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Lower bounds on eigenfunctions and fractal uncertainty principle Dyatlov, Semyon

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Let $(M,g)$ be a compact Riemannian manifold and $\Omega\subset M$ a nonempty open set. Take an $L^2$ normalized eigenfunction $u$ of the Laplacian on $M$ with eigenvalue $\lambda^2$. What lower bounds can we get on the mass $m_\Omega(u)=\int_\Omega |u|^2$ There are two well-known bounds for general $M$: (a) $m_\Omega(u)\geq ce^{-C\lambda}$, following from unique continuation estimates, and (b) $m_\Omega(u)\geq c$, where $c>0$ is independent of $\lambda$, assuming that $\Omega$ intersects every sufficiently long geodesic (this is known as the geometric control condition). In general one cannot improve on the bound (a) for arbitrary $\Omega$, as illustrated by Gaussian beams on the round sphere. I will present a recent result which establishes the frequency-independent lower bound (b) for any choice of $\Omega$ when $M$ is a surface of constant negative curvature. This bound has numerous applications, such as control for the Schrödinger equation, exponential decay of damped waves, and the full support property of semiclassical measures. The proof uses the chaotic nature of the geodesic flow on $M$. The key new ingredient is a recently established fractal uncertainty principle, which states that no function can be localized close to a fractal set in both position and frequency. This talk is based on joint works with Jean Bourgain, Long Jin, and Joshua Zahl.

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