Abstract
We establish Evans–Krylov estimates for certain nonconvex fully nonlinear elliptic and parabolic equations by exploiting partial Legendre transformations. The equations under consideration arise in part from the study of the “pluriclosed flow” introduced by Streets and Tian (Int Math Res Not 16:3101–3133, 2010).
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J. Streets and M. Warren gratefully acknowledge support from the NSF via DMS-1301864 and DMS-1161498, respectively.
Appendix: Evolution equations
Appendix: Evolution equations
In this section we prove the crucial subsolution properties for the matrix W along the real and complex twisted Monge–Ampère equations. The results are contained in Lemmas 6.3 and 6.1. We directly prove the case of complex variables first, which consists of lengthy calculations and applications of the Cauchy-Schwarz inequality. Again we note that these monotonicity properties are suggested by the discussion of Legendre transformations in Sect. 2. A similar direct calculation can yield the case of real variables, but we suppress this as it is lengthy and nearly identical to the complex case. Instead we show that the real case follows by formally extending variables and appealing to the complex setting.
Lemma 6.1
Let \(u_t\) be a solution to (1.4) such that \(u_t \in \mathcal E^{k,l}_U\) for all t. Then
$$\begin{aligned} \left( \frac{\partial }{\partial t}- \mathcal L\right) \frac{\partial u}{\partial t}&= 0. \end{aligned}$$
(6.1)
Also,
$$\begin{aligned} \left( \frac{\partial }{\partial t}- \mathcal L\right) W = Q, \end{aligned}$$
where
$$\begin{aligned} Q_{\alpha _z \bar{\alpha }_z}&= - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q \alpha _z} u_{z_r \bar{z}_s \bar{\alpha }_z} + u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q \alpha _z} u_{z_r \bar{z}_s \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z}\nonumber \\&\quad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_p} u_{z_r \bar{z}_s \bar{w}_q} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} \nonumber \\&\quad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_l} u_{z_r \bar{z}_s \bar{\alpha }_z}\nonumber \\&\quad + u^{\bar{z}_b z_a} \left[ - u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_b} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} + u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z \bar{z}_b} \right. \nonumber \\&\quad - u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} \nonumber \\&\quad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\nonumber \\&\quad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z}\nonumber \\&\quad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z \bar{z}_b}\nonumber \\&\quad \left. + u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z z_a} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z z_a} \right] \nonumber \\&\quad - u^{\bar{w}_b w_a} \left[ - u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} \right. \nonumber \\&\quad \left. + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} \right. \nonumber \\&\quad \left. + u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z w_a} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z w_a} \right] \end{aligned}$$
(6.2)
$$\begin{aligned} Q_{\alpha _w \bar{\alpha }_w}&= - u^{\bar{\alpha }_w w_k} u^{\bar{w}_l \alpha _w} u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_k} u_{z_r \bar{z}_s \bar{w}_l} \nonumber \\&\quad +\, u^{\bar{z}_l z_k} u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \bar{z}_l} u^{\bar{w}_q w_j} u_{w_j \bar{w}_k z_k} u^{\bar{w}_k \alpha _w}\nonumber \\&\quad +\, u^{\bar{z}_l z_k} u^{\bar{\alpha }_w w_j} u_{w_j \bar{w}_k z_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_l} u^{\bar{w}_q \alpha _w}\nonumber \\&\quad -\, u^{\bar{w}_l w_k} u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \bar{w}_l} u^{\bar{w}_q w_j} u_{w_j \bar{w}_r w_k} u^{\bar{w}_r \alpha _w}, \end{aligned}$$
(6.3)
$$\begin{aligned} Q_{\alpha _z \alpha _w}&= - u^{\bar{z}_b z_a} u^{\bar{z}_d z_c} u_{z_a \bar{z}_d \alpha _z} u_{z_c \bar{z}_b \bar{w}_k} u^{\bar{w}_k \alpha _w} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u^{\bar{w}_p \alpha _w} u^{\bar{z}_b z_a} u^{\bar{z}_d z_c} u_{z_a \bar{z}_d w_l} u_{z_c \bar{z}_b \bar{w}_p}\nonumber \\&\quad - u^{\bar{z}_b z_a} \left[ - u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_b} u^{\bar{w}_q \alpha _w} - u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q \alpha _w} \right. \nonumber \\&\quad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q \alpha _w}\nonumber \\&\quad \left. + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s \alpha _w} \right] \nonumber \\&\quad + u^{\bar{w}_b w_a} \left[ - u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q \alpha _w}\right. \nonumber \\&\quad \left. + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q \alpha _w}\right] , \end{aligned}$$
(6.4)
$$\begin{aligned} Q_{\bar{\alpha }_z \bar{\alpha }_w} =\bar{Q}_{\alpha _z \alpha _w} \end{aligned}$$
(6.5)
Proof
First we prove (6.1).
