Workshop on Geometric Analysis
日時: 2005年 2月 21日(月)〜 2月24日(木)  場所: 東北大学 理学研究科 数学棟 518号室

    Miguel Abreu (Instituto Superior Tecnico) Mon. 10:40-11:40 &  Mon. 2:30-3:30              
"Toric Kahler metrics in symplectic coordinates"
Abstract: I will motivate and describe a parametrization for toric Kahler metrics
on symplectic 2n-manifolds (and orbifolds) equipped with a Hamiltonian
n-torus action, using symplectic coordinates and potentials (instead of
the more common complex analogues). The relevant properties of this
approach will be discussed and illustrated with several explicit
    I will then concentrate on the constant and extremal (in the sense of
Calabi) scalar curvature equation, motivating its special form in this
setting, describing explicit interesting solutions and discussing some
of its analytical properties.
    Almost all of what will be presented in these two talks is the result of
separate work by Guillemin, Abreu and Donaldson.

    Jingyi Chen (University of British Columbia) Mon. 4:00-5:00 & Tue. 10:40-11:40                 
"Some geometric PDEs arising from calibrations"
Abstract: In the first talk, I will discuss mean curvature flows for 2-dimensional
symplectic surface in a Kaehler-Eistein surface and Lagangian submanifolds
in a Calabi-Yau n-fold. In the second talk, I will discuss maps between
hyperkaehler manifolds, which satisfy a first order system and are related to
harmonic maps and minimal surfaces.

Richard Schoen (Stanford University) Tue. 2:30-3:30 & Tue. 4:00-5:00

"Minimal Submanifolds I: Lagrangian volume minimization and special
lagrangian submanifolds"
Abstract: In high codimension there are special classes of
volume minimizing submanifolds which one would like to construct in
general situations. These include special lagrangian submanifolds. In this
talk we will describe this class of submanifolds and survey the progress
on existence questions via variational methods, mean curvature flow, and
gluing methods. We will discuss some of the many open problems in this

"Minimal Submanifolds II: On the isoperimetric inequality for minimal

Abstract: A longstanding conjecture concerning the geometry of minimal
surfaces is that the sharp isoperimetric inequality (relating the area of
the surface and its boundary length) should hold in general. In this talk
we will survey this problem and present some new methods of attack which
solve the problem in general situations. The methods involve a careful
analysis of flat cone metrics which are related in a natural way to the
minimal surface.

Peter Topping (University of Warwick) Wed. 10:40-11:40 & Wed. 2:30-3:30

" Ricci flow I: A selected survey."
Abstract: I'll describe what the Ricci flow is, and some of the
basic theory, including a few ideas of Perelman which are
interesting from a geometric analysis point of view.
(Aimed at a general audience who know a bit of Riemannian

"Ricci flow II: Singular initial configurations."
Abstract: I'll show how one of the ideas from the 1st talk can be
used to prove a compactness theorem for Ricci flows, which
gives us some unconventional existence theorems.

Seiki Nishikawa (Tohoku University) Wed. 4:00-5:00

"Harmonic maps into complex Finsler manifolds"

Jiaping Wang (University of Minnesota) Thurs. 10:40-11:40 & Thurs. 2:30-3:30

"Function theory and its applications"
Abstract: Function theory has been successfully applied to
derive various geometric and topological information of Riemannian
manifolds. In the two talks, I plan to explain some of these
aspects and mention a few open problems.

Futoshi Takahashi (Tohoku University) Thurs. 4:00-5:00

"On an isoperimetric inequality for mapping with remainder term"