375 research outputs found
Recent Progress on Ricci Solitons
Ricci solitons are natural generalizations of Einstein metrics. They are also
special solutions to Hamilton's Ricci flow and play important roles in the
singularity study of the Ricci flow. In this paper, we survey some of the
recent progress on Ricci solitons.Comment: 32 pages; to appear in Proceedings of International Conference on
Geometric Analysis (Taipei, July 2007
Geometry of Complete Gradient Shrinking Ricci Solitons
We survey some of the recent progress on complete gradient shrinking Ricci
solitons, including the classifications in dimension three and asymptotic
behavior of potential functions as well as volume growths of geodesic balls in
higher dimensions. This article is written for the conference proceedings
dedicated to Yau's 60th birthday.Comment: 16 pages; updated versio
The K\"ahler-Ricci flow on Fano manifolds
In this lecture notes, we aim at giving an introduction to the K\"ahler-Ricci
flow (KRF) on Fano manifolds. It covers some of the developments of the KRF in
its first twenty years (1984-2003), especially an essentially self-contained
exposition of Perelman's uniform estimates on the scalar curvature, the
diameter, and the Ricci potential function for the normalized K\"ahler-Ricci
flow (NKRF), including the monotonicity of Perelman's \mu-entropy and
\kappa-noncollapsing theorems for the Ricci flow on compact manifolds.
The Notes is based on a mini-course on KRF delivered at University of
Toulouse III in February 2010, a talk on Perelman's uniform estimates for NKRF
at Columbia University's Geometry and Analysis Seminar in Fall 2005, and
several conference talks, including "Einstein Manifolds and Beyond" at CIRM
(Marseille - Luminy, fall 2007), "Program on Extremal K\"ahler Metrics and
K\"ahler-Ricci Flow" at the De Giorgi Center (Pisa, spring 2008), and "Analytic
Aspects of Algebraic and Complex Geometry" at CIRM (Marseille - Luminy, spring
2011).Comment: v.2: corrected a number of typos and added the proof of Theorem 2.3
on preserving positive orthogonal bisectional curvature. To appear as a book
chapter in An Introduction to the K\"ahler-Ricci Flow, Lecture Notes in
Mathematics, vol. 2086, Springer, 201
Existence of Gradient Kahler-Ricci Solitons
This is the original paper appeared in the book "Elliptic and Parabolic
Methods in Geometry (Minneapolis, MN,1994), A K Peters, Wellesley, MA, (1996)"
(p.1-16), except with a few minor modifications as described at the end of the
paper (on p.14). Due to frequent requests for the article, we decided to post
it on the arXiv
On Quantum de Rham Cohomology Theory
We define quantum exterior product wedge_h and quantum exterior differential
d_h on Poisson manifolds (of which symplectic manifolds are an important class
of examples). Quantum de Rham cohomology, which is a deformation quantization
of de Rham cohomology, is defined as the cohomology of d_h. We also define
quantum Dolbeault cohomology. A version of quantum integral on symplectic
manifolds is considered and the correspoding quantum Stokes theorem is proved.
We also derive quantum hard Lefschetz theorem. By replacing d by d_h and wedge
by wedge_h in the usual definitions, we define many quantum analogues of
important objects in differential geometry, e.g. quantum curvature. The quantum
characteristic classes are then studied along the lines of classical Chern-Weil
theory. Quantum equivariant de Rham cohomology is defined in the similar
fashion.Comment: 8 pages, AMSLaTe
Matrix Li-Yau-Hamilton estimates for the heat equation on Kaehler manifolds
We proved a matrix Li-Yau-Hamilton type gradient estimates for the positive
solutin of the heat equation on complete Kaehler manifolds with nonnegative
bisectional curvature. As a consequence we obtain a comparison theorem for the
distance function under this curvature assumption
On second variation of Perelman's Ricci shrinker entropy
In this paper we provide a detailed proof of the second variation formula,
essentially due to Richard Hamilton, Tom Ilmanen and the first author, for
Perelman's -entropy. In particular, we correct an error in the stability
operator stated in Theorem 6.3 of [2]. Moreover, we obtain a necessary
condition for linearly stable shrinkers in terms of the least eigenvalue and
its multiplicity of certain Lichnerowicz type operator associated to the second
variation.Comment: 13 pages; final version; to appear in Math. An
Degenerate Chern-Weil Theory and Equivariant Cohomology
We develop a Chern-Weil theory for compact Lie group action whose generic
stabilizers are finite in the framework of equivariant cohomology. This
provides a method of changing an equivariant closed form within its
cohomological class to a form more suitable to yield localization results. This
work is motivated by our work on reproving wall crossing formulas in
Seiberg-Witten theory, where the Lie group is the circle. As applications, we
derive two localization formulas of Kalkman type for G = SU(2) or SO(3)-actions
on compact manifolds with boundary. One of the formulas is then used to yield a
very simple proof of a localization formula due to Jeffrey-Kirwan in the case
of G = SU(2) or SO(3).Comment: 23 pages, AMSLaTe
Frobenius Manifold Structure on Dolbeault Cohomology and Mirror Symmetry
We construct a differential Gerstenhaber-Batalin-Vilkovisky algebra from
Dolbeault complex of any close Kaehler manifold, and a Frobenius manifold
structure on Dolbeault cohomology.Comment: 10 pages, AMS LaTe
DGBV Algebras and Mirror Symmetry
We describe some recent development on the theory of formal Frobenius
manifolds via a construction from differential Gerstenhaber-Batalin-Vilkovisk
(DGBV) algebras and formulate a version of mirror symmetry conjecture: the
extended deformation problems of the complex structure and the Poisson
structure are described by two DGBV algebras; mirror symmetry is interpreted in
term of the invariance of the formal Frobenius manifold structures under
quasi-isomorphism.Comment: 11 pages, to appear in Proceedings of ICCM9
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