26. CJM 2010 (vol 62 pp. 1264)
 Chen, Jingyi; Fraser, Ailana

Holomorphic variations of minimal disks with boundary on a Lagrangian surface
Let $L$ be an oriented Lagrangian submanifold in an $n$dimensional
KÃ¤hler manifold~$M$. Let $u \colon D \to M$ be a minimal immersion
from a disk $D$ with $u(\partial D) \subset L$ such that $u(D)$ meets
$L$ orthogonally along $u(\partial D)$. Then the real dimension of
the space of admissible holomorphic variations is at least
$n+\mu(E,F)$, where $\mu(E,F)$ is a boundary Maslov index; the minimal
disk is holomorphic if there exist $n$ admissible holomorphic
variations that are linearly independent over $\mathbb{R}$ at some
point $p \in \partial D$; if $M = \mathbb{C}P^n$ and $u$ intersects
$L$ positively, then $u$ is holomorphic if it is stable, and its
Morse index is at least $n+\mu(E,F)$ if $u$ is unstable.
Categories:58E12, 53C21, 53C26 

27. CJM 2010 (vol 62 pp. 1082)
 Godinho, Leonor; SousaDias, M. E.

The Fundamental Group of $S^1$manifolds
We address the problem of computing the fundamental
group of a symplectic $S^1$manifold for nonHamiltonian actions on
compact manifolds, and for Hamiltonian actions on noncompact
manifolds with a proper moment map. We generalize known results for
compact manifolds equipped with a Hamiltonian $S^1$action. Several
examples are presented to illustrate our main results.
Categories:53D20, 37J10, 55Q05 

28. CJM 2010 (vol 62 pp. 975)
 Bjorndahl, Christina; Karshon, Yael

Revisiting TietzeNakajima: Local and Global Convexity for Maps
A theorem of Tietze and Nakajima, from 1928, asserts that
if a subset $X$ of $\mathbb{R}^n$ is closed, connected, and locally convex,
then it is convex.
We give an analogous ``local to global convexity" theorem
when the inclusion map of $X$ to $\mathbb{R}^n$ is replaced by a map
from a topological space $X$ to $\mathbb{R}^n$ that satisfies
certain local properties.
Our motivation comes from the CondevauxDazordMolino proof
of the AtiyahGuilleminSternberg convexity theorem in symplectic geometry.
Categories:53D20, 52B99 

29. CJM 2010 (vol 62 pp. 1037)
30. CJM 2009 (vol 62 pp. 52)
 Deng, Shaoqiang

An Algebraic Approach to Weakly Symmetric Finsler Spaces
In this paper, we introduce a new algebraic notion, weakly symmetric
Lie algebras, to give an algebraic description of an
interesting class of homogeneous RiemannFinsler spaces, weakly symmetric
Finsler spaces. Using this new definition, we are able to give a
classification of weakly symmetric Finsler spaces with dimensions $2$
and $3$. Finally, we show that all the nonRiemannian reversible weakly
symmetric Finsler spaces we find are nonBerwaldian and with vanishing
Scurvature. This means that reversible nonBerwaldian Finsler spaces
with vanishing Scurvature may exist at large. Hence the generalized
volume comparison theorems due to Z. Shen are valid for a rather large
class of Finsler spaces.
Keywords:weakly symmetric Finsler spaces, weakly symmetric Lie algebras, Berwald spaces, Scurvature Categories:53C60, 58B20, 22E46, 22E60 

31. CJM 2009 (vol 62 pp. 320)
 Jerrard, Robert L.

Some Rigidity Results Related to MongeâAmpÃ¨re Functions
The space of MongeAmpÃ¨re functions, introduced by J. H. G. Fu, is
a space of rather rough functions in which the map $u\mapsto \operatorname{Det} D^2
u$ is well defined and weakly continuous with respect to a natural
notion of weak convergence. We prove a rigidity theorem for
Lagrangian integral currents that allows us to extend the original
definition of MongeAmpÃ¨re functions. We also
prove that if a MongeAmpÃ¨re function $u$ on a bounded set
$\Omega\subset\mathcal{R}^2$ satisfies the equation $\operatorname{Det} D^2 u=0$ in a
particular weak sense, then the graph of $u$ is a developable surface,
and moreover $u$ enjoys somewhat better regularity properties than an
arbitrary MongeAmpÃ¨re function of $2$ variables.
Categories:49Q15, 53C24 

