34,178 research outputs found
Davies type estimate and the heat kernel bound under the Ricci flow
We prove a Davies type double integral estimate for the heat kernel
under the Ricci flow. As a result, we give an affirmative answer
to a question proposed by Chow etc.. Moreover, we apply the Davies type
estimate to provide a new proof of the Gaussian upper and lower bounds of
which were first shown by Chau-Tam-Yu.Comment: conditions refined, some errors correcte
The second variation of the Ricci expander entropy
We compute the second variation of the Ricci expander entropy and briefly
discuss the linear stability of compact negative Einstein manifolds
On rigidity of gradient K\"ahler-Ricci solitons with harmonic Bochner tensor
In this paper, we prove that complete gradient steady K\"ahler-Ricci solitons
with harmonic Bochner tensor are necessarily K\"ahler-Ricci flat, i.e.,
Calabi-Yau, and that complete gradient shrinking (or expanding) K\"ahler-Ricci
solitons with harmonic Bochner tensor must be isometric to a quotient of
, where is a K\"ahler-Einstein manifold with
positive (or negative) scalar curvature.Comment: minor errors correcte
Ricci solitons on Sasakian manifolds
We show that a Sasakian metric which also satisfies the gradient Ricci
soliton equation is necessarily Einstein.Comment: 4 page
Light pseudoscalar meson and doubly charmed baryon scattering lengths with heavy diquark-antiquark symmetry
We adopt the heavy baryon chiral perturbation theory (HBChPT) to calculate
the scattering lengths of up to , where
is the pseudoscalar mesons. The recoil effect and the mass splitting
between the spin- and spin- doubly charmed baryons are
included. In order to give the numerical results, we construct the chiral
Lagrangians with heavy diquark-antiquark (HDA) symmetry in a formally covariant
approach. Then, we relate the low energy constants (LECs) of the doubly charmed
baryons to those of mesons. The LECs for the
scattering are estimated in two scenarios, fitting lattice QCD results and
using the resonance saturation model. The chiral convergence of the first
scenario is not good enough due to the the large strange quark mass and the
presence of the possible bound states, virtual states and resonance. The final
results for two scenarios are consistent with each other. The interaction for
the , ,
, ,
and channels
are attractive. The most attractive channel may
help to form the partner states of the () in the
doubly heavy sector.Comment: 12 pages, 2 figur
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
New Volume Comparison results and Applications to degeneration of Riemannian metrics
We consider a condition on the Ricci curvature involving vector fields, which
is broader than the Bakry-\'Emery Ricci condition. Under this condition volume
comparison, Laplacian comparison, isoperimetric inequality and gradient bounds
are proven on the manifold.
Specializing to the Bakry-\'Emery Ricci curvature condition, we initiate an
approach to work on the original manifold, which yields, under a weaker than
usual assumption, the results mentioned above for the {\it original manifold}.
These results are different from most well known ones in the literature where
the conclusions are made on the weighted manifold instead.
Applications on convergence and degeneration of Riemannian metrics under this
curvature condition are given. To this effect, in particular for the
Bakry-\'Emery Ricci curvature condition, the gradient of the potential function
is allowed to have singularity of order close to while the traditional
method of weighted manifolds allows bounded gradient. This approach enables us
to extend some of the results in the papers \cite{Co}, \cite{ChCo2},
\cite{zZh}, \cite{TZ} and \cite{WZ}. The condition also covers general Ricci
solitons instead of just gradient Ricci solitons.Comment: 45 page
Aronson-B\'enilan estimates for the fast diffusion equation under the Ricci flow
We study the fast diffusion equation (FDE) with a linear forcing term under
the Ricci flow on complete manifolds with bounded curvature and nonnegative
curvature operator. We prove Aronson-B\'enilan and Li-Yau-Hamilton type
differential Harnack estimates for positive solutions of the FDE. In addition,
we use similar method to prove certain Li-Yau-Hamilton estimates for the heat
equation and conjugate heat equation which extend those obtained by X. Cao and
R. Hamilton, X. Cao, and S. Kuang and Q. Zhang to noncompact setting
Aronson-B\'enilan estimates for the porous medium equation under the Ricci flow
In this paper we study the porous medium equation (PME) coupled with the
Ricci flow on complete manifolds with bounded nonnegative curvature operator.
In particular, we derive Aronson-B\'enilan and Li-Yau-Hamilton type
differential Harnack estimates for positive solutions to the PME, with a linear
forcing term, under the Ricci flow.Comment: Minor changes to the abstract and remark 1.
Bounds on harmonic radius and limits of manifolds with bounded Bakry-\'Emery Ricci curvature
Under the usual condition that the volume of a geodesic ball is close to the
Euclidean one or the injectivity radii is bounded from below, we prove a lower
bound of the harmonic radius for manifolds with bounded
Bakry-\'Emery Ricci curvature when the gradient of the potential is bounded.
Under these conditions, the regularity that can be imposed on the metrics under
harmonic coordinates is only , where and is the
dimension of the manifolds. This is almost 1 order lower than that in the
classical harmonic coordinates under bounded Ricci
curvature condition [And]. The loss of regularity induces some difference in
the method of proof, which can also be used to address the detail of
convergence in the classical case.
Based on this lower bound and the techniques in [ChNa2] and [WZ], we extend
Cheeger-Naber's Codimension 4 Theorem in [ChNa2] to the case where the
manifolds have bounded Bakry-\'Emery Ricci curvature when the gradient of the
potential is bounded. This result covers Ricci solitons when the gradient of
the potential is bounded.
During the proof, we will use a Green's function argument and adopt a linear
algebra argument in [Bam]. A new ingradient is to show that the diagonal
entries of the matrices in the Transformation Theorem are bounded away from 0.
Together these seem to simplify the proof of the Codimension 4 Theorem, even in
the case where Ricci curvature is bounded.Comment: 35pages; modified after suggestions by a number of peopl
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