Expand all Collapse all | Results 1 - 25 of 28 |
1. CJM Online first
Geometric Spectra and Commensurability The work of Reid, Chinburg-Hamilton-Long-Reid,
Prasad-Rapinchuk, and the author with Reid have demonstrated that
geodesics or totally geodesic submanifolds can sometimes be used to
determine the commensurability class of an arithmetic manifold. The
main results of this article show that generalizations of these
results to other arithmetic manifolds will require a wide range of
data. Specifically, we prove that certain incommensurable arithmetic
manifolds arising from the semisimple Lie groups of the form
$(\operatorname{SL}(d,\mathbf{R}))^r \times
(\operatorname{SL}(d,\mathbf{C}))^s$ have the same commensurability
classes of totally geodesic submanifolds coming from a fixed
field. This construction is algebraic and shows the failure of
determining, in general, a central simple algebra from subalgebras
over a fixed field. This, in turn, can be viewed in terms of forms of
$\operatorname{SL}_d$ and the failure of determining the form via certain classes of
algebraic subgroups.
Keywords:arithmetic groups, Brauer groups, arithmetic equivalence, locally symmetric manifolds Category:20G25 |
2. CJM 2013 (vol 66 pp. 141)
Existence of Taut Foliations on Seifert Fibered Homology $3$-spheres This paper concerns the problem of existence of taut foliations among $3$-manifolds.
Since the contribution of David Gabai,
we know that closed $3$-manifolds with non-trivial second homology group
admit a taut foliation.
The essential part of this paper focuses on Seifert fibered homology $3$-spheres.
The result is quite different if they are integral or rational but non-integral homology $3$-spheres.
Concerning integral homology $3$-spheres, we can see that all but the $3$-sphere and the PoincarÃ© $3$-sphere admit a taut foliation.
Concerning non-integral homology $3$-spheres,
we prove there are infinitely many which admit a taut foliation, and infinitely many without taut foliation.
Moreover, we show that the geometries do not determine the existence of taut foliations
on non-integral Seifert fibered homology $3$-spheres.
Keywords:homology 3-spheres, taut foliation, Seifert-fibered 3-manifolds Categories:57M25, 57M50, 57N10, 57M15 |
3. 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 |
4. 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 |
5. CJM 2012 (vol 65 pp. 621)
On Surfaces in Three Dimensional Contact Manifolds In this paper, we introduce two notions on a surface in a contact
manifold. The first one is called degree of transversality (DOT) which
measures the transversality between the tangent spaces of a surface
and the contact planes. The second quantity, called curvature of
transversality (COT), is designed to give a comparison principle for
DOT along characteristic curves under bounds on COT. In particular,
this gives estimates on lengths of characteristic curves assuming COT
is bounded below by a positive constant.
We show that surfaces with constant COT exist and we classify all graphs in the Heisenberg group with vanishing COT. This is accomplished by showing that the equation for graphs with zero COT can be decomposed into two first order PDEs, one of which is the backward invisicid Burgers' equation. Finally we show that the p-minimal graph equation in the Heisenberg group also has such a decomposition. Moreover, we can use this decomposition to write down an explicit formula of a solution near a regular point. Keywords:contact manifolds, subriemannian manifolds, surfaces Category:35R03 |
6. 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 |
7. CJM 2010 (vol 63 pp. 436)
Simplicial Complexes and Open Subsets of Non-Separable LF-Spaces
Let $F$ be a non-separable LF-space homeomorphic to
the direct sum $\sum_{n\in\mathbb{N}} \ell_2(\tau_n)$,
where $\aleph_0 < \tau_1 < \tau_2 < \cdots$.
It is proved that
every open subset $U$ of $F$ is homeomorphic to the product $|K| \times F$
for some locally finite-dimensional simplicial complex $K$ such that
every vertex $v \in K^{(0)}$ has the star $\operatorname{St}(v,K)$
with $\operatorname{card} \operatorname{St}(v,K)^{(0)} < \tau = \sup\tau_n$
(and $\operatorname{card} K^{(0)} \le \tau$),
and, conversely, if $K$ is such a simplicial complex,
then the product $|K| \times F$ can be embedded in $F$ as an open set,
where $|K|$ is the polyhedron of $K$ with the metric topology.
Keywords:LF-space, open set, simplicial complex, metric topology, locally finite-dimensional, star, small box product, ANR, $\ell_2(\tau)$, $\ell_2(\tau)$-manifold, open embedding, $\sum_{i\in\mathbb{N}}\ell_2(\tau_i)$ Categories:57N20, 46A13, 46T05, 57N17, 57Q05, 57Q40 |
8. CJM 2009 (vol 62 pp. 242)
A Second Order Smooth Variational Principle on Riemannian Manifolds We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.