$$\begin{aligned} \frac{\partial }{\partial t}\left( \frac{\partial u}{\partial t} \right)&= \frac{\partial }{\partial t}\left( \log \det u_{\alpha _z \bar{\alpha }_z} - \log \det (-u_{\alpha _w \bar{\alpha }_w}) \right) \\&= u^{\bar{z}_b z_a} \left( \frac{\partial u}{\partial t} \right) _{z_a \bar{z}_b} - u^{\bar{w}_b w_a} \left( \frac{\partial u}{\partial t} \right) _{w_a \bar{w}_b}\\&= \mathcal L\frac{\partial u}{\partial t}. \end{aligned}$$
Next we establish (6.3). We start by computing partial derivatives
$$\begin{aligned} \left( \log \det u_{z \bar{z}} \right) _{,\alpha \beta }&= \left( u^{\bar{z}_q z_p} u_{z_p \bar{z}_q \alpha } \right) _{,\beta }= u^{\bar{z}_q z_p} u_{z_p \bar{z}_q \alpha \beta } - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q \alpha } u_{z_r \bar{z}_s \beta },\nonumber \\ \left( \log \det (- u_{yy}) \right) _{,\alpha \beta }&= \left( u^{\bar{w}_q w_p} u_{w_p \bar{w}_q \alpha } \right) _{,\beta }\nonumber \\&= u^{\bar{w}_q w_p} u_{w_p \bar{w}_q \alpha \beta } - u^{\bar{w}_q w_r} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q \alpha } u_{w_r \bar{w}_s \beta }, \end{aligned}$$
(6.6)
Using this we compute
$$\begin{aligned} \frac{\partial }{\partial t}u^{\bar{\alpha }_w \alpha _w}&= - u^{\bar{\alpha }_w w_k} \left( \frac{\partial }{\partial t}u \right) _{w_k \bar{w}_l} u^{\bar{w}_l \alpha _w}\nonumber \\&= - u^{\bar{\alpha }_w w_k} \left( \log \det u_{z \bar{z}} - \log \det (- u_{w \bar{w}}) \right) _{w_k \bar{w}_l} u^{\bar{w}_l \alpha _w}\nonumber \\&= u^{\bar{\alpha }_w w_k} u^{\bar{w}_l \alpha _w} \left( u^{\bar{w}_q w_p} u_{w_p \bar{w}_q w_k \bar{w}_l} - u^{\bar{w}_q w_r} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q w_k} u_{w_r \bar{w}_s \bar{w}_l}\right. \nonumber \\&\ \left. \quad - u^{\bar{z}_q z_p} u_{z_p \bar{z}_q w_k \bar{w}_l} + u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_k} u_{z_r \bar{z}_s \bar{w}_l} \right) . \end{aligned}$$
(6.7)
Also we compute the partial derivatives
$$\begin{aligned} u^{\bar{\alpha }_w \alpha _w}_{\alpha \beta }&= - \left( u^{\bar{\alpha }_w w_j} u_{w_j \bar{w}_k \alpha } u^{\bar{w}_k \alpha _w} \right) _{,\beta }\nonumber \\&= u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \beta } u^{\bar{w}_q w_j} u_{w_j \bar{w}_k \alpha } u^{\bar{w}_k \alpha _w} - u^{\bar{\alpha }_w w_j} u_{w_j \bar{w}_k \alpha \beta } u^{\bar{w}_k \alpha _w}\nonumber \\&\quad + u^{\bar{\alpha }_w w_j} u_{w_j \bar{w}_k \alpha } u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \beta } u^{\bar{w}_q \alpha _w}. \end{aligned}$$
(6.8)
Thus we have
$$\begin{aligned} \mathcal L(u^{\bar{\alpha }_w \alpha _w})&= u^{\bar{z}_l z_k} (u^{\bar{\alpha }_w \alpha _w})_{,z_k \bar{z}_l} - u^{\bar{w}_k w_l} (u^{\bar{\alpha }_w \alpha _w})_{,w_k \bar{w}_l}\nonumber \\&= u^{\bar{z}_l z_k} \left( u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \bar{z}_l} u^{\bar{w}_q w_j} u_{w_j \bar{w}_k z_k} u^{\bar{w}_k \alpha _w} \right. \nonumber \\&\left. \quad -u^{\bar{\alpha }_w w_j} u_{w_j \bar{w}_k z_k \bar{z}_l} u^{\bar{w}_k \alpha _w} + u^{\bar{\alpha }_w w_j} u_{w_j \bar{w}_k z_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_l} u^{\bar{w}_q \alpha _w}\right) \nonumber \\&\quad - u^{\bar{w}_l w_k} \left( u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \bar{w}_l} u^{\bar{w}_q w_j} u_{w_j \bar{w}_r w_k} u^{\bar{w}_r \alpha _w} \right. \nonumber \\&\quad \left. -u^{\bar{\alpha }_w w_j} u_{w_j \bar{w}_r w_k \bar{w}_l} u^{\bar{w}_r \alpha _w} + u^{\bar{\alpha }_w w_j} u_{w_j \bar{w}_r w_k} u^{\bar{w}_r w_p} u_{w_p \bar{w}_q \bar{w}_l} u^{\bar{w}_q \alpha _w}\right) . \end{aligned}$$
