TY - JOUR
TI - Isoperimetric Inequalities for the Magnetic Neumann and Steklov Problems with Aharonov–Bohm Magnetic Potential
UR - https://link.springer.com/article/10.1007/s12220-022-01001-2
LA - en
AU - Colbois, B.
AU - Provenzano, L.
AU - Savo, A.
PY - 2022
DA - 14.9
T2 - The Journal of Geometric Analysis
VL - 32
SP - 1
EP - 38
ER -
TY - JOUR
TI - Neumann Eigenvalues of the Biharmonic Operator on Domains: Geometric Bounds and Related Results
LA - en
AU - Colbois, B.
PY - 2022
DA - 4.5
T2 - J. Geom. Anal.
VL - 32
SP - 1
EP - 58
ER -
TY - JOUR
TI - Conformal upper bounds for the eigenvalues of the p-Laplacian
LA - en
AU - Colbois, B.
AU - Provenzano, L.
PY - 2021
DA - 4.12
T2 - J. Lond. Math. Soc. (2)
VL - 104
SP - 2128
EP - 2147
ER -
TY - JOUR
TI - Sharp Steklov upper bound for submanifolds of revolution
LA - en
AU - Colbois, B.
AU - Verma, S.
PY - 2021
DA - 10.5
T2 - Journal of Geometric Analysis
IS - 11
VL - 31
SP - 11214
EP - 11225
ER -
TY - JOUR
TI - Upper bounds for Steklov eigenvalues of submanifolds in Euclidean space via the intersection index
LA - en
AU - Colbois, B.
AU - Gittins, K. .
PY - 2021
DA - 18.6
T2 - Differential Geometry and its Applications
VL - 78
SP - 1
EP - 21
ER -
TY - JOUR
TI - Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains
UR - https://link.springer.com/article/10.1007%2Fs10455-021-09759-4
LA - en
AU - Colbois, B.
AU - Savo, A.
PY - 2021
DA - 15.3
T2 - Ann. Global Anal. Geom.
VL - 60
SP - 1
EP - 18
ER -
TY - JOUR
TI - Lower bounds for the first eigenvalue of the Laplacian with zero magnetic field in planar domains
LA - en
AU - Colbois, B.
AU - Savo, A.
PY - 2021
DA - 26.3
T2 - Journal of Functional Analysis
VL - 281
SP - 1
EP - 32
ER -
TY - JOUR
TI - The Steklov and Laplacian spectra of Riemannian manifolds with boundary
UR - https://doi.org/10.1016/j.jfa.2019.108409
LA - en
AU - Colbois, B.
AU - Girouard, A.
AU - Hassannezhad, A.
PY - 2020
DA - 1.4
AB - Given two compact Riemannian manifolds $M_1$ and $M_2$ such that their respective boundaries $\Sigma_1$ and $\Sigma_2$ admit neighbourhoods $\Omega_1$ and $\Omega_2$ which are isometric, we prove the existence of a constant $C$ such that $|\sigma_k(M_1)-\sigma_k(M_2)|\leq C$ for each $k\in\N$. The constant $C$ depends only on the geometry of $\Omega_1\cong\Omega_2$. This follows from a quantitative relationship between the Steklov eigenvalues $\sigma_k$ of a compact Riemannian manifold $M$ and the eigenvalues $\lambda_k$ of the Laplacian on
its boundary. Our main result states that the difference $|\sigma_k-\sqrt{\lambda_k}|$ is bounded above by a constant which depends on the geometry of $M$ only in a neighbourhood of its boundary.
The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant $C$ is given explicitly in terms of bounds on the geometry of $\Omega_1\cong\Omega_2$.
T2 - Journal of Functional Analysis
IS - 6
VL - 278
SP - 1
EP - 32
ER -
TY - JOUR
TI - Hypersurfaces with prescribed boundary and small Steklov eigenvalues
UR - https://doi.org/10.4153/S000843951900050X
LA - en
AU - Colbois, B.
AU - Girouard, A.
AU - Métras, A.
PY - 2020
DA - 17.1
AB - iven a smooth compact hypersurface $M$ with boundary $\Sigma=\partial M$, we prove the existence of a sequence $M_j$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $\sigma_k(M_j)$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_j$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$, while the principal curvatures of the boundary remain unchanged.
T2 - Canadian Mathematics Bulletin
VL - 63
SP - 46
EP - 57
ER -
TY - JOUR
TI - Compact manifolds with fixed boundary and large Steklov eigenvalues
UR - https://doi.org/10.1090/proc/14426
LA - en
AU - Colbois, B.
AU - El Soufi, A.
AU - Girouard, A.
PY - 2019
DA - 22.8
AB - Let $(M,g)$ be a compact Riemannian manifold with boundary. Let $b>0$ be the number of connected components of its boundary. For manifolds of dimension $\geq 3$, we prove that for $j=b+1$ it is possible to obtain an arbitrarily large Steklov eigenvalue $\sigma_j(M,e^\delta g)$ using a conformal perturbation $\delta\in C^\infty(M)$ which is supported in a thin neighbourhood of the boundary, with $\delta=0$ on the boundary. For $j\leq b$, it is also possible to obtain arbitrarily large eigenvalues, but the conformal factor must spread throughout the interior of $M$. This is in stark contrast with the situation for the eigenvalues of the Laplace operator, for which the supremum is bounded in each fixed conformal class. In fact, when working in a fixed conformal class, it is known that the volume of $(M,e^\delta g)$ has to tend to infinity in order for some $\sigma_j$ to become arbitrarily large. We also prove that it is possible to obtain large eigenvalues while keeping different boundary components arbitrarily close to each others, by constructing a convenient Riemannian submersion.
