# Yamabe Memorial: Past symposia

## Speakers, abstracts, and materials

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2022

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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.

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2018

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Friday, September 28 to Sunday, September 30, 2018

#### The 9th Symposium: Moduli in Algebraic Geometry

#### Speakers

- Dan Abramovich
- Robert Friedman
- Jun Li
- Shigeru Mukai
- Aaron Pixton
- Yukinobu Toda
- Chenyiang Xu
- Alina Marian

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2016

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Friday September, 30 to Sunday, October 2, 2016

#### The 8th Symposium: Symplectic Geometry and Complex Geometry

#### Speakers:

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

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2014

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Friday, October 17 to Sunday, 19, 2014

#### The 7th Symposium: Current Topics in Three-Manifolds

#### Speakers:

- Ian Agol
- Mladen Bestvina
- Ursula Hamenstaedt
- Jeremy Khan
- Ciprian Manolescu
- Vlad Markovic
- Mahan Mj
- Stefano Vidussi

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2012

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Friday, October 5 to Sunday, October 7, 2012

#### The 6^{th} Symposium: Geometric Analysis

#### Speakers

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

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2010

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Friday to Sunday, October 8–10, 2010

#### The 5th Symposium: Geometry and Low-Dimensional Topology

#### Speakers

- Toby Colding
- Kenji Fukaya
- David Gabai
- Ian Hambleton
- Claude LeBrun
- Melissa Liu
- Yi Ni
- Ron Stern

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2008

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Friday to Sunday, September 26–28, 2008

#### The 4th Symposium: Geometry and Analysis

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

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2006

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Friday to Sunday, September 15–17, 2006

#### The 3rd Symposium: Geometry & Symplectic Topology

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

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2004

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Friday to Sunday, September 17–19, 2004

#### The 2nd Symposium: Geometry & Physics

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

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2002

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Friday to Sunday, September 20–22, 2002

#### The 1st Symposium: Geometry & Analysis

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