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Pierre Jammes

Publications and source records attributed to Pierre Jammes.

At least 19 recordsLinked to original sources
arXiv2021

Systole and small eigenvalues of hyperbolic surfaces

Let $S$ be a closed orientable hyperbolic surface with Euler characteristic$χ$, and let $λ_k(S)$ be the $k$-th positive eigenvalue for the Laplacian on $S$. According to famous result of Otal and Rosas, $λ_{-χ}>\frac14$. In this article, we prove that if thesystole of $S$ is greater than 3,46, then $λ_{-χ-1}>\frac14$.This inequality is also true for geometrically finite orientable hyperbolic surfaces without cusps with the same assumption on the systole.

math.DG↗
arXiv2016

Petite valeurs propres des fibrés principaux en tores

Let M^n be a compact n-dimensional principal T^k-bundle. We consider collapsings of M on N=M/T^k such that the diameter and sectional curvature of M satisfy diam(M)<d and |K(M)|<a, and give examples of collapsings for all k such that the first non-zero eigenvalue of Laplacian acting on 1-forms and 2-forms of M are bounded above by c(M).inj(M)^2k. Moreover, we prove that the first non-zero eigenvalue of 1-form Laplacian of all T^k-bundle M over N is bounded below by c(n,d,a,N).Vol(M)^2 and c.inj(M)^2k when M collapses on N.

math.DG↗
arXiv2016

Multiplicité du spectre de Steklov sur les surfaces et nombre chromatique

We prove several results about the multiplicity of the first Steklov eigenvalues on compact surfaces with boundary. We improve some bounds on the multiplicity, especially for the first eigenvalue, and we prove they are sharp on some surfaces of small genus. In a previous article, we defined a new chromatic invariant of surfaces with boundary and conjectured that this invariant is related to the bound on the first eigenvalue. In the present article, we study this invariant, and prove that the conjecture is true when the known bound is sharp.

math.DG↗
arXiv2015

The supremum of conformally covariant eigenvalues in a conformal class

Let (M,g) be a compact Riemannian manifold of dimension >2. We show that there is a metric h conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with repect to h is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a spin manifold of dimension >1.

math.DG↗
arXiv2015

Spectre et géométrie conforme des variétés compactes à bord

We prove that on any compact manifold $M^n$ with boundary, there exist a conformal class $C$ such that for any riemannian metric $g\in C$, $λ_1(M^n,g)Vol(M^n,g)^{2/n}< n.Vol(S^n,g_{\textrm{can}})^{2/n}$ and $σ_1(M,g,ρ)\mathcal M(\partial M)Vol(M)^{\frac{2-n}n}<n.Vol(S^n,g_{\textrm{can}})^{2/n}$, where $λ_1(M^n,g)$ denotes the first positive eigenvalue of the Neumann laplacian on $(M,g)$, $σ_1(M,g,ρ)$ the first positive Steklov eigenvalue for the density $ρ$ on $\partial M$, and $\mathcal M(\partial M)=\int_{\partial M}ρdv_g$. The proof relies on a handle decomposition of the manifold. We also prove that the conformal volume of $(M,C)$ is $Vol(S^n,g_{\textrm{can}})$, and that the Friedlander-Nadirashvili and the Möbius volume of $M$ are equal to those of the sphere. If $M$ is a domain in a space form, $C$ is the conformal class of the canonical metric.

math.DG↗
arXiv2014

Tight polyhedral embeddings and relative chromatic number of surfaces with boundary

The relative chromatic number $c\_0(S)$ of a compact surface $S$ with boundary is defined as the supremum of the chromatic numbers of graphs embedded in $S$ with all vertices on $\partial S$. This topological invariant was introduced for the study of the multiplicity of the first Steklov eigenvalue of $S$. In this article, we show that $c\_0(S)$ is also relevant for the study of tight polyhedral embeddings of $S$ byproving two results. The first one is that if there is a tight polyhedral embedding of $S$ in $\R^n$ which is not contained in a hyperplane, then $n\leq c\_0(S)-1$. The second result is that this inequality is sharp for surfaces of small genus.

math.GT↗
arXiv2014

Prescription du spectre de Steklov dans une classe conforme

On any compact manifold of dimension $n\geq3$ with boundary, we prescibe any finite part of the Steklov spectrum whithin a given conformal class. In particular, we prescribe the multiplicity of the first eigenvalues. On a compact surface with boundary, we show that the multiplicity of the $k$-th eigenvalue is bounded independently of the metric. On the disk, we give more precise results : the multiplicity of the first and second positive eigenvalues are at most 2 and 3 respectively. For the Steklov-Neumann problem on the disk, we prove that the multiplicity of the $k$-th positive eigenvalue is at most $k+1$.

