Related papers: Looping directions and integrals of eigenfunctions…

By methods of stochastic analysis on Riemannian manifolds, we develop two approaches to determine an explicit constant $c(D)$ for an $n$-dimensional compact manifold $D$ with boundary such that $\frac{\lambda}{n}\,\|\phi\|_{\infty} \leq…

We discuss the geometry of Laplacian eigenfunctions $-\Delta \phi = \lambda \phi$ on compact manifolds $(M,g)$ and combinatorial graphs $G=(V,E)$. The 'dual' geometry of Laplacian eigenfunctions is well understood on $\mathbb{T}^d$…

We study the eigenvalues of the magnetic Schroedinger operator associated with a magnetic potential A and a scalar potential q, on a compact Riemannian manifold M, with Neumann boundary conditions if the boundary is not empty. We obtain…

Let $(\Omega, g)$ be a real analytic Riemannian manifold with real analytic boundary $\partial \Omega$. Let $\psi_{\lambda}$ be an eigenfunction of the Dirichlet-to-Neumann operator $\Lambda$ of $(\Omega, g, \partial \Omega)$ of eigenvalue…

We consider linear combinations of eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold $(M,g)$ and investigate a density property of their zero sets. More precisely, let $f=\sum_{k=1}^m a_k…

Let (M,g) be a n-dimensional compact Riemannian manifold. We consider the magnetic deformations of semiclassical Schrodinger operators on M for a family of magnetic potentials that depends smoothly on $k$ parameters $u$, for $k \geq n$, and…

Let M be a compact n-dimensional manifold, $n\ge 2$, with metric g(t) evolving by the Ricci flow $\partial g_{ij}/\partial t=-2R_{ij}$ in (0,T) for some $T\in\Bbb{R}^+\cup\{\infty\}$ with $g(0)=g_0$. Let $\lambda_0(g_0)$ be the first…

Let $e_\l(x)$ be an eigenfunction with respect to the Dirichlet Laplacian $\Delta_N$ on a compact Riemannian manifold $N$ with boundary: $\Delta_N e_\l=\l^2 e_\l$ in the interior of $N$ and $e_\l=0$ on the boundary of $N$. We show the…

In this note, we prove the following generalization of a theorem of Shi and Tam \cite{ShiTam02}: Let $(\Omega, g)$ be an $n$-dimensional ($n \geq 3$) compact Riemannian manifold, spin when $n>7$, with non-negative scalar curvature and mean…

We consider a compact Riemannian manifold $M$ (possibly with boundary) and $\Sigma \subset M\setminus \partial M$ an interior hypersurface (possibly with boundary). We study observation and control from $\Sigma$ for both the wave and heat…

We establish a global rigidity theorem for Riemannian metrics without conjugate points on three-manifolds of the form $M = \Sigma \times S^1$, where $\Sigma$ is a compact orientable surface of genus at least 2. The main result states that…

For a closed Riemannian orbifold $O$, we compare the spectra of the Laplacian, acting on functions or differential forms, to the Neumann spectra of the orbifold with boundary given by a domain $U$ in $O$ whose boundary is a smooth manifold.…

Given a compact surface of revolution with Laplace-beltrami operator $\Delta$, we consider the spectral projector $P_{\lambda,\delta}$ on a polynomially narrow frequency interval $[\lambda-\delta,\lambda + \delta]$, which is associated to…

We view all smooth metrics $g$ on a closed surface $\Sigma$ through their Nash isometric embeddings $f_g: (\Sigma,g) \rightarrow (\mathbb{S}^{\tilde{n}}, \tilde{g})$ into a standard sphere of large, but fixed, dimension $\tilde{n}$. We…

We are concerned with an harmonic analysis in Hilbert spaces $L^2(\mu)$, where $\mu$ is a probability measure on $\br^n$. The unifying question is the presence of families of orthogonal (complex) exponentials $e_\lambda(x) = \exp(2\pi i…

Let $\Omega\subset\mathbb{R}^d$ be any open set. We consider solutions of $H\psi_\lambda=\lambda \psi_\lambda$, $\lambda\in\mathbb{C}$, where $H$ is an $m$th order complex constant-coefficient elliptic partial differential operator. We…

A typical path integral on a manifold, $M$ is an informal expression of the form \frac{1}{Z}\int_{\sigma \in H(M)} f(\sigma) e^{-E(\sigma)}\mathcal{D}\sigma, \nonumber where $H(M)$ is a Hilbert manifold of paths with energy $E(\sigma) <…

Suppose that $M$ is a compact Riemannian manifold with boundary and $u$ is an $L^2$-normalized Dirichlet eigenfunction with eigenvalue $\lambda$. Let $\psi$ be its normal derivative at the boundary. Scaling considerations lead one to expect…

Motivated by the work of Li and Mantoulidis, we study singular metrics which are uniformly Euclidean $(L^\infty)$ on a compact manifold $M^n$ ($n\ge 3$) with negative Yamabe invariant $\sigma(M)$. It is well-known that if $g$ is a smooth…

If $(M,g)$ is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order $o(\lambda)$ saturating sup-norm estimates. In particular, it gives optimal conditions…