A Compactness Principle for Maximizing Smooth Functions over Toroidal Geodesics

Stefan Steinerberger

Let $f\in C^2({\mathbb{T}}^2)$ have mean value 0 and consider $$ \sup_{γ~{\tiny \mbox{closed geodesic}}}{~~~ \frac{1}{|γ|} \left| \int_γ{ f ~~d\mathcal{H}^1}\right| },$$ where $\gamma$ r…