The goal of the paper is to prove an exact representation formula for
the Laplacian of the distance (and more generally for an arbitrary
-Lipschitz
function) in the framework of metric measure spaces satisfying Ricci curvature
lower bounds in a synthetic sense (more precisely in essentially nonbranching
-spaces).
Such a representation formula makes apparent the classical upper bounds together
with lower bounds and a precise description of the singular part. The exact
representation formula for the Laplacian of a general 1-Lipschitz function holds also
(and seems new) in a general complete Riemannian manifold.
We apply these results to prove the equivalence of
and a dimensional Bochner inequality on signed distance functions.
Moreover we obtain a measure-theoretic splitting theorem for
infinitesimally Hilbertian, essentially nonbranching spaces satisfying
.
Keywords
Ricci curvature, optimal transport, Laplacian comparison,
distance function