast***120 Vincent Hall Lounge*

**9-10am **—** Groups homology classes, and Plateau's problem **(Antoine Song, University of California Berkeley)*16 Vincent Hall*

Abstract: 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.

**10-10:30am **—** Coffee break***120 Vincent Hall Lounge*

**10:30-11:30am — Translators for Mean Curvature Flow** (Brian White, Stanford University)*16 Vincent Hall*

## History

Professor Hidehiko Yamabe (1924-1960) came to Minnesota from IAS as an assistant professor in 1956. During his four years here he worked on a number of diverse topics in geometry and analysis: he initiated the problem of whether, given a Riemannian metric on a smooth compact manifold of dimension > 2, there exists a conformally equivalent metric of constant scalar curvature; he made contributions to Hilbert's 5th problem and various diffusion problems. Yamabe died in 1960, at age 37, of a massive stroke, said to have been brought on by war injuries as a civilian. This tragedy happened soon after he was appointed a full professor at Northwestern University. Yamabe's widow and two young daughters were left virtually penniless as Yamabe no longer had a pension from Minnesota and hadn't yet gained one from Northwestern. Faculty of both Northwestern and Minnesota raised a small amount of money to help the family, who then moved back to Japan.

The Yamabe Memorial Lecture was initiated jointly with Northwestern about 1962. It consisted of one lecture by a prominent mathematician each year, to alternate between Northwestern (odd numbered years) and Minnesota (even years). It was soon supported by a small subsidy from Yamabe's widow (this initial donation was later augmented in Minnesota by significant faculty contributions resulting in the present endowment called the Yamabe Fund). The distinguished speakers brought in year after year established the Lectures as an important event in the mathematical calendar, especially in the Midwest.

Past speakers include Neil Trudinger, Eugenio Calabi, Rick Schoen, Shizuo Kakutani, Craig Evans, Walter Rudin, Robert Hardt, Katsumi Nomizu, Fred Gehring, Richard Hamilton, Peter Sarnak, Jeff Cheeger, Shing-Tung Yau, Terence Tao, Igor Rodnianski, Luis Caffarelli, Tai-Ping Liu, Sergiu Klainerman and Simon Brendle.

## Hidehiko & Etsuko Yamabe

## Hidehiko, Etsuko & Kimiko Yamabe

## Hidehiko Yamabe (right)

## Past symposia

## The 1^{st} Symposium:

Geometry & Analysis

Friday to Sunday, September 20–22, 2002

Speakers: Hubert Bray, Ben Chow, Richard Hamilton, Peter Li, Fang-Hua Lin, Richard Schoen, Gang Tian, and Brian White

## The 2^{nd }Symposium:

Geometry & Physics

Friday to Sunday, September 17–19, 2004

Speakers: Robert Bryant, Sheldon Katz, Kefeng Liu, Duong Phong, Paul Seidel, Isadore M. Singer, Karen Uhlenbeck, and Shing-Tung Yau

## The 3^{rd }Symposium

Geometry & Symplectic Topology

Friday to Sunday, September 15–17, 2006

Speakers: Denis Auroux, Yasha Eliashberg, Ron Fintushel, Yongbin Ruan, Peter Ozsváth,Helmut Hofer, Dusa McDuff, and Mikio Furuta

## The 4^{th} Symposium:

Geometry and Analysis

Friday to Sunday, September 26–28, 2008

Speakers: Simon Brendle, Alice Chang, Gerhard Huisken, Ngaiming Mok, Leon Simon, Yum-Tong Siu, Neil Trudinger, and Burkhard Wilking

## The 5^{th} Symposium:

Geometry and Low-Dimensional Topology

Friday to Sunday, October 8–10, 2010

Speakers: Toby Colding, Kenji Fukaya, David Gabai, Ian Hambleton, Claude LeBrun, Melissa Liu, Yi Ni, and Ron Stern

## The 6^{th} Symposium:

Geometric Analysis

Friday to Sunday, October 5–7, 2012

Speakers: Huaidong Cao,Ben Weinkove, Benson Farb, Jean-Pierre Demailly, Robert Hardt, Conan Leung, Misha Kapovich, and Natasa Sesum

## The 7^{th} Symposium:

Current Topics in Three-Manifolds

Friday to Sunday, October 17–19, 2014

Speakers: Ian Agol, Mladen Bestvina, Ursula Hamenstaedt, Jeremy Khan, Ciprian Manolescu, Vlad Markovic, Mahan Mj, and Stefano Vidussi

## The 8^{th} Symposium:

Symplectic Geometry and Complex Geometry

Friday to Sunday, September 30–October 2, 2016

Speakers: Paul Biran, Mark McLean, Emmy Murphy, Kaoru Ono, Li-Sheng Tseng, Claire Voisin, Chris Wendl, and Sai Kee Yeung

## The 9^{th} Symposium:

Moduli in Algebraic Geometry

Friday, September 28–Sunday, September 30, 2018

Speakers: Dan Abramovich, Robert Friedman, Jun Li, Shigeru Mukai, Aaron Pixton, Yukinobu Toda, Chenyiang Xu, and Alina Marian