32. CJM 2009 (vol 62 pp. 3)
 Anchouche, Boudjemâa

On the Asymptotic Behavior of Complete KÃ¤hler Metrics of Positive Ricci Curvature
Let $( X,g) $ be a complete noncompact KÃ¤hler manifold, of
dimension $n\geq2,$ with positive Ricci curvature and of standard type
(see the definition below). N. Mok proved that $X$ can be
compactified, \emph{i.e.,} $X$ is biholomorphic to a quasiprojective
variety$.$ The aim of this paper is to prove that the $L^{2}$
holomorphic sections of the line bundle $K_{X}^{q}$ and the volume
form of the metric $g$ have no essential singularities near the
divisor at infinity. As a consequence we obtain a comparison between
the volume forms of the KÃ¤hler metric $g$ and of the FubiniStudy
metric induced on $X$. In the case of $\dim_{\mathbb{C} }X=2,$ we
establish a relation between the number of components of the divisor
$D$ and the dimension of the groups $H^{i}( \overline{X},
\Omega_{\overline{X}}^{1}( \log D) )$.
Categories:53C55, 32A10 

33. CJM 2009 (vol 61 pp. 1357)
 Shen, Zhongmin

On a Class of Landsberg Metrics in Finsler Geometry
In this paper, we study a long existing open problem on Landsberg
metrics in Finsler geometry. We consider Finsler metrics defined by a
Riemannian metric and a $1$form on a manifold. We show that a
\emph{regular} Finsler metric in this form is Landsbergian if and only if it
is Berwaldian. We further show that there is a twoparameter family of
functions, $\phi=\phi(s)$, for which there are a Riemannian metric
$\alpha$ and a $1$form $\beta$ on a manifold $M$ such that the scalar
function $F=\alpha \phi (\beta/\alpha)$ on $TM$ is an almost regular
Landsberg metric, but not a Berwald metric.
Categories:53B40, 53C60 

34. CJM 2009 (vol 61 pp. 1201)
 Arvanitoyeorgos, Andreas; Dzhepko, V. V.; Nikonorov, Yu. G.

Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups
A Riemannian manifold $(M,\rho)$ is called Einstein if the metric
$\rho$ satisfies the condition \linebreak$\Ric (\rho)=c\cdot \rho$ for some
constant $c$. This paper is devoted to the investigation of
$G$invariant Einstein metrics, with additional symmetries,
on some homogeneous spaces $G/H$ of classical groups.
As a consequence, we obtain new invariant Einstein metrics on some
Stiefel manifolds $\SO(n)/\SO(l)$.
Furthermore, we show that for any positive integer $p$ there exists a
Stiefel manifold $\SO(n)/\SO(l)$
that admits at least $p$
$\SO(n)$invariant Einstein metrics.
Keywords:Riemannian manifolds, homogeneous spaces, Einstein metrics, Stiefel manifolds Categories:53C25, 53C30 

35. CJM 2009 (vol 61 pp. 721)
 Calin, Ovidiu; Chang, DerChen; Markina, Irina

SubRiemannian Geometry on the Sphere $\mathbb{S}^3$
We discuss the subRiemannian
geometry induced by two noncommutative
vector fields which are left invariant
on the Lie group $\mathbb{S}^3$.
Keywords:noncommutative Lie group, quaternion group, subRiemannian geodesic, horizontal distribution, connectivity theorem, holonomic constraint Categories:53C17, 53C22, 35H20 

36. CJM 2009 (vol 61 pp. 740)
 Caprace, PierreEmmanuel; Haglund, Frédéric

On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings
Given a complete CAT(0) space $X$ endowed with a geometric action of a group $\Gamma$, it is known that if
$\Gamma$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We
prove the converse of this statement in the special case where $X$ is a convex subcomplex of the CAT(0)
realization of a Coxeter group $W$, and $\Gamma$ is a subgroup of $W$. In particular a convex cocompact subgroup
of a Coxeter group is Gromovhyperbolic if and only if it does not contain a free abelian group of rank 2. Our
result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits
buildings.
Keywords:Coxeter group, flat rank, $\cat0$ space, building Categories:20F55, 51F15, 53C23, 20E42, 51E24 

37. CJM 2009 (vol 61 pp. 641)
 Maeda, Sadahiro; Udagawa, Seiichi

Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions
For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form
$\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we
show that if the mean curvature vector of $M^n$ is parallel and the
sectional curvature $K$ of $M^n$ satisfies some inequality, then
the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is
parallel and our manifold $M^n$ is a space form.
Keywords:space forms, parallel isometric immersions, isotropic immersions, totally umbilic, Veronese manifolds, sectional curvatures, parallel mean curvature vector Categories:53C40, 53C42 

38. CJM 2008 (vol 60 pp. 1201)
 Bahuaud, Eric; Marsh, Tracey

HÃ¶lder Compactification for Some Manifolds with Pinched Negative Curvature Near Infinity
We consider a complete noncompact Riemannian manifold $M$ and give
conditions on a compact submanifold $K \subset M$ so that the outward
normal exponential map off the boundary of $K$ is a diffeomorphism
onto $\MlK$. We use this to compactify $M$ and show that pinched
negative sectional curvature outside $K$ implies $M$ has a
compactification with a welldefined H\"older structure independent of
$K$. The H\"older constant depends on the ratio of the curvature
pinching. This extends and generalizes a 1985 result of Anderson and
Schoen.
Category:53C20 

