1. CJM Online first
 Klartag, Bo'az; Kozma, Gady; Ralli, Peter; Tetali, Prasad

Discrete curvature and abelian groups
We study a natural discrete Bochnertype inequality on graphs,
and explore its merit as a notion of ``curvature'' in discrete
spaces.
An appealing feature of this discrete version of the socalled
$\Gamma_2$calculus (of BakryÃmery) seems to be that it is
fairly
straightforward to compute this notion of curvature parameter
for
several specific graphs of interest  particularly, abelian
groups, slices of the hypercube, and the symmetric group under
various sets of generators.
We further develop this notion by deriving Busertype inequalities
(Ã la Ledoux), relating functional and isoperimetric constants
associated with a graph.
Our derivations provide a tight bound on the Cheeger constant
(i.e., the edgeisoperimetric constant) in terms of
the spectral gap, for graphs with nonnegative curvature, particularly,
the class of abelian Cayley graphs  a result of independent
interest.
Keywords:Ricci curvature, graph theory, abelian groups Categories:53C21, 57M15 

2. CJM Online first
 Sadykov, Rustam

The Weak bprinciple: Mumford Conjecture
In this note we introduce and study a new class of maps called
oriented colored broken submersions. This is the simplest class
of maps that satisfies a version of the bprinciple and in dimension
$2$ approximates the class of oriented submersions well in the
sense that
every oriented colored broken submersion of dimension $2$ to
a closed simply connected manifold is bordant to a submersion.
We show that the MadsenWeiss theorem (the standard Mumford Conjecture)
fits a general setting of the bprinciple. Namely, a version
of the bprinciple for
oriented colored broken submersions together with the Harer
stability theorem and MillerMorita theorem implies the MadsenWeiss
theorem.
Keywords:generalized cohomology theories, fold singularities, hprinciple, infinite loop spaces Categories:55N20, 53C23 

3. CJM 2015 (vol 67 pp. 1091)
 Mine, Kotaro; Yamashita, Atsushi

Metric Compactifications and Coarse Structures
Let $\mathbf{TB}$ be the category of totally bounded, locally
compact metric spaces
with the $C_0$ coarse structures. We show that if $X$ and $Y$
are in $\mathbf{TB}$ then $X$ and $Y$ are coarsely equivalent
if and only if their Higson coronas are homeomorphic. In fact,
the Higson corona functor gives an equivalence of categories
$\mathbf{TB}\to\mathbf{K}$, where $\mathbf{K}$ is the category
of compact metrizable spaces. We use this fact to show that the
continuously controlled coarse structure on a locally compact
space $X$ induced by some metrizable compactification $\tilde{X}$
is determined only by the topology of the remainder $\tilde{X}\setminus
X$.
Keywords:coarse geometry, Higson corona, continuously controlled coarse structure, uniform continuity, boundary at infinity Categories:18B30, 51F99, 53C23, 54C20 

4. CJM Online first
 Kawakami, Yu

Functiontheoretic Properties for the Gauss Maps of Various Classes of Surfaces
We elucidate the geometric background of functiontheoretic properties
for the Gauss maps of
several classes of immersed surfaces in threedimensional space
forms, for example, minimal surfaces in Euclidean threespace, improper affine spheres in the affine threespace, and constant
mean curvature one surfaces and flat surfaces in hyperbolic threespace. To achieve this purpose, we prove an optimal curvature bound
for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for
the Gauss maps of these classes of surfaces.
Keywords:Gauss map, minimal surface, constant mean curvature surface, front, ramification, omitted value, the Ahlfors island theorem, unicity theorem. Categories:53C42, 30D35, 30F45, 53A10, 53A15 

5. CJM 2013 (vol 66 pp. 1413)
 Zhang, Xi; Zhang, Xiangwen

Generalized KÃ¤hlerEinstein Metrics and Energy Functionals
In this paper, we consider a generalized
KÃ¤hlerEinstein equation on KÃ¤hler manifold $M$. Using the
twisted $\mathcal K$energy introduced by Song and Tian, we show
that the existence of generalized KÃ¤hlerEinstein metrics with
semipositive twisting $(1, 1)$form $\theta $ is also closely
related to the properness of the twisted $\mathcal K$energy
functional. Under the condition that the twisting form $\theta $ is
strictly positive at a point or $M$ admits no nontrivial Hamiltonian
holomorphic vector field, we prove that the existence of generalized
KÃ¤hlerEinstein metric implies a MoserTrudinger type inequality.
Keywords:complex MongeAmpÃ¨re equation, energy functional, generalized KÃ¤hlerEinstein metric, MoserTrudinger type inequality Categories:53C55, 32W20 

6. CJM 2013 (vol 66 pp. 783)
 Izmestiev, Ivan

Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the HilbertEinstein Functional
The paper is centered around a new proof of the infinitesimal rigidity
of convex polyhedra. The proof is based on studying derivatives of the
discrete HilbertEinstein functional on the space of "warped
polyhedra" with a fixed metric on the boundary.
The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas.
In the spherical and in the hyperbolicde Sitter space, there is a perfect duality between the HilbertEinstein functional and the volume, as well as between both kinds of rigidity.
We review some of the related work and discuss directions for future research.
Keywords:convex polyhedron, rigidity, HilbertEinstein functional, Minkowski theorem Categories:52B99, 53C24 

