Conformal boundary rigidity for simple Finsler metrics

Abstract: In this paper we prove that simple Finsler manifolds are conformally stable. Given a simple Finsler manifold and a class of conformal factors, we characterize the singularity of their induced boundary distance functions which allows us to define an appropiate $H<sup>2$ norm. We then obtain a stability estimate with respect to this Sobolev norm and the $L<sup>2$ norm on the class of the conformal factors. To prove the main theorems, we adapt the classical integration by parts technique employed by Mukhometov \cite{Muhometov} to the Finsler setting.