T2 - Proc. Amer. Math. Soc.
IS - 9
VL - 147
SP - 3813
EP - 3827
ER -
TY - JOUR
TI - Steklov Eigenvalues of Submanifolds with Prescribed Boundary in Euclidean Space
UR - https://doi.org/10.1007/s12220-018-0063-x
KW - Steklov problem, Euclidean space, prescribed boundary, manifolds, hypersurfaces of revolution
LA - en
AU - Colbois, B.
AU - Girouard, A.
AU - Gittins, K.
PY - 2019
DA - 4.4
AB - We obtain upper and lower bounds for Steklov eigenvalues of
submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: we prove
that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is Steklov isospectral to the disk.
T2 - The Journal of Geometric Analysis
IS - 2
VL - 29
SP - 1811
EP - 1834
ER -
TY - JOUR
TI - Spectrum of the Laplacian with weights
UR - https://doi.org/10.1007/s10455-018-9621-5
KW - eigenvalue, Laplacian, density, Cheeger inequality, upper bounds
LA - en
AU - Colbois, B.
AU - El Soufi, A.
PY - 2019
DA - 4.3
AB - Given a compact Riemannian manifold $(M,g)$ and two positive functions $\rho$ and $\sigma$, we are interested in the eigenvalues of the Dirichlet energy functional weighted by $\sigma$, with respect to the $L^2$ inner product weighted by $\rho$. Under some regularity conditions on $\rho$ and $\sigma$, these eigenvalues are those of the operator
$-\rho^{-1} \mbox{div}(\sigma \nabla u)$
with Neumann conditions on the boundary if $\partial M\ne \emptyset$.
We investigate the effect of the weights on eigenvalues and
discuss the existence of lower and upper bounds under the condition that the total mass is preserved.
T2 - Annals of Global Analysis and Geometry
IS - 2
VL - 55
SP - 149
EP - 180
ER -
TY - JOUR
TI - The Steklov spectrum and coarse discretizations of manifolds with boundary
UR - https://dx.doi.org/10.4310/PAMQ.2018.v14.n2.a3
LA - en
AU - Colbois, B.
AU - Girouard, A.
AU - Raveendran, B.
PY - 2018
DA - 22.8
AB - Given $\kappa, r_0>0$ and $n\in\N$, we consider the class
$\mathcal{M}=\mathcal{M}(\kappa,r_0,n)$ of compact $n$-dimensional
Riemannian manifolds with cylindrical boundary, Ricci curvature
bounded below by $-(n-1)\kappa$ and injectivity radius bounded below
by $r_0$ away from the boundary. For a manifold $M\in\mathcal{M}$ we introduce a notion of
discretization, leading to a graph with boundary which is roughly
isometric to $M$, with constants depending only on $\kappa,r_0,n$. In
this context, we prove a uniform spectral comparison inequality
between the Steklov eigenvalues of a manifold $M\in\mathcal{M}$ and
those of its discretization. Some applications to the construction of
sequences of surfaces with boundary of fixed length and with
arbitrarily large
Steklov spectral gap $\sigma_2-\sigma_1$ are given. In particular, we obtain such a
sequence for surfaces with connected boundary. The applications are
based on the construction of graph-like surfaces which are obtained from
sequences of graphs with good expansion properties.
T2 - Pure and Applied Mathematics Quarterly,
IS - 2
VL - 14
SP - 357
EP - 392
ER -
TY - JOUR
TI - Lower bounds for the first eigenvalue of the magnetic Laplacian
UR - https://doi.org/10.1016/j.jfa.2018.02.012
LA - en
AU - Colbois, B.
AU - Savo, A.
PY - 2018
DA - 17.5
AB - We consider a Riemannian cylinder $\Omega$ endowed with a closed potential $1$-form $A$ and study the magnetic Laplacian $\Delta_A$ with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate.
T2 - Journal of Functional Analysis
IS - 10
VL - 274
SP - 2818
EP - 2845
ER -
TY - JOUR
TI - Eigenvalues of elliptic operators with density
UR - https://doi.org/10.1007/s00526-018-1307-0
LA - en
AU - Colbois, B.
AU - Provenzano, L.
PY - 2018
DA - 17.5
AB - We consider eigenvalue problems for elliptic operators of arbitrary order $2m$ subject to Neumann boundary conditions on bounded domains of the Euclidean $N$-dimensional space. We study the dependence of the eigenvalues upon variations of mass density. In particular we discuss the existence and characterization of upper and lower bounds under both the condition that the total mass is fixed and the condition that the $L^{\frac{N}{2m}}$-norm of the density is fixed. We highlight that the interplay between the order of the operator and the space dimension plays a crucial role in the existence of eigenvalue bounds.
T2 - Calc. Var. Partial Differential Equations
VL - 57
SP - 1
EP - 35
ER -
TY - CHAP
TI - The spectrum of the Laplacian: A geometric approach
T2 - Geometric and Computational Spectral Theory
CY - Centre de Recherches Mathématiques, Université de Montréal
AU - Colbois, B.
