Related papers: Constructing Riemannian metrics with prescribed no…

In this paper we study the problem of prescribing Dirichlet eigenvalues on an arbitrary compact manifold $M$ of dimension $n\geq 3$ with a non-empty smooth boundary $\partial M$. We show that for any finite increasing sequence of real…

Given a closed symplectic manifold (M,\omega) of dimension greater than 2, we consider all Riemannian metrics on M, which are compatible with the symplectic structure \omega. For each such metric, we look at the first eigenvalue \lambda_1…

Let $(M,g,\sigma)$ be a compact Riemmannian surface equipped with a spin structure $\sigma$. For any metric $\tilde{g}$ on $M$, we denote by $\mu\_1(\tilde{g})$ (resp. $\lambda\_1(\tilde{g})$) the first positive eigenvalue of the Laplacian…

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: For any Riemannian metric $g$ on the Klein bottle $\mathbb{K}$ one has $$\lambda\_1 (\mathbb{K}, g) A (\mathbb{K}, g)\le 12 \pi…

We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function…

We investigate in this paper the existence of a metric which maximizes the first eigenvalue of the Laplacian on Riemannian surfaces. We first prove that, in a given conformal class, there always exists such a maximizing metric which is…

We show that zero is not an eigenvalue of the conformal Laplacian for generic Riemannian metrics. We also discuss non-compactness for sequences of metrics with growing number of negative eigenvalues of the conformal Laplacian.

In this note we partially answer a question posed by Colbois, Dryden, and El Soufi. Consider the space of constant-volume Riemannian metrics on a connected manifold M which are invariant under the action of a discrete Lie group G. We show…

We obtain the sharp version of the uncertainty principle recently introduced in [47], and improved by [13], relating the size of the zero set of a continuous function having zero mean and the optimal transport cost between the mass of the…

In this paper, we consider the nodal set of a bi-harmonic function $u$ on an $n$ dimensional $C^{\infty}$ Riemannian manifold $M$, that is, $u$ satisfies the equation $\triangle_M^2u=0$ on $M$, where $\triangle_M$ is the Laplacian operator…

Let $M$ denote a compact, connected Riemannian manifold of dimension $n\in{\mathbb N}$. We assume that $ M$ has a smooth and connected boundary. Denote by $g$ and ${\rm d}v_g$ respectively, the Riemannian metric on $M$ and the associated…

In this work we study the asymptotics of the fractional Laplacian as $s\to 0^+$ on any complete Riemannian manifold $(M,g)$, both of finite and infinite volume. Surprisingly enough, when $M$ is not stochastically complete this asymptotics…

On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions $\{\phi_{\lambda}\}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2…

In this paper, we investigate critical points of the Laplacian's eigenvalues considered as functionals on the space of Riemmannian metrics or a conformal class of metrics on a compact manifold. We obtain necessary and sufficient conditions…

Let $\Sigma$ be an oriented compact hypersurface in the round sphere $\mathbb{S}^n$ or in the flat torus $\mathbb{T}^n$, $n\geq 3$. In the case of the torus, $\Sigma$ is further assumed to be contained in a contractible subset of…

The \emph{flat deformation theorem} states that given a semi-Riemannian analytic metric $g$ on a manifold, locally there always exists a two-form $F$, a scalar function $c$, and an arbitrarily prescribed scalar constraint depending on the…

This paper investigates the asymptotic behavior of the principal eigenvalue $\lambda(s)$, as $s\to+\infty$, for the following elliptic eigenvalue problem \begin{equation*}\label{E} -\Delta_{M}u-s\langle \nabla_M f, \nabla_M u\rangle_g +c…

For a bounded domain $\Omega$ with a piecewise smooth boundary in a complete Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal…

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…

The eigenfunctions of the Laplacian are a central object from the realms of analytic number theory to geometric analysis. We prove that H\"ormander $L^2$-$L^{\infty}$ estimates are equivalent to restriction estimates to small geodesic…