In Fall 2009, we will meet on Tuesdays at 11:00am in Room 6495,
unless otherwise noted. The organizers of this seminar are Dan Lee
and Marcello Lucia.
Please email Dan at Dan.Lee2(NoSpamPlease)qc.cuny.edu to schedule a guest
The CUNY Graduate Center is located at 365 Fifth Avenue at 34th Street, diagonally across the street
from the Empire State Building, just two blocks from Penn Station (NYC).
Past Seminar Schedules and Abstracts:
Tuesday, 9/8 at 11am (8404): Kei Kondo (Tokai University)
Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below, II and III
Abstract: In this talk, I would like to talk about finite topology and properties of a complete open Riemannian manifold with
a base point whose radial curvature (at the base point) is bounded from below by that of a non-compact model surface of
revolution which admits a finite total curvature: finite topological type, exhaustion properties of Busemann functions,
which generalize results in my last year talk at the CUNY, and the diffeomorphism theorem via the model volume growth.
Notice the radial curvature may change signs ``wildly''. All results in this talk are from recent collaborative works with
Professor Minoru Tanaka.
Tuesday, 9/15 at 11am (8404): Kei Kondo (Tokai University)
Toponogov's comparison theorem for open triangles
Abstract: As you know, the Toponogov comparison theorem has produced many great classical results,
e.g., the maximal diameter theorem (Toponogov), the structure theorem with positive sectional curvature (Gromoll--Meyer),
and the soul theorem with non-negative sectional curvature (Cheeger--Gromoll). The comparison theorem enables
us to make use of some techniques originating from Euclidean geometry such as drawing a circle or a geodesic triangle,
and joining two points by a minimal geodesic segments. Such techniques are very powerful in the comparison geometry.
Very recently, I generalized the comparison theorem to a complete Riemannian manifold with smooth convex boundary
in collaborative works with Professor Tanaka. Here a geodesic triangle is replaced by an open (geodesic) triangle standing on
the boundary of the manifold, and a model surface is replaced by the universal covering surface of a cylinder of revolution with
totally geodesic boundary. In this talk, I would like to talk about our Toponogov comparison theorem and its applications.
Tuesday, 9/22 at 11am (6495): Valentino Tosatti (Columbia
Degenerations of Calabi-Yau metrics
Abstract: We are interested in the behaviour of families of Ricci-flat
metrics on a compact Calabi-Yau manifold, with Kahler classes
approaching the boundary of the Kahler cone. We will give an overview of
some examples and results concerning this quesion, and explain the
connections to the theory of degenerate complex Monge-Ampere equations and
to birational geometry.
Tuesday, 9/29: no seminar
CUNY Classes follow Monday schedule.
Tuesday, 10/6 at 11am (6495): Nan Li (Rutgers University)
Bounding Geometry of Loops in Alexandrov Spaces
Abstract: Two basic geometric invariants of a loop in an n-dimensional
Alexandrov space X with curvature \geq k is the length and the total
turning angle (which measures the closeness from being a geodesic). We
show that either the length or the total turning angle can be bounded from
below in terms of n, k, the diameter and the rough volume of X. This
generalizes and improves some well known results.
Tuesday, 10/20 at 11am (6495): Mark Stern (Duke University)
The Geometry of Stable Yang-Mills connections
Abstract: I will discuss the geometry of bundles with connections
minimizing the Yang-Mills energy. I will focus on the relation with
instantons on 4 dimensional homogeneous spaces, on Calabi-Yau 3 folds,
and on other manifolds of special holonomy.
Tuesday, 10/27 at 11am (6495): Lan-Hsuan Huang (Columbia
Rigidity results on hemispheres
Abstract: In this talk, we will discuss the curvature condition which
characterizes the hemisphere for hypersurfaces in Euclidean or hyperbolic
Tuesday, 11/3 at 11am (6495): Jeff Streets (Princeton
Existence of complete conformal metrics of negative Ricci
curvature on manifolds with boundary
Abstract: We show that on compact manifolds with boundary there exists
a solution to the sigma-k Ricci curvature Dirichlet problem. In
particular, there exist conformal metrics of negative Ricci curvature
preserving the boundary metric. Furthermore we solve the "infinite
boundary data" Dirichlet problem and show that these metrics are complete.
Finally, after finding precise asymptotics for the solutions in this case,
we define an invariant capturing the existence of Poincare-Einstein
- Tuesday, 11/10 at 11am (6495): Thomas Poole (Stony Brook
Existence of local isometric immersions: a nonlinear PDE
Abstract: An isometric immersion from a Riemannian manifold into
Euclidean space satisfies a first order nonlinear PDE. Depending on the
relative dimension between the Riemannian manifold and the Euclidean
space, this PDE can be either over determined, under determined or
determined. In this talk I will focus on the determined case and discuss
how the curvature tensor determines the type of PDE (be it elliptic,
hyperbolic etc.) I will also talk about Bryant, Griffiths and Yang's
result on the existence of local isometric immersions from 3-dim
manifolds, whose curvature tensor does not vanish at a point, into 6-dim
Euclidean space. Time permitting, I will discuss an extension of this
result to the case where the curvature tensor does vanish.
Tuesday, 11/17: no seminar
Workshop on General Relativity at Stony Brook this week
Tuesday, 11/24 at 11am (6495): Sergei Artamoshin (CUNY Graduate
Geometric interpretation of the 2-D Poisson Kernel and its
Abstract: Hermann Schwarz, while studying complex analysis,
introduced the geometric interpretation for the Poisson kernel in 1890. In
my talk we define the geometric interpretation in a multidimensional space
and use it as a tool to obtaim some results related to the Euclidean and
hyperbolic geometries. For example, we obtain an integral identity
involving the 2-dimentional Poisson kernel; obtain some non-trivial
inequalities; see that any-dimensional Poisson kernel can be obtained as
the result of this integral identity, which leads to the new approach to
the Dirichelt Poroblem in a ball or in its exterior.
As another result of this integral identity, we obtain One Radius Theorem
saying that any two radial eigenfunctions of a hyperbolic laplacian
assuming the value 1 at the origin can not assume any other common value
for all radiuses within some interval [0, p], where the length of this
interval depends only on the location of the eigenvalues on the complex
plane and does not depend on the distance between these two eigenvalues.