1.
Preface
Peter Li has been the founding editor of our journal Communications in Analysis and Geometry.
2.
Dehn-Thurston Coordinates for Curves on Surfaces - Luo
, Feng; Strong
, Richard
We prove that the geometric intersection numbers between two
proper 1-dimensional submanifolds satisfy a Cauchy type inequality
expressed in terms of the Dehn-Thurston coordinate. As an
application, we reestablish the fundamental results in the theory
of measured laminations on surfaces.
4.
On the Asymptotic Scalar Curvature Ratio of
Complete Type I-like Ancient Solutions to the Ricci
Flow on Noncompact 3-manifolds - Chow
, Bennett; Lu
, Peng
Complete noncompact Riemannian manifolds with nonnegative sectional
curvature arise naturally in the Ricci flow when one takes the limits of
dilations about a singularity of a solution of the Ricci flow on a compact
3-manifold [H-95a]. To analyze the singularities in the Ricci flow one needs
to understand these manifolds in depth. There are three invariants, asymptotic
scalar curvature ratio, asymptotic volume ratio and aperture, that have
been used to study the geometry of these manifolds at infinity.
5.
Maps With Prescribed Tension Fields - Chen
, Wenyi; Jost
, Jörgen
We consider maps into a Riemannian manifold of nonpositive sectional
curvature with prescribed tension field. We derive a priori
estimates and solve a Dirichlet problem.
6.
Kähler-Ricci Flow and the Poincaré-Lelong
Equation - Ni
, Lei; Tam
, Luen-Fai
In [M-S-Y], Mok-Siu-Yau studied complete Kähler manifolds with nonnegative
holomorphic bisectional curvature by solving the Poincaré-Lelong
equation ¶
? - 1? \overscore ?u = Ric (0.1) ¶
where Ric is the Ricci form of the manifold. In [M-S-Y], the authors solved
(0.1) under the assumptions that the manifold is of maximal volume growth
and the scalar curvature decays quadratically. On the other hand, in a series
of papers of W.-X. Shi [Sh2-4], Kähler-Ricci flow ¶
{? \over
?t}
??\overscore ? = - R?\overscore ? (0.2) ¶
has been studied extensively and important applications were given. In
[N1] and [N-S-T], the Poincaré-Lelong
equation has been solved under more
general conditions than in [M-S-Y]. The conditions...
7.
Geometric Estimates for the Logarithmic
Fast Diffusion Equation - Daskalopoulos
, P.; Hamilton
, R.
We consider solutions u of the logarithmic fast diffusion equation
{?u \over
?t }
= ?logu (1.1)
on the plane R2, with initial data f ? 0 of finite mass. ? denotes the
Euclidean Laplace operator
¶
? = ?2
?x2 + ?2
?y2 ¶
with respect to the standard metric ds2 = dx2+dy2.
8.
Brownian Motion on a Submanifold - Stroock
, Daniel W.
Given a submanifold M of a Riemannian manifold N, we give two
different constructions of Brownian motion on M: one by "projection"
onto M of the Brownian motion on Nand the other by a
more intrinsic approach. The two procedures lead to very different
ways in which vectors are transported along Brownian paths.
9.
Global Existence of the m-equivariant Yang-Mills
Flow in Four Dimensional Spaces - Hong
, Min-Chun; Tian
, Gang
The use of non-linear parabolic equations (the heat flow method) to find
solutions of corresponding elliptic equations goes back to Eells-Sampson in
1964. In their seminal paper [ES], Eells and Sampson introduced the heat
flow for harmonic maps to establish the existence of smooth harmonic maps
from a compact Riemmanian manifold into a Riemmanian manifold having
non-positive section curvature. In general, the heat flow for harmonic maps
even on two dimensional manifolds may develop singularity at finite time (cf.
[CDY]). Struwe [St1] established the existence of the unique global weak
solution, which is smooth with exception of at most finitely many points, to
the heat flow for harmonic maps...
10.
Convex Hypersurfaces of Prescribed Weingarten
Curvatures - Sheng
, Weimin; Trudinger
, Neil; Wang
, Xu-Jia
In this paper we study the existence of closed convex hypersurfaces
in the Euclidean space ?n+1 with a Weingarten curvature
prescribed as a function of their unit normal.
11.
On A-twisted Moduli Stack for Curves from
Witten's Gauged Linear Sigma Models - Liu
, Chien-Hao; Liu
, Kefeng; Yau
, Shing-Tung
Witten's gauged linear sigma model [Wi1] is one of the universal
frameworks or structures that lie behind stringy dualities. Its Atwisted
moduli space at genus 0 case has been used in the Mirror
Principle [L-L-Y] that relates Gromov-Witten invariants and mirror
symmetry computations. In this paper the A-twisted moduli
stack for higher genus curves is defined and systematically studied.
It is proved that such a moduli stack is an Artin stack. For genus
0, it has the A-twisted moduli space of [M-P] as the coarse moduli
space. The detailed proof of the regularity of the collapsing morphism
by Jun Li in [L-L-Y : I and II] can be viewed...
12.
Potential Functions and Actions of Tori on Kähler Manifolds - Burns
, D.; Guillemin
, V.
Let M be a Kähler manifold equipped with a free Hamiltonian action of
the standard n-torus, T with moment map, ? : M ? ?n. For ? ? ?n the
symplectic quotient
M? = ? -1(?)/T
inherits from M a Kähler structure, and in the first part of this paper
we will describe what the Kähler form and Ricci form look like locally on
coordinate patches in M?. Then in the second part of this paper we will
discuss some global implications of these results. This will include ¶
1. A Kählerian proof of the Duistermaat-Heckman theorem. ¶
2. A formula, due to Biquard and Gauduchon, for the Kähler...
13.
