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1. 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 |
2. 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 |
3. 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 |
4. 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 |