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1. CJM Online first

Folha, Abigail; Penafiel, 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 surfaceCategories:53C42, 53C30

2. 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 coversCategories:53C07, 14D20

3. CJM Online first

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 $\mu$-based-Lebesgue-space $L^{n/\beta}_\mu$ 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 perimeterCategories:52A38, 53A15, 53A30

4. CJM 2016 (vol 69 pp. 220)

Zheng, Tao
 The Chern-Ricci Flow on Oeljeklaus-Toma Manifolds We study the Chern-Ricci flow, an evolution equation of Hermitian metrics, on a family of Oeljeklaus-Toma (OT-) manifolds which are non-KÃ¤hler compact complex manifolds with negative Kodaira dimension. We prove that, after an initial conformal change, the flow converges, in the Gromov-Hausdorff sense, to a torus with a flat Riemannian metric determined by the OT-manifolds themselves. Keywords:Chern-Ricci flow, Oeljeklaus-Toma manifold, Calabi-type estimate, Gromov-Hausdorff convergenceCategories:53C44, 53C55, 32W20, 32J18, 32M17

5. 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 frontsCategories:57R45, 53A05, 53A55

6. CJM 2016 (vol 68 pp. 655)

 Discrete Curvature and Abelian Groups We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of curvature'' in discrete spaces. An appealing feature of this discrete version of the so-called $\Gamma_2$-calculus (of Bakry-Ãmery) 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 Buser-type 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 edge-isoperimetric 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 groupsCategories:53C21, 57M15

7. CJM 2015 (vol 68 pp. 463)

 The Weak b-principle: 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 b-principle 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 Madsen-Weiss theorem (the standard Mumford Conjecture) fits a general setting of the b-principle. Namely, a version of the b-principle for oriented colored broken submersions together with the Harer stability theorem and Miller-Morita theorem implies the Madsen-Weiss theorem. Keywords:generalized cohomology theories, fold singularities, h-principle, infinite loop spacesCategories:55N20, 53C23

8. 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 infinityCategories:18B30, 51F99, 53C23, 54C20

9. 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 degenerationCategories:53D30, 14H60

10. CJM 2015 (vol 67 pp. 1411)

Kawakami, Yu
 Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. 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

11. CJM 2014 (vol 67 pp. 107)

Chang, Jui-En; Xiao, Ling
 The Weyl Problem With Nonnegative Gauss Curvature In Hyperbolic Space In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form $H$ and extend the result to $n$-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions. Categories:53A99, 5J15, 58J05

12. CJM 2013 (vol 66 pp. 1413)

Zhang, Xi; Zhang, Xiangwen
 Generalized KÃ¤hler--Einstein Metrics and Energy Functionals In this paper, we consider a generalized KÃ¤hler-Einstein 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¤hler-Einstein metrics with semi-positive 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¤hler-Einstein metric implies a Moser-Trudinger type inequality. Keywords:complex Monge--AmpÃ¨re equation, energy functional, generalized KÃ¤hler--Einstein metric, Moser--Trudinger type inequalityCategories:53C55, 32W20

13. CJM 2013 (vol 66 pp. 783)

Izmestiev, Ivan
 Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert-Einstein 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 Hilbert-Einstein 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 hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein 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, Hilbert-Einstein functional, Minkowski theoremCategories:52B99, 53C24

14. 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 physicsCategory:53D18

15. 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 two-parameter 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 one-parameter 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 Dajczer-type 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

16. CJM 2012 (vol 65 pp. 266)

Bérard, Vincent
 Les applications conforme-harmoniques 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 $n$ non-linÃ©aire qui est covariante conforme par rapport Ã  la variÃ©tÃ© de dÃ©part, on les appelle les applications conforme-harmoniques. 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

17. 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 systemsCategories:70S05, 70S10, 53C80

18. 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 Berger-Kazdan inequality and SantalÃ³'s formula in Finsler geometry. Being based on these, we derive a Berger-Kazdan type comparison theorem and a Croke type isoperimetric inequality for Finsler manifolds. Keywords:Finsler manifold, Berger-Kazdan inequality, Berger-Kazdan comparison theorem, SantalÃ³'s formula, Croke's isoperimetric inequalityCategories:53B40, 53C65, 52A38

19. CJM 2012 (vol 65 pp. 634)

Mezzetti, Emilia; Miró-Roig, 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 $(d-1)$-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 threefoldCategories:13E10, 14M25, 14N05, 14N15, 53A20

20. 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 PDEsCategories:53Z05, 35Q99

21. 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 two-point 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, almost-positive $c$-curvatureCategories:53C35, 53C21, 53C26, 49N60

22. CJM 2012 (vol 65 pp. 553)

Godinho, Leonor; Sousa-Dias, M. E.
 Addendum and Erratum to "The Fundamental Group of $S^1$-manifolds" This paper provides an addendum and erratum to L. Godinho and M. E. Sousa-Dias, "The Fundamental Group of $S^1$-manifolds". Canad. J. Math. 62(2010), no. 5, 1082--1098. Keywords:symplectic reduction; fundamental groupCategories:53D19, 37J10, 55Q05

23. CJM 2012 (vol 65 pp. 467)

Wilson, Glen; Woodward, Christopher T.
 Quasimap Floer Cohomology for Varying Symplectic Quotients We show that quasimap Floer cohomology for varying symplectic quotients resolves several puzzles regarding displaceability of toric moment fibers. For example, we (i) present a compact Hamiltonian torus action containing an open subset of non-displaceable orbits and a codimension four singular set, partly answering a question of McDuff, and (ii) determine displaceability for most of the moment fibers of a symplectic ellipsoid. Keywords:Floer cohomology, Hamiltonian displaceabilityCategory:53Dxx

24. 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 two-step 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 non-Riemannian 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, two-step nilpotent Lie groupsCategories:22E46, 53C30

25. CJM 2011 (vol 64 pp. 778)

Calvaruso, Giovanni; Fino, Anna
 Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces We study the geometry of non-reductive $4$-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable. Keywords:non-reductive homogeneous spaces, pseudo-Riemannian metrics, Ricci solitons, Einstein-like metricsCategories:53C21, 53C50, 53C25
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