LA - en
PY - 2017
AB - These notes correspond to a 4-hour lecture and one exercise session given in Montréal in June 2015 during the summer school "Geometric and Computational Spectral Theory". The goal was to introduce the subject, that is to present various aspects of the spectrum of the Laplacian on a compact Riemannian manifold from a geometric viewpoint, and also to prepare the audience for some of the following lectures.
PB - Contemp. Math. 700
ER -
TY - JOUR
TI - Eigenvalues of the Laplacian on a compact manifold with density
UR - https://dx.doi.org/10.4310/CAG.2015.v23.n3.a6
LA - en
AU - Colbois, B.
AU - El Soufi, A.
AU - Savo, A.
PY - 2015
DA - 1.3
T2 - Communications in Analysis and Geometry
IS - 3
VL - 23
SP - 639
EP - 670
ER -
TY - JOUR
TI - Two properties of volume growth entropy in Hilbert geometry
UR - https://doi.org/10.1007/s10711-013-9934-2
LA - en
AU - Colbois, B.
AU - Verovic, P.
PY - 2014
DA - 9.11
AB - The aim of this paper is to provide two examples in Hilbert geometry which show that volume growth entropy is not always a limit on the one hand, and that it may vanish for a non-polygonal domain in the plane on the other hand.
T2 - Geometriae Dedicata
IS - 1-2
VL - 173
SP - 163
EP - 175
ER -
TY - JOUR
TI - Extremal eigenvalues of the Laplacian on Euclidean domains
UR - https://doi.org/10.1007/s00209-014-1325-3
LA - en
AU - Colbois, B.
AU - El Soufi, A.
PY - 2014
DA - 1.10
AB - We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part, we study sequences of extremal eigenvalues of the Laplace-Beltrami operator on closed surfaces of unit area.
T2 - Math Zeitschrift
IS - 1-2
VL - 278
SP - 529
EP - 546
ER -
TY - JOUR
TI - Laplacian and spectral gap in regular Hilbert geometries
UR - https://projecteuclid.org/euclid.tmj/1412783204
LA - en
AU - Barthelmé, T.
AU - Colbois, B.
AU - Crampon, M.
AU - Verovic, P.
PY - 2014
DA - 19.9
AB - We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with C2 boundaries. We show that for an n-dimensional geometry, the spectral gap is bounded above by (n−1)2/4, which we prove to be the infimum of the essential spectrum. We also construct examples of convex sets with arbitrarily small eigenvalues.
T2 - Tohoku Math. J.
VL - 66
SP - 377
EP - 407
ER -
TY - JOUR
TI - The spectral gap of graphs and Steklov eigenvalues on surfaces
LA - en
AU - Colbois, B.
AU - Girouard, A.
PY - 2014
DA - 1.2
T2 - Electronic Research Announcements in Mathematical Sciences
VL - 21
SP - 19
EP - 27
ER -
TY - JOUR
TI - Uniform stability of the Dirichlet spectrum for rough perturbations
LA - en
AU - Colbois, B.
AU - Girouard, A.
AU - Iversen, M.
PY - 2013
DA - 29.10
T2 - Journal of Spectral Theory
IS - 4
VL - 3
SP - 575
EP - 599
ER -
TY - JOUR
TI - Isoperimetric control of the spectrum of a compact hypersurface
LA - en
AU - Colbois, B.
AU - El Soufi, A.
AU - Girouard, A.
PY - 2013
DA - 2.10
T2 - Journal für die Reine und Angewandte Mathematik
VL - 683
SP - 49
EP - 66
ER -
TY - JOUR
TI - Eigenvalue control for a Finsler--Laplace operator
LA - en
AU - Barthelmé, T.
AU - Colbois, B.
PY - 2013
DA - 1.5
T2 - Annals of global analysis and geometry
IS - 1
VL - 44
SP - 43
EP - 72
ER -
TY - JOUR
TI - Involutive isometries, eigenvalue bounds and a spectral property of Clifford tori
LA - en
AU - Colbois, B.
AU - Savo, A.
PY - 2012
DA - 17.2
T2 - Indiana University Mathematics Journal
IS - 1
VL - 61
SP - 337
EP - 357
ER -
TY - JOUR
TI - Hilbert geometry for convex polygonal domains
LA - en
AU - Colbois, B.
AU - Vernicos, C.
AU - Verovic, P.
PY - 2011
DA - 21.1
T2 - Journal of Geometry
IS - 1-2
VL - 100
SP - 37
EP - 64
ER -
TY - JOUR
TI - Isoperimetric control of the Steklov spectrum
KW - Dirichlet--to--Neumann map, Steklov eigenvalues, upper bounds, isoperimetric ratio
LA - en
AU - Colbois, B.
AU - El Soufi, A.
AU - Girouard, A.
PY - 2011
DA - 21.6
AB - We prove that the normalized Steklov eigenvalues of a bounded domain in a complete Riemannian manifold are bounded above in terms of the inverse of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniforml bounded
above. On a compact surface with boundary, we obtain uniform bounds for the normalized Steklov eigenvalues in terms of the genus.
We also establish a relationship between the Steklov eigenvalues of a
domain and the eigenvalues of the Laplace-Beltrami operator on its boundary hypersurface.
T2 - J. Funct. Anal.
IS - 5
VL - 261
SP - 1384
EP - 1399
ER -
TY - JOUR
TI - Large eigenvalues and concentration
KW - Eigenvalues, upper bounds, Laplace type operators, concentration.
