Bounds for Restrictions of Laplace Eigenfunctions (17rit687)

Arriving in Banff, Alberta Sunday, October 15 and departing Sunday October 22, 2017


(Stanford University)

(McGill University)

(University of North Carolina, Chapel Hill)


\subsection{Objectives} The goal of the workshop is to work on obtaining lower and upper bounds for the $L^2$-mass $\|\phi_{\leb}\|_{L^2(H)}$ of the restrictions of $(\phi_{\leb})$ to a curve $H \subset M$ in the limit ${\leb} \to +\infty$. An understanding of these bounds will have important applications to the study of nodal sets. We plan to start by studying the case in which $(\phi_\leb)$ is any sequence of eigenfunctions that are quantum ergodic near $H$. We do not wish to require the geodesic flow to be ergodic, and the only assumption on $H$ that we believe we need is that it cannot be a segment of a geodesic. In the following, we say that an $L^2$-normalized sequence of functions $(\phi_\leb)$ is quantum ergodic (QE) near $H$} if there exists a tubular neighborhood $U_H \supset H$ such that for any $a \in S^{0}(T^*M),$ with $ \pi ( \text{supp \, a )\subset U_H,$ \[ \lim_{\leb \to + \infty} \langle Op_{\leb}(a) \phi_\leb, \phi_\leb \rangle_{L^2(U_H)} = c_n \int_{B^*U_H} a(x,\eta) dx d\eta,\] where $c_n$ is some positive constant and $dxd\eta$ is the canonical measure on $T^*M$. Our main conjecture is the following.

\begin{conjecture} \label{T: lower bounds} Let $(M,g)$ be a compact Riemannian surface with no boundary, and let $H\subset M$ be a smooth curve that is not a segment of a geodesic. Let $(\phi_\leb)$ be a sequence of $L^2$-normalized eigenfunctions that is locally QE near $H.$ Then, there exist positive constants $C_1$ and $\leb_1$ so that \[C_1 \leq \| \phi_\leb \|_{L^2(H)} ,\] for all $\leb \geq \leb_1.$ \end{conjecture}

Our second companion conjecture for restriction upper bounds is the following.

\begin{conjecture} \label{T: upper bounds} Let $(\phi_{\lambda})$ be a locally QE sequence of $L^2$-normalized eigenfunctions near a closed curve $H$ with everywhere non-vanishing geodesic curvature. Then, there exist positive constants $C_2$ and $\leb_2$ so that \[ \| \phi_\leb \|_{L^2(H)} \leq C_2,\] for all $\leb \geq \leb_2.$ \end{conjecture}

We note that the assumption on the curvature of $H$ in Conjecture \ref{T: lower bounds} is crucial due to the existence of Riemannian surfaces $(M,g)$ and geodesics $H \subset M$ for which there are infinitely many eigenfunctions $(\phi_\leb)$ with $H \subset \phi_\leb^{-1}(0)$. For example, this occurs when $H$ is the equator in the $2$-sphere and $(\phi_\leb)$ is any sequence of odd spherical harmonics. Indeed, the same is true for any surface $M$ that has a $Z_2$ orientation reversing isometric involution where $H$ is a geodesic that is fixed under the involution.

It is easy to see, for example on the disk, that there cannot be any natural conditions on a curve guaranteeing a uniform lower bound like that in Conjecture \ref{T: lower bounds} for all eigenfunctions. However, it is reasonable to expect that if a curve persists in the zero set of infinitely many eigenfunctions than it must have very rigid geometry. Our final goal is to study unique continuation off of curves and hypersurfaces. As in the quantum ergodic case, the most natural assumption is that $H$ is not a segment of a geodesic. However, we wish to start with a less degenerate condition and conjecture \begin{conjecture}\label{T:Goodness} Let $(\phi_{\lambda})$ be a sequence of $L^2$ normalized eigenfunctions and $H$ a curve with everywhere non-vanishing geodesic curvature. Then there exist positive constants $C, c,$ and $\leb_3$ such that \begin{equation} \label{e:good}\|\phi_\leb\|_{L^2(H)}\geq ce^{-C\lambda}, \end{equation} for all $\leb\geq \leb_3$. \end{conjecture}

