2
$\begingroup$

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 $\lambda$)?

$\endgroup$
1
  • $\begingroup$ 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. $\endgroup$ Nov 30, 2022 at 13:13

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.