ICMS

Geometric analysis

Jun 20, 2011 - Jun 24, 2011

ICMS 15 South College Street

Organisers

Name	Institution
Huisken, Gerhard	Max Planck Institute for Gravitational Physics
Neves, Andre	Imperial College London
Topping, Peter	University of Warwick

This workshop in geometric analysis focussing particularly on geometric flows presents a rare opportunity for young researchers to meet and collaborate with world-leading experts in the field. The academic focus of the meeting will be on the “hard analysis” side of the subject. The interplay between geometry and analysis has produced some of the most fruitful ideas in mathematics in recent years.

The three main objectives of this workshop are:

• to bring together established experts in geometric flows with young UK researchers
• to foster an environment where further advances in the area of hard analysis of geometry can occur
• to create a better environment for young UK researchers in the area and to consolidate the promising position of the UK in geometric analysis

The daily programme of the workshops gives plenty of opportunities for informal interactions amongst the participants.

Speakers
Sigurd Angenent (University of Winsconsin, USA)
Richard Bamler (Princeton University and Stanford University, USA)
Simon Brendle (Stanford University, USA)
Tobias Colding (MIT, USA)
Mihalis Dafermos (Cambridge, UK)
Simon Donaldson (Imperial, UK)
Klaus Ecker (Frei Universitaet Berlin, Germany)
Ernst Kuwert (Universitaet Freiburg, Germany)
Camillo De Lellis (Universitaet Zuerich, Switzerland)
Tom Ilmanen (ETH-Zuerich, Switzerland)
Fernando Marques (IMPA, Brazil)
William Minicozzi (Johns Hopkins University, USA)
Tristan Riviere (ETH-Zurich, Switzerland)
Richard Schoen (Stanford University, USA)
Felix Schulze (Frei Universitaet Berlin, Germany)
Michael Struwe (ETH, Switzerland)
Natasa Sesum (Rutgers University, USA)
Brian White (Stanford University, USA)
Neshan Wickramasekera (Cambridge, UK)

Arrangements

Participation
Participation is by invitation only. The workshop will begin on Monday morning 20 June and finish at lunchtime on Friday afternoon 24 June 2011.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and, if so, how to apply for one. If you do require a visa, ICMS can (on request) provide a signed invitation letter.

Venue
The workshop will be held at ICMS 15 South College Street  Edinburgh EH8 9AA. The venue is open as follows:
Monday 20 June 09.00 - 19.00
Tuesday 21 June 08.30 - 18.30
Wednesday 22 June 08.30 - 17.00
Thursday 23 June 08.30 - 18.30
Friday 24 June 08.30 - 17.00

Wireless Access
At 15 South College Street
ICMS at 15 South College Street, has wireless access throughout. On arrival at Registration you will be given instructions and a code for accessing the wireless network.  For those without laptops, there are a number of computers available for you to check your emails. Delegates are welcome to make use of the public areas at 15 South College Street throughout the week.

Travel
Information about travel to the UK and Edinburgh is available here. Lothian buses charge 1.20 GBP for a single journey and 3.00 GBP for a day ticket. Please note that the exact fare is required and no change is given.

A taxi directly from the airport will cost approximately 18.00 to 20.00 GBP to the city centre for a one-way journey.

From the airport, you can take the airport bus Airlink to the city centre. It will cost you 3.50 GBP for a single ticket, or 6.00 GBP return. More details on the airport bus Airlink is available here. The bus will take you to Waverley Bridge (next to Waverley Railway Station).

If travelling by train, please note that Edinburgh has two railway stations - Waverley Railway Station being the main station and closest to the workshop venue at ICMS and George Square. If you alight at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge map. The second railway station is called Haymarket and is at the West End of the city centre.

Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies that may occur on your trip.

Accommodation
If your invitation included the offer of accommodation, then ICMS will arrange this and details will be in your letter of invitation. ICMS has reserved rooms for all delegates in an hotel and University of Edinburgh en suite accommodation. All accommodation is no more than a twenty minute walk from ICMS. Please indicate on your booking form if you wish to take up the offer of these rooms or make your own arrangements.

Meals and Refreshments
Morning and afternoon refreshments will be provided on each day of the workshop.

A buffet lunch will be provided at ICMS in the Chapterhouse on the ground floor of 15 South College Street on Monday, Tuesday and Thursday.

The workshop dinner will take place on Thursday.

The workshop grant will cover the cost of this catering.

Registration
Registration will take place on Monday 20 June at ICMS, 15 South College Street Edinburgh EH8 9AA

Conference coordinator
Questions about the workshop should be sent to  the conference coordinator, Helene Frossling (contact details here)

Programme

Programme published 27 May 2011. 