LA - en
AU - Colbois, B.
AU - Savo, A.
PY - 2011
DA - 21.4
T2 - Pacific J. Math.
IS - 2
VL - 249
SP - 271
EP - 290
ER -
TY - JOUR
TI - Hilbert domains that admit a quasi-isometric embedding into Euclidean space
LA - en
AU - Colbois, B.
AU - Verovic, P.
PY - 2011
DA - 21.12
T2 - Adv. Geom.
IS - 3
VL - 11
SP - 465
EP - 470
ER -
TY - JOUR
TI - Eigenvalue estimate for the rough Laplacian on differential forms
KW - Rough Laplacian, eigenvalue estimate, differential forms, Weyl law
LA - en
AU - Colbois, B.
AU - Maerten, D.
PY - 2010
DA - 21.2
AB - We study the spectrum of the rough Laplacian acting on differential
forms on a compact Riemannian manifold (M,g).
We first construct on M metrics of volume 1 whose spectrum is as large as desired.
Then, provided that the Ricci curvature of g is bounded below,
we relate the spectrum of the rough Laplacian on 1--forms to the spectrum of the Laplacian on functions, and derive some upper bound in agreement with the asymptotic Weyl law.
T2 - Manuscripta Math.
IS - 3-4
VL - 132
SP - 399
EP - 413
ER -
TY - JOUR
TI - Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds
KW - Laplacian, eigenvalue, upper bound, submanifold
LA - en
AU - Colbois, B.
AU - Dryden, E. B.
AU - El Soufi, A.
PY - 2010
DA - 21.1
AB - We give upper bounds for the eigenvalues of the La-place-Beltrami operator of a compact m-dimensional submanifold M of R^{m+p}. Besides the dimension and the volume of the submanifold and the order of the eigenvalue, these bounds depend on either the maximal number of intersection points of M with a p-plane in a generic position (transverse to M), or an invariant which measures the concentration
of the volume of M in R^{m+p}. These bounds are asymptotically optimal in the sense of the Weyl law. On the other hand, we show that even for hypersurfaces (i.e., when p=1), the first positive eigenvalue cannot be controlled only in terms of the volume, the dimension and (for m>2) the differential structure.
T2 - Bull. Lond. Math. Soc.
IS - 1
VL - 42
SP - 96
EP - 108
ER -
TY - JOUR
TI - Some smooth Finsler deformations of hyperbolic surfaces
KW - Finsler geometry, Riemannian geometry, Rigidity results, MINIMAL ENTROPY, RIGIDITY THEOREMS, SPACES, CURVATURE, METRICS
LA - en
AU - Colbois, B.
AU - Newberger, F.
AU - Verovic, P.
PY - 2009
DA - .
AB - Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectrum rigidity and boundary rigidity all fail to extend to the Finsler category.
T2 - Annals of Global Analysis and Geometry
IS - 2
VL - 35
SP - 191
EP - 226
ER -
TY - JOUR
TI - Eigenvalue pinching on convex domains in space forms
KW - LAPLACIAN, STABILITY
LA - en
AU - Aubry, E.
AU - Bertrand, J.
AU - Colbois, B.
PY - 2009
DA - .
AB - In this paper, we show that the convex domains of H-n which are almost extremal for the Faber-Krahn or the Payne-Polya-Weinberger inequalities are close to geodesic balls. Our proof is also valid in other space forms and allows us to recover known results in R-n and S-n.
T2 - Transactions of the American Mathematical Society
IS - 1
VL - 361
SP - 1
EP - 18
ER -
TY - JOUR
TI - Eigenvalues estimate for the Neumann problem of a bounded domain
KW - Neumann spectrum, upper bound, Weyl law, metric geometry, METRICS
LA - en
AU - Colbois, B.
AU - Maerten, D.
PY - 2008
DA - 21.12
AB - In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a domain Omega in a given complete ( not compact a priori) Riemannian manifold ( M, g). For this, we use test functions for the Rayleigh quotient subordinated to a family of open sets constructed in a general metric way, interesting for itself. As applications, we prove that if the Ricci curvature of ( M, g) is bounded below Ric(g) >= -( n - 1) a(2), a >= 0, then there exist constants A(n) > 0, B-n > 0 only depending on the dimension, such that lambda(k)(Omega)
T2 - Journal of Geometric Analysis
IS - 4
VL - 18
SP - 1022
EP - 1032
ER -
TY - JOUR
TI - Area of ideal triangles and Gromov hyperbolicity in Hilbert Geometry
LA - en
AU - Colbois, B.
AU - Vernicos, C.
AU - Verovic, P.
PY - 2008
DA - .
T2 - Illinois Journal of Mathematics
IS - 1
VL - 52
SP - 319
EP - 343
ER -
TY - JOUR
TI - Extremal g-invariant eigenvalues of the Laplacian of g-invariant metrics
KW - laplacian, eigenvalue, invariant, extremal metric, upper bound, 1ST EIGENVALUE, RIEMANNIAN-MANIFOLDS, MINIMAL IMMERSIONS, CONFORMAL, CLASS, SURFACES, SPECTRUM, BOUNDS
LA - en
AU - Colbois, B.
AU - Dryden, E. B.
AU - El Soufi, A.
