  We derive and study supergravity BPS flow equations for M5 or D3 branes
wrapping a Riemann surface. They take the form of novel geometric flows
intrinsically defined on the surface. Their dual field-theoretic interpretation
suggests the existence of solutions interpolating between an arbitrary metric
in the UV and the constant-curvature metric in the IR. We confirm this
conjecture with a rigorous global existence proof.
