Let $X$ be a compact hyperbolic surface, and $\gamma$ a closed geodesic on it. Consider the integral $$\int_\gamma f(x)\, dl(x)$$ where $f$ is a (normalized) Laplace eigenfunction on $X$. Intuitively, if the length $l$ of $\gamma$ is large while the eigenvalue $\lambda$ is moderate, then one would expect this integral to be small. Did anyone found bounds for this integral (in terms of $l$ and $\lambda$)?
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$\begingroup$ A long geodesic can spend entire life at the set where $f$ is close to its maximum. Also, you may go around fix geodesic many times, so increasing the length does not give a better estimate. $\endgroup$– Anton PetruninCommented Nov 30, 2022 at 13:13
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