(6.9)
Putting together (6.7) and (6.9) yields
$$\begin{aligned}&\left( \frac{\partial }{\partial t}- \mathcal L\right) W_{\alpha _w \bar{\alpha }_w}\\&\quad = - u^{\bar{\alpha }_w w_k} u^{\bar{w}_l \alpha _w} \left( - u^{\bar{w}_q w_r} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q w_k} u_{w_r \bar{w}_s \bar{w}_l} + u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_k} u_{z_r \bar{z}_s \bar{w}_l} \right) \\&\qquad + u^{\bar{z}_l z_k} \left( u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \bar{z}_l} u^{\bar{w}_q w_j} u_{w_j \bar{w}_k z_k} u^{\bar{w}_k \alpha _w} + u^{\bar{\alpha }_w w_j} u_{w_j \bar{w}_k z_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_l} u^{\bar{w}_q \alpha _w} \right) \\&\qquad - u^{\bar{w}_l w_k} \left( u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \bar{w}_l} u^{\bar{w}_q w_j} u_{w_j \bar{w}_r w_k} u^{\bar{w}_r \alpha _w} + u^{\bar{\alpha }_w w_j} u_{w_j \bar{w}_r w_k} u^{\bar{w}_r w_p} u_{w_p \bar{w}_q \bar{w}_l} u^{\bar{w}_q \alpha _w} \right) \\&\quad = - u^{\bar{\alpha }_w w_k} u^{\bar{w}_l \alpha _w} u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_k} u_{z_r \bar{z}_s \bar{w}_l} + u^{\bar{z}_l z_k} u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \bar{z}_l} u^{\bar{w}_q w_j} u_{w_j \bar{w}_k z_k} u^{\bar{w}_k \alpha _w}\\&\qquad + u^{\bar{z}_l z_k} u^{\bar{\alpha }_w w_j} u_{w_j \bar{w}_k z_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_l} u^{\bar{w}_q \alpha _w} - u^{\bar{w}_l w_k} u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \bar{w}_l} u^{\bar{w}_q w_j} u_{w_j \bar{w}_r w_k} u^{\bar{w}_r \alpha _w}, \end{aligned}$$
finishing the proof of (6.3). Next we establish (6.2). First we compute using (6.6)
$$\begin{aligned} \frac{\partial }{\partial t}u_{\alpha _z \bar{\alpha }_z}&= \left( \log \det u_{\alpha _z \bar{\alpha }_z} - \log \det (- u_{\alpha _w \bar{\alpha }_w}) \right) _{\alpha _z \bar{\alpha }_z}\\&= u^{\bar{z}_q z_p} u_{z_p \bar{z}_q \alpha _z \bar{\alpha }_z} - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q \alpha _z} u_{z_r \bar{z}_s \bar{\alpha }_z}\\&\quad - u^{\bar{w}_q w_p} u_{w_p \bar{w}_q \alpha _z \bar{\alpha }_z} + u^{\bar{w}_q w_r} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q \alpha _z} u_{w_r \bar{w}_s \bar{\alpha }_z}. \end{aligned}$$
Also
$$\begin{aligned} \mathcal Lu_{\alpha _z \bar{\alpha }_z} = u^{\bar{z}_q z_p} u_{\alpha _z \bar{\alpha }_z z_p \bar{z}_q} - u^{\bar{w}_q w_p} u_{\alpha _z \bar{\alpha }_z w_p \bar{w}_q}. \end{aligned}$$
Thus
$$\begin{aligned} \left( \frac{\partial }{\partial t}- \mathcal L\right) u_{\alpha _z \bar{\alpha }_z} = - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q \alpha _z} u_{z_r \bar{z}_s \bar{\alpha }_z} + u^{\bar{w}_q w_r} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q \alpha _z} u_{w_r \bar{w}_s \bar{\alpha }_z}. \end{aligned}$$
(6.10)
To compute the next term we first differentiate using (6.6)
