137 research outputs found
K\"ahler Ricci flow with vanished Futaki invariant
We study the convergence of the K\"ahler-Ricci flow on a compact K\"ahler
manifold with positive first Chern class and vanished Futaki
invariant on . As the application we establish a criterion for
the stability of the K\"ahler-Ricci flow (with perturbed complex structure)
around a K\"ahler-Einstein metric with positive scalar curvature, under certain
local stable condition on the dimension of holomorphic vector fields. In
particular this gives a stability theorem for the existence of
K\"ahler-Einstein metrics on a K\"ahler manifold with possibly nontrivial
holomorphic vector fields.Comment: 23 pages; Remark 4.1 changed due to the comments of Valentino Tosatt
Degeneration of K\"ahler-Ricci solitons
Let be a Gromov-Hausdorff limit of -dimensional closed shrinking
K\"ahler-Ricci solitons with uniformly bounded volumes and Futaki invariants.
We prove that off a closed subset of codimension at least 4, Y is a smooth
manifold satisfying a shrinking K\"ahler-Ricci soliton equation. A similar
convergence result for K\"ahler-Ricci flow of positive first Chern class is
also obtained.Comment: 24 page
Convergence of K\"ahler-Ricci flow on lower dimensional algebraic manifolds of general type
In this paper, we prove that the -norm of Ricci curvature is uniformly
bounded along a K\"ahler-Ricci flow on any minimal algebraic manifold. As an
application, we show that on any minimal algebraic manifold of general type
and with dimension , any solution of the normalized K\"ahler-Ricci flow
converges to the unique singular K\"ahler-Einstein metric on the canonical
model of in the Cheeger-Gromov topology
Relative volume comparison of Ricci Flow and its applications
In this paper, we derive a relative volume comparison estimate along Ricci
flow and apply it to studying the Gromov-Hausdorff convergence of
K\"ahler-Ricci flow on a minimal manifold. This new estimate generalizes
Perelman's no local collapsing estimate and can be regarded as an analogue of
the Bishop-Gromov volume comparison for Ricci flow.Comment: 28 pages; minor change in the proof of Lemma 3.
Regularity of K\"ahler-Ricci flow
In this short note we announce a regularity theorem for K\"ahler-Ricci flow
on a compact Fano manifold (K\"ahler manifold with positive first Chern class)
and its application to the limiting behavior of K\"ahler-Ricci flow on Fano
3-manifolds. Moreover, we also present a partial estimate to the
K\"ahler-Ricci flow under the regularity assumption, which extends previous
works on K\"ahler-Einstein metrics and shrinking K\"ahler-Ricci solitons (cf.
\cite{Ti90}, \cite{DoSu12}, \cite{Ti12}, \cite{PSS12}). The detailed proof will
appear in \cite{TiZh13}
Regularity of K\"ahler-Ricci flows on Fano manifolds
In this paper, we will establish a regularity theory for the K\"ahler-Ricci
flow on Fano -manifolds with Ricci curvature bounded in -norm for some
. Using this regularity theory, we will also solve a long-standing
conjecture for dimension 3. As an application, we give a new proof of the
Yau-Tian-Donaldson conjecture for Fano 3-manifolds. The results have been
announced in \cite{TiZh12b}.Comment: Proof of results announced in arXiv:1304.2651v
Non-singular solutions to the normalized Ricci flow equation
In this paper we study non-singular solutions of Ricci flow on a closed
manifold of dimension at least 4. Amongst others we prove that, if M is a
closed 4-manifold on which the normalized Ricci flow exists for all time t>0
with uniformly bounded sectional curvature, then the Euler characteristic . Moreover, the 4-manifold satisfies one of the following \noindent
(i) M is a shrinking Ricci solition;
\noindent (ii) M admits a positive rank F-structure;
\noindent (iii) the Hitchin-Thorpe type inequality holds 2\chi (M)\ge
3|\tau(M)| where (resp. ) is the Euler characteristic
(resp. signature) of M.Comment: 23 page
Non-singular solutions of normalized Ricci flow on noncompact manifolds of finite volume
The main result of this paper shows that, if is a complete
non-singular solution of the normalized Ricci flow on a noncompact 4-manifold
of finite volume, then the Euler characteristic number .
Moreover, , there exist a sequence times , a
double sequence of points and domains
with satisfying the followings:
[(i)] \dist_{g(t_k)}(p_{k,l_1},p_{k,l_2})\to\infty as , for any
fixed ; [(ii)] for each , converges
in the sense to a complete negative Einstein manifold
when ; [(iii)]
\Vol_{g(t_{k})}(M\backslash\bigcup_{l=1}^{N}U_{k,l})\to0 as
Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-\'{E}mery Ricci curvature
In this paper, we prove two generalized versions of the Cheeger-Gromoll
splitting theorem via the non-negativity of the Bakry-\'Emery Ricci curavture
on complete Riemannian manifolds.Comment: 14 page
Supremum of Perelman's entropy and K\"ahler-Ricci flow on a Fano manifold
In this paper, we extend the method in [TZhu5] to study the energy level
of Perelman's entropy for K\"ahler-Ricci flow on a
Fano manifold. Consequently, we first compute the supremum of
in K\"ahler class under an assumption that the modified Mabuchi's
K-energy defined in [TZhu2] is bounded from below. Secondly, we
give an alternative proof to the main theorem about the convergence of
K\"ahler-Ricci flow in [TZhu3].Comment: 28 page
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