EDUCATION:
1991-1997.
Courant Institute of Mathematical
Sciences
1987 - 1991.
College of Arts and Science, NYU
EXPERIENCE:
1999-2001.
Lehman
College, C.U.N.Y., Assistant Professor (tenure track)
1996-1997.
Harvard University,
Lecturer
1993-1996:
Courant Institute of Mathematical
Sciences, Graduate Fellow
1997 - 1998.
ICM 1998 Travel Grant
1991 - 1996.
Courant Institute of Mathematical Sciences
1987-1991.
College of Arts and Science, NYU
2000-2001. Dartmouth College, Colloquium
C.U.N.Y. Graduate Center, Differential Geometry Seminar,
Bronx Community College, C.U.N.Y., Differential Geometry Seminar
A.M.S. Spring 2001 Eastern Sectional Meeting
A.M.S. Fall 2000 Eastern Sectional Meeting
1999-2000. AMS/SCAND 2000, Denmark, Special Session on Differential Geometry
C.U.N.Y. Graduate Center Topology Seminar
AMS/MAA Joint Meeting, Special Session on Geometric Analysis
Association for Women in Mathematics Workshop
Indiana University Purdue University at Indianapolis, Colloquium
1998-1999. U. Maryland College Park, and Harvard, Differential Geometry Seminars
International Congress of Mathematicians, Berlin
Aarhus Geometry/Topology Conference, Denmark
1997-1998. AMS/MAA Joint Meeting, Special Session of Laplace Operator
1996-1997. Yale University Topology Seminar
Dartmouth College Differential Geometry Seminar
Rutgers University Differential Geometry Seminar
Johns Hopkins University Analysis and Geometry Seminar
Lehigh University Conference on Geometry and Topology
1995-1996 Massachusetts Institute of Technology and City University of New York
Lehigh University Geometry/Topology Conference.
1992-1995. Introductory Lecture Series, NYU
PAPERS AND PREPRINTS:
In this paper, we consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and, in particular, manifolds with nonnegative Ricci curvature. Busemann functions, which are the limits of distance functions about points approaching infinity, have been used in the past by S.T.Yau to prove that noncompact manifolds with nonnegative Ricci curvature have at least linear volume growth. We first prove a volume comparison theorem for sets covered by the flows of the gradient of a Busemann function. We then apply this theorem to study manifolds with the given lower Ricci curvature bounds and minimal volume growth, in particular, manifolds with nonnegative Ricci curvature and linear volume growth. We prove that the level sets of Busemann functions in such manifolds are compact and the diameters of the levels grow at most linearly. Thus a Busemann function is an exhaustion. We close with examples of manifolds in which the diameter of a Busemann function grows logarithmically. These constructions have metrics of doubly-warped product type for which the warping functions are solutions to an integrable ordinary differential equation.
The Rigidity and Almost Rigidity of Manifolds with Minimal Volume Growth, Communications in Analysis and Geometry, Vol 8, No. 1, 159-212, January 2000.
We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manifolds with nonnegative and quadratically decaying lower Ricci curvature bounds have minimal volume growth. Compact regions in such manifolds are shown to be asymptotically close to warped products in the Gromov-Hausdorff topology.
Manifolds with nonnegative Ricci curvature and linear volume growth are shown to have regions which are asymptotically close to being isometric products. The proofs involve a careful analysis of the Busemann functions on these manifolds using the recently developed Cheeger-Colding Almost Rigidity Theory. In addition, we show that the diameters of the level sets of Busemann functions in such manifolds grow sublinearly.
Harmonic Functions on Manifolds with Nonnegative Ricci Curvature and Linear Volume Growth, Pacific Journal of Mathematics, Vol 192, No 1, 183-189, January 2000.
Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this result to prove that a complete noncompact manifold with nonnegative Ricci curvature has at least linear volume growth. In this paper, we prove the following theorem concerning harmonic functions on these manifolds.
Theorem: Let M be a complete noncompact manifold with nonnegative Ricci curvature and at most linear volume growth. If there exists a nonconstant harmonic function, f, of polynomial growth of any given degree q, then manifold splits isometrically, M= N x R.