$$\begin{aligned}&\frac{\partial }{\partial t}\left( u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z} \right) \nonumber \\&\quad = \left( \frac{\partial }{\partial t}u \right) _{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} \left( \frac{\partial }{\partial t}u \right) _{w_p \bar{w}_q} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\nonumber \\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} \left( \frac{\partial }{\partial t}u_{w_l \bar{\alpha }_z} \right) \nonumber \\&\quad = \left( u^{\bar{z}_q z_p} u_{z_p \bar{z}_q \alpha _z \bar{w}_k} - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q \alpha _z} u_{z_r \bar{z}_s \bar{w}_k} - u^{\bar{w}_b w_a} u_{w_a \bar{w}_b \alpha _z \bar{w}_k} \right. \nonumber \\&\qquad \left. + u^{\bar{w}_d w_c} u^{\bar{w}_b w_a} u_{w_a \bar{w}_d \alpha _z} u_{w_c \bar{w}_b \bar{w}_k} \right) u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z}\nonumber \\&\qquad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} \left( u^{\bar{z}_q z_p} u_{z_p \bar{z}_q w_p \bar{w}_q} - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_p} u_{z_r \bar{z}_s \bar{w}_q} \right. \nonumber \\&\qquad \left. - u^{\bar{w}_b w_a} u_{w_a \bar{w}_b w_p \bar{w}_q} + u^{\bar{w}_d w_c} u^{\bar{w}_b w_a} u_{w_a \bar{w}_d w_p} u_{w_c \bar{w}_b \bar{w}_q} \right) u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\nonumber \\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} \left( u^{\bar{z}_q z_p} u_{z_p \bar{z}_q w_l \bar{\alpha }_z} - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_l} u_{z_r \bar{z}_s \bar{\alpha }_z} - u^{\bar{w}_b w_a} u_{w_a \bar{w}_b w_l \bar{\alpha }_z} \right. \nonumber \\&\qquad \left. + u^{\bar{w}_d w_c} u^{\bar{w}_b w_a} u_{w_a \bar{w}_d w_l} u_{w_c \bar{w}_b \bar{\alpha }_z} \right) . \end{aligned}$$
(6.11)
Next we compute the partial derivatives
$$\begin{aligned}&\left( u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z}\right) _{,\alpha \beta }\\&\quad = \left( u_{\alpha _z \bar{w}_k \alpha } u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \alpha } u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z \alpha } \right) _{,\beta }\\&\quad = u_{\alpha _z \bar{w}_k \alpha \beta } u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z} - u_{\alpha _z \bar{w}_k \alpha } u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \beta } u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} + u_{\alpha _z \bar{w}_k \alpha } u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z \beta }\\&\qquad - u_{\alpha _z \bar{w}_k \beta } u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \alpha } u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \beta } u^{\bar{w}_s w_p} u_{w_p \bar{w}_q \alpha } u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\\&\qquad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \alpha \beta } u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \alpha } u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \beta } u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z}\\&\qquad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \alpha } u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z \beta }\\&\qquad + u_{\alpha _z \bar{w}_k \beta } u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z \alpha } - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \beta } u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z \alpha } + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z \alpha \beta }. \end{aligned}$$
Thus we have