7. CJM 2013 (vol 66 pp. 400)
 Mendonça, Bruno; Tojeiro, Ruy

Umbilical Submanifolds of $\mathbb{S}^n\times \mathbb{R}$
We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of
$\mathbb{S}^n\times \mathbb{R}$, extending the classification of umbilical surfaces
in $\mathbb{S}^2\times \mathbb{R}$ by Souam and Toubiana as well as the local
description of umbilical hypersurfaces in $\mathbb{S}^n\times \mathbb{R}$ by Van der
Veken and Vrancken. We prove that, besides small spheres in a slice,
up to isometries of the ambient space they come in a twoparameter
family of rotational submanifolds
whose substantial codimension is either one or two and whose profile
is a curve in a totally geodesic $\mathbb{S}^1\times \mathbb{R}$ or $\mathbb{S}^2\times
\mathbb{R}$, respectively, the former case arising in a oneparameter
family. All of them are diffeomorphic to a sphere, except for a single
element that is diffeomorphic to Euclidean space. We obtain explicit
parametrizations of all such submanifolds. We also study more general
classes of submanifolds of $\mathbb{S}^n\times \mathbb{R}$ and $\mathbb{H}^n\times \mathbb{R}$. In
particular, we give a complete description of all submanifolds in
those product spaces
for which the tangent component of a unit vector field spanning the
factor $\mathbb{R}$ is an eigenvector of all shape operators. We show that
surfaces with parallel mean curvature vector in $\mathbb{S}^n\times \mathbb{R}$ and
$\mathbb{H}^n\times \mathbb{R}$ having this property are rotational surfaces, and use
this fact to improve some recent results by Alencar, do Carmo, and
Tribuzy.
We also obtain a Dajczertype reduction of codimension theorem for
submanifolds of $\mathbb{S}^n\times \mathbb{R}$ and $\mathbb{H}^n\times \mathbb{R}$.
Keywords:umbilical submanifolds, product spaces $\mathbb{S}^n\times \mathbb{R}$ and $\mathbb{H}^n\times \mathbb{R}$ Categories:53B25, 53C40 

8. CJM 2012 (vol 65 pp. 266)
 Bérard, Vincent

Les applications conformeharmoniques
Sur une surface de Riemann, l'Ã©nergie d'une application Ã valeurs dans une variÃ©tÃ© riemannienne est une fonctionnelle invariante conforme, ses points critiques sont les applications harmoniques. Nous proposons ici un analogue en dimension supÃ©rieure, en construisant une fonctionnelle invariante conforme pour les applications entre deux variÃ©tÃ©s riemanniennes, dont la variÃ©tÃ© de dÃ©part est de dimension $n$ paire. Ses points critiques satisfont une EDP elliptique d'ordre $n$ nonlinÃ©aire qui est covariante conforme par rapport Ã la variÃ©tÃ© de dÃ©part, on les appelle les applications conformeharmoniques. Dans le cas des fonctions, on retrouve l'opÃ©rateur GJMS, dont le terme principal est une puissance $n/2$ du laplacien. Quand $n$ est impaire, les mÃªmes idÃ©es permettent de montrer que le terme constant dans le dÃ©veloppement asymptotique de l'Ã©nergie d'une application asymptotiquement harmonique sur une variÃ©tÃ© AHE est indÃ©pendant du choix du reprÃ©sentant de l'infini conforme.
Categories:53C21, 53C43, 53A30 

9. CJM 2012 (vol 65 pp. 1164)
 Vitagliano, Luca

Partial Differential Hamiltonian Systems
We define partial differential (PD in the following), i.e., field
theoretic analogues of Hamiltonian systems on abstract symplectic
manifolds and study their main properties, namely, PD Hamilton
equations, PD Noether theorem, PD Poisson bracket, etc.. Unlike in
standard multisymplectic approach to Hamiltonian field theory, in our
formalism, the geometric structure (kinematics) and the dynamical
information on the ``phase space''
appear as just different components of one single geometric object.
Keywords:field theory, fiber bundles, multisymplectic geometry, Hamiltonian systems Categories:70S05, 70S10, 53C80 

10. CJM 2012 (vol 65 pp. 1401)
 Zhao, Wei; Shen, Yibing

A Universal Volume Comparison Theorem for Finsler Manifolds and Related Results
In this paper, we establish a universal volume comparison theorem
for Finsler manifolds and give the BergerKazdan inequality and
SantalÃ³'s formula in Finsler geometry. Being based on these, we
derive a BergerKazdan type comparison theorem and a Croke type
isoperimetric inequality for Finsler manifolds.
Keywords:Finsler manifold, BergerKazdan inequality, BergerKazdan comparison theorem, SantalÃ³'s formula, Croke's isoperimetric inequality Categories:53B40, 53C65, 52A38 