 

Monday 20 June

10.00 - 10.55	Registration and coffee/tea
10.55 - 11.00	Welcome, Health&Safety information
11.00 - 12.00	Brian White (Stanford University) Curvature blow-up in embedded minimal surfaces
12.00 - 14.00	Lunch served in the Chapterhouse
14.00 - 15.00	Ernst Kuwert (Universität Freiburg) W^{2,2} conformal immersions
15.00 - 15.30	Coffee and tea served in the Chapterhouse
15.30 - 16.30	Mihalis Dafermos (Univeristy of Cambridge) The black hole stability problem: a mathematical perspective
16.30 - 17.30	Michael Eichmair (MIT) Isoperimetric Structure of Initial Data Sets
17.30 - 19.00	Wine reception in the Chapterhouse

Tuesday 21 June

09.30 - 10.30	Simon Brendle (Stanford University) Hypersurfaces of constant mean curvature in Schwarzschild space
10.30 - 11.00	Coffee and tea served in the Chapterhouse
11.00 - 12.00	Fernando Coda Marques (IMPA) Rigidity of min-max minimal spheres in three-manifolds
12.00 - 14.00	Lunch served in the Chapterhouse
14.00 - 15.00	Simon Donaldson (Imperial College London) Kahler-Einstein metrics and the Chow invariant
15.00 - 15.30	Coffee and tea served in the Chapterhouse
15.30 - 16.30	Richard Schoen (Stanford University) An eigenvalue problem and minimal surfaces in the ball
16.30- 17.30	Tobias Colding (MIT) Singularities and dynamics of mean curvature flow I

Wednesday 22 June

09.00 - 10.00	William Minicozzi (Johns Hopkins University) Singularities and dynamics of mean curvature flow II
10.00 - 10.30	Coffee and tea served in the Chapterhouse
10.30 - 11.30	Neshan Wickramasekera (University of Cambridge) Regularity of stable minimal hypersurfaces
11.30 - 12.30	Tom Ilmanen (ETH-Zuerich) Singularity formation and resolution for n=2 mean curvature flow

Thursday 23 June

09.30 - 10.30	Michael Struwe (ETH-Zuerich) Well-posedness for a critical nonlinear wave equation in 2 space dimensions
10.30 - 11.00	Coffee and tea served in the Chapterhouse
11.00 - 12.00	Tristan Riviere (ETH-Zuerich) Bubble-neck decomposition for conformally invariant problems
12.00 - 14.00	Lunch served in the Chapterhouse
14.00 - 15.00	Sigurd Angenent (University of Wisconsin) Uniqueness and nonuniqueness in mean curvature flow
15.00 - 15.30	Coffee and tea served in the Chapterhouse
15.30 - 16.30	Natasa Sesum (Rutgers University) Classification of Yamabe solitons
16.30 - 17.30	Richard Bamler (Princeton Universiy) Stability of hyperbolic manifolds with cusps under Ricci flow
19.00	Workshop dinner

Friday 24 June

09.00 - 10.00	Felix Schulze (Freie Universität Berlin) Expanding solitons with non-negative curvature operator coming out of cones
10.00 - 10.30	Coffee and tea served in the Chapterhouse
10.30 - 11.30	Klaus Ecker (Freie Universität Berlin) On entropy and Harnack type expressions for geometric flows
11.30 - 12.30	Camillo DeLellis (University of Zuerich) A remark on Almgren's center manifold

 

Presentations:

