Short-Time Existence: The DeTurck Trick
Section 9 — Part III: Analytic Tools
The Parabolicity Problem
The Ricci flow equation is not strictly parabolic. The diffeomorphism invariance of the Riemann curvature tensor introduces a null space in the linearisation, preventing direct application of standard parabolic PDE theory.
The linearisation of the Ricci tensor about a metric gives:
\[ -2\,\delta R_{ij} \;=\; \Delta_L h_{ij} + \nabla_i V_j + \nabla_j V_i \]
where the Lichnerowicz Laplacian is:
\[ (\Delta_L h)_{ij} \;=\; \Delta h_{ij} - 2\,R_{ikjl}\,h^{kl} + R_i^{\;k}\,h_{kj} + R_j^{\;k}\,h_{ki} \]
The \( \nabla_i V_j + \nabla_j V_i \) terms encode the diffeomorphism degeneracy and destroy strict parabolicity.
DeTurck Vector Field
DeTurck's idea (1983) is to fix a background metric \( \tilde{g} \) and define a vector field that absorbs the diffeomorphism degeneracy:
\[ W^k \;=\; g^{pq}\!\left(\Gamma^k_{pq}(g) - \Gamma^k_{pq}(\tilde{g})\right) \]
This is the trace of the difference of Christoffel symbols, which is a genuine tensor (unlike the Christoffel symbols themselves).
Ricci–DeTurck Flow
The modified equation, the Ricci–DeTurck flow (RDT), adds the Lie derivative term:
\[ \partial_t g_{ij} \;=\; -2\,R_{ij} + \nabla_i W_j + \nabla_j W_i \]
This equation is strictly parabolic: the principal symbol is now a positive-definite Laplacian on symmetric 2-tensors. Standard theory gives:
- Short-time existence: For any smooth initial metric on a closed manifold, the RDT flow has a unique smooth solution on some time interval \( [0, T) \).
- Recovering Ricci flow: Let \( \varphi_t \) be the one-parameter family of diffeomorphisms generated by \( -W \). Then \( \varphi_t^* g(t) \) solves the original Ricci flow equation.
Theorem (DeTurck, 1983)
For any smooth Riemannian metric \( g_0 \) on a closed manifold \( M \), the Ricci flow \( \partial_t g = -2\,\mathrm{Ric}(g) \) has a unique smooth solution on some interval \( [0, T) \) with \( g(0) = g_0 \).