PY - 2008
DA - 21.12
AB - The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S-2 endowed with S-1-invariant metrics, we consider the subsequence lambda(G)(k) of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If. G has dimension at least 1, we show that the functional lambda(G)(k) admits no extremal metric under volume-preserving G-invariant deforma- tions. If, moreover, M has dimension at least three, then the functional lambda(G)(k) is unbounded when restricted to any conformal class of G-invariant metrics of fixed volume. As a special case of this, we can consider the standard 0(n)-action on S-n; however, if we also require the metric to be induced by an embedding of S-n in Rn+1, we get an optimal upper bound on lambda(G)(k).
T2 - Mathematische Zeitschrift
IS - 1
VL - 258
SP - 29
EP - 41
ER -
TY - JOUR
TI - A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space
KW - spectrum, Laplacian, pinching results, hypersurfaces, POSITIVE RICCI CURVATURE, MANIFOLDS, DIAMETER, SUBMANIFOLDS
LA - en
AU - Colbois, B.
AU - Grosjean, J. F.
PY - 2007
DA - .
AB - In this paper, we give pinching theorems for the first nonzero eigenvalue lambda(1) (M) of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of M is I then, for any epsilon > 0, there exists a constant C, depending on the dimension n of M and the L-infinity-norm of the mean curvature H, so that if the L-2p-norm parallel to H parallel to(2p) (p >= 2) of H satisfies n parallel to H parallel to(2)(2p)-C-epsilon < lambda(1) (M), then the Hausdorff-distance between M and a round sphere of radius (n/lambda(1) (M))(1/2) is smaller than epsilon. Furthermore, we prove that if C is a small enough constant depending on n and the L-infinity-norm of the second fundamental form, then the pinching condition n parallel to H parallel to(2)(2p)-C < lambda(1) (M) implies that M is diffeomorphic to an n-dimensional sphere.
T2 - Commentarii Mathematici Helvetici
IS - 1
VL - 82
SP - 175
EP - 195
ER -
TY - JOUR
TI - Les géométries de Hilbert sont à géométrie locale bornée
KW - GROMOV-HYPERBOLIC SPACES, EMBEDDINGS
LA - fr
AU - Colbois, B.
AU - Vernicos, C.
PY - 2007
DA - 21.12
AB - We prove that the Hilbert geometry of a convex domain in R-n has bounded local geometry, i.e., for a given radius, all balls are bilipschitz to a euclidean domain of R-n. As a consequence, if the Hilbert geometry is also Gromov hyperbolic, then the bottom of its spectrum is strictly positive. We also give a counter exemple in dimension three wich shows that the reciprocal is not true for non plane Hilbert geometries.
T2 - Annales de l'Institut Fourier
IS - 4
VL - 57
SP - 1359
EP - 1375
ER -
TY - JOUR
TI - Eigenvalues of the laplacian acting on p-forms and metric conformal deformations
KW - Laplacian, p-forms, eigenvalue, conformal deformations, 1ST EIGENVALUE, GAP
LA - en
AU - Colbois, B.
AU - El Soufi, A.
PY - 2006
DA - .
AB - Let (M, g) be a compact connected orientable Riemannian manifold of dimension n >= 4 and let lambda(k,p)(g) be the k-th positive eigenvalue of the Laplacian. Delta g,p = dd* + d* d acting on differential forms of degree p on M. We prove that the metric g can be conformally deformed to a metric g', having the same volume as g, with arbitrarily large lambda 1, p(g') for all p is an element of [2,n-2]. Note that for the other values of p, that is p = 0, 1, n-1 and n, one can deduce from the literature that, for all k > 0, the k-th eigenvalue lambda(k,p) is uniformly bounded on any conformal class of metrics of fixed volume on M. For p = 1, we show that, for any positive integer N, there exists a metric g(N) conformal to g such that, for all k
T2 - Proceedings of the American Mathematical Society
IS - 3
VL - 134
SP - 715
EP - 721
ER -
TY - JOUR
TI - Bas du spectre et delta-hyperbolicité en géométrie de Hilbert plane
LA - fr
AU - Colbois, B.
AU - Vernicos, C.
PY - 2006
DA - 21.12
AB - We prove that the Hilbert geometry of a convex domain in the plane is Gromov hyperbolic, if, and only if, the bottom of its spectrum is not zero.
T2 - Bulletin de La Société Mathématique de France
IS - 3
VL - 134
SP - 357
EP - 381
ER -
TY - JOUR
TI - Capacité et inégalité de Faber-Krahn dans Rn
KW - capacity, eigenvalue estimates, convergence of the spectrum, DIRICHLET EIGENVALUES, DOMAINS, HOLES
LA - fr
AU - Bertrand, J.
AU - Colbois, B.
PY - 2006
DA - 18.4
AB - In this paper, we define a new capacity which allows us to control the behaviour of the Dirichlet spectrum of a compact Riemannian manifold with boundary, with "small" subsets (which may intersect the boundary) removed. This result generalises a classical result of Rauch and Taylor ("the crushed ice theorem"). In the second part, we show that the Dirichlet spectrum of a sequence of bounded Euclidean domains converges to the spectrum of a ball with the same volume, if the first eigenvalue of these domains converges to the first eigenvalue of a ball.
T2 - Journal of Functional Analysis
IS - 1
VL - 232
SP - 1
EP - 28
ER -
TY - JOUR
TI - Hilbert geometry for strictly convex domains
KW - convex sets, Finsler spaces, metric geometry
LA - en
AU - Colbois, B.
