An introduction to conformal Ricci flow

Abstract

We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the conformal Ricci flow equations because of the role that conformal geometry plays in constraining the scalar curvature and because these equations are the vector field sum of a conformal flow equation and a Ricci flow equation. These new equations are given byfor a dynamically evolving metric g and a scalar non-dynamical field p. The conformal Ricci flow equations are analogous to the Navier–Stokes equations of fluid mechanics,Because of this analogy, the time-dependent scalar field p is called a conformal pressure and, as for the real physical pressure in fluid mechanics that serves to maintain the incompressibility of the fluid, the conformal pressure serves as a Lagrange multiplier to conformally deform the metric flow so as to maintain the scalar curvature constraint. The equilibrium points of the conformal Ricci flow equations are Einstein metrics with Einstein constant . Thus, the term measures the deviation of the flow from an equilibrium point and acts as a nonlinear restoring force. The conformal pressure p ⩾ 0 is zero at an equilibrium point and strictly positive otherwise. The constraint force −pg acts pointwise orthogonally to the nonlinear restoring force and conformally deforms g so that the scalar curvature is preserved. A variety of properties of the conformal Ricci flow equation are discussed, including the reduced conformal Ricci flow equation, the use of a nonlinear Trotter product formula to prove local existence and uniqueness, a variational formulation using a quasi-gradient of the Yamabe functional, strictly monotonically decreasing global and local volume results and applications to 3-manifold geometry. The geometry of the conformal Ricci flow is discussed, leading to the remarkable analytic fact that the constraint force does not lose derivatives and thus analytically the conformal Ricci equation is a bounded perturbation of the classical unnormalized Ricci equation. Lastly, we discuss potential applications to Perelman's proposed implementation of Hamilton's programme to prove Thurston's 3-manifold geometrization conjectures.