CUNY Metric Geometry 2019

January 14-15, 2019 at the CUNY Graduate Center

Department of Mathematics, 4214:03

organized by Christina Sormani (Lehman and CUNYGC) and Sajjad Lakzian (Fordham)

Schedule: Jan 14 in room 4214:03

(updated when Prof Strichartz cancelled)

10:00 -11:00am Conrad Plaut ``Length Spectra, with and without Length''
11:00-12:00 Discussions
12:00-1:00 Lunch
1:00-1:30 Changliang Wang ``Perelman's functionals on compact manifolds with isolated conical singularities''
2:00-2:30 Anusha Krishnan ``Ricci flow on cohomogeneity one manifolds''
3:00-3:30 Wenchuan Tian ``A Compactness Theorem for Rotationally Symmetric Riemannian Manifolds with Positive Scalar Curvature''
4:00-7:00 Discussions

Abstracts:

Conrad Plaut ``Length Spectra, with and without Length''

Abstract: I’ll begin with a quick overview of the work of Gurvich-Gvishiani from 1987, which is somewhat overlooked and was the first paper in which resistance metrics on finite graphs were actually shown to be metrics. The paper shows a fascinating connection between resistance metrics and geodesic (shortest length) metrics and a host of other interesting “transportation” metrics on finite graphs. In my view this is motivation for metric geometers to be interested in resistance metrics, which since the early 1990’s have been studied on self-similar fractals considered as limits of finite graphs, lead by Kigami, Strichartz and others. However, these metrics on fractals seem to be polar opposites of geodesic metrics: they have no (non-constant) rectifiable curves! The rest of the talk will be devoted to development of new spectra related to the length spectrum of Riemannian manifolds (lengths of closed geodesics) and showing how these spectra along with other classics such as the Covering Spectrum can be equivalently defined without the notion of lengths of curves. This opens questions concerning the relationship between these spectra and the spectrum of the Laplacian, which is defined on fractals with resistance metrics.

Changliang Wang ``Perelman's functionals on compact manifolds with isolated conical singularities'

Abstract: We extend the theory of the Perelman's functionals on compact smooth manifolds to compact manifolds with isolated conical singularities. For the lambda-functional, this is essentially an eigenvalue problem for a Schrodinger operator with singular potential. We obtain a certain asymptotic behavior of eigenfunctions near the singularities. This asymptotic behavior plays an important role for deriving the variation formulas of the lambda-functional and other applications. Moreover, we show that the infimum of the W-functional on compact manifolds with isolated conical singularities is finite, and the minimizing function exists. We also obtain a certain asymptotic behavior for the minimizing function near the singularities. This is a joint work with Professor Xianzhe Dai.

Anusha Krishnan ``Ricci flow on cohomogeneity one manifolds''

Abstract: Abstract: The Ricci flow is the geometric PDE \frac{\partial g}{\partial t} = -2Ric_g for evolving a Riemannian metric with time. We study the Ricci flow in the context of cohomogeneity one manifolds, specifically, a manifold (M,g) acted on isometrically by a Lie group G such that the orbit space M/G is a closed interval [0,L]. We prove that under some hypotheses on the isotropy groups of the G-action, a certain canonical form for the metric is preserved under the flow. A geometric consequence is that a geodesic orthogonal to all the G-orbits in the initial metric g(0), remains a geodesic (up to reparametrization) in the evolving metric g(t).

Wenchuan Tian ``A Compactness Theorem for Rotationally Symmetric Riemannian Manifolds with Positive Scalar Curvature''

Abstract: Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with
nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have nonnegative generalized scalar curvature and Euclidean tangent cones almost everywhere. In this paper we prove this conjecture for sequences of rotationally symmetric warped product manifolds. We show that the limit spaces have $H^1_{loc}$ warping function that has nonnegative scalar curvature in a weak sense, and have Euclidean tangent cones almost everywhere.

Senior Speaker: (60 min talk)

Conrad Plaut (University of Tennessee)

Junior Speakers: (30 min talks) (staying for the week to collaborate)

Anusha Krishnan (University of Pennsylvania)

Wenchuan Tian (Michigan State University)

Changliang Wang (Max Plank Institute for Mathematics in Bonn)

Additional Participants:

Demetre Kazaras (SCGP)

funded in part by Prof Sormani’s NSF grant DMS 1612049