3,729 research outputs found
Weinstein Conjecture and GW Invariants
In this paper, we establish a general relationship between the nonvanishing
of GW invariants with the existence of the closed orbits of a Hamiltonian
system. As an application, we completely solved the stabilized Weinstein
conjecture
Stability of pairs
This is essentially an expository note based on S. Paul's works on the
stability of pairs. Its connection to K-stability will be also discussed.Comment: 15 page
On uniform K-stability of pairs
In this paper, we discuss stable pairs, which were first studied by S. Paul,
and give a proof for a result I learned from him. As a consequence, we will
show that the K-stability implies the CM-stability.Comment: Add an appendix which contains a new proof of our main theorem by Yan
Li and Xiaohua Zhu. This proof follows the approach suggested in Section 6
and is more elementar
Geometry and nonlinear analysis
Nonlinear analysis has played a prominent role in the recent developments in
geometry and topology. The study of the Yang-Mills equation and its cousins
gave rise to the Donaldson invariants and more recently, the Seiberg-Witten
invariants. Those invariants have enabled us to prove a number of striking
results for low dimensional manifolds, particularly, 4-manifolds. The theory of
Gromov-Witten invariants was established by using solutions of the
Cauchy-Riemann equation. These solutions are often refered as
pseudo-holomorphic maps which are special minimal surfaces studied long in
geometry. It is certainly not the end of applications of nonlinear partial
differential equations to geometry. In this talk, we will discuss some recent
progress on nonlinear partial differential equations in geometry. We will be
selective, partly because of my own interest and partly because of recent
applications of nonlinear equations. There are also talks in this ICM to cover
some other topics of geometric analysis by R. Bartnik, B. Andrew, P. Li and
X.X. Chen, etc
On the structure of almost Einstein manifolds
In this paper, we study the structure of the limit space of a sequence of
almost Einstein manifolds, which are generalizations of Einstein manifolds.
Roughly speaking, such manifolds are the initial manifolds of some normalized
Ricci flows whose scalar curvatures are almost constants over space-time in the
-sense, Ricci curvatures are bounded from below at the initial time. Under
the non-collapsed condition, we show that the limit space of a sequence of
almost Einstein manifolds has most properties which is known for the limit
space of Einstein manifolds. As applications, we can apply our structure
results to study the properties of K\"ahler manifolds.Comment: 40 pages, 3 figure
The log term of Szego Kernel
In this paper, we study the relations between the log term of the Szeg\"o
kernel of the unit circle bundle of the dual line bundle of an ample line
bundle over a compact K\"ahlermanifold. We proved a local rigidity theorem. The
result is related to the classical Ramadanov Conjecture.Comment: We corrected a typo in the title in this versio
Compactness results for triholomorphic maps
We consider triholomorphic maps from an almost hyper-Hermitian manifold
into a hyperK\"ahler manifold . This means
that satisfies a quaternionic del-bar equation. We work under
the assumption that is locally strongly approximable in by smooth
maps: then such maps are almost stationary harmonic (when is
hyperK\"ahler as well, then stationary harmonic). We show that in this more
general situation the classical -regularity result still holds. We
then address compactness issues for a weakly converging sequence of strongly approximable triholomorphic maps
with uniformly bounded Dirichlet energies.
The blow up analysis leads, as in the usual stationary setting, to the
existence of a rectifiable blow-up set of codimension , away from
which the sequence converges strongly. The defect measure encodes the loss of energy in the limit;
we prove that for a.e. point on the value of is given by the
sum of energies of a (finite) number of smooth non-constant holomorphic bubbles
(here the holomorphicity is understood w.r.t. a complex structure on
that depends on the chosen point on ). In the case that
is hyperK\"ahler this result was established by C. Y. Wang (2003)
with a different proof; we rely on Lorentz space estimates. By means of a
calibration and a homological argument we further prove that for each portion
of contained in a Lipschitz graph we
find a unique alm. compl. st. on that makes the portion
pseudoholomorphic and smooth, with constant; moreover the bubbles
originating at points of such a smooth piece are holomorphic for a common
complex structure.Comment: Revised version, Thm 1.3 improved, Section 7 adde
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.
Virtual Manifolds and Localization
In this paper, we explore the virtual technique that is very useful in
studying moduli problem from differential geometric point of view. We introduce
a class of new objects "virtual manifolds/orbifolds", on which we develop the
integration theory. In particular, the virtual localization formula is
obtained.Comment: 23 page
Geometric Structures of Collapsing Riemannian Manifolds II: N*-bundles and Almost Ricci Flat Spaces
In this paper we study collapsing sequences M_{i}-> X of Riemannian manifolds
with curvature bounded or bounded away from a controlled subset. We introduce a
structure over X which in an appropriate sense is dual to the N-structure of
Cheeger, Fukaya and Gromov. As opposed to the N-structure, which live over the
M_{i} themselves, this structure lives over X and allows for a convenient
notion of global convergence as well as the appropriate background structure
for doing analysis on X. This structure is new even in the case of uniformly
bounded curvature and as an application we give a generalization of Gromov's
Almost Flat Theorem and prove new Ricci pinching theorems which extend those
known in the noncollapsed setting. There are also interesting topological
consequences to the structure.Comment: 36 page
- β¦