1. CJM Online first
 Cushing, David; Liu, Shiping; Peyerimhoff, Norbert

BakryÃmery Curvature Functions on Graphs
We study local properties of the BakryÃmery curvature function
$\mathcal{K}_{G,x}:(0,\infty]\to \mathbb{R}$ at a vertex $x$ of a graph
$G$
systematically. Here $\mathcal{K}_{G,x}(\mathcal{N})$ is defined as the optimal
curvature lower bound $\mathcal{K}$ in the BakryÃmery curvaturedimension
inequality $CD(\mathcal{K},\mathcal{N})$ that $x$ satisfies. We provide upper
and
lower bounds for the curvature functions, introduce fundamental
concepts like curvature sharpness and $S^1$out regularity,
and
relate the curvature functions of $G$ with various spectral
properties of (weighted) graphs constructed from local structures
of
$G$. We prove that the curvature functions of the Cartesian
product
of two graphs $G_1,G_2$ are equal to an abstract product of
curvature functions of $G_1,G_2$. We explore the curvature
functions
of Cayley graphs and many particular (families of) examples.
We
present various conjectures and construct an infinite increasing
family of $6$regular graphs which satisfy $CD(0,\infty)$ but
are
not Cayley graphs.
Keywords:BakryÃmery curvature, curvaturedimension inequality, Cayley graph, Cartesian product Categories:05C50, 52C99, 53A40 

2. CJM Online first
 Chang, DerChen; Chang, ShuCheng; Han, Yingbo; Tie, Jingzhi

A CR Analogue of Yau's Conjecture On Pseudoharmonic Functions of Polynomial Growth
In this paper, we first derive the CR volume doubling property,
CR Sobolev
inequality, and mean value inequality. We then apply them to
prove the CR
analogue of Yau's conjecture on the space consisting of all pseudoharmonic
functions of polynomial growth of degree at most $d$ in a complete
noncompact
pseudohermitian $(2n+1)$manifold. As a byproduct, we obtain
the CR analogue
of volume growth estimate and Gromov precompactness theorem.
Keywords:CR Bochner formula, heat kernel, subgradient estimate, Liouvile theorem, CR volume doubling property, CR Sobolev inequality, mean value inequality Categories:32V05, 32V20, 53C56 

3. CJM 2018 (vol 70 pp. 943)
 Yuan, Rirong

On a Class of Fully Nonlinear Elliptic Equations containing Gradient Terms on Compact Hermitian Manifolds
In this paper we study a class of second order fully nonlinear
elliptic equations
containing gradient terms on compact Hermitian manifolds and
obtain a priori estimates under
proper assumptions close to optimal.
The analysis developed here should
be useful to deal with other Hessian equations containing gradient
terms in other contexts.
Keywords:Sasakian manifold, Hermitian manifold, subsolution, extra concavity condition, fully nonlinear elliptic equation containing gradient term on complex manifold Categories:35J15, 53C55, 53C25, 35J25 

4. CJM 2017 (vol 69 pp. 1292)
 Folha, Abigail; Peñafiel, Carlos

Weingarten Type Surfaces in $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$
In this article, we study complete surfaces $\Sigma$, isometrically
immersed in the product space $\mathbb{H}^2\times\mathbb{R}$ or
$\mathbb{S}^2\times\mathbb{R}$
having positive extrinsic curvature $K_e$. Let $K_i$ denote the
intrinsic curvature of $\Sigma$. Assume that the equation $aK_i+bK_e=c$
holds for some real constants $a\neq0$, $b\gt 0$ and $c$. The main
result of this article state that when such a surface is a topological
sphere it is rotational.
Keywords:Weingarten surface, extrinsic curvature, intrinsic curvature, height estimate, rotational Weingarten surface Categories:53C42, 53C30 

5. CJM 2016 (vol 68 pp. 1096)
 Smith, Benjamin H.

Singular $G$Monopoles on $S^1\times \Sigma$
This article provides an account of the functorial correspondence
between irreducible singular $G$monopoles on $S^1\times \Sigma$
and $\vec{t}$stable meromorphic pairs on $\Sigma$.
A theorem of B. Charbonneau and J. Hurtubise
is thus generalized here from unitary to arbitrary
compact, connected gauge groups. The required distinctions and
similarities for unitary versus arbitrary gauge are clearly outlined
and many parallels are drawn for easy transition. Once the correspondence
theorem is complete, the spectral decomposition is addressed.
Keywords:connection, curvature, instanton, monopole, stability, Bogomolny equation, Sasakian geometry, cameral covers Categories:53C07, 14D20 