Harmonic functions of polynomial growth have been an object of study for some time. Until recently it was not known whether the space of harmonic functions of polynomial growth of a given degree on a manifold with nonnegative Ricci curvature was finite dimensional. Atsushi Kasue proved this result with various additional assumptions. Tobias Colding and Bill Minicozzi have recently proven that this space is indeed finite dimensional with no additional assumptions. With our stronger condition of linear volume growth, we are able to prove that this space is only one dimensional directly, using a gradient estimate of Yau and previous results by the author.
Nonnegative Ricci Curvature, Small Linear Diameter Growth and Finite Generation of Fundamental Groups, to appear in the Journal of Differential Geometry
In 1968, Milnor conjectured that any complete noncompact manifold, M, with nonnegative Ricci curvature has a finitely generated fundamnetal group. In this paper, we prove there exists a universal contant, S, depending only on the dimension, such that the following theorems hold.
Theorem I: If M has small linear diameter growth, diam(r)<4Sr, then the manifold has a finitely generated fundamental group.
Theorem II: If M has an infinitely generated fundamental group, then it has a tangent cone at infinity, Y, which does not have a pole at its base point. In fact if Z is a length space such that the Gromov-Hausdorff distance from Z to Y is less than S/4, then the base point of Z is not a pole.
Corollary: Let M be a complete noncompact manifold with nonnegative Ricci curvature. If M has linear volume growth then it has a finitely generated fundamental group.
These theorems are proven using two main lemmas. The first creates a sequence cut points in M by choosing a special sequence of generators of the fundamental group. The second uniformly estimates how cut these points are using Abresch and Gromoll's Excess Theorem.
On Loops Representing Elements of the Fundamental Group of a Complete Manifold with Nonnegative Ricci Curvature, preprint March 1999.
This paper concerns complete noncompact manifolds with nonnegative Ricci curvature. Roughly, we say that M has the loops to infinity property if given any noncontractible closed curve, C, and given any compact set, K, there exists a closed curve contained in M\K which is homotopic to C.
The main theorems in this paper are the following.
Theorem I: If M has positive Ricci curvature then it has the loops to infinity property.
Theorem II: If M has nonnegative Ricci curvature then it either has the loops to infinity property or it is isometric to a flat normal bundle over a compact totally geodesic submanifold and its double cover is split.
Theorem III: Let M be a complete riemannian manifold with the loops to infinity property along some ray starting at a point, p. Let D containing p be a precompact region with smooth boundary and S be any connected component of the boundary containing a point, q, on the ray. Then the map from the fundamental group of S based at q to the fundamental group of Cl(D) based at p induced by the inclusion map is onto.
The Codimension One Homology of a Complete Manifold with Nonnegative Ricci Curvature, joint with Zhongmin Shen, preprint September 1999, to appear in the American Journal of Mathematics.
In this paper we prove that a complete noncompact manifold with nonnegative Ricci curvature has a trivial codimension one homology unless it is a split or flat normal bundle over a compact totally geodesic submanifold. In particular, we prove the conjecture that a complete noncompact manifold with positive Ricci curvature has a trivial codimension one integer homology. We also have a corollary stating when the codimension two integer homology of such a manifold is torsion free.
Hausdorff Convergence and Universal Covers , joint with Guofang Wei, to appear in the Transactions of the American Mathematical Society.
We prove that if Y is the Gromov-Hausdorff limit of a sequence of compact manifolds M_i of dimension n with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then Y has a universal cover. We then show that, for i sufficiently large, the fundamental group of M_i has a surjective homeomorphism onto the group of deck transforms of Y . Finally, in the non-collapsed case where the M_i have an additional uniform lower bound on volume, we prove that the kernels of these surjective maps are finite with a uniform bound on their cardinality.
A number of theorems are also proven concerning the limits of covering spaces and their deck transforms when the M_i are only assumed to be compact length spaces with a uniform upper bound on diameter.