Presentation Details
Angenent, Sigurd
Uniqueness and nonuniqueness in mean curvature flow
View Abstract Hide Abstract
I will discuss some examples and phenomena that appear when solving MCF with smooth or singular initial data
Bamler, Richard
Stability of hyperbolic manifolds with cusps under Ricci flow
View Abstract Hide Abstract
We show that every hyperbolic manifold of finite volume and dimension greater or equal to 3 is stable under rescaled Ricci flow, i.e. that every small perturbation of the hyperbolic metric flows back to the hyperbolic metric again. Here we do not need to make any decay assumptions on this perturbation. As we will see, the main difficulty in the proof comes from a weak stability of the cusps which has to do with the existence of certain cusp deformations. We will overcome this weak stability by using a new analytical method developed by Koch and Lamm.
Brendle, Simon
Hypersurfaces of constant mean curvature in Schwarzschild space
View Abstract Hide Abstract
We describe a uniqueness theorem for hypersurfaces with constant mean curvature in the Schwarzschild manifold. This result generalizes the classical Alexandrov theorem in Euclidean space.
DeLellis, Camillo
A remark on Almgren's center manifold
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The construction of the center manifold is probably the most technical and difficult part of Almgren's celebrated work on the regularity of area-minimizing currents with codimension larger than $1$. Roughly speaking, in the neighborhood of a point of branching, the center manifold is close to the average of the sheets. As pointed out by Almgren in the introduction to his work, a remarkable corollary of his construction is a direct proof (i.e. without using the nonparametric theory of minimal surfaces) of $C^{3,alpha}$ regularity in neighborhoods of points where no branching occurs. In this talk we show how to prove this corollary in a rather straightforward way, hoping that this will be the starting step of a much cleaner proof of Almgren's full result.
Donaldson, Simon
Kahler-Einstein metrics and the Chow invariant
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The main point of the talk will be to decribe how to go from differential geometric estimates on the volume of the set of points with small "curvature radius" in a Kahler-Einstein manifold to estimates for the Chow invariant, which is related to algebraic geometry. If time allows I will also discuss variants of the situation for metrics with cone singularities along a divisor.
Ecker, Klaus
On entropy and Harnack type expressions for geometric flows
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In this talk we will outline how Hamilton's Harnack quantity for mean curvature flow and also the Li-Yau-Hamilton Harnack quantity arise in calculations along the lines of Perelman, when his entropy is augmented by a boundary term. Potential applications of the resulting formulas will be explained. The formulas raise some iteresting questions even about the linear heat equation.
Eichmair, Michael
Isoperimetric Structure of Initial Data Sets
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I will present recent joint work with Jan Metzger. A basic question in mathematical relativity is how geometric properties of an asymptotically flat manifold (or initial data set) encode information about the physical properties of the space time that it is embedded in. For example, the square root of the area of the outermost minimal surface of an initial data with non-negative scalar curvature provides a lower bound for the "mass" of its associated space time, as was conjectured by Penrose and proven by Bray and Huisken-Ilmanen. Other special surfaces that have been studied in this context include stable constant mean curvature surfaces and isoperimetric surfaces. I will explain why positive mass works to the effect that large stable constant mean curvature surfaces are always isoperimetric. This answers a question of Bray's and complements the results by Huisken-Yau and Qing-Tian on the "global uniqueness problem for stable CMC surfaces" in initial data sets with positive scalar curvature.
Kuwert, Ernst
W^{2,2} conformal immersions
View Abstract Hide Abstract
We study the weak convergence of sequences of conformally immersed, compact Riemann surfaces with fixed genus and bounded Willmore energy (joint work with Yuxiang Li)
Marques, Fernando Coda
Rigidity of min-max minimal spheres in three-manifolds
View Abstract Hide Abstract
In this talk we will consider min-max minimal surfaces in three-manifolds and describe some rigidity results. For instance, we will discuss the proof that any metric on a 3-sphere which has scalar curvature greater than or equal to 6 and is not round must have an embedded minimal sphere of area strictly smaller than $4pi$ and index at most one. We will also mention some sharp estimates for the width in the case of positive Ricci curvature. The proofs use Ricci flow. This is joint work with Andre Neves.
Minicozzi, William
Singularities and dynamics of mean curvature flow
View Abstract Hide Abstract
I will talk about joint work with Toby Colding on mean curvature flow. The main topic will be dynamical stability of singularities under the flow.
Schoen, Richard
An eigenvalue problem and minimal surfaces in the ball
View Abstract Hide Abstract
We consider the spectrum of the Dirichlet-Neumann map. This is the spectrum of the operator which sends a function on the boundary of a manifold to the normal derivative of its harmonic extension. Along with the Dirichlet and Neumann spectrum, this eigenvalue problem has been classically studied. We show how the question of finding surfaces with fixed boundary length and largest first eigenvalue is connected to the study of minimal surfaces in the ball which meet the boundary orthogonally (free boundary solutions). We describe some conjectures and results on determining optimal surfaces. This approach also leads to the solution of problems on the geometry of this class of minimal submanifolds such as optimal isoperimetric inequalities. This is joint work with Ailana Fraser.
Schulze, Felix
Expanding solitons with non-negative curvature operator coming out of cones
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We report on joint work with M. Simon, in which we show that a Ricci flow of any complete Riemannian manifold without boundary with bounded non-negative curvature operator and non-zero asymptotic volume ratio exists for all time and has constant asymptotic volume ratio. We show that there is a limit solution, obtained by scaling down this solution at a fixed point in space, which is an expanding soliton coming out of the asymptotic cone at infinity.
Sesum, Natasa
Classification of Yamabe solitons
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In this joint work with Daskalopoulos we classify locally conformally flat Yamabe solitons with positive sectional curvature. We first provide the classification of rotationally symmetric ones with no curvature assumptions and then we show that all locally conformally flat Yamabe solitons with positive sectional curvature need to be rotationally symmetric.
Struwe, Michael
Well-posedness for a critical nonlinear wave equation in 2 space dimensions
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We show that the Cauchy problem for the equation utt-Δu+ ueu2=0 on R × R2 related to the Trudinger-Moser embedding admits a global smooth solution for arbitrary smooth initial data. Previously, Ibrahim, Majdoub, and Masmoudi had obtained well-posedness of this equation for smooth Cauchy data with small energy; moreover, in 2009 I had shown global well-posedness in the case of radial symmetry. The resolution of the general case requires a new approach, which also involves a subtle improvement of the Trudinger-Moser inequality.
Wickramasekera, Neshan
Regularity of Stable Minimal Hypersurfaces
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The focus of the lecture will be on the question of regularity of minimal hypersurfaces on a smooth Riemannian manifold. In its full generality---that is, assuming only that the hypersurface is a critical point of the area functional---very little is known concerning this question. The talk will focus on recent developments in the theory for stable minimal hypersurfaces, presenting what amounts to a complete theory giving embeddedness of the hypersurface away from a small singular set under a geometric structural condition. The lectures will conclude with a discussion of the situation in the absence of this structural condition.