On Dimension Reduction in the Kähler-Ricci Flow - Cao
, H. D.
We extend the method of dimension reduction of Hamilton for the
Ricci flow to the Kähler-Ricci flow. In the case of complex dimension
n = 2, we prove a dimension reduction theorem for complete
translating Kähler-Ricci solitons with nonnegative bisectional curvature.
For n > 2, we also prove a dimension reduction theorem
for complete ancient solutions of the Kähler-Ricci flow with nonnegative
bisectional curvature under a finiteness assumption on the
Chern number cn1
.
14.
The Futaki Invariant and the Mabuchi Energy of a
Complete Intersection - Phong
, D. H.; Sturm
, Jacob
Let M be a compact complex Kähler manifold. If c1(M) = 0
or if c1(M) < 0,
then it is known by the work of Yau [Y78] and Yau, Aubin [Y78], [A78] that
M has a Kähler-Einstein metric. If c1(M) > 0, then there are obstructions
to the existence of such a metric, and here the guiding conjecture is that
formulated by Yau in [Y93], which says that M has a Kähler-Einstein metric
if and only if M is stable in the sense of geometric invariant theory. ¶
An important obstruction to the existence of Kähler-Einstein metric is
the invariant of Futaki [F83], which is a map F...
15.
1+1 Wave Maps into Symmetric Spaces - Terng
, Chuu-Lian; Uhlenbeck
, Karen
We explain how to apply techniques from integrable systems
to construct 2k-soliton homoclinic wave maps from the periodic
Minkowski space S1 x R1 to a compact Lie group, and more generally
to a compact symmetric space. We give a correspondence
between solutions of the -1 flow equation associated to a compact
Lie group G and wave maps into G. We use Bäcklund transformations
to construct explicit 2k-soliton breather solutions for
the -1 flow equation and show that the corresponding wave maps
are periodic and homoclinic. The compact symmetric space G/K
can be embedded as a totally geodesic submanifold of G via the
Cartan embedding. We prescribe the constraint condition for...
16.
Local splitting structures on nonpositively curved
manifolds and semirigidity in dimension 3 - Cao
, Jianguo; Cheeger
, Jeff; Rong
, Xiaochun
Let Mn denote a closed Riemannian manifold with nonpositive sectional
curvature. Let Xn denote a closed smooth manifold which admits an F- structure, \frak F.
If there exists
f : Xn ? Mn with
nonzero degree, then Mn has a local splitting structure S: 1) The
universal covering space with the pull-back metric, has a locally
finite covering by closed convex subsets, each of which splits isometrically
as a product with nontrivial Euclidean factor. 2) This
collection of sets and splittings are invariant under the group of
covering transformations. 3) The projection to Mn of any flat (i.e.
Euclidean slice) of Sis a closed immersed submanifold. The structures,
\frak F, S,...
17.
Non-integral Toroidal Dehn Surgeries - McA. Gordon
, C.; Luecke
, John
If we perform a non-trivial Dehn surgery on a hyperbolic knot in the 3-
sphere, the result is usually a hyperbolic 3-manifold. However, there are
exceptions: there are hyperbolic knots with surgeries that give lens spaces
[1], small Seifert fiber spaces [2], [5], [7], [19], and toroidal manifolds, that
is, manifolds containing (embedded) incompressible tori [6], [7]. In particular,
Eudave-Muñoz [6] has explicitly described an infinite family of hyperbolic
knots k(??, m, n, p), each of which has a specific half-integral toroidal
surgery. (These are the only known examples of non-trivial, non-integral,
non-hyperbolic surgeries on hyperbolic knots.) Here we show that these
knots are the only hyperbolic knots with...
18.
Existence and Compactness of Minimizers of the
Yamabe Problem on Manifolds with Boundary - Araújo
, Henrique
We show existence of minimizers of the Yamabe functional on a
compact Riemannian manifold with boundary (M,g), of dimension
n ? 3, restricted to the set of all metrics conformal to g and
satisfying aV + bA = 1, where V and A are the volume of M
and area of ?M, respectively, when a and b are positive real numbers
and when the infimum of the functional on that set is stricly
less than the corresponding quantity on the standard Euclidean
half-sphere. This shows that for such manifolds we can deform g
conformally to obtain a metric with constant scalar curvature R
and constant mean curvature h on...
19.
Quasiconvex Foliations and Asymptotically Flat
Metrics of Non-negative Scalar Curvature - Smith
, Brian; Weinstein
, Gilbert
We prove that a broad subset of the space of asymptotically flat
Riemannian metrics of nonnegative scalar curvature on R3 is connected
using a new method for prescribing scalar curvature that
generalizes a method developed by Bartnik for quasi-spherical metrics.
20.
Connected Sums of Special Lagrangian
Submanifolds - Lee
, Dan A.
Let M1 and M2 be special Lagrangian submanifolds of a compact
Calabi-Yau manifold X that intersect transversely at a single point.
We can then think of M1 ? M2 as a singular special Lagrangian
submanifold of X with a single isolated singularity. We investigate
when we can regularize M1 ? M2 in the following sense: There exists
a family of Calabi-Yau structures X? on X and a family of special
Lagrangian submanifolds M? of X? such that M? converges to
M1 ? M2 and X? converges to the original Calabi-Yau structure on
X. We prove that a regularization exists in two important cases:
(1) when dimC X = 3,...
1.