39. CJM 2008 (vol 60 pp. 822)
 Kuwae, Kazuhiro

Maximum Principles for Subharmonic Functions Via Local SemiDirichlet Forms
Maximum principles for subharmonic
functions in the framework of quasiregular local semiDirichlet
forms admitting lower bounds are presented.
As applications, we give
weak and strong maximum principles
for (local) subsolutions of a second order elliptic
differential operator on the domain of Euclidean space under conditions on coefficients,
which partially generalize the results by Stampacchia.
Keywords:positivity preserving form, semiDirichlet form, Dirichlet form, subharmonic functions, superharmonic functions, harmonic functions, weak maximum principle, strong maximum principle, irreducibility, absolute continuity condition Categories:31C25, 35B50, 60J45, 35J, 53C, 58 

40. CJM 2008 (vol 60 pp. 572)
 Hitrik, Michael; Sj{östrand, Johannes

NonSelfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point
This is the third in a series of works devoted to spectral
asymptotics for nonselfadjoint
perturbations of selfadjoint $h$pseudodifferential operators in dimension 2, having a
periodic classical flow. Assuming that the strength $\epsilon$
of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$
(and may sometimes reach even smaller values), we
get an asymptotic description of the eigenvalues in rectangles
$[1/C,1/C]+i\epsilon [F_01/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point
value of the flow average of the leading perturbation.
Keywords:nonselfadjoint, eigenvalue, periodic flow, branching singularity Categories:31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40 

41. CJM 2008 (vol 60 pp. 457)
 Teplyaev, Alexander

Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure
We define sets with finitely ramified cell structure, which are
generalizations of postcrit8cally finite selfsimilar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local selfsimilarity, and allow countably many cells
connected at each junction point.
In particular, we consider postcritically infinite fractals.
We prove that if Kigami's resistance form
satisfies certain assumptions, then there exists a weak Riemannian metric
such that the energy can be expressed as the integral of the norm squared
of a weak gradient with respect to an energy measure.
Furthermore, we prove that if such a set can be homeomorphically represented
in harmonic coordinates, then for smooth functions the weak gradient can be
replaced by the usual gradient.
We also prove a simple formula for the energy measure Laplacian in harmonic
coordinates.
Keywords:fractals, selfsimilarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18 

42. CJM 2008 (vol 60 pp. 443)
43. CJM 2007 (vol 59 pp. 1245)
 Chen, Qun; Zhou, ZhenRong

On Gap Properties and Instabilities of $p$YangMills Fields
We consider the
$p$YangMills functional
$(p\geq 2)$
defined as
$\YM_p(\nabla):=\frac 1 p \int_M \\rn\^p$.
We call critical points of $\YM_p(\cdot)$ the $p$YangMills
connections, and the associated curvature $\rn$ the $p$YangMills
fields. In this paper, we prove gap properties and instability theorems for $p$YangMills
fields over submanifolds in $\mathbb{R}^{n+k}$ and $\mathbb{S}^{n+k}$.
Keywords:$p$YangMills field, gap property, instability, submanifold Categories:58E15, 53C05 

44. CJM 2007 (vol 59 pp. 981)
 Jiang, Yunfeng

The ChenRuan Cohomology of Weighted Projective Spaces
In this paper we study the ChenRuan cohomology ring of weighted
projective spaces. Given a weighted projective space ${\bf
P}^{n}_{q_{0}, \dots, q_{n}}$, we determine all of its twisted
sectors and the corresponding degree shifting numbers. The main
result of this paper is that the obstruction bundle over any
3\nobreakdashmulti\sector is a direct sum of line bundles which we use to
compute the orbifold cup product. Finally we compute the
ChenRuan cohomology ring of weighted projective space ${\bf
P}^{5}_{1,2,2,3,3,3}$.
Keywords:ChenRuan cohomology, twisted sectors, toric varieties, weighted projective space, localization Categories:14N35, 53D45 

45. CJM 2007 (vol 59 pp. 845)
 Schaffhauser, Florent

Representations of the Fundamental Group of an $L$Punctured Sphere Generated by Products of Lagrangian Involutions
In this paper, we characterize unitary representations of $\pi:=\piS$ whose
generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially)
can be decomposed as products of two Lagrangian involutions
$u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. Our main result is that such
representations are exactly the elements of the fixedpoint set of an
antisymplectic involution defined on the moduli space
$\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixedpoint set is
nonempty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use
the quasiHamiltonian description of the symplectic structure of $\Mod$ and
give conditions on an involution defined on a quasiHamiltonian $U$space
$(M, \w, \mu\from M \to U)$ for it to induce an antisymplectic involution on
the reduced space $M/\!/U := \mu^{1}(\{1\})/U$.
Keywords:momentum maps, moduli spaces, Lagrangian submanifolds, antisymplectic involutions, quasiHamiltonian Categories:53D20, 53D30 

