
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES 
201314
Fields
Geometric Analysis Colloquium
at the Fields Institute
Organizing
Committee:
Spyros Alexakis (Toronto), Walter Craig (Fields &
McMaster)
Spiro Karigiannis (Waterloo), McKenzie Wang (McMaster)



UPCOMING SEMINARS 
Friday
April 4
Location:
Room 230 
2:00  3:00 p.m.
Jeff Viaclovsky (Wisconsin)
Critical metrics on connected sums of Einstein fourmanifolds (slides)
I will discuss a gluing procedure designed to obtain canonical metrics
on connected sums of Einstein fourmanifolds. The main application
is an existence result, using two wellknown Einstein manifolds as
building blocks: the FubiniStudy metric on CP^2, and the product
metric on S^2 x S^2. Using these metrics in various gluing configurations,
critical metrics are found on connected sums for a specific Riemannian
functional, which depends on the global geometry of the factors. This
is joint work with Matt Gursky.
3:30  4:30 p.m.
Jeff Streets (University of California  Irvine)
On the singularity formation in fourthorder curvature flows
(slides)
The L2 norm of the Riemannian curvature tensor is a natural energy
to associate to a Riemannian manifold, especially in dimension 4.
A natural path for understanding the structure of this functional
and its minimizers is via its gradient flow, the "L2 flow."
This is a quasilinear fourth order parabolic equation for a Riemannian
metric, which one might hope shares behavior in common with the YangMills
flow. We verify this idea by exhibiting structural results for finite
time singularities of this flow resembling results on YangMills flow.
We also exhibit a new shorttime existence statement for the flow
exhibiting a lower bound for the existence time purely in terms of
a measure of the volume growth of the initial data. As corollaries
we establish new compactness and diffeomorphism finiteness theorems
for fourmanifolds generalizing known results to ones with have effectively
minimal hypotheses/dependencies. These results all rely on a new technique
for controlling the growth of distances along a geometric flow, which
is especially wellsuited to the L2 flow.

PAST SEMINARS 
Friday
March 7
Room 230
The first lecture will start at 2:00 p.m. . and the second lecture
at 3:30 p.m.

Albert Chau (UBC)
The Kahler Ricci flow on complete Kahler manifolds with unbounded
curvature
Consider the Kahler Ricci flow on a complete noncompact Kahler manifold
(M, g). A classical result of W.X. Shi says that the flow has a short
time solution if g has bounded curvature. This talk addresses the
problem of finding a solution when g has unbounded curvature. I will
begin by discussing some a priori estimates and general existence
results for the flow. I will then discuss applications in the case
g is a U(n) invariant Kahler metric on C^n. The talk is based on joint
work with L.F. Tam and K.F Li.
Tommy Murphy (McMaster University)
Spectral geometry of toric Einstein Manifolds
The eigenvalues of the Laplacian encode fundamental geometric information
about a Riemannian metric. As an example of their importance, I will
discuss how they arose in work of Cao, Hamilton and Illmanan concerning
stability of Einstein manifolds. I will then outline joint work with
Stuart Hall which allows us to make progress on these problems for
Einstein metrics with large symmetry groups. We calculate bounds on
the first nonzero eigenvalue for certain HermitianEinstein four
manifolds. Similar ideas allow us estimate to the spectral gap (the
distance between the first and second nonzero
eigenvalues) for any toric KaehlerEinstein manifold M in terms of
the polytope associated to M.

Friday
February 14
Stewart Library

2:00 p.m.
Mihalis Dafermos (Princeton and Cambridge)
The stability of the schwarzschild solution to gravitational perturbations
3:30 p.m.
Grigorios Fournodavlos (Toronto)
Singular Ricci Solutions and their Stability under the Ricci Flow
We introduce certain spherically symmetric singular Ricci solitons
and study their stability under the Ricci flow from a dynamical PDE
point of view. The solitons in question exist for all dimensions $n$+1
≥ 3, and all have a point singularity where the curvature blows
up. Their evolution under the Ricci flow is in sharp contrast to the
evolution of their smooth counterparts. In particular, the family
of diffeomorphisms associated with the Ricci flow "pushes away"
from the singularity causing the evolving soliton to open up immediately
becoming an incomplete (but nonsingular) metric. We prove a local
wellposedness result for the Ricci flow in a natural class of initial
data close to the singular ones, which in particular implies that
the "opening up" of the singularity persists for the perturbations
as well.
** joint with Spyros Alexakis and Dezhong Chen.
Please note the first lecture will take place from 2:003:00 p.m. and
the second lecture will take place from 3:304:30 p.m.

Monday, Jan. 20
Room 230
Fields Institute

2:00 p.m.
Sergiu Klainerman, Princeton
Are Black Holes real
I will give a survey on some of the most basic issues inthe mathematical
theory of Black Holes: Rigidity, Stability and collapse.
3:30 p.m.
Reza Seyyedali, Waterloo
Extremal metrics on ruled manifolds
Consider a compact Kahler manifold with extremal Kahler metric and
a Mumford stable holomorphic bundle over it. We show that, if the
holomorphic vector field defining the extremal Kahler metric is liftable
to the bundle and if the bundle is relatively stable with respect
to the action of automorphisms of the manifold, then there exist extremal
Kahler metrics on the projectivization of the dual vector bundle.
This is joint work with Zhiqin Lu.

Dec. 13
Stewart Library 
Richard Bamler, Stanford
There are finitely many surgeries in Perelman's Ricci flow
Although the Ricci flow with surgery has been used by Perelman to
solve the Poincaré and Geometrization Conjectures, some of
its basic properties are still unknown. For example it has been an
open question whether the surgeries eventually stop to occur (i.e.
whether there are finitely many surgeries) and whether the full geometric
decomposition of the underlying manifold is exhibited by the flow
as $t \to \infty$.
In this talk I will show that the number of surgeries is indeed finite
and that the curvature is globally bounded by $C t^{1}$ for large
$t$. Using this curvature bound it is possible to give a more precise
picture of the longtime behavior of the flow.
Arick Shao, Toronto
Unique continuation from infinity for linear waves
We prove various unique continuation results from infinity for linear
waves on asymptotically flat spacetimes. Assuming vanishing of the
solution to infinite order on suitable parts of future and past null
infinities, we derive that the solution must vanish in an open set
in the interior. The parts of infinity where we must impose a vanishing
condition depend strongly on the background geometry; in particular,
for backgrounds with positive mass (such as Schwarzschild or Kerr),
the required assumptions are much weaker than in Minkowski spacetime.
The results are nearly optimal in many respects, and they can be considered
as analogues of uniqueness from infinity results for secondorder
elliptic operators. This work is partly motivated by questions in
general relativity.
This is joint work with Spyros Alexakis and Volker Schlue.
Please note the first talk is schedule to start at 2:00 p.m. and the
second talk will start at 3:30 p.m.

Nov. 15
*2:00 p.m
Room 230
* note revised time

Speakers:
2:00 p.m Pengfei Guan, McGill
3:30 p.m. Ben Weinkove, Northwestern University
The MongeAmpere equation for (n1)plurisubharmonic functions
A C^2 function on C^n is called (n1)plurisubharmonic in the sense
of HarveyLawson if the sum of every n1 eigenvalues of its complex
Hessian is nonnegative. We show that the associated MongeAmpere equation
can be solved on any compact Kahler manifold. As a consequence we
prove the existence of solutions to an equation of FuWangWu, giving
CalabiYau theorems for balanced, Gauduchon and strongly Gauduchon
metrics on compact Kahler manifolds.

Oct. 4
1:00 p.m 
Speaker:
Niky Kamran, McGill 

