Macroscopic scalar curvature and the smoothing method.




In a recent paper L. Guth proved the following theorem : if an hyperbolic manifold endowed with an auxiliary Riemannian metric has sufficently small volume in comparison to the hyperbolic one, then we can find in its universal cover a ball of radius 1 whose volume is bigger than an hyperbolic ball of radius 1. This theorem is strongly related to a conjecture by R. Schoen, and is conjectured to hold for any radius. In this talk, I will present new results around this topic obtained in collaboration with S. Karam by using a technique due to Gromov and which deserves to be better known : the smoothing method.