The Conjugate Linearized Ricci Flow on Closed 3-Manifolds

Understand

We characterize the conjugate linearized Ricci flow and the associated backward heat kernel on closed three--manifolds of bounded geometry.

• We discuss their properties, and introduce the notion of Ricci flow conjugated constraint sets which characterizes a way of Ricci flow averaging metric dependent geometrical data.
• We also provide an integral representation of the Ricci flow metric itself and of its Ricci tensor in terms of the heat kernel of the conjugate linearized Ricci flow.
• These results, which readily extend to closed n-dimensional manifolds, yield for various conservation laws, monotonicity and asymptotic formulas for the Ricci flow and its linearization.