Expand all Collapse all | Results 1 - 25 of 70 |
1. 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 |
2. CJM 2012 (vol 65 pp. 879)
A Space of Harmonic Maps from the Sphere into the Complex Projective Space Guest-Ohnita and Crawford have shown the path-connectedness of the
space of harmonic maps from $S^2$ to $\mathbf{C} P^n$
of a fixed degree and energy.It is well-known that the $\partial$ transform is defined on this space.
In this paper,we will show that the space is decomposed into mutually disjoint connected subspaces on which
$\partial$ is homeomorphic.
Keywords:harmonic maps, harmonic sequences, gluing Categories:58E20, 58D15 |
3. CJM 2012 (vol 65 pp. 1255)
Variations of Integrals in Diffeology We establish the formula for the variation of
integrals of differential forms on cubic chains, in the
context of diffeological spaces. Then, we establish the diffeological version of Stoke's
theorem, and we apply that to get the diffeological variant of the
Cartan-Lie formula. Still in the context of Cartan-De-Rham calculus
in diffeology, we
construct a Chain-Homotopy Operator $\mathbf K$ we apply it here to
get the homotopic invariance of De Rham cohomology for
diffeological spaces. This is the Chain-Homotopy Operator which used in
symplectic diffeology to construct the Moment Map.
Keywords:diffeology, differential geometry, Cartan-De-Rham calculus Categories:58A10, 58A12, 58A40 |
4. CJM 2012 (vol 65 pp. 544)
Iterated Integrals and Higher Order Invariants We show that higher order invariants of smooth functions can be
written as linear combinations of full invariants times iterated
integrals.
The non-uniqueness of such a presentation is captured in the kernel of
the ensuing map from the tensor product. This kernel is computed
explicitly.
As a consequence, it turns out that higher order invariants are a free
module of the algebra of full invariants.
Keywords:higher order forms, iterated integrals Categories:14F35, 11F12, 55D35, 58A10 |
5. CJM 2011 (vol 64 pp. 924)
Rectifiability of Optimal Transportation Plans The regularity of solutions to optimal transportation problems has become
a hot topic in current research. It is well known by now that the optimal measure
may not be concentrated on the graph of a continuous mapping unless both the transportation
cost and the masses transported satisfy very restrictive hypotheses (including sign conditions
on the mixed fourth-order derivatives of the cost function).
The purpose of this note is to show that in spite of this,
the optimal measure is supported on a Lipschitz manifold, provided only
that the cost is $C^{2}$ with non-singular mixed second derivative.
We use this result to provide a simple proof that solutions to Monge's
optimal transportation problem satisfy a change of variables equation
almost everywhere.
Categories:49K20, 49K60, 35J96, 58C07 |
6. CJM 2011 (vol 63 pp. 721)
Isoresonant Complex-valued Potentials and Symmetries Let $X$ be a connected Riemannian manifold such that the resolvent of
the free Laplacian $(\Delta-z)^{-1}$, $z\in\mathbb{C} \setminus
\mathbb{R}^+$, has a meromorphic continuation
through $\mathbb{R}^+$. The poles of this continuation are called
resonances. When $X$ has some symmetries, we construct complex-valued
potentials, $V$, such that the resolvent of $\Delta+V$, which has also
a meromorphic continuation, has the same resonances with
multiplicities as the free Laplacian.
Categories:31C12, 58J50 |
7. 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 |
8. 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 |
9. CJM 2010 (vol 62 pp. 1325)
On Some Explicit Constructions of Finsler Metrics with Scalar Flag Curvature
We give an explicit construction of polynomial (\emph{of
arbitrary degree}) $(\alpha,\beta)$-metrics with scalar flag curvature
and determine their scalar flag curvature. These Finsler metrics
contain all non-trivial projectively flat $(\alpha,\beta)$-metrics of
constant flag curvature.
Keywords:Finsler metric, scalar curvature, projective flatness Category:58E20 |
10. CJM 2009 (vol 62 pp. 52)
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 Riemann--Finsler 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 non-Riemannian reversible weakly
symmetric Finsler spaces we find are non-Berwaldian and with vanishing
S-curvature. This means that reversible non-Berwaldian Finsler spaces
with vanishing S-curvature 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, S-curvature Categories:53C60, 58B20, 22E46, 22E60 |
11. 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 |
12. CJM 2009 (vol 61 pp. 548)
Fundamental Tone, Concentration of Density, and Conformal Degeneration on Surfaces We study the effect of two types of degeneration of a Riemannian
metric on the first eigenvalue of the Laplace operator on
surfaces. In both cases we prove that the first eigenvalue of the
round sphere is an optimal asymptotic upper bound. The first type of
degeneration is concentration of the density to a point within a
conformal class. The second is degeneration of the
conformal class to the boundary of the moduli space on the torus and
on the Klein bottle. In the latter, we follow the outline proposed
by N. Nadirashvili in 1996.
Categories:35P, 58J |
13. CJM 2008 (vol 60 pp. 1336)
Moving Frames for Lie Pseudo--Groups We propose a new, constructive theory of moving frames for Lie
pseudo-group actions on submanifolds. The moving frame provides an
effective means for determining complete systems of differential
invariants and invariant differential forms, classifying their
syzygies and recurrence relations, and solving equivalence and
symmetry problems arising in a broad range of applications.
Categories:58A15, 58A20, 58H05, 58J70 |
14. CJM 2008 (vol 60 pp. 1149)
Conjugate Reciprocal Polynomials with All Roots on the Unit Circle We study the geometry, topology and Lebesgue measure of the set of
monic conjugate reciprocal polynomials of fixed degree with all
roots on the unit circle. The set of such polynomials of degree $N$
is naturally associated to a subset of $\R^{N-1}$. We calculate
the volume of this set, prove the set is homeomorphic to the $N-1$
ball and that its isometry group is isomorphic to the dihedral
group of order $2N$.