AU - Verovic, P.
PY - 2004
DA - .
AB - We prove in this paper that the Hilbert geometry associated with a bounded open convex domain C in R-n whose boundary partial derivativeC is a C-2 hypersuface with nonvanishing Gaussian curvature is bi-Lipschitz equivalent to the n-dimensional hyperbolic space H-n. Moreover, we show that the balls in such a Hilbert geometry have the same volume growth entropy as those in H-n.
T2 - Geometriae Dedicata
IS - 1
VL - 105
SP - 29
EP - 42
ER -
TY - JOUR
TI - L'aire des triangles idéaux en géométrie de Hilbert
LA - fr
AU - Colbois, B.
AU - Vernicos, C.
AU - Verovic, P.
PY - 2004
DA - 20.12
T2 - L'enseignement mathématique
IS - 2
VL - 50
SP - 203
EP - 237
ER -
TY - SLIDE
TI - Spectre conforme et métriques extrémales
LA - fr
AU - Colbois, B.
PY - 2004
DA - 20.12
ER -
TY - JOUR
TI - Extremal eigenvalues of the Laplacian in a conformal class of metrics: The 'conformal spectrum'
KW - Laplacian, eigenvalue, conformal metric, universal lower bound, MINIMAL IMMERSIONS, 1ST EIGENVALUE, SURFACES, CONJECTURE
LA - en
AU - Colbois, B.
AU - El Soufi, A.
PY - 2003
DA - 21.12
AB - Let M be a compact connected manifold of dimension n endowed with a conformal class C of Riemannian metrics of volume one. For any integer k greater than or equal to 0, we consider the conformal invariant.c k( C) defined as the supremum of the k-th eigenvalue lambda(k)(g) of the Laplace-Beltrami operator Delta(g), where g runs over C. First, we give a sharp universal lower bound for lambda(k)(c)(C) extending to all k a result obtained by Friedlander and Nadirashvili for k = 1. Then, we show that the sequence {lambda(k)(c)(C)}, that we call 'conformal spectrum', is strictly increasing and satisfies, For Allk greater than or equal to 0, lambda(k+1)(c)(C)(n/2)-lambda(k)(c)(C)(n/2) greater than or equal to n(n/2) omega(n), where omega(n) is the volume of the n-dimensional standard sphere. When M is an orientable surface of genus gamma, we also consider the supremum zeta(k)(top) (gamma) of lambda(k)(g) over the set of all the area one Riemannian metrics on M, and study the behavior of lambda(k)(top)(gamma) in terms of gamma.
T2 - Annals of Global Analysis and Geometry
IS - 4
VL - 24
SP - 337
EP - 349
ER -
TY - JOUR
TI - On the optimality of J. Cheeger and P. Buser inequalities
KW - eigenvalue, Laplacian, Cheeger constant, asymptotic behaviour
LA - en
AU - Colbois, B.
AU - Matei, A. M.
PY - 2003
DA - .
AB - We study the relationship between the first eigenvalue of the Laplacian and Cheeger constant when the Cheeger constant converges to zero, in the case of compact Riemannian manifolds and of finite graphs. (C) 2003 Elsevier B.V. All rights reserved.
T2 - Differential Geometry and Its Applications
IS - 3
VL - 19
SP - 281
EP - 293
ER -
TY - JOUR
TI - Asymptotic estimates of the first eigenvalue of the p-Laplacian
KW - p-Laplacian, first eigenvalue, asymptotic estimates, RIEMANN SURFACES, GRAPHS
LA - en
AU - Colbois, B.
AU - Matei, A. M.
PY - 2003
DA - .
AB - ,We consider a 1-parameter family of hyperbolic surfaces M(t) of genus v which degenerate as t --> 0 and we obtain a precise estimate of lambda(1,p)(t), the first eigenvalue of the p-Laplacian (p > 1) on M(t). In some cases we also give a precise estimate of the first eigenfunctions. As a direct application, we obtain that the quotient (lambda1,q)(1/q)/(lambda1,p)(1/p) (which is invariant under scaling of the metric) is unbounded even on the set of Riemannian manifolds with constant sectional curvature. This is to our knowledge, the first example of a family of manifolds with this property. To prove our results we use in an essential way the geometry of hyperbolic surfaces which is very well known. We show that an eigenfunction for lambda(1,p)(t) of L-p norm one is almost constant in the L-p sense (as t --> 0) on the parts of M(t) with large injectivity radius, and we estimate precisely its p-energy on the parts with small injectivity radius.
T2 - Advanced Nonlinear Studies
IS - 2
VL - 3
SP - 207
EP - 217
ER -
TY - JOUR
TI - Curvature, Harnack's inequality, and a spectral characterization of nilmanifolds
KW - nilmanifolds, Laplacian, Harnack inequality, RIEMANNIAN-MANIFOLDS
LA - en
AU - Aubry, E.
AU - Colbois, B.
AU - Ghanaat, P.
AU - Ruh, E.
PY - 2003
DA - .
AB - For closed n-dimensional Riemannian manifolds M with almost nonnegative Ricci curvature, the Laplacian on one-forms is known to admit at most n small eigenvalues. If there are n small eigenvalues, or if M is orientable and has n - 1 small eigenvalues, then M is diffeomorphic to a nilmanifold, and the metric is almost left invariant. We show that our results are optimal for n greater than or equal to 4.