46. CJM 2006 (vol 58 pp. 600)
 MartinezMaure, Yves

Geometric Study of Minkowski Differences of Plane Convex Bodies
In the Euclidean plane $\mathbb{R}^{2}$, we define the Minkowski difference
$\mathcal{K}\mathcal{L}$ of two arbitrary convex bodies $\mathcal{K}$,
$\mathcal{L}$ as a rectifiable closed curve $\mathcal{H}_{h}\subset \mathbb{R}
^{2}$ that is determined by the difference $h=h_{\mathcal{K}}h_{\mathcal{L}
} $ of their support functions. This curve $\mathcal{H}_{h}$ is
called the
hedgehog with support function $h$. More generally, the object of hedgehog
theory is to study the BrunnMinkowski theory in the vector space of
Minkowski differences of arbitrary convex bodies of Euclidean space $\mathbb{R}
^{n+1}$, defined as (possibly singular and selfintersecting) hypersurfaces
of $\mathbb{R}^{n+1}$. Hedgehog theory is useful for: (i)
studying convex bodies by splitting them into a sum in order to reveal their
structure; (ii) converting analytical problems into
geometrical ones by considering certain real functions as support
functions.
The purpose of this paper is to give a detailed study of plane
hedgehogs, which constitute the basis of the theory. In particular:
(i) we study their length measures and solve the extension of the
ChristoffelMinkowski problem to plane hedgehogs; (ii) we
characterize support functions of plane convex bodies among support
functions of plane hedgehogs and support functions of plane hedgehogs among
continuous functions; (iii) we study the mixed area of
hedgehogs in $\mathbb{R}^{2}$ and give an extension of the classical Minkowski
inequality (and thus of the isoperimetric inequality) to hedgehogs.
Categories:52A30, 52A10, 53A04, 52A38, 52A39, 52A40 

47. CJM 2006 (vol 58 pp. 282)
 Fels, M. E.; Renner, A. G.

Nonreductive Homogeneous PseudoRiemannian Manifolds of Dimension Four
A method, due to \'Elie Cartan, is used to give an algebraic
classification of the nonreductive homogeneous pseudoRiemannian
manifolds of dimension four. Only one case with Lorentz signature can
be Einstein without having constant curvature, and two cases with
$(2,2)$ signature are Einstein of which one is Ricciflat. If a
fourdimensional nonreductive homogeneous pseudoRiemannian manifold
is simply connected, then it is shown to be diffeomorphic to
$\reals^4$. All metrics for the simply connected nonreductive
Einstein spaces are given explicitly. There are no nonreductive
pseudoRiemannian homogeneous spaces of dimension two and none of
dimension three with connected isotropy subgroup.
Keywords:Homogeneous pseudoRiemannian, Einstein space Category:53C30 

48. CJM 2006 (vol 58 pp. 381)
 Jakobson, Dmitry; Nadirashvili, Nikolai; Polterovich, Iosif

Extremal Metric for the First Eigenvalue on a Klein Bottle
The first eigenvalue of the Laplacian on a surface can be viewed
as a functional on the space of Riemannian metrics of a given
area. Critical points of this functional are called extremal
metrics. The only known extremal metrics are a round sphere, a
standard projective plane, a Clifford torus and an equilateral
torus. We construct an extremal metric on a Klein bottle. It is a
metric of revolution, admitting a minimal isometric embedding into
a sphere ${\mathbb S}^4$ by the first eigenfunctions. Also, this
Klein bottle is a bipolar surface for Lawson's
$\tau_{3,1}$torus. We conjecture that an extremal metric for the
first eigenvalue on a Klein bottle is unique, and hence it
provides a sharp upper bound for $\lambda_1$ on a Klein bottle of
a given area. We present numerical evidence and prove the first
results towards this conjecture.
Keywords:Laplacian, eigenvalue, Klein bottle Categories:58J50, 53C42 

49. CJM 2006 (vol 58 pp. 362)
 Goldin, R. F.; Martin, S.

Cohomology Pairings on the Symplectic Reduction of Products
Let $M$ be the product of two compact Hamiltonian
$T$spaces $X$ and $Y$. We present a formula for evaluating
integrals on the symplectic reduction of $M$ by the diagonal $T$
action. At every regular value of the moment map for $X\times Y$, the
integral is the convolution of two distributions associated to the
symplectic reductions of $X$ by $T$ and of $Y$ by $T$. Several
examples illustrate the computational strength of this relationship.
We also prove a linear analogue which can be used to find cohomology
pairings on toric orbifolds.
Category:53D20 

50. CJM 2006 (vol 58 pp. 262)