Categories:11C08, 28A75, 15A52, 54H10, 58D19 |
15. CJM 2008 (vol 60 pp. 822)
Maximum Principles for Subharmonic Functions Via Local Semi-Dirichlet Forms Maximum principles for subharmonic
functions in the framework of quasi-regular local semi-Dirichlet
forms admitting lower bounds are presented.
As applications, we give
weak and strong maximum principles
for (local) subsolutions of a second order elliptic
differential operator on the domain of Euclidean space under conditions on coefficients,
which partially generalize the results by Stampacchia.
Keywords:positivity preserving form, semi-Dirichlet form, Dirichlet form, subharmonic functions, superharmonic functions, harmonic functions, weak maximum principle, strong maximum principle, irreducibility, absolute continuity condition Categories:31C25, 35B50, 60J45, 35J, 53C, 58 |
16. CJM 2008 (vol 60 pp. 572)
Non-Selfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point This is the third in a series of works devoted to spectral
asymptotics for non-selfadjoint
perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, having a
periodic classical flow. Assuming that the strength $\epsilon$
of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$
(and may sometimes reach even smaller values), we
get an asymptotic description of the eigenvalues in rectangles
$[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point
value of the flow average of the leading perturbation.
Keywords:non-selfadjoint, eigenvalue, periodic flow, branching singularity Categories:31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40 |
17. CJM 2008 (vol 60 pp. 457)
Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure We define sets with finitely ramified cell structure, which are
generalizations of post-crit8cally finite self-similar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local self-similarity, and allow countably many cells
connected at each junction point.
In particular, we consider post-critically infinite fractals.
We prove that if Kigami's resistance form
satisfies certain assumptions, then there exists a weak Riemannian metric
such that the energy can be expressed as the integral of the norm squared
of a weak gradient with respect to an energy measure.
Furthermore, we prove that if such a set can be homeomorphically represented
in harmonic coordinates, then for smooth functions the weak gradient can be
replaced by the usual gradient.
We also prove a simple formula for the energy measure Laplacian in harmonic
coordinates.
Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18 |
18. CJM 2008 (vol 60 pp. 297)
Transitive Factorizations in the Hyperoctahedral Group The classical Hurwitz enumeration problem has a presentation in terms of
transitive factorizations in the symmetric group. This presentation suggests
a generalization from type~$A$ to other
finite reflection groups and, in particular, to type~$B$.
We study this generalization both from a combinatorial and a geometric
point of view, with the prospect of providing a means of understanding more
of the structure of the moduli spaces of maps with an $\gS_2$-symmetry.
The type~$A$ case has been well studied and connects Hurwitz numbers
to the moduli space of curves. We conjecture an analogous setting for the
type~$B$ case that is studied here.
Categories:05A15, 14H10, 58D29 |
19. CJM 2008 (vol 60 pp. 241)
Semi-Classical Wavefront Set and Fourier Integral Operators Here we define and prove some properties of the semi-classical
wavefront set. We also define and study semi-classical Fourier
integral operators and prove a generalization of Egorov's theorem to
manifolds of different dimensions.
Keywords:wavefront set, Fourier integral operators, Egorov theorem, semi-classical analysis Categories:35S30, 35A27, 58J40, 81Q20 |
20. 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 |
21. CJM 2007 (vol 59 pp. 943)
A Weighted $L^2$-Estimate of the Witten Spinor in Asymptotically Schwarzschild Manifolds We derive a weighted $L^2$-estimate of the Witten spinor in
a complete Riemannian spin manifold~$(M^n, g)$ of non-negative scalar curvature
which is asymptotically Schwarzschild.
The interior geometry of~$M$ enters this estimate only
via the lowest eigenvalue of the square of the Dirac
operator on a conformal compactification of $M$.
Categories:83C60, 35Q75, 35J45, 58J05 |
22. 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 |
23. CJM 2007 (vol 59 pp. 712)
Jet Modules In this paper we classify indecomposable modules for the Lie algebra
of vector fields on a torus that admit a compatible action of the algebra
of functions. An important family of such modules is given by spaces of jets
of tensor fields.
Categories:17B66, 58A20 |
24. CJM 2007 (vol 59 pp. 225)
Harmonic Analysis on Metrized Graphs This paper studies the Laplacian operator on a metrized graph, and its
spectral theory.
Keywords:metrized graph, harmonic analysis, eigenfunction Categories:43A99, 58C40, 05C99 |
25. CJM 2006 (vol 58 pp. 381)
Extremal Metric for the First Eigenvalue on a Klein Bottle The first eigenvalue of the Laplacian on a surface can be viewed
as a functional on the space of Riemannian metrics of a given
area. Critical points of this functional are called extremal
metrics. The only known extremal metrics are a round sphere, a
standard projective plane, a Clifford torus and an equilateral
torus. We construct an extremal metric on a Klein bottle. It is a
metric of revolution, admitting a minimal isometric embedding into
a sphere ${\mathbb S}^4$ by the first eigenfunctions. Also, this
Klein bottle is a bipolar surface for Lawson's
$\tau_{3,1}$-torus. We conjecture that an extremal metric for the
first eigenvalue on a Klein bottle is unique, and hence it
provides a sharp upper bound for $\lambda_1$ on a Klein bottle of
a given area. We present numerical evidence and prove the first
results towards this conjecture.
Keywords:Laplacian, eigenvalue, Klein bottle Categories:58J50, 53C42 |