6. CJM 2016 (vol 69 pp. 873)
 Xiao, Jie; Ye, Deping

Anisotropic Sobolev Capacity with Fractional Order
In this paper, we introduce the anisotropic
Sobolev capacity with fractional order and develop some basic
properties for this new object. Applications to the theory of
anisotropic fractional Sobolev spaces are provided. In particular,
we give geometric characterizations for a nonnegative Radon
measure $\mu$ that naturally induces an embedding of the anisotropic
fractional Sobolev class $\dot{\Lambda}_{\alpha,K}^{1,1}$ into
the $\mu$basedLebesguespace $L^{n/\beta}_\mu$ with $0\lt \beta\le
n$. Also, we investigate the anisotropic fractional $\alpha$perimeter.
Such a geometric quantity can be used to approximate the anisotropic
Sobolev capacity with fractional order. Estimation on the constant
in the related Minkowski inequality, which is asymptotically
optimal as $\alpha\rightarrow 0^+$, will be provided.
Keywords:sharpness, isoperimetric inequality, Minkowski inequality, fractional Sobolev capacity, fractional perimeter Categories:52A38, 53A15, 53A30 

7. CJM 2016 (vol 69 pp. 220)
 Zheng, Tao

The ChernRicci Flow on OeljeklausToma Manifolds
We study the ChernRicci flow, an evolution equation of Hermitian
metrics, on a family of OeljeklausToma (OT) manifolds which
are nonKÃ¤hler compact complex manifolds with negative Kodaira
dimension. We prove that, after an initial conformal change,
the flow converges, in the
GromovHausdorff sense, to a torus with a flat Riemannian metric
determined by the OTmanifolds themselves.
Keywords:ChernRicci flow, OeljeklausToma manifold, Calabitype estimate, GromovHausdorff convergence Categories:53C44, 53C55, 32W20, 32J18, 32M17 

8. CJM 2016 (vol 68 pp. 445)
 Martins, Luciana de Fátima; Saji, Kentaro

Geometric Invariants of Cuspidal Edges
We give a normal form of the cuspidal edge
which uses only diffeomorphisms on the source
and isometries on the target.
Using this normal form, we study differential
geometric invariants of
cuspidal edges which determine them up to order three.
We also
clarify relations between these invariants.
Keywords:cuspidal edge, curvature, wave fronts Categories:57R45, 53A05, 53A55 

9. CJM 2016 (vol 68 pp. 655)
 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 

10. CJM 2015 (vol 68 pp. 463)
 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 

11. 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 

12. CJM 2015 (vol 67 pp. 1109)
 Nohara, Yuichi; Ueda, Kazushi

Goldman Systems and Bending Systems
We show that the moduli space
of parabolic bundles on the projective line
and the polygon space are isomorphic,
both as complex manifolds
and symplectic manifolds equipped with structures of completely integrable systems,
if the stability parameters are
small.
Keywords:toric degeneration Categories:53D30, 14H60 

13. CJM 2015 (vol 67 pp. 1411)
 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 

14. CJM 2014 (vol 67 pp. 107)
15. 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 

16. 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 

17. CJM 2013 (vol 66 pp. 31)
 Bailey, Michael

Symplectic Foliations and Generalized Complex Structures
We answer the natural question: when is a transversely holomorphic
symplectic foliation induced by a generalized complex structure? The
leafwise symplectic form and transverse complex structure determine an
obstruction class in a certain cohomology, which vanishes if and only
if our question has an affirmative answer. We first study a component
of this obstruction, which gives the condition that the leafwise
cohomology class of the symplectic form must be transversely
pluriharmonic. As a consequence, under certain topological
hypotheses, we infer that we actually have a symplectic fibre bundle
over a complex base. We then show how to compute the full obstruction
via a spectral sequence. We give various concrete necessary and
sufficient conditions for the vanishing of the obstruction.
Throughout, we give examples to test the sharpness of these
conditions, including a symplectic fibre bundle over a complex base
which does not come from a generalized complex structure, and a
regular generalized complex structure which is very unlike a
symplectic fibre bundle, i.e., for which nearby leaves are not
symplectomorphic.
Keywords:differential geometry, symplectic geometry, mathematical physics Category:53D18 

18. 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 

19. 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 

20. 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 

21. 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 

22. CJM 2012 (vol 65 pp. 634)
 Mezzetti, Emilia; MiróRoig, Rosa M.; Ottaviani, Giorgio

Laplace Equations and the Weak Lefschetz Property
We prove that $r$ independent homogeneous polynomials of the same degree $d$
become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety
whose $(d1)$osculating spaces have dimension smaller than expected. This gives an equivalence
between an algebraic notion (called Weak Lefschetz Property)
and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case,
some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture.
Keywords:osculating space, weak Lefschetz property, Laplace equations, toric threefold Categories:13E10, 14M25, 14N05, 14N15, 53A20 

23. CJM 2012 (vol 65 pp. 655)
 Shemyakova, E.

Proof of the Completeness of Darboux Wronskian Formulae for Order Two
Darboux Wronskian formulas allow to construct Darboux transformations,
but Laplace transformations, which are Darboux transformations of
order one
cannot be represented this way.
It has been a long standing problem on what are other exceptions. In
our previous work we proved that among transformations of total
order one there are no other exceptions. Here we prove that for
transformations of total order two there are no exceptions at all.
We also obtain a simple explicit invariant description of all possible
Darboux Transformations of total order two.
Keywords:completeness of Darboux Wronskian formulas, completeness of Darboux determinants, Darboux transformations, invariants for solution of PDEs Categories:53Z05, 35Q99 

24. 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 

25. CJM 2012 (vol 65 pp. 553)