T2 - Annals of Global Analysis and Geometry
IS - 3
VL - 23
SP - 227
EP - 246
ER -
TY - JOUR
TI - Une inégalité du type Payne-Polya-Weinberger pour le laplacien brut
KW - rough Laplacian, spectrum, Riemannian bundle
LA - fr
AU - Colbois, B.
PY - 2003
DA - .
AB - Let us consider a riemannian vector bundle E with compact basis (M, g) and the rough laplacian (&UDelta;) over bar associated to a connection D on E. We prove that the eigenvalues of (&UDelta;) over bar are bounded above by a function of the first eigenvalue and of the geometry of (M, g), but independently of the choice of the connection D.
T2 - Proceedings of the American Mathematical Society
IS - 12
VL - 131
SP - 3937
EP - 3944
ER -
TY - JOUR
TI - Rigidity of Hilbert metrics
KW - FINSLER
LA - en
AU - Colbois, B.
AU - Verovic, P.
PY - 2002
DA - .
AB - We study the groups of isometries for Hilbert metrics on bounded open convex domains in R-n and show that if C is such a set with a strictly convex boundary, the Hilbert geometry is asymptotically Riemannian at infinity. As a consequence of this result, we prove there are no Hausdorff quotients of C by isometry subgroups with finite volume except when partial derivativeC is an ellipsoid.
T2 - Bulletin of the Australian Mathematical Society
IS - 1
VL - 65
SP - 23
EP - 34
ER -
TY - RPRT
TI - Curvature and a spectral characterization of nilmanifolds
AU - Colbois, B.
AU - Ghanaat, P.
AU - Ruh, E.
PY - 2000
DA - 20.12
PB - Université de Neuchâtel
CY - Neuchâtel
LA - en
ER -
TY - SLIDE
TI - Une caractérisation spectrale des nilvariétés
LA - fr
AU - Colbois, B.
PY - 2000
DA - 20.12
ER -
TY - JOUR
TI - Petites valeurs propres des p-formes différentielles et classe d'Euler des S1-fibrés
KW - COLLAPSING RIEMANNIAN-MANIFOLDS, SPECTRAL SEQUENCES, LAPLACIAN, CURVATURE
LA - fr
AU - Colbois, B.
AU - Courtois, G.
PY - 2000
DA - 21.12
AB - Let M(n, a, d) be the set of compact oriented Riemannian manifolds (M, g) of dimension n whose sectional curvature K-g and diameter d(g) satisfy \K-g\ less than or equal to a and d(g) less than or equal to d. Let M(n, a, d, rho) be the subset of M(n, a, d) of those manifolds (M, g) such that the injectivity radius is greater than or equal to rho. if (M, g) is an element of M(n + 1, a, d) and (N, h) is an element of M(n, a', d') are sufficiently close in the sense of Gromov-Hausdorff, M is a circle bundle over N according to a theorem of K. Fukaya. When the Gromov-Hausdorff distance between (M, g) and (N, h) is small enough, we show that there exists m(p) - b(p)(N) + b(p-1) (N) - b(p)(M) small eigenvalues of the Laplacian acting on differential p-forms on M, 1 < p < n + 1, where b(p) denotes the p-th Betti number. We give uniform bounds of these eigenvalues depending on the Euler class of the circle bundle S-1 --> M --> N and the Gromov-Hausdorff distance between (M, g) and (N, h). (C) 2000 Editions scientifiques et medicales Elsevier SAS.
T2 - Annales Scientifiques de l'Ecole Normale Supérieure
IS - 5
VL - 33
SP - 611
EP - 645
ER -
TY - RPRT
TI - Curvature and Gradient Estimates for Eigenforms of the Laplacian
AU - Colbois, B.
AU - Ghanaat, P.
AU - Ruh, E.
PY - 1999
DA - 20.12
PB - Université de Savoie
CY - Chambéry
LA - en
ER -
TY - JOUR
TI - Sur la multiplicité de la première valeur propre de l'opérateur de Schrödinger avec champ magnétique sur la sphère S2
LA - fr
AU - Besson, G.
AU - Colbois, B.
AU - Courtois, G.
PY - 1998
DA - .
AB - The purpose of this text is to study the first eigenvalue of Schrodinger operator with magnetic field on the 2-sphere and to show that its multiplicity can be arbitrarily high. We also show that this multiplicity is bounded in terms of the curvature of the corresponding connection. This answers a question asked by Y. Colin de Verdiere.
T2 - Transactions of the American Mathematical Society
IS - 1
VL - 350
SP - 331
EP - 345
ER -
TY - JOUR
TI - Metrische Geometrie
LA - en
AU - Colbois, B.
PY - 1996
DA - .
T2 - El. Mathematik
IS - 51
SP - 133
EP - 144
ER -
TY - SLIDE
TI - Le spectre du laplacien agissant sur les p-formes différentielles
LA - fr
AU - Colbois, B.
PY - 1996
DA - 20.12
ER -
TY - JOUR
TI - Spectre du laplacien sur les p-formes différentielles et écrasement d'anses
KW - MANIFOLDS, OPERATOR, HANDLES
LA - fr
AU - Anné, C.
AU - Colbois, B.
PY - 1995
DA - .
T2 - Mathematische Annalen
IS - 3
VL - 303
SP - 545
EP - 573
ER -
TY - JOUR
TI - Riemannian Metrics with large lambda(1)
KW - LAPLACIAN, SPECTRUM
LA - en
AU - Colbois, B.
