Let M be a Riemannian n-manifold with a metric such that the manifold is Ahlfors-regular. We also assume either non-negative Ricci curvature, or that the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. We characterize BMO-functions u:M→R by a Carleson measure condition of their σ-harmonic extension U:M×(0,∞)→R
Nearby in the stack
. We make crucial use of a
T(b)
theorem proved by Hofmann, Mitrea, Mitrea, and Morris. As an application we show that the famous theorem of Coifman--Lions--Meyer--Semmes holds in this class of manifolds: Jacobians of
W1,n
-maps from
M
to
Rn
can be estimated against BMO-functions, which now follows from the arguments for commutators recently proposed by Lenzmann and the second-named author using only harmonic extensions, integration by parts, and trace space characterizations.