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1. CJM Online first
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 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 |
3. 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 |
4. 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 |
5. 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 |
6. 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 |
7. 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 |
8. 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 |
9. 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 |
10. 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 |
11. 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 |
12. 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 |
13. CJM 2005 (vol 57 pp. 506)
Reverse Hypercontractivity for Subharmonic Functions Contractivity and hypercontractivity properties of semigroups
are now well understood when the generator, $A$, is a Dirichlet form
operator.
It has been shown that in some holomorphic function spaces the
semigroup operators, $e^{-tA}$, can be bounded {\it below} from
$L^p$ to $L^q$ when $p,q$ and $t$ are suitably related.
We will show that such lower boundedness occurs also in spaces
of subharmonic functions.
Keywords:Reverse hypercontractivity, subharmonic Categories:58J35, 47D03, 47D07, 32Q99, 60J35 |
14. CJM 2005 (vol 57 pp. 251)
Some New Results on $L^2$ Cohomology of Negatively Curved Riemannian Manifolds The present paper is concerned with the study of the $L^2$ cohomology
spaces of negatively curved manifolds. The first half presents a
finiteness and vanishing result obtained under some curvature
assumptions, while the second half identifies a class of metrics
having non-trivial $L^2$ cohomology for degree equal to the half
dimension of the space. For the second part we rely on the existence
and regularity properties of the solution for the heat equation for
forms.
Category:58J50 |
15. CJM 2005 (vol 57 pp. 225)
Unbounded Fredholm Operators and Spectral Flow We study the gap (= ``projection norm'' = ``graph distance'') topology
of the space of all (not necessarily bounded) self-adjoint Fredholm
operators in a separable Hilbert space by the Cayley transform and
direct methods. In particular, we show the surprising result that
this space is connected in contrast to the bounded case. Moreover, we
present a rigorous definition of spectral flow of a path of such
operators (actually alternative but mutually equivalent definitions)
and prove the homotopy invariance. As an example, we discuss operator
curves on manifolds with boundary.
Categories:58J30, 47A53, 19K56, 58J32 |
16. CJM 2005 (vol 57 pp. 99)
Second Order Operators on a Compact Lie Group We describe the structure of the space of second order elliptic
differential operators on a homogenous bundle over a compact Lie
group. Subject to a technical condition, these operators are
homotopic to the Laplacian. The technical condition is further
investigated, with examples given where it holds and others where
it does not. Since many spectral invariants are also homotopy
invariants, these results provide information about the invariants
of these operators.
Categories:58J70, 43A77 |
17. CJM 2004 (vol 56 pp. 638)
Multisymplectic Reduction for Proper Actions We consider symmetries of the Dedonder equation arising from
variational problems with partial derivatives. Assuming a proper
action of the symmetry group, we identify a set of reduced equations
on an open dense subset of the domain of definition of the fields
under consideration. By continuity, the Dedonder equation is
satisfied whenever the reduced equations are satisfied.
Keywords:Dedonder equation, multisymplectic structure, reduction,, symmetries, variational problems Categories:58J70, 35A30 |
18. CJM 2004 (vol 56 pp. 590)
The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary We study the Riemannian Laplace-Beltrami operator $L$ on a Riemannian
manifold with Heisenberg group $H_1$ as boundary. We calculate the heat
kernel and Green's function for $L$, and give global and small time
estimates of the heat kernel. A class of hypersurfaces in this
manifold can be regarded as approximations of $H_1$. We also restrict
$L$ to each hypersurface and calculate the corresponding heat kernel
and Green's function. We will see that the heat kernel and Green's
function converge to the heat kernel and Green's function on the
boundary.
Categories:35H20, 58J99, 53C17 |
19. CJM 2002 (vol 54 pp. 1086)
Combinatorics of the Heat Trace on Spheres We present a concise explicit expression for the heat trace
coefficients of spheres. Our formulas yield certain combinatorial
identities which are proved following ideas of D.~Zeilberger. In
particular, these identities allow to recover in a surprising way
some known formulas for the heat trace asymptotics. Our approach is
based on a method for computation of heat invariants developed in [P].
Categories:05A19, 58J35 |
20. CJM 2000 (vol 52 pp. 695)
Correspondences, von Neumann Algebras and Holomorphic $L^2$ Torsion Given a holomorphic Hilbertian bundle on a compact complex manifold, we
introduce the notion of holomorphic $L^2$ torsion, which lies in the
determinant line of the twisted $L^2$ Dolbeault cohomology and
represents a volume element there. Here we utilise the theory of
determinant lines of Hilbertian modules over finite von~Neumann
algebras as developed in \cite{CFM}. This specialises to the
Ray-Singer-Quillen holomorphic torsion in the finite dimensional case.
We compute a metric variation formula for the holomorphic $L^2$
torsion, which shows that it is {\it not\/} in general independent of
the choice of Hermitian metrics on the complex manifold and on the
holomorphic Hilbertian bundle, which are needed to define it. We
therefore initiate the theory of correspondences of determinant lines,
that enables us to define a relative holomorphic $L^2$ torsion for a
pair of flat Hilbertian bundles, which we prove is independent of the
choice of Hermitian metrics on the complex manifold and on the flat
Hilbertian bundles.
Keywords:holomorphic $L^2$ torsion, correspondences, local index theorem, almost KÃ¤hler manifolds, von~Neumann algebras, determinant lines Categories:58J52, 58J35, 58J20 |