\subsection{Importance, relevance and timeliness}

The problem of obtaining lower bounds for $\|\phi_{\leb}\|_{L^2(H)}$ is quite challenging and has only been attempted in very specific settings. It has profound applications to the study of the nodal sets of high energy eigenfunctions. In [TZ} where the authors show that if $\Omega \subset \reals^2$ is a bounded domain with piece-wise real analytic boundary and $(\phi_{\leb})$ is a sequence of Neumann eigenfunctions, then there exists $C>0$ for which $\|\phi_{\leb}\|_{L^2(\partial \Omega)} \geq e^{-C{\leb}}$ as ${\leb} \to +\infty$. Curves satisfying such exponential lower bound (such as $H=\partial \Omega$), are said to be good}. This is a concept that arises frequently when bounding from above the number of zeros of $\phi_{\leb}$ along $H$. Indeed, on real analytic compact surfaces, the goodness condition on $H$ is needed to prove the sharp upper bound $\#\{ \phi_{\leb}^{-1}(0) \cap H\} =O({\leb})$, see \cite{CT}. For the purpose of obtaining upper bounds on $\#\{ \phi_{\leb}^{-1}(0) \cap H\}$, it is proved in \cite{Jun} that horocycles lying inside compact hyperbolic surfaces are good curves. With the same goal, the authors in \cite{ET} proved that if $\Omega \subset \reals^2$ is a bounded convex domain with piecewise real analytic boundary and $(\phi_{\leb})$ is a sequence of quantum ergodic (QE) Neumann eigenfunctions, then any real analytic closed curve with strictly positive geodesic curvature is good. One can view the restriction lower bounds in this proposal as a natural closed manifold analogue of the lower bounds in Theorem 1.3 of \cite{ET}. However, the restriction lower bounds for curved $H$'s and QE eigenfunction sequences $(\phi_{\lambda})$ give $\| \phi_{\lambda} \|_{L^2(H)} \gtrapprox 1$ whereas Theorem 1.3 in \cite{ET} only implies the much weaker goodness estimate $\| \phi_{\lambda} \|_{L^2(H) \gtrapprox e^{-C \lambda]$ for some $C=C(H,\Omega)>0.$

The exponential lower bound was improved in [GRS} in the case in which $H$ is a closed horocycle lying inside an arithmetic surface and the eigenfunctions $(\phi_{\leb})$ are even Maass cusps forms. In this case the authors prove that for every $\ep>0$ there exists $C_\ep$ so that $\|\phi_{\leb}\|_{L^2(H)} \geq C_\ep {\leb}^{-\ep}$ as ${\leb} \to +\infty$. An even stronger lower bound was obtained in \cite{BR} for the flat torus. In \cite{BR} the authors prove that for any sequence $(\phi_\leb)$ of Laplace eigenfunctions there exists a constant $C>0$ for which $\|\phi_{\leb}\|_{L^2(H)} \geq C$ as ${\leb]\to +\infty$, provided $H$ has non-vanishing geodesic curvature.

As for upper bounds, the universal estimates in [BGT} give $\|\phi_\leb\|_{L^2(H)}=O(\leb^{1/4})$ when $H \subset M$ is any curve, and $\|\phi_\leb\|_{L^2(H)}=O(\leb^{1/6})$ when $H$ has non-vanishing curvature. These upper bounds where slightly improved by a $\log(\leb)^{-1}$ factor in \cite{Che} for negatively curved surfaces. In some specific cases, the improvement is polynomial in $\lambda.$ For example, on flat tori in \cite{BR}, it is shown that $\|\phi_\leb\|_{L^2(H)}=O(1)$ when $H$ has non-vanishing geodesic curvature. In \cite{GRS}, the authors prove that for any $\ep>0$ there exists $C_\ep>0$ for which $\|\phi_\leb\|_{L^2(H)}\leq C_\ep \leb^{\ep]$ when $H$ is a closed horocycle inside an arithmetic surface and the eigenfunctions $(\phi_h)$ are even Maass cusps forms.

One expects to obtain uniform lower and upper bounds for $\|\phi_{\leb}\|_{L^2(H)} $ in cases where the eigenfunctions are equidistributed along $H$. Indeed, it follows from [TZ,DZ} that if $(M,g)$ has ergodic geodesic flow and $H$ has a `zero measure of microlocal symmetry', then there exists a density one subsequence $(\phi_{\leb_j})$ of the set of Laplace eigenfunctions for which $\|\phi_{\leb_j}\|_{L^2(H)} \to C$ as $j \to \infty$. The microlocal asymmetry assumption on $H$ in \cite{TZ} is generic; but it is quite difficult to check and has only been established for geodesic circles, closed geodesics and closed horocycles inside certain hyperbolic surfaces \cite{TZ}. The existence of a limit for $\|\phi_{\leb_j}\|_{L^2(H)}$ hinges on the assumption that the geodesic flow is ergodic. In particular, this assumption gives the existence of a quantum ergodic sequence from which $(\phi_{\leb_j])$ is built.

Conjecture \ref{T: lower bounds} and Conjectures \ref{T: upper bounds} are of course consistent with the QER theorem in [TZ}. However, there are two important differences that should be pointed out: (i) under the curvature assumption on $H$, the results in the conjectures should hold for all} sequences of eigenfunctions provided they are quantum ergodic (QE) near $H$ (there are no exceptional subsequences) and (ii) the statements in the conjectures make no a priori assumption on the geometry of $(M,g)$; we only require that $H$ have positive geodesic curvature and that the eigenfunctions sequence be locally QE near $H$. Finally, in Conjecture \ref{T:Goodness, the result should hold for {\em all] sequences of eigenfunctions even if they are not quantum ergodic.