math.DG↗
arXiv2012

Sur la multiplicité des valeurs propres du laplacien de Witten

On any compact manifold of dimension greater than 4, we prescribe the volume and any finite part of the spectrum of the Witten Laplacian acting on $p$-form for $0<p<n$. In particular, we prescribe the multiplicity of the first eigenvalues. On 3-dimensional manifolds, we give examples of multiple first eigenvalue for 1-forms, whose multiplicity depends on the maximal genus of embedded surfaces all of whose 1-cohomology is induced by the cohomology of the manifold. In particular, this multiplicity is at least 3.

math.DG↗
arXiv2012

Minoration du spectre des variétés hyperboliques de dimension 3

Let $M$ be a compact hyperbolic 3-manifold of diameter $d$ and volume $\leq V$. If $μ_i(M)$ denotes the $i$-th egenvalue of the Hodge laplacian acting on coexact 1-forms of $M$, we prove that $μ_1(M)\geq \frac c{d^3e^{2kd}}$ and $μ_{k+1}(M)\geq \frac c{d^2}$, where $c>0$ depends only on $V$, and $k$ is the number of connected component of the thin part of $M$. Moreover, we prove that for any finite volume hyperbolic 3-manifold $M_\infty$ with cusps, there is a sequence $M_i$ of compact fillings of $M_\infty$ of diameter $d_i\to+\infty$ such that $μ_1(M_i)\geq \frac c{d_i^2}$.

math.DG↗
arXiv2011

Un théorème de la masse positive pour le problème de Yamabe en dimension paire

Let $(M,g)$ be a compact conformally flat manifold of dimension $n\geq4$ with positive scalar curvature. According to a positive mass theorem by Schoen and Yau, the constant term in the development of the Green function of the conformal Laplacian is positive if $(M,g)$ is not conformally equivalent to the sphere. On spin manifolds, there is an elementary proof of this fact by Ammann and Humbert, based on a proof of Witten. Using differential forms instead of spinors, we give an elementary proof on even dimensional manifolds, without any other topological assumption.

math.DG↗
arXiv2010

Effondrement, spectre et propriétés diophantiennes des flots riemanniens

Let F be a riemannian flow on a closed manifold M. We study the behavior of the first eigenvalues of the Hodge Laplacian acting on differential forms under adiabatic collapsing of the flow. We show that the number of small eigenvalues is related to the basic cohomology of F, and give spectral criteria for the vanishing of the Álvarez class and the Euler class of F. We also define a diophantine invariant of the flow wich is related to the asymptotical behavior of the small eigenvalues. An appendix is devoted to arithmetic properties of riemannian flows.

math.DG↗
arXiv2008

Première valeur propre du laplacien, volume conforme et chirurgies

We define a new differential invariant a compact manifold by $V_{\mathcal M}(M)=\inf_g V_c(M,[g])$, where $V_c(M,[g])$ is the conformal volume of $M$ for the conformal class $[g]$, and prove that it is uniformly bounded above. The main motivation is that this bound provides a upper bound of the Friedlander-Nadirashvili invariant defined by $\inf_g\sup_{\tilde g\in[g]}λ_1(M,\tilde g)\Vol(M,\tilde g)^{\frac 2n}$. The proof relies on the study of the behaviour of $V_{\mathcal M}(M)$ when one performs surgeries on $M$.

math.DG↗
arXiv2007

Prescription du spectre du laplacien de Hodge-de Rham dans une classe conforme

For any compact manifold of dimension n>=5, we prescribe the volume and any finite part of the spectrum of the Hodge Laplacian acting on diffential forms of degree 1<p<n-1 (exept for p=n/2 if n is even), within a given conformal class. When n<5 and when p=0,1,n-1,n, and p=n/2 if n is even, this simultaneous prescription of the volume, the spectrum and the conformal class is known to be impossible.

math.DG↗
arXiv2006

Construction de valeurs propres doubles du laplacien de Hodge-de Rham

On any compact manifold of dimension greater than 3, we exhibit a metric whose first positive eigenvalue for the Laplacian acting on p-form is of multiplicity 2. As a corollary, we prescribe the volume and any finite part of the spectrum of the Hodge Laplacian with multiplicity 1 or 2.

math.DG↗