Almost Hermitian Ricci flow

Abstract

We introduce a new curvature flow which matches with the Ricci flow on metrics and preserves the almost Hermitian condition. This enables us to use Ricci flow to study almost Hermitian manifolds.

Appendix

Lemma 4.4

For $\left( M^n, J ,\omega \right)$ almost Hermitian, the vector field defined in (4.12) is

$$\begin{aligned} Z^p \equiv g^{uk} \left( {\overline{\Gamma }}_{ku}^p- \Gamma _{ku}^p \right) + \vartheta ^p. \end{aligned}$$

Proof

Starting with (4.12) we combute

$$\begin{aligned} \begin{aligned} Z^p&= \omega ^{kl} \left( {\overline{\nabla }}_k J_l^p \right) \\&= \omega ^{kl} \left( \partial _k J_l^p \right) -\omega ^{kl} {\overline{\Gamma }}_{kl}^m J_m^p + \omega ^{kl} {\overline{\Gamma }}_{km}^p J^m_l \\&= \omega ^{kl} \left( \nabla _k J_l^p + \Upsilon _{kl}^u J_u^p - \Upsilon _{ku}^p J_l^u \right) -{\overline{\Gamma }}_{kl}^m J_m^p \omega ^{kl} + {\overline{\Gamma }}_{km}^p g^{mk} \\&= \omega ^{kl}\Upsilon _{kl}^u J_u^p - \Upsilon _{ku}^p g^{uk}+ {\overline{\Gamma }}_{km}^p g^{mk}. \end{aligned} \end{aligned}$$

(4.16)

Let’s investigate these terms. For the first, by Lemma 2.7,

$$\begin{aligned} \omega ^{kl}\Upsilon _{kl}^u J_u^p&= \omega ^{kl}\left( \Gamma _{kl}^u - \Theta _{kl}^u \right) J_u^p \\&= 0. \end{aligned}$$

For the next term

$$\begin{aligned} \Upsilon ^p_{ku} g^{uk}&= \left( \Gamma ^p_{ku} - \Theta _{ku}^p \right) g^{uk} \\&= g^{uk} \Gamma ^p_{ku} + \vartheta ^p. \end{aligned}$$

Inserting these into (4.16) yields the result. $\square$

We now justify (2.3) above regarding the type decompositions of $TM_{\otimes 3}$ by determining the corresponding projections into $TM_{\otimes 3}^{1,2+2,1}$ and $TM_{\otimes 3}^{3,0+0,3}$.

Lemma 4.5

For $(M^n, \omega , J)$ almost Hermitian,

$$\begin{aligned} F^{3,0+0,3}_{ijk}&\equiv \frac{1}{4} \left( F_{ijk} -J_j^b J_k^c F_{ibc} - J_i^a J_k^c F_{ajc} - J_i^a J_j^b F_{abk} \right) , \\ F^{2,1+1,2}_{ijk}&\equiv \frac{3}{4}F_{ijk} + \frac{1}{4} \left( J_j^b J_k^c F_{ibc} + J_i^a J_k^c F_{ajc} + J_i^a J_j^b F_{abk} \right) . \end{aligned}$$

Proof

For a vector field $X \in TM$, define the projections

$$\begin{aligned} \Pi _{1,0} X \triangleq \frac{1}{2} \left( X + iJ X \right) , \qquad \Pi _{0,1} X \triangleq \frac{1}{2} \left( X - iJX \right) . \end{aligned}$$

With this we have that

$$\begin{aligned}&\left( \Pi _{3,0} F \right) \left( X,Y,Z \right) \\&\quad = F \left( \Pi _{1,0} X, \Pi _{1,0} Y, \Pi _{1,0} Z \right) \\&\quad = \frac{1}{8} F \left( X,Y,Z \right) + i \frac{1}{8} F\left( X,Y,JZ \right) + i \frac{1}{8} F \left( X,JY,Z \right) + \left( i \right) ^2 \frac{1}{8} F \left( X,JY,JZ \right) \\&\qquad \quad + i \frac{1}{8} F \left( JX,Y,Z \right) + \left( i \right) ^2 \frac{1}{8} F \left( JX,Y,JZ \right) \\&\qquad \quad + \left( i \right) ^2 \frac{1}{8} F \left( JX,JY,Z \right) + \left( i \right) ^3 \frac{1}{8} F \left( JX,JY,JZ \right) \\&\quad = \frac{1}{8} F \left( X,Y,Z \right) + i \frac{1}{8} F\left( X,Y,JZ \right) + i \frac{1}{8} F \left( X,JY,Z \right) - \frac{1}{8} F \left( X,JY,JZ \right) \\&\qquad \quad + i \frac{1}{8} F \left( JX,Y,Z \right) - \frac{1}{8} F \left( JX,Y,JZ \right) -\frac{1}{8} F \left( JX,JY,Z \right) \\ {}&\quad \qquad - i \frac{1}{8} F \left( JX,JY,JZ \right) . \end{aligned}$$

Similarly, we have that

$$\begin{aligned}&\left( \Pi _{0,3} F \right) \left( X,Y,Z \right) \\&\quad = F \left( \Pi _{0,1} X, \Pi _{0,1} Y, \Pi _{0,1} Z \right) \\&\quad = \frac{1}{8} F \left( X,Y,Z \right) - i \frac{1}{8} F\left( X,Y,JZ \right) - i \frac{1}{8} F \left( X,JY,Z \right) + \left( i \right) ^2 \frac{1}{8} F \left( X,JY,JZ \right) \\&\qquad \quad - i \frac{1}{8} F \left( JX,Y,Z \right) + \left( i \right) ^2 \frac{1}{8} F \left( JX,Y,JZ \right) \\&\qquad \quad + \left( i \right) ^2 \frac{1}{8} F \left( JX,JY,Z \right) - \left( i \right) ^3 \frac{1}{8} F \left( JX,JY,JZ \right) \\&\quad = \frac{1}{8} F \left( X,Y,Z \right) - i \frac{1}{8} F\left( X,Y,JZ \right) - i \frac{1}{8} F \left( X,JY,Z \right) - \frac{1}{8} F \left( X,JY,JZ \right) \\&\qquad \quad - i \frac{1}{8} F \left( JX,Y,Z \right) - \frac{1}{8} F \left( JX,Y,JZ \right) - \frac{1}{8} F \left( JX,JY,Z \right) \\&\quad \qquad + i \frac{1}{8} F \left( JX,JY,JZ \right) . \end{aligned}$$

Therefore we have that

$$\begin{aligned}&\left( \Pi _{3,0} F + \Pi _{0,3} F \right) \left( X,Y,Z \right) \\&\quad = \frac{1}{4} \left( F \left( X,Y,Z \right) - F \left( X,JY,JZ \right) - F \left( JX,Y,JZ \right) - F \left( JX,JY,Z \right) \right) . \end{aligned}$$

Now we compute

$$\begin{aligned}&\left( \Pi _{2,1}F + \Pi _{1,2} F \right) \left( X,Y,Z \right) \\&\quad = \left( F \left( X,Y,Z \right) - \left( \Pi _{3,0} F + \Pi _{0,3} F \right) \right) \left( X,Y,Z \right) \\&\quad = \frac{3}{4} F \left( X,Y,Z \right) + \frac{1}{4} \left( F \left( X,JY,JZ \right) + F \left( JX,Y,JZ \right) + F \left( JX,JY,Z \right) \right) . \end{aligned}$$

Converting to coordinates yields the result. $\square$