Related papers: Unimodal value distribution of Laplace eigenfuncti…

Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording to the laws of geometric…

We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of…

Let $C$ be a configuration of $n$ ovals in $\mathbb{S}^2$. We show that there is a Riemannian metric $g$ over $\mathbb{S}^2$ with a Laplacian eigenfunction whose zero set is $C$, and the corresponding eigenvalue is the $k$-th eigenvalue for…

In this paper, we study the spectrum of the weighted Laplacian (also called Bakry-Emery or Witten Laplacian) $L_\sigma$ on a compact, connected, smooth Riemannian manifold $(M,g)$ endowed with a measure $\sigma dv_g$. First, we obtain upper…

On a compact Riemannian manifold $M$ of dimension $n$, we consider $n$ eigenfunctions of the Laplace operator $\Delta $ with eigenvalue $\lambda$. If $M$ is homogeneous under a compact Lie group preserving the metric then we prove that the…

We study random Morse functions on a Riemann manifold $(M^m,g)$ defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric $g$. The randomness is determined by a fixed Schwartz function $w$ and a…

Let $\Delta_M$ be the Laplace operator on a compact $n$-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions $u:\Delta u + \lambda u =0$. In dimension $n=2$ we refine the Donnelly-Fefferman estimate…

Let $(M^n,g)$ be a complete Riemannian manifold which is not isometric to $\mathbb{R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set $\mathcal{G}\subset…

These notes present a recent approach to study the high-frequency eigenstates of the Laplacian on compact Riemannian manifolds of negative sectional curvature. The main result is a lower bound on the Kolmogorov-Sinai entropy of the…

We prove that, given any knot $\gamma$ in a compact 3-manifold M, there exists a Riemannian metric on M such that there is a complex-valued eigenfunction u of the Laplacian, corresponding to the first nontrivial eigenvalue, whose nodal set…

We study the absolute continuity with respect to the Lebesgue measure of the distribution of the nodal volume associated with a smooth, non-degenerated and stationary Gaussian field $(f(x), {x \in \mathbb R^d})$. Under mild conditions, we…

We consider a random Gaussian model of Laplace eigenfunctions on the hemisphere satisfying the Dirichlet boundary conditions along the equator. For this model we find a precise asymptotic law for the corresponding zero density functions, in…

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 any generalised smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying…

We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3= \mathbb{R}^3/ \mathbb{Z}^3$ ($3$-dimensional 'arithmetic random waves'). We prove that, as the multiplicity of the eigenspace…

In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface \Sigma in a compact n-manifold M, there is a Riemannian…

We consider a compact Riemannian manifold with boundary with a certain class of critical singular Riemannian metrics that are singular at the boundary. The corresponding Laplace-Beltrami operator can be seen as a Grushin-type operator plus…

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 $(M,g)$ be a $n-$dimensional, compact Riemannian manifold. We define the frequency scale $\lambda$ of a function $f \in C^{0}(M)$ as the largest number such that $\left\langle f, \phi_k \right\rangle =0$ for all Laplacian eigenfunctions…

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…