25,063 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
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
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
On dimension reduction in the K\"ahler-Ricci flow
We consider dimension reduction for solutions of the K\"ahler-Ricci flow with
nonegative bisectional curvature. When the complex dimension , we prove an
optimal dimension reduction theorem for complete translating K\"ahler-Ricci
solitons with nonnegative bisectional curvature. We also prove a general
dimension reduction theorem for complete ancient solutions of the
K\"ahler-Ricci flow with nonnegative bisectional curvature on noncompact
complex manifolds under a finiteness assumption on the Chern number .Comment: 15 pages, Late
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
Jet production in black-hole X-ray binaries and active galactic nuclei: mass feeding and advection of magnetic fields
Relativistic jets are observed only in the low/hard and intermediate states
of X-ray binaries (XRBs), and are switched off in the thermal state, but they
appear to be present in both low-luminosity and luminous active galactic nuclei
(AGNs). It is widely believed that strong large-scale magnetic fields is a
crucial ingredient in jet production; such fields can be attained only through
efficient advection from the outer disc. We suggest that geometrically thin
accretion discs with magnetic outflows are present in luminous radio-loud AGNs;
this is likely because the interstellar medium provides both mass and
sufficient magnetic flux to the outer disc. Most angular momentum of such disc
is removed by the outflows, and the radial velocity of the disc is
significantly increased compared to viscous drift velocity. This facilitates
efficient magnetic field advection through the disc to produce a strong field
near the black hole in luminous AGNs, which helps launch relativistic jets. In
XRBs, the magnetic fields of the gas from companion stars are too weak to drive
outflows from outer discs. Jets are therefore switched off in the thermal state
due to inefficient magnetic field advection in the disc.Comment: 8 pages, accepted by MNRA
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
Identification of Two Frobenius Manifolds In Mirror Symmetry
We identify two Frobenius manifolds obtained from two different differential
Gerstenhaber-Batalin-Vilkovisky algebras on a compact Kaehler manifold. One is
constructed on the Dolbeault cohomology, and the other on the de Rham
cohomology. Our result can be considered as a generalization of the
identification of the Dolbeault cohomology ring with the complexified de Rham
cohomology ring on a Kaehler manifold.Comment: 12 pages, AMS LaTe
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