Participants

Name	Institution
Angenent, Sigurd	University of Wisconsin
Bamler, Richard	Princeton University
Bellettini, Costante	ETH
Blatt, Simon	ETH Zuerich
Bourni, Theodora	Freie Universitat, Berlin
Brendle, Simon	Stanford University
Buzano, Maria	University of Oxford
Cabezas-Rivas, Esther	Muenster University
Colding, Tobias	MIT
Dafermos, Mihalis	Univeristy of Cambridge
DeLellis, Camillo	Universitaet Zuerich
Donaldson, Simon	Imperial College London
Ecker, Klaus	Freie Universitaet Berlin
Eichmair, Michael	MIT
Foscolo, Lorenzo	Imperial College London
Fraser, Ailana	University of British Columbia
Giesen, Gregor	University of Warwick
Hall, Stuart	University of Manchester
Haskins, Mark	Imperial College London
Haslhofer, Robert	ETH Zurich
He, Xinyu	University of Warwick
Head, John	Albert Einstein Institute
Hein, Hans-Joachim	Imperial College London
Helmensdorfer, Sebastian	University of Warwick
Hopper, Christopher	The University of Oxford
Hornung, Peter	University of Bath
Huisken, Gerhard	Max Planck Institute for Gravitational Physics
Hwang, Seungsu	Chung-Ang University
Ilmanen, Tom	ETH-Zuerich
Klingenberg, Wilhelm	Durham University
Kovalev, Alexei	University of Cambridge
Kuwert, Ernst	Universitaet Freiburg
Lambert, Ben	Durham University
Lee, Hojoo	Seoul National University
Lotay, Jason	University College London
Mantegazza, Carlo	Scuola Normale Superiore di Pisa
Marquardt, Thomas	Max Planck Institute for Gravitational Physics
Marques, Fernando Coda	IMPA
Micallef, Mario	University of Warwick
Minicozzi, William	Johns Hopkins University
Moore, Kristen	Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
Moraru, Vlad	University of Warwick
Moser, Roger	University of Bath
Müller, Reto	Scuola Normale Superiore di Pisa
Munn, Mike	City University of New York
Neves, Andre	Imperial College London
Nguyen, Huy	University of Warwick
Nordström, Johannes	Imperial College London
Pyo, Juncheol	KIAS
Rivière, Tristan	ETH-Zurich
Rupflin, Melanie	University of Warwick
Schikorra, Armin	ETH Zuerich
Schnuerer, Oliver	University of Konstanz
Schoen, Richard	Stanford University
Schulze, Felix	Frei Universitaet Berlin
Schwetlick, Hartmut	University of Bath
Sesum, Natasa	Rutgers University
Sharp, Ben	University of Warwick
Sinestrari, Carlo	Universita' di Roma "Tor Vergata"
Singer, Michael	University of Edinburgh
Smith, Graham	CRM, Barcelona
Spotti, Cristiano	Imperial College London
Struwe, Michael	ETH-Zuerich
Sun, Song	Imperial College London
Topping, Peter	University of Warwick
Walpuski, Thomas	Imperial College London
White, Brian	Stanford University
Wickramasekera, Neshan	Univeristy of Cambridge
Yun, Gabjin	Myong Ji University
Zhu, Miaomiao	University of Warwick