$$\begin{aligned}&\mathcal L( u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z})\nonumber \\&\quad = u^{\bar{z}_b z_a} \left( u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z}\right) _{,z_a \bar{z}_b} - u^{\bar{w}_b w_a} \left( u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z}\right) _{,w_a \bar{w}_b}\nonumber \\&\quad = u^{\bar{z}_b z_a} \left[ u_{\alpha _z \bar{w}_k z_a \bar{z}_b} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z} - u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_b} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\right. \nonumber \\&\qquad + u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z \bar{z}_b} - u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\nonumber \\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}- u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a \bar{z}_b} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\nonumber \\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z}- u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z \bar{z}_b}\nonumber \\&\qquad \left. + u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z z_a} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z z_a} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z z_a \bar{z}_b} \right] \nonumber \\&\qquad - u^{\bar{w}_b w_a} \left[ u_{\alpha _z \bar{w}_k w_a \bar{w}_b} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z} - u_{\alpha _z \bar{w}_k w_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{w}_b} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\right. \nonumber \\&\qquad + u_{\alpha _z \bar{w}_k w_a} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z \bar{w}_b}- u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\nonumber \\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}- u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a \bar{w}_b} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} \nonumber \\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z}\nonumber \\&\qquad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z \bar{w}_b}+ u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z w_a}\nonumber \\&\qquad \left. - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z w_a} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z w_a \bar{w}_b} \right] . \end{aligned}$$
(6.12)
Putting together (6.10), (6.11) and (6.12) yields
$$\begin{aligned}&\left( \frac{\partial }{\partial t}- \mathcal L\right) W_{\alpha _z \bar{\alpha }_z}\\&\quad = - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q \alpha _z} u_{z_r \bar{z}_s \bar{\alpha }_z} + u^{\bar{w}_q w_r} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q \alpha _z} u_{w_r \bar{w}_s \bar{\alpha }_z}\\&\qquad - \left( - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q \alpha _z} u_{z_r \bar{z}_s \bar{w}_k} + u^{\bar{w}_d w_c} u^{\bar{w}_b w_a} u_{w_a \bar{w}_d \alpha _z} u_{w_c \bar{w}_b \bar{w}_k} \right) u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z}\\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} \left( - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_p} u_{z_r \bar{z}_s \bar{w}_q} + u^{\bar{w}_d w_c} u^{\bar{w}_b w_a} u_{w_a \bar{w}_d w_p} u_{w_c \bar{w}_b \bar{w}_q} \right) \\&\qquad \times u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}- u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} \left( - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_l} u_{z_r \bar{z}_s \bar{\alpha }_z}\right. \\&\qquad \left. + u^{\bar{w}_d w_c} u^{\bar{w}_b w_a} u_{w_a \bar{w}_d w_l} u_{w_c \bar{w}_b \bar{\alpha }_z} \right) \\&\qquad + u^{\bar{z}_b z_a} \left[ - u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_b} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} + u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z \bar{z}_b} \right. \\&\qquad - u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z \bar{z}_b}\\&\qquad \left. + u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z z_a} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z z_a} \right] \\&\qquad - u^{\bar{w}_b w_a} \left[ - u_{\alpha _z \bar{w}_k w_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{w}_b} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} + u_{\alpha _z \bar{w}_k w_a} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z \bar{w}_b} \right. \\&\qquad - u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} \\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z}\\&\qquad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z \bar{w}_b}\\&\qquad \left. + u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z w_a} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z w_a} \right] \\&\quad =: \sum _{i=1}^{24} A_i. \end{aligned}$$
We observe that \(A_2 + A_{18} = A_4 + A_{17} = A_6 + A_{21} = A_8 + A_{22} = 0\), and hence
$$\begin{aligned}&\left( \frac{\partial }{\partial t}- \mathcal L\right) W_{\alpha _z \bar{\alpha }_z}\\&\quad = - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q \alpha _z} u_{z_r \bar{z}_s \bar{\alpha }_z} + u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q \alpha _z} u_{z_r \bar{z}_s \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z}\\&\qquad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_p} u_{z_r \bar{z}_s \bar{w}_q} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_l} u_{z_r \bar{z}_s \bar{\alpha }_z}\\&\qquad + u^{\bar{z}_b z_a} \left[ - u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_b} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} + u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z \bar{z}_b} \right. \\&\qquad - u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z \bar{z}_b}\\&\qquad \left. + u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z z_a} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z z_a} \right] \\&\qquad - u^{\bar{w}_b w_a} \left[ - u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\right. \\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} \\&\qquad \left. + u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z w_a} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z w_a} \right] , \end{aligned}$$
completing the proof of (6.2). Next we establish (6.4). Using (6.6) we compute
$$\begin{aligned}&\frac{\partial }{\partial t}u_{\alpha _z \bar{w}_k} u^{\bar{w}_k \alpha _w}\nonumber \\&\quad = \left( \frac{\partial }{\partial t}u \right) _{\alpha _z \bar{w}_k} u^{\bar{w}_k \alpha _w} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} \left( \frac{\partial }{\partial t}u \right) _{w_l \bar{w}_p} u^{\bar{w}_p \alpha _w}\nonumber \\&\quad = \left( u^{\bar{z}_b z_a} u_{z_a \bar{z}_b \alpha _z \bar{w}_k} - u^{\bar{w}_b w_a} u_{w_a \bar{w}_b \alpha _z \bar{w}_k} \right. \nonumber \\&\qquad \left. - u^{\bar{z}_b z_a} u^{\bar{z}_d z_c} u_{z_a \bar{z}_d \alpha _z} u_{z_c \bar{z}_b \bar{w}_k} + u^{\bar{w}_b