94 research outputs found
Space of K\"ahler metrics (IV)--On the lower bound of the K-energy
We partially confirm an old conjecture of Donaldson that if there exists a
cscK metrics in a given K\"ahler class, then there is no degenerated geodesic
ray which is tamed by a bounded ambient geometry unless it parallels to a
holomorphic line consists of cscK metrics only. We also prove that for simple
test configuration where the central fibre has a cscK metric, the K energy
functionals in the nearby fibre must also have a uniform lower bound in its
underlying K\"ahler class
Calabi flow in Riemann surfaces revisited: A new point of view
In this paper, we observe a set of functionals of metrics which are all
decrease under the Calabi flow and have uniform lower bound along the flow,
which give rise to a set of integral estimates on the curvature flow. Using
these estimates, together with weak compactness we obtained in previous papers
[8] and [10], we prove the long term existence and convergence of the Calabi
flow. Thus give a new proof to Chruscial's theorem. The set of simple ideas of
global integral estimates and concentration compactness should have further
implications in other heat flow problems.Comment: 22 page
On the lower bound of energy functional E_1 (I)-- a stability theorem on the Kaehler Ricci flow
In the present paper, we prove a stability theorem for the Kaehler Ricci flow
near the infimum of the functional E_1 under the assumption that the initial
metric has Ricci > -1 and |Riem| bounded. At present stage, our main theorem
still need a topological assumption (1.2) which we hope to be removed in
subsequent papers. The underlying moral is: if a Kaehler metric is sufficiently
closed to a Kaehler Einstein metric, then the Kaehler Ricci flow converges to
it. The present work should be viewed as a first step in a more ambitious
program of deriving the existence of Kaehler Einstein metrics with an arbitrary
energy level, provided that this energy functional has a uniform lower bound in
this Kaehler class.Comment: 23 page
Weak Limits of Riemannian Metrics in Surfaces with integral Curvature Bound
In this paper, we study the weak compactness of the set of conformal metrics
in any Riemann surface without boundary whose
Calabi energy and area are uniformly bounded. We prove that for any sequence
of such metrics, there alwasy exists a subsequence which converges in
H\sp{2,2}_\sb{loc} everywhere except a finite number of bubble points. Blowup
analysis near bubble shows that the bubble on bubble phenomenon occurs. The
limit metric gives rise to a tree structure decomposition, where each node in
the tree represents a limit metric of a subsequence at that stage while the
edge of the tree structure represents the neck on the process of blowing up. We
also show that the number of the nodes which have more than three edges
attached is finite.Comment: 40 pages, 5 figure
Recent progress in K\"ahler geometry
In recent years, there are many progress made in K\"ahler geometry. In
particular, the topics related to the problems of the existence and uniqueness
of extremal K\"ahler metrics, as well as obstructions to the existence of such
metrics in general K\"ahler manifold. In this talk, we will report some recent
developments in this direction. In particular, we will discuss the progress
recently obtained in understanding the metric structure of the infinite
dimensional space of Kaehler potentials, and their applications to the problems
mentioned above. We also will discuss some recent on Kaehler Ricci flow
Remarks on Kahler Ricci Flow
We study some estimates along the Kahler Ricci flow on Fano manifolds. Using
these estimates, we show the convergence of Kahler Ricci flow directly if the
-invariant of the canonical class is greater than .
Applying these convergence theorems, we can give a flow proof of Calabi
conjecture on such Fano manifolds. In particular, the existence of Kahler
Einstein metrics on a lot of Fano surfaces can be proved by flow method. Note
that this geometric conclusion (based on the same assumption) was established
earlier via elliptic method by G. Tian. However, a new proof based on Kahler
Ricci flow should be still interesting in its own right.Comment: We note an overlap with the paper of Rubinstein [Ru1]. We add more
referenc
Gravitational instantons with faster than quadratic curvature decay (I)
In this paper, we study gravitational instantons (i.e., complete hyperk\"aler
4-manifolds with faster than quadratic curvature decay).
We prove three main theorems:
1.Any gravitational instanton must have known end----ALE, ALF, ALG or ALH.
2.In ALG and ALH-non-splitting cases, it must be biholomorphic to a compact
complex elliptic surface minus a divisor. Thus, we confirm a long-standing
question of Yau in ALG and ALH cases.
3.In ALF-D_k case, it must have an O(4)-multiplet
Gravitational instantons with faster than quadratic curvature decay (II)
This is our second paper in a series to study gravitational instantons, i.e.
complete hyperk\"aler 4-manifolds with faster than quadratic curvature decay.
We prove two main theorems:
1.The asymptotic rate of gravitational instantons to the standard models can
be improved automatically.
2.Any ALF-D_k gravitational instanton must be the
Cherkis-Hitchin-Ivanov-Kapustin-Lindstr\"om-Ro\v{c}ek metric.Comment: We add a corollary and the applications, correct the asymptotic rate
of the multi-Taub-NUT metri
Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K\"ahler metrics
We prove that constant scalar curvature K\"ahler metric "adjacent" to a fixed
K\"ahler class is unique up to isomorphism. This extends the uniqueness theorem
of Donaldson and Chen-Tian, and formally fits into the infinite dimensional
G.I.T picture described by Donaldson. We prove that the Calabi flow near a cscK
metric exists globally and converges uniformly to a cscK metric in a polynomial
rate. Viewed in a K\"ahler class, the Calabi flow is also shown to be
asymptotic to a smooth geodesic ray at infinity. This latter fact is also
interesting in the finite dimensional analogue, where we show that the downward
gradient flow of the Kempf-Ness function in a semi-stable orbit is asymptotic
to the direction of optimal degeneration.Comment: 2 Figures. Theorem 5.9 added, providing an analytic proof of a result
due to X. Chen and G. Szekelyhidi on the lower bound of K energy on a
deformation of cscK manifold; Proof of Theorem 4.4 corrected; More details
supplied in the end of Section 6; References update
Space of K\"ahler metrics (V)-- K\"ahler quantization
We prove the convergence of geodesic distance during the quantization of the
space of K\"ahler potentials. As applications, this provides alternative proofs
of certain inequalities about the K-energy functional in the projective case.Comment: 25 pages. Corrected typos. Added section 5
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