Keywords:smooth variational principle, Riemannian manifold Categories:58E30, 49J52, 46T05, 47J30, 58B20 |
9. CJM 2009 (vol 62 pp. 218)
The General Definition of the Complex Monge--AmpÃ¨re Operator on Compact KÃ¤hler Manifolds We introduce a wide subclass ${\mathcal F}(X,\omega)$ of
quasi-plurisubharmonic functions in a compact KÃ¤hler manifold, on
which the complex Monge-AmpÃ¨re operator is well defined and the
convergence theorem is valid. We also prove that ${\mathcal F}(X,\omega)$
is a convex cone and includes all quasi-plurisubharmonic functions
that are in the Cegrell class.
Keywords:complex Monge--AmpÃ¨re operator, compact KÃ¤hler manifold Categories:32W20, 32Q15 |
10. CJM 2009 (vol 61 pp. 1201)
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 |
11. CJM 2009 (vol 61 pp. 641)
Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions |
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 |
12. CJM 2008 (vol 60 pp. 658)
Inverse Pressure Estimates and the Independence of Stable Dimension for Non-Invertible Maps We study the case of an Axiom A holomorphic non-degenerate
(hence non-invertible) map $f\from\mathbb P^2
\mathbb C \to \mathbb P^2 \mathbb C$, where $\mathbb P^2 \mathbb C$
stands for the complex
projective space of dimension 2. Let $\Lambda$ denote a basic set for
$f$ of unstable index 1, and $x$ an arbitrary point of $\Lambda$; we
denote by $\delta^s(x)$ the Hausdorff dimension of $W^s_r(x) \cap
\Lambda$, where $r$ is some fixed positive number and $W^s_r(x)$ is
the local stable manifold at $x$ of size $r$; $\delta^s(x)$ is called
\emph{the stable dimension at} $x$. Mihailescu and
Urba\'nski introduced a notion of inverse topological pressure,
denoted by $P^-$, which takes into consideration preimages of points.
Manning and McCluskey study the case of hyperbolic diffeomorphisms on
real surfaces and give formulas for Hausdorff dimension. Our
non-invertible situation is different here since the local unstable
manifolds are not uniquely determined by their base point, instead
they depend in general on whole prehistories of the base points. Hence
our methods are different and are based on using a sequence of inverse
pressures for the iterates of $f$, in order to give upper and lower
estimates of the stable dimension. We obtain an estimate of the
oscillation of the stable dimension on $\Lambda$. When each point $x$
from $\Lambda$ has the same number $d'$ of preimages in $\Lambda$,
then we show that $\delta^s(x)$ is independent
of $x$; in fact $\delta^s(x)$ is shown to be equal in this case with
the unique zero of the map $t \to P(t\phi^s - \log d')$. We also
prove the Lipschitz continuity of the stable vector spaces over
$\Lambda$; this proof is again different than the one for
diffeomorphisms (however, the unstable distribution is not always
Lipschitz for conformal non-invertible maps). In the end we include
the corresponding results for a real conformal setting.
Keywords:Hausdorff dimension, stable manifolds, basic sets, inverse topological pressure Categories:37D20, 37A35, 37F35 |
13. CJM 2007 (vol 59 pp. 1245)
On Gap Properties and Instabilities of $p$-Yang--Mills Fields We consider the
$p$-Yang--Mills 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 $p$-Yang--Mills
connections, and the associated curvature $\rn$ the $p$-Yang--Mills
fields. In this paper, we prove gap properties and instability theorems for $p$-Yang--Mills
fields over submanifolds in $\mathbb{R}^{n+k}$ and $\mathbb{S}^{n+k}$.
Keywords:$p$-Yang--Mills field, gap property, instability, submanifold Categories:58E15, 53C05 |
14. CJM 2007 (vol 59 pp. 845)
Representations of the Fundamental Group of an $L$-Punctured Sphere Generated by Products of Lagrangian Involutions |
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 fixed-point set of an
anti-symplectic involution defined on the moduli space
$\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixed-point set is
non-empty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use
the quasi-Hamiltonian description of the symplectic structure of $\Mod$ and
give conditions on an involution defined on a quasi-Hamiltonian $U$-space
$(M, \w, \mu\from M \to U)$ for it to induce an anti-symplectic involution on
the reduced space $M/\!/U := \mu^{-1}(\{1\})/U$.
