Readings for Prof Gang Tian

Please see attachments below

Readings for Prof Richard Schoen

In these lectures we will discuss an extremal eigenvalue problem for metrics on a

fixed smooth surface and its connection to minimal immersions of the surface

into the sphere or the ball with appropriate boundary conditions. There are only a small 

number of surfaces for which extremal metrics can be shown to exist and

a smaller number for which they can be explicitly described. The outline of the

lecture content is:

 

1) We will introduce the problem and describe its connection with harmonic maps and minimal surfaces. 

We will discuss the role played by the conformal geometry

of the sphere and ball. References for this lecture are the paper of Fraser and the speaker 

<http://arxiv.org/abs/0912.5392>, and the paper of Li and Yau

<http://www.doctoryau.com/papers/NewConformalInvariant.pdf>.

 

2) We will discuss the existence and regularity question for extremal metrics. As with other two dimensional 

problems this has two parts, the first is to control the conformal structures of metrics near the maximum, and 

the second is to construct and analyze a weak extremal metric. For closed surfaces, techniques for this problem 

were introduced by Nadirashvili <http://www.mat.univie.ac.at/~esiprpr/esi193.pdf>. See also the recent paper 

of Kokarev <http://arxiv.org/pdf/1103.2448.pdf>. We will describe a new method which appears to simplify 

and generalize some of this work to surfaces with boundary.

 

3) We will discuss the geometry of the extremal metrics on genus zero surfaces with arbitrarily many boundary 

components and on the M\"obius band. This will involve proving certain uniqueness theorems for minimal 

surfaces in the ball, and analyzing the limit of extremal minimal surfaces as the number of boundary 

components goes to infinity. Part of this work has analogues for minimal surfaces in the sphere which can be 

found in Montiel and Ros <www.springerlink.com/index/M53515207707P754.pdf> and in

El Soufi, Giacomini, and Jazar

<http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1159281066>.

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Readings for Prof Ben Andrews

See attachment below or go to drop box : https://www.dropbox.com/sh/41juohp044vg9wt/T99825BAOu?m

Preliminary readings for Prof Sethian

(1) People might visit my web page--http://www.math.berkeley.edu/~sethian
  (2) There is a book on the subject
          "Level Set Methods and Fast Marching Methods":
           Cambridge University Press, 1999
  (3) There is a review article in Acta Mathematica:

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1770436

 

Readings for Prof. Robert McCann's lectures are available here.

 Also see attachment below: glimpse.pdf

It is also linked as publication # [52] at

http://www.math.toronto.edu/mccann/publications

Specific references to publications on that page relevant to each of my three lectures:

Overview: [47]

Lecture 1 (Tuesday): [52]

Lecture 2 (Thursday): [42] [48]

Lecture 3 (Friday): [49]

 

Readings for Prof. Michael Struwe:

The topics of my three lectures will be as follows:

1) Partial regularity for harmonic maps:

In this first lecture I will outline in detail the gauge-theoretic approach to the partial regularity
results for stationary harmonic maps from an m-dimensional ball to a closed Riemannian manifold
that Tristan Riviere and I developed in our paper 
Rivière, Tristan; Struwe, Michael Partial regularity for harmonic maps and related problems.
Comm. Pure Appl. Math.61 (2008), no. 4,451–463.

2) The heat flow of harmonic maps:

In my second lecture I quickly review the existence of unique, partially regular weak solutions with
non-increasing energy to the heat flow of harmonic mapsfrom a closed Riemann surface to a closed
Riemannian manifold, as shown in my paper
Struwe, Michael On the evolution of harmonic mappings of Riemannian surfaces.
Comment. Math. Helv.60 (1985), no. 4,558–581.

These results are also covered in Chapter 3.6 of my book
Struwe, Michael Variationalmethods.Applications to nonlinear partial differential equations and
Hamiltonian systems. Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.
A Series of Modern Surveys in Mathematics], 34. Springer-Verlag, Berlin,2008.
Participants who wish to become familiar with this topic ahead of time may find this reference
more easily accessible.

3) Uniquenes of weak solutions:

In my final lecture I first recall the examples of Peter Topping and Michiel Bertsch et al. on "reverse
bubbling" and non-uniqueness for the harmonic map heat flow. I then explain the result of Melanie Rupflin,
showing how the gauge-theoretic approach to regularity can be employed to show that the weak solutions
to the harmonic map heat flow introduced in my second lecture are unique inthe a-priori much larger class
of weak solutions with finite energy unless reverse bubbling occurs, as had been conjectured by Peter Topping.
 
The material for this lecture can be found in the following papers:
Bertsch, Michiel; Dal Passo, Roberta; van der Hout, Rein: Nonuniqueness for the heat flow of harmonic maps
on the disk. Arch. Ration. Mech. Anal.161 (2002), no. 2, 93–112.
Moser, Roger Regularity for the approximated harmonic map equation and application to the heat flow for
harmonic maps.Math. Z.243 (2003), no. 2,263–289.
Rupflin, Melanie: An improved uniqueness result for the harmonic map flow in two dimensions.
Calc. Var. Partial Differential (2008), no. 3,329–341.
Topping, Peter: Reverse bubbling and nonuniqueness in the harmonic map flow.
Int. Math. Res. Not.2002, no. 10,505–520.

AttachmentsSize
GeometricFlowsBook_Ch2.pdf1.23 MB
glimpse.pdf217.5 KB
proc-for-calabi.pdf182.28 KB
proc-for-cheeger.pdf249.81 KB