Deformative Magnetic Marked Length Spectrum Rigidity

Abstract

Let $M$ be a closed surface and let $\{g_s|s\in (-\hat{I}\mu ,\hat{I}\mu )\}$ be a smooth one-parameter family of Riemannian metrics on $M$. Also let $\{\hat{I}{º}_s:M\to \mathbb{R}|s\in (-\hat{I}\mu ,\hat{I}\mu )\}$ be a smooth one-parameter family of functions on $M$. Then the family $\{(g_s,\hat{I}{º}_s)|s\in (-\hat{I}\mu ,\hat{I}\mu )\}$ gives rise to a family of magnetic flows on $TM$. We show that if the magnetic curvatures are negative for $s\in (-\hat{I}\mu ,\hat{I}\mu )$ and the lengths of each periodic orbit remains constant as the parameter $s$ varies, then there exists a smooth family of diffeomorphisms $\{f_s:M\to M|s\in (-\hat{I}\mu ,\hat{I}\mu )\}$ such that $f_s^{\ast }(g_s)=g_0$ and $f_s^{\ast }(\hat{I}{º}_s)=\hat{I}{º}_0$. This generalizes a result of Guillemin and Kazhdan to the setting of magnetic flows.

v3: 10 pages, corrected errors in Corollary 3.2 and Lemma 3.4. Version 2 could still be useful for those who want Carlemann estimates for magnetic flows. v2: 17 pages, incorporated referee comments. To appear in "Bulletin of the London Mathematical Society."