Keywords:momentum maps, moduli spaces, Lagrangian submanifolds, anti-symplectic involutions, quasi-Hamiltonian Categories:53D20, 53D30 |
15. CJM 2007 (vol 59 pp. 742)
Geometry and Spectra of Closed Extensions of Elliptic Cone Operators We study the geometry of the set of closed extensions of index $0$ of
an elliptic differential cone operator and its model operator in
connection with the spectra of the extensions, and we give a necessary
and sufficient condition for the existence of rays of minimal growth
for such operators.
Keywords:resolvents, manifolds with conical singularities, spectral theor, boundary value problems, Grassmannians Categories:58J50, 35J70, 14M15 |
16. CJM 2007 (vol 59 pp. 36)
Classification of Ding's Schubert Varieties: Finer Rook Equivalence K.~Ding studied a class of Schubert varieties $X_\lambda$
in type A partial
flag manifolds, indexed by
integer partitions $\lambda$ and in bijection
with dominant permutations. He observed that the
Schubert cell structure of $X_\lambda$ is indexed by maximal rook
placements on the Ferrers board $B_\lambda$, and that the
integral cohomology groups $H^*(X_\lambda;\:\Zz)$, $H^*(X_\mu;\:\Zz)$ are
additively isomorphic exactly when the Ferrers boards $B_\lambda, B_\mu$
satisfy the combinatorial condition of \emph{rook-equivalence}.
We classify the varieties $X_\lambda$ up to isomorphism, distinguishing them
by their graded cohomology rings with integer coefficients. The crux of our approach
is studying the nilpotence orders of linear forms in
the cohomology ring.
Keywords:Schubert variety, rook placement, Ferrers board, flag manifold, cohomology ring, nilpotence Categories:14M15, 05E05 |
17. CJM 2005 (vol 57 pp. 1314)
Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra In 1999 V. Arnol'd introduced the local contact algebra: studying the
problem of classification of singular curves in a contact space, he
showed the existence of the ghost of the contact structure (invariants
which are not related to the induced structure on the curve). Our
main result implies that the only reason for existence of the local
contact algebra and the ghost is the difference between the geometric
and (defined in this paper) algebraic restriction of a $1$-form to a
singular submanifold. We prove that a germ of any subset $N$ of a
contact manifold is well defined, up to contactomorphisms, by the
algebraic restriction to $N$ of the contact structure. This is a
generalization of the Darboux-Givental' theorem for smooth
submanifolds of a contact manifold. Studying the difference between
the geometric and the algebraic restrictions gives a powerful tool for
classification of stratified submanifolds of a contact manifold. This
is illustrated by complete solution of three classification problems,
including a simple explanation of V.~Arnold's results and further
classification results for singular curves in a contact space. We
also prove several results on the external geometry of a singular
submanifold $N$ in terms of the algebraic restriction of the contact
structure to $N$. In particular, the algebraic restriction is zero if
and only if $N$ is contained in a smooth Legendrian submanifold of
$M$.
Keywords:contact manifold, local contact algebra,, relative Darboux theorem, integral curves Categories:53D10, 14B05, 58K50 |
18. CJM 2005 (vol 57 pp. 871)
Hermitian Yang-_Mills--Higgs Metrics on\\Complete KÃ¤hler Manifolds In this paper, first, we will investigate the
Dirichlet problem for one type of vortex equation, which
generalizes the well-known Hermitian Einstein equation. Secondly,
we will give existence results for solutions of these vortex
equations over various complete noncompact K\"ahler manifolds.
Keywords:vortex equation, Hermitian Yang--Mills--Higgs metric,, holomorphic vector bundle, KÃ¤hler manifolds Categories:58E15, 53C07 |
19. CJM 2005 (vol 57 pp. 771)
The Resolvent of Closed Extensions of Cone Differential Operators We study closed extensions $\underline A$ of
an elliptic differential operator $A$ on a manifold with conical
singularities, acting as an unbounded operator on a weighted $L_p$-space.
Under suitable conditions we show that the resolvent
$(\lambda-\underline A)^{-1}$ exists
in a sector of the complex plane and decays like $1/|\lambda|$ as
$|\lambda|\to\infty$. Moreover, we determine the structure of the resolvent
with enough precision to guarantee existence and boundedness of imaginary
powers of $\underline A$.
As an application we treat the Laplace--Beltrami operator for a metric with
straight conical degeneracy and describe domains yielding
maximal regularity for the Cauchy problem $\dot{u}-\Delta u=f$, $u(0)=0$.
