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If $(M,g)$ is a compact Riemannian surface then the integrals of $L^2(M)$-normalized eigenfunctions $e_j$ over geodesic segments of fixed length are uniformly bounded. Also, if $(M,g)$ has negative curvature and $\gamma(t)$ is a geodesic…

Motivated by problems of mathematical physics (quantum chaos) questions of equidistribution of eigenfunctions of the Laplace operator on a Riemannian manifold have been studied by several authors. We consider here, in analogy with…

We prove the existence of optimal metrics for a wide class of combinations of Laplace eigenvalues on closed orientable surfaces of any genus. The optimal metrics are explicitely related to Laplace minimal eigenmaps, defined as branched…

We give sharp limiting case Hardy inequalities on the sphere $\mathbb{S}^{2}$ and show that their optimal constants are unattainable by any $f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}$. The singularity of the problem is related to…

The real symplectic Stiefel manifold is the manifold of symplectic bases of symplectic subspaces of a fixed dimension. It features in a large variety of applications in physics and engineering. In this work, we study this manifold with the…

For a smooth $k$-dimensional submanifold $\Sigma$ of a $d$-dimensional compact Riemannian manifold $M$, we extend the $L^p(\Sigma)$ restriction bounds of Burq-G\'erard-Tzvetkov -- originally proved for individual Laplace--Beltrami…

Let $f \in C^2(\mathbb{T}^2)$ have mean value 0 and consider $$ \sup_{\gamma~{\tiny \mbox{closed geodesic}}}{~~~ \frac{1}{|\gamma|} \left| \int_{\gamma}{ f ~~d\mathcal{H}^1}\right| },$$ where $\gamma$ ranges over all closed geodesics…

It has been known since the work of Avakumov\'ic, H\"ormander and Levitan that, on any compact smooth Riemannian manifold, if $-\Delta_g \psi_\lambda = \lambda \psi_\lambda$, then $\|\psi_\lambda\|_{L^\infty} \leq C \lambda^{\frac{d-1}{4}}…

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 article, we prove an eigenvalue pinching theorem for the first eigenvalue of the Laplacian on compact hypersurfaces in a sphere. Let $(M^n,g)$ be a closed, connected and oriented Riemannian manifold isometrically immersed by $\phi$…

This paper concerns the concentration of Dirichlet eigenfunctions of the Laplacian on a compact two-dimensional Riemannian manifold with strictly geodesically concave boundary. We link three inequalities which bound the concentration in…

We show that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging…

We show that one can obtain improved $L^4$ geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge's strategy in proving improved critical $L^p$…

Let $u$ be an eigenfunction of the Laplacian on a compact manifold with boundary, with Dirichlet or Neumann boundary conditions, and let $-\lambda^2$ be the corresponding eigenvalue. We consider the problem of estimating the maximum of $u$…

The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If $X$ is a random point on a manifold $M$ and $f$ is an…

We study the problem of estimating the $L^2$ norm of Laplace eigenfunctions on a compact Riemannian manifold $M$ when restricted to a hypersurface $H$. We prove mass estimates for the restrictions of eigenfunctions $\phi_h$, $(h^2 \Delta -…

We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite…

We prove that for each positive integer $N$ the set of smooth, zero degree maps $\psi\colon\mathbb{S}^2\to \mathbb{S}^2$ which have the following three properties: (1) there is a unique minimizing harmonic map $u\colon \mathbb{B}^3\to…

Let $\phi$ be an $L^2$-normalized spherical vector in an everywhere unramified cuspidal automorphic representation of $\mathrm{PGL}_n$ over $\mathbb{Q}$ with Laplace eigenvalue $\lambda_{\phi}$. We establish explicit estimates for various…

For $2\leq p<4$, we study the $L^p$ norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact $2$-dimensional Riemannian manifolds. Burq, G\'erard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and…