Variation of Laplace spectra of compact "nearly" hyperbolic surfaces

We use the time real analyticity of Ricci flow proved by Kotschwar to extend a result in ~\cite{B}, namely, we prove that the Laplace spectra of negatively curved compact surfaces having same genus $\gamma \geq 2$, same area and same curvature bounds vary in a "controlled way", of which we give a quantitative estimate (Theorem 1.1 below). We also observe how said real analyticity can lead to unexpected conclusions about spectral properties of generic metrics on a compact surface of genus $\gamma \geq 2$ (Proposition 1.5 below).Comment: 7 pages, comments welcome

Boundedness of spectral multipliers of generalized Laplacians on compact manifolds with boundary

Consider a second order, strongly elliptic negative semidefinite differential operator $L$ (maybe a system) on a compact Riemannian manifold $\overline{M}$ with smooth boundary, where the domain of $L$ is defined by a coercive boundary condition. Classically known results, and also recent work in \cite{DOS} and \cite{DM} establish sufficient conditions for $L^\infty-\text{BMO}_L$ continuity of $\varphi(\sqrt{A})$, where $\varphi \in S^0_1(\mathbb{R})$, and $A$ is a suitable elliptic operator. Using a variant of the Cheeger-Gromov-Taylor functional calculus due to \cite{MMV}, and short time bounds on the integral kernel of $e^{tL}$ due to \cite{G}, we prove that a variant of such sufficient conditions holds for our operator $L$.Comment: 20 pages, updated version, comments most welcome

On maximizing the fundamental frequency of the complement of an obstacle

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain satisfying a Hayman-type asymmetry condition, and let $D$ be an arbitrary bounded domain referred to as "obstacle". We are interested in the behaviour of the first Dirichlet eigenvalue $\lambda_1(\Omega \setminus (x+D))$. First, we prove an upper bound on $\lambda_1(\Omega \setminus (x+D))$ in terms of the distance of the set $x+D$ to the set of maximum points $x_0$ of the first Dirichlet ground state $\phi_{\lambda_1} > 0$ of $\Omega$. In short, a direct corollary is that if \begin{equation} \mu_\Omega := \max_{x}\lambda_1(\Omega \setminus (x+D)) \end{equation} is large enough in terms of $\lambda_1(\Omega)$, then all maximizer sets $x+D$ of $\mu_\Omega$ are close to each maximum point $x_0$ of $\phi_{\lambda_1}$. Second, we discuss the distribution of $\phi_{\lambda_1(\Omega)}$ and the possibility to inscribe wavelength balls at a given point in $\Omega$. Finally, we specify our observations to convex obstacles $D$ and show that if $\mu_\Omega$ is sufficiently large with respect to $\lambda_1(\Omega)$, then all maximizers $x+D$ of $\mu_\Omega$ contain all maximum points $x_0$ of $\phi_{\lambda_1(\Omega)}$.Comment: 6 pages, comments most welcome

Polyhedral billiards, eigenfunction concentration and almost periodic control

We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called ``pockets''. We prove there are only finitely many immersed periodic tubes missing the pockets and moreover establish a new quantitative estimate for the lengths of such tubes. This extends well-known results in dimension $2$. We then apply these dynamical results to prove a quantitative Laplace eigenfunction mass concentration near the pockets of convex polyhedral billiards. As a technical tool for proving our concentration results on irrational polyhedra, we establish a control-theoretic estimate on a product space with an almost-periodic boundary condition. This extends previously known control estimates for periodic boundary conditions, and seems to be of independent interest.Comment: 32 pages, a few sections reorganised and a few results adde

A Spectral Gap Estimate and Applications

We consider the Schr\"odinger operator -\frac{d^2}{d x^2} + V \qquad \mbox{on an interval}~~[a,b]~\mbox{with Dirichlet boundary conditions}, where $V$ is bounded from below and prove a lower bound on the first eigenvalue $\lambda_1$ in terms of sublevel estimates: if $w_V(y) = |I_y|,\text{ where } I_y := \left\{ x \in [a,b]: V(x) \leq y \right\},$ then $$\lambda_1 \geq \frac{1}{250} \min_{y > \min V}{\left(\frac{1}{w_V(y)^2} + y\right)}.$$ The result is sharp up to a universal constant if $\left\{ x \in [a,b]: V(x) \leq y \right\}$ is an interval for the value of $y$ solving the minimization problem. An immediate application is as follows: let $\Omega \subset \mathbb{R}^2$ be a convex domain with inradius $\rho$ and diameter $D$ and let $u:\Omega \rightarrow \mathbb{R}$ be the first eigenfunction of the Laplacian $-\Delta$ on $\Omega$ with Dirichlet boundary conditions on $\partial \Omega$. We prove $$\| u \|_{L^{\infty}} \lesssim \frac{1}{\rho^{}} \left( \frac{\rho}{D} \right)^{1/6} \|u\|_{L^2},$$ which answers a question of van den Berg in the special case of two dimensions