A Compactness Principle for Maximizing Smooth Functions over Toroidal Geodesics

TL;DR

This paper proves the existence of maximizers for a smooth function's average over closed geodesics on a torus and bounds the geodesic length in terms of the function's smoothness, with implications for broader geometric settings.

Contribution

It establishes the attainment of supremum averages over closed geodesics and provides bounds on geodesic length based on function smoothness, a novel geometric-analytic connection.

Findings

Supremum of average over closed geodesics is always attained.

Bound on geodesic length in terms of function smoothness.

Sharp bounds for trigonometric polynomial cases.

Abstract

Let $f \in C^2(\mathbb{T}^2)$ have mean value 0 and consider $$ \sup_{\gamma~{\tiny \mbox{closed geodesic}}}{~~~ \frac{1}{|\gamma|} \left| \int_{\gamma}{ f ~~d\mathcal{H}^1}\right| },$$ where $\gamma$ ranges over all closed geodesics $\gamma:\mathbb{S}^1 \rightarrow \mathbb{T}^2$ and $|\gamma|$ denotes their length. We prove that this supremum is always attained. Moreover, we can bound the length of the geodesic $\gamma$ attaining the supremum in terms of \textit{smoothness} of the function: for all $s \geq 2$, $$ |\gamma|^{s} \lesssim_s \left( \max_{|\alpha| = s}{ \| \partial_{\alpha} f \|_{L^{1}(\mathbb{T}^2)}} \right) \| \nabla f \|_{L^2}^{} \|f\|_{L^2}^{-2}.$$ We also prove a sharp bound for trigonometric polynomials. This seems like an interesting phenomenon. We do not know at which level of generality it holds or whether versions or variants of it could be established in other settings (hyperbolic surfaces, groups,...).