AU - Dodziuk, J.
PY - 1994
DA - 21.12
AB - We show that every compact smooth manifold of three or more dimensions carries a Riemannian metric of volume one and arbitrarily large first eigenvalue of the Laplacian.
T2 - Proceedings of the American Mathematical Society
IS - 3
VL - 122
SP - 905
EP - 906
ER -
TY - JOUR
TI - Tubes and eigenvalues for negatively curved manifolds
KW - CUSPS, EIGENVALUES, LAPLACIAN, NEGATIVE CURVATURE, TUBES, COMPARISON-THEOREMS, CURVATURE, VOLUME, SPACES
LA - en
AU - Buser, P.
AU - Colbois, B.
AU - Dodziuk, J.
PY - 1993
DA - .
AB - We investigate the structure of the spectrum near zero for the Laplace operator on a complete negatively curved Riemannian manifold M. If the manifold is compact and its sectional curvatures K satisfy 1 less-than-or-equal-to K < 0, we show that the smallest positive eigenvalue of the Laplacian is bounded below by a constant depending only on the volume of M. Our result for a complete manifold of finite volume with sectional curvatures pinched between -a2 and -1 asserts that the number of eigenvalues of the Laplacian between 0 and (n -1)2/4 is bounded by a constant multiple of the volume of the manifold with the constant depending on a and the dimension only.
T2 - Journal of Geometric Analysis
IS - 1
VL - 3
SP - 1
EP - 26
ER -
TY - JOUR
TI - Opérateur de Hodge-Laplace sur des variétés compactes privées d'un nombre fini de boules
KW - DIFFERENTIAL FORMS, PRINCIPLE, HANDLES, TORSION
LA - fr
AU - Anné, C.
AU - Colbois, B.
PY - 1993
DA - .
T2 - Journal of Functional Analysis
IS - 1
VL - 115
SP - 190
EP - 211
ER -
TY - CONF
TI - Introduction au laplacien. Rencontre de théorie spectrale et géométrie
AU - Colbois, B.
LA - fr
PY - 1991
DA - 20.12
C2 - 2012
T2 - Rencontre de théorie spectrale et géométrie
CY - Aussois
ER -
TY - JOUR
TI - Convergence de variétés et convergence du spectre du Laplacien
KW - EIGENVALUES
LA - fr
AU - Colbois, B.
AU - Courtois, G.
PY - 1991
DA - 21.12
T2 - Annales Scientifiques de l'Ecole Normale Supérieure
IS - 4
VL - 24
SP - 507
EP - 518
ER -
TY - CONF
TI - Small Eigenvalues of the Laplacian on Negatively Curved Manifolds
AU - Buser, P.
AU - Colbois, B.
AU - Dodziuk, J.
LA - en
PY - 1990
DA - 21.7
C2 - 1993
T2 - AMS Summer Institute on Differential Geometry
CY - University of Callifornia
SP - 95
ER -
TY - SLIDE
TI - Small eigenvalues of the Laplacian on negatively curved manifolds
LA - en
AU - Colbois, B.
PY - 1990
DA - 20.12
ER -
TY - JOUR
TI - A note on the 1st nonzero eigenvalue of the laplacian acting on p-forms
LA - en
AU - Colbois, B.
AU - Courtois, G.
PY - 1990
DA - .
T2 - Manuscripta Mathematica
IS - 2
VL - 68
SP - 143
EP - 160
ER -
TY - JOUR
TI - Les valeurs propres inférieures à 1/4 des surfaces de Riemann de petit rayon d'injectivité
LA - fr
AU - Colbois, B.
AU - Courtois, G.
PY - 1989
DA - .
T2 - Commentarii Mathematici Helvetici
IS - 3
VL - 64
SP - 349
EP - 362
ER -
TY - RPRT
TI - Les petites valeurs propres des variétés hyperboliques de dimension 3
AU - Colbois, B.
AU - Courtois, G.
PY - 1989
DA - 20.12
PB - Institut Fourier
CY - Grenoble
LA - fr
ER -
TY - JOUR
TI - Sur la multiplicité de la première valeur propre d'une surface de Riemann à courbure constante
LA - fr
AU - Colbois, B.
AU - De Verdiere, Y. C.
PY - 1988
DA - .
T2 - Commentarii Mathematici Helvetici
IS - 2
VL - 63
SP - 194
EP - 208
ER -
TY - THES
TI - Sur la multiplicité de la première valeur propre non nulle du laplacien des surfaces à courbure -1
LA - fr
AU - Colbois, B.
PY - 1987
PB - Lausanne
CY - Lausanne
M3 - Doctorat
ER -
TY - JOUR
TI - A propos de la multiplicité de la première valeur propre du laplacien d'une surface de Riemann
LA - fr
AU - Burger, M.
AU - Colbois, B.
PY - 1985
DA - 21.12
T2 - Comptes Rendus de l'Académie des Sciences Serie I-Mathématique
IS - 8
VL - 300
SP - 247
EP - 249
ER -
TY - JOUR
TI - Petites valeurs propres du laplacien sur une surface de Riemann compacte et graphes
LA - fr
AU - Colbois, B.
PY - 1985
DA - 21.12
T2 - Comptes Rendus de l'Académie des Sciences Série I-Mathématique
IS - 20
VL - 301
SP - 927
EP - 930
ER -