A very partial answer: by the results of Richard Bamler, bounds on Ricci curvature (which is less than what you are requiring here) and diameter (more than you are requiring), implies that the metric is Lipschitz close to an Einstein metric. In three dimension, this means that the length of the homotopy class is close to that of the hyperbolic manifold. In higher dimension, if your manifold is (topologically) hyperbolic, then the Einstein metric is actually hyperbolic (Leung, 1997), so same as above. 

PS Of course, if you have a diameter bound, then your lengths are bounded below in the two-dimensional case (by the collar lemma, if you like).

Negatively curved metrics minimizing the length of a homotopy class of simple closed curves
Is there a way to matching the code of ESRI WKID with EPSG code? I'm trying to find the coordinate system of PCS Name = CGRS_1993_LTM. Which has WKID:102319 code. But I donn't know the EPSG code.