Keywords:Manifolds with conical singularities, resolvent, maximal regularity Categories:35J70, 47A10, 58J40 |
20. CJM 2004 (vol 56 pp. 776)
Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices We explicitly describe
the best approximation in
geodesic submanifolds of positive definite matrices
obtained from involutive
congruence transformations on the
Cartan-Hadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of
positive definite matrices.
An explicit calculation for the minimal distance
function from the geodesic submanifold
${\mathrm{Sym}}(p,{\mathbb R})^{++}\times
{\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in
${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is
given in terms of metric and
spectral geometric means, Cayley transform, and Schur
complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$
Keywords:Matrix approximation, positive, definite matrix, geodesic submanifold, Cartan-Hadamard manifold,, best approximation, minimal distance function, global tubular, neighborhood theorem, Schur complement, metric and spectral, geometric mean, Cayley transform Categories:15A48, 49R50, 15A18, 53C3 |
21. CJM 2002 (vol 54 pp. 1254)
Effective Actions of the Unitary Group on Complex Manifolds We classify all connected $n$-dimensional complex manifolds admitting
effective actions of the unitary group $U_n$ by biholomorphic
transformations. One consequence of this classification is a
characterization of $\CC^n$ by its automorphism group.
Keywords:complex manifolds, group actions Categories:32Q57, 32M17 |
22. CJM 2001 (vol 53 pp. 715)
Differential Structure of Orbit Spaces We present a new approach to singular reduction of Hamiltonian systems
with symmetries. The tools we use are the category of differential
spaces of Sikorski and the Stefan-Sussmann theorem. The former is
applied to analyze the differential structure of the spaces involved
and the latter is used to prove that some of these spaces are smooth
manifolds.
Our main result is the identification of accessible sets of the
generalized distribution spanned by the Hamiltonian vector fields of
invariant functions with singular reduced spaces. We are also able
to describe the differential structure of a singular reduced space
corresponding to a coadjoint orbit which need not be locally closed.
Keywords:accessible sets, differential space, Poisson algebra, proper action, singular reduction, symplectic manifolds Categories:37J15, 58A40, 58D19, 70H33 |
23. CJM 2001 (vol 53 pp. 780)
Seiberg-Witten Invariants of Lens Spaces We show that the Seiberg-Witten invariants of a lens space determine
and are determined by its Casson-Walker invariant and its
Reidemeister-Turaev torsion.
Keywords:lens spaces, Seifert manifolds, Seiberg-Witten invariants, Casson-Walker invariant, Reidemeister torsion, eta invariants, Dedekind-Rademacher sums Categories:58D27, 57Q10, 57R15, 57R19, 53C20, 53C25 |
24. CJM 2001 (vol 53 pp. 278)
Darboux Transformations for the KP Hierarchy in the Segal-Wilson Setting In this paper it is shown that inclusions inside the Segal-Wilson
Grassmannian give rise to Darboux transformations between the
solutions of the $\KP$ hierarchy corresponding to these planes. We
present a closed form of the operators that procure the transformation
and express them in the related geometric data. Further the
associated transformation on the level of $\tau$-functions is given.
Keywords:KP hierarchy, Darboux transformation, Grassmann manifold Categories:22E65, 22E70, 35Q53, 35Q58, 58B25 |
25. CJM 2001 (vol 53 pp. 212)
Group Actions and Codes A $\mathbb{Z}_2$-action with ``maximal number of isolated fixed
points'' ({\it i.e.}, with only isolated fixed points such that
$\dim_k (\oplus_i H^i(M;k)) =|M^{\mathbb{Z}_2}|, k = \mathbb{F}_2)$
on a $3$-dimensional, closed manifold determines a binary self-dual
code of length $=|M^{\mathbb{Z}_2}|$. In turn this code determines
the cohomology algebra $H^*(M;k)$ and the equivariant cohomology
$H^*_{\mathbb{Z}_2}(M;k)$. Hence, from results on binary self-dual
codes one gets information about the cohomology type of $3$-manifolds
which admit involutions with maximal number of isolated fixed points.
In particular, ``most'' cohomology types of closed $3$-manifolds do
not admit such involutions. Generalizations of the above result are
possible in several directions, {\it e.g.}, one gets that ``most''
cohomology types (over $\mathbb{F}_2)$ of closed $3$-manifolds do
not admit a non-trivial involution.
Keywords:Involutions, $3$-manifolds, codes Categories:55M35, 57M60, 94B05, 05E20 |