Evolution Equations for Curvature Tensors
Section 10 — Part III: Analytic Tools
Scalar Curvature Evolution
The scalar curvature satisfies a reaction–diffusion equation under Ricci flow:
\[ \partial_t R \;=\; \Delta R + 2\,|R_{ij}|^2 \]
Since \( |R_{ij}|^2 \geq 0 \), the maximum principle immediately implies that the minimum of the scalar curvature is non-decreasing along the flow. In particular, if \( R \geq 0 \) initially, it remains so for all time.
Ricci Tensor Evolution
The Ricci tensor evolves by the Lichnerowicz Laplacian:
\[ \partial_t R_{ij} \;=\; \Delta_L R_{ij} \]
Here \( \Delta_L \) is the Lichnerowicz Laplacian, which differs from the rough Laplacian by lower-order curvature terms. This equation shows that the Ricci tensor diffuses, with curvature acting as a source.
Full Riemann Tensor Evolution
The full Riemann curvature tensor satisfies:
\[ \partial_t R_{ijkl} \;=\; \Delta R_{ijkl} + Q(R_{ijkl}) \]
where the quadratic reaction term is:
\[ Q_{ijkl} \;=\; 2\bigl(B_{ijkl} - B_{ijlk} - B_{iljk} + B_{ikjl}\bigr) \]
with \( B_{ijkl} = R_{ipjq}\,R^{p\;\;q}_{\ k\ l} \). The tensor \( Q \) encodes all the nonlinear interaction of curvature with itself.
Algebraic Curvature Operator
Viewing the Riemann tensor as an operator on two-forms \( \mathcal{R} : \Lambda^2 TM \to \Lambda^2 TM \), the evolution takes the elegant form:
\[ \partial_t \mathcal{R} \;=\; \Delta \mathcal{R} + \mathcal{R}^2 + \mathcal{R}^{\#} \]
where \( \mathcal{R}^2 \) is the square of the operator and \( \mathcal{R}^{\#} \) is the Lie algebra square (adjoint square) coming from the Lie bracket on \( \mathfrak{so}(n) \).
This formulation is the key to Hamilton's maximum principle for systems: a convex cone in the space of algebraic curvature operators is preserved by the flow if and only if it is preserved by the ODE \( \tfrac{d}{dt}\mathcal{R} = \mathcal{R}^2 + \mathcal{R}^{\#} \). This gives preserved curvature conditions (e.g. non-negative curvature operator, non-negative isotropic curvature, PIC).