This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also provide a more invariant view on the approach used in the above-mentioned paper.
"Area minimizers and boundary rigidity of almost hyperbolic metrics." Duke Math. J. 162 (7) 1205 - 1248, 15 May 2013. https://doi.org/10.1215/00127094-2142529