11. CJM 2012 (vol 65 pp. 757)
 Delanoë, Philippe; Rouvière, François

Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved
The squared distance curvature is a kind of twopoint curvature the
sign of which turned out crucial for the smoothness of optimal
transportation maps on Riemannian manifolds. Positivity properties of
that new curvature have been established recently for all the simply
connected compact rank one symmetric spaces, except the Cayley
plane. Direct proofs were given for the sphere, an indirect one
via the Hopf fibrations) for the complex and quaternionic
projective spaces. Here, we present a direct proof of a property
implying all the preceding ones, valid on every positively curved
Riemannian locally symmetric space.
Keywords:symmetric spaces, rank one, positive curvature, almostpositive $c$curvature Categories:53C35, 53C21, 53C26, 49N60 

12. CJM 2012 (vol 65 pp. 66)
 Deng, Shaoqiang; Hu, Zhiguang

On Flag Curvature of Homogeneous Randers Spaces
In this paper we give an explicit formula for the flag curvature of
homogeneous Randers spaces of Douglas type and apply this formula to
obtain some interesting results. We first deduce an explicit formula
for the flag curvature of an arbitrary left invariant Randers metric
on a twostep nilpotent Lie group. Then we obtain a classification of
negatively curved homogeneous Randers spaces of Douglas type. This
results, in particular, in many examples of homogeneous nonRiemannian
Finsler spaces with negative flag curvature. Finally, we prove a
rigidity result that a homogeneous Randers space of Berwald type whose
flag curvature is everywhere nonzero must be Riemannian.
Keywords:homogeneous Randers manifolds, flag curvature, Douglas spaces, twostep nilpotent Lie groups Categories:22E46, 53C30 

13. CJM 2011 (vol 64 pp. 778)
 Calvaruso, Giovanni; Fino, Anna

Ricci Solitons and Geometry of Fourdimensional Nonreductive Homogeneous Spaces
We study the geometry of nonreductive $4$dimensional homogeneous
spaces. In particular, after describing their LeviCivita connection
and curvature properties, we classify homogeneous Ricci solitons on
these spaces, proving the existence of shrinking, expanding and steady
examples. For all the nontrivial examples we find, the Ricci operator
is diagonalizable.
Keywords:nonreductive homogeneous spaces, pseudoRiemannian metrics, Ricci solitons, Einsteinlike metrics Categories:53C21, 53C50, 53C25 

14. CJM 2011 (vol 64 pp. 44)
 Carvalho, T. M. M.; Moreira, H. N.; Tenenblat, K.

Surfaces of Rotation with Constant Mean Curvature in the Direction of a Unitary Normal Vector Field in a Randers Space
We consider the Randers space $(V^n,F_b)$ obtained by perturbing the Euclidean metric by a translation, $F_b=\alpha+\beta$, where $\alpha$ is the Euclidean metric and $\beta$ is a $1$form with norm $b$, $0\leq b\lt 1$. We introduce the concept of a hypersurface with constant mean curvature in the direction of a unitary normal vector field. We obtain the ordinary differential equation that characterizes the rotational surfaces $(V^3,F_b)$ of constant mean curvature (cmc) in the direction of a unitary normal vector field. These equations reduce to the classical equation of the rotational cmc surfaces in Euclidean space, when $b=0$. It also reduces to the equation that characterizes the minimal rotational surfaces in $(V^3,F_b)$ when $H=0$, obtained by M. Souza and K. Tenenblat. Although the differential equation depends on the choice of the normal direction, we show that both equations determine the same rotational surface, up to a reflection. We also show that the round cylinders are cmc surfaces in the direction of the unitary normal field. They are generated by the constant solution of the differential equation. By considering the equation as a nonlinear dynamical system, we provide a qualitative analysis, for $0\lt b\lt \frac{\sqrt{3}}{3}$. Using the concept of stability and considering the linearization around the single equilibrium point (the constant solution), we verify that the solutions are locally asymptotically stable spirals. This is proved by constructing a Lyapunov function for the dynamical system and by determining the basin of stability of the equilibrium point. The surfaces of rotation generated by such solutions tend asymptotically to one end of the cylinder.
Keywords:Finsler spaces, Randers spaces, mean curvature, Liapunov functions Category:53C20 

15. CJM 2010 (vol 63 pp. 55)
 Chau, Albert; Tam, LuenFai; Yu, Chengjie

Pseudolocality for the Ricci Flow and Applications
Perelman established a differential LiYauHamilton
(LYH) type inequality for fundamental solutions of the conjugate
heat equation corresponding to the Ricci flow on compact manifolds.
As an application of the LYH inequality,
Perelman proved a pseudolocality result for the Ricci flow on
compact manifolds. In this article we provide the details for the
proofs of these results in the case of a complete noncompact
Riemannian manifold. Using these results we prove that under
certain conditions, a finite time singularity of the Ricci flow
must form within a compact set. The conditions are satisfied by
asymptotically flat manifolds. We also prove a long time existence
result for the K\"ahlerRicci flow on complete nonnegatively curved K\"ahler
manifolds.
Categories:53C44, 58J37, 35B35 

16. 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 

17. CJM 2010 (vol 62 pp. 1037)
18. 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 

19. 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 

20. 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 

21. 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 

22. 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 

23. 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 

24. 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 

25. 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 
