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1. CJM Online first
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 spaces Categories:55N20, 53C23 |
2. CJM 2014 (vol 67 pp. 107)
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 |
3. CJM 2013 (vol 66 pp. 1413)
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 inequality Categories:53C55, 32W20 |
4. CJM 2013 (vol 66 pp. 783)
Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert-Einstein Functional |
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 theorem Categories:52B99, 53C24 |
5. CJM 2013 (vol 66 pp. 31)
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 |
6. CJM 2013 (vol 66 pp. 400)
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 |
7. CJM 2012 (vol 65 pp. 266)
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 |
8. CJM 2012 (vol 65 pp. 1164)
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 |
9. CJM 2012 (vol 65 pp. 1401)
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 inequality Categories:53B40, 53C65, 52A38 |
10. CJM 2012 (vol 65 pp. 634)
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 threefold Categories:13E10, 14M25, 14N05, 14N15, 53A20 |
11. CJM 2012 (vol 65 pp. 655)
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 |
12. CJM 2012 (vol 65 pp. 757)
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 $c$-curvature Categories:53C35, 53C21, 53C26, 49N60 |
13. CJM 2012 (vol 65 pp. 553)
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 group Categories:53D19, 37J10, 55Q05 |
14. CJM 2012 (vol 65 pp. 467)
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 displaceability Category:53Dxx |
15. CJM 2012 (vol 65 pp. 66)
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 groups Categories:22E46, 53C30 |
16. CJM 2011 (vol 64 pp. 778)
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 metrics Categories:53C21, 53C50, 53C25 |
17. CJM 2011 (vol 64 pp. 991)
Poisson Brackets with Prescribed Casimirs We consider the problem of constructing Poisson brackets on smooth
manifolds $M$ with prescribed Casimir functions. If $M$ is of even
dimension, we achieve our construction by considering a suitable
almost symplectic structure on $M$, while, in the case where $M$ is
of odd dimension, our objective is achieved by using a convenient
almost cosymplectic structure. Several examples and applications are
presented.
Keywords:Poisson bracket, Casimir function, almost symplectic structure, almost cosymplectic structure Categories:53D17, 53D15 |
18. CJM 2011 (vol 64 pp. 44)
Surfaces of Rotation with Constant Mean Curvature in the Direction of a Unitary Normal Vector Field in a Randers Space |
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 |
19. CJM 2011 (vol 63 pp. 878)
The Toric Geometry of Triangulated Polygons in Euclidean Spac Speyer and Sturmfels associated GrÃ¶bner toric
degenerations $\mathrm{Gr}_2(\mathbb{C}^n)^{\mathcal{T}}$
of $\mathrm{Gr}_2(\mathbb{C}^n)$ with each
trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations
induce toric
degenerations $M_{\mathbf{r}}^{\mathcal{T}}$ of $M_{\mathbf{r}}$, the
space of $n$ ordered, weighted (by $\mathbf{r}$) points on the projective line.
Our goal in this paper is to give a
geometric (Euclidean polygon) description of the toric fibers
and describe the action of the
compact part of the torus
as "bendings of polygons".
We prove the conjecture of Foth and Hu that
the toric fibers are homeomorphic
to the spaces defined by Kamiyama and Yoshida.
Categories:14L24, 53D20 |
20. CJM 2011 (vol 63 pp. 938)
AV-Courant Algebroids and Generalized CR Structures We construct a generalization of Courant algebroids that are
classified by the third cohomology group $H^3(A,V)$, where $A$ is a
Lie Algebroid, and $V$ is an $A$-module. We see that both Courant
algebroids and $\mathcal{E}^1(M)$ structures are examples of
them. Finally we introduce generalized CR structures on a manifold,
which are a generalization of generalized complex structures, and show
that every CR structure and contact structure is an example of a
generalized CR structure.
Category:53D18 |
21. CJM 2010 (vol 63 pp. 55)
Pseudolocality for the Ricci Flow and Applications
Perelman established a differential Li--Yau--Hamilton
(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\"ahler--Ricci flow on complete nonnegatively curved K\"ahler
manifolds.
Categories:53C44, 58J37, 35B35 |
22. CJM 2010 (vol 62 pp. 1264)
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 |
23. CJM 2010 (vol 62 pp. 1082)
The Fundamental Group of $S^1$-manifolds
We address the problem of computing the fundamental
group of a symplectic $S^1$-manifold for non-Hamiltonian actions on
compact manifolds, and for Hamiltonian actions on non-compact
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 |
24. CJM 2010 (vol 62 pp. 975)
Revisiting Tietze-Nakajima: 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 Condevaux--Dazord--Molino proof
of the Atiyah--Guillemin--Sternberg convexity theorem in symplectic geometry.
Categories:53D20, 52B99 |
25. CJM 2010 (vol 62 pp. 1037)
Riemann Extensions of Torsion-Free Connections with Degenerate Ricci Tensor
{Correspondence} between torsion-free connections with {nilpotent skew-symmetric curvature operator} and IP Riemann
extensions is shown. Some consequences are derived in the study of
four-dimensional IP metrics and locally homogeneous affine surfaces.
Keywords:Walker metric, Riemann extension, curvature operator, projectively flat and recurrent affine connection Categories:53B30, 53C50 |