w_a} u^{\bar{w}_d w_c} u_{w_a \bar{w}_d \alpha _z} u_{w_c \bar{w}_b \bar{w}_k} \right) u^{\bar{w}_k \alpha _w}\nonumber \\&\qquad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u^{\bar{w}_p \alpha _w} \left( u^{\bar{z}_b z_a} u_{z_a \bar{z}_b w_l \bar{w}_p} - u^{\bar{w}_b w_a} u_{w_a \bar{w}_b w_l \bar{w}_p} \right. \nonumber \\&\qquad \left. - u^{\bar{z}_b z_a} u^{\bar{z}_d z_c} u_{z_a \bar{z}_d w_l} u_{z_c \bar{z}_b \bar{w}_p} + u^{\bar{w}_b w_a} u^{\bar{w}_d w_c} u_{w_a \bar{w}_d w_l} u_{w_c \bar{w}_b \bar{w}_p} \right) . \end{aligned}$$
(6.13)
Next we compute partial derivatives
$$\begin{aligned}&\left( u_{\alpha _z \bar{w}_k} u^{\bar{w}_k \alpha _w} \right) _{,\mu \rho } =\ \left( u_{\alpha _z \bar{w}_k \mu } u^{\bar{w}_k \alpha _w} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \mu } u^{\bar{w}_q \alpha _w} \right) _{,\rho }\nonumber \\&\quad = u_{\alpha _z \bar{w}_k \mu \rho } u^{\bar{w}_k \alpha _w} - u_{\alpha _z \bar{w}_k \mu } u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \rho } u^{\bar{w}_q \alpha _w}\nonumber \\&\qquad - u_{\alpha _z \bar{w}_k \rho } u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \mu } u^{\bar{w}_q \alpha _w} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \rho } u^{\bar{w}_s w_p} u_{w_p \bar{w}_q \mu } u^{\bar{w}_q \alpha _w}\nonumber \\&\qquad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \mu \rho } u^{\bar{w}_q \alpha _w} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \mu } u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \rho } u^{\bar{w}_s \alpha _w}. \end{aligned}$$
(6.14)
Using this we compute
$$\begin{aligned}&\mathcal L\left( u_{\alpha _z \bar{w}_k} u^{\bar{w}_k \alpha _w} \right) \nonumber \\&\quad = u^{\bar{z}_b z_a} \left( u_{\alpha _z \bar{w}_k} u^{\bar{w}_k \alpha _w} \right) _{z_a \bar{z}_b} - u^{\bar{w}_b w_a} \left( u_{\alpha _z \bar{w}_k} u^{\bar{w}_k \alpha _w} \right) _{w_a \bar{w}_b}\nonumber \\&\quad = u^{\bar{z}_b z_a} \left[ u_{\alpha _z \bar{w}_k z_a \bar{z}_b} u^{\bar{w}_k \alpha _w} - u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_b} u^{\bar{w}_q \alpha _w} \right. \nonumber \\&\qquad - u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q \alpha _w} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q \alpha _w}\nonumber \\&\qquad \left. - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a \bar{z}_b} u^{\bar{w}_q \alpha _w} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s \alpha _w} \right] \nonumber \\&\qquad - u^{\bar{w}_b w_a} \left[ u_{\alpha _z \bar{w}_k w_a \bar{w}_b} u^{\bar{w}_k \alpha _w} - u_{\alpha _z \bar{w}_k w_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{w}_b} u^{\bar{w}_q \alpha _w} \right. \nonumber \\&\qquad - u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q \alpha _w} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q \alpha _w}\nonumber \\&\qquad \left. - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a \bar{w}_b} u^{\bar{w}_q \alpha _w} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s \alpha _w} \right] . \end{aligned}$$
(6.15)
Combining (6.13) and (6.15) yields
$$\begin{aligned}&\left( \frac{\partial }{\partial t}- \mathcal L\right) W_{\alpha _z \alpha _w}\\&\quad = \left( - u^{\bar{z}_b z_a} u^{\bar{z}_d z_c} u_{z_a \bar{z}_d \alpha _z} u_{z_c \bar{z}_b \bar{w}_k} + u^{\bar{w}_b w_a} u^{\bar{w}_d w_c} u_{w_a \bar{w}_d \alpha _z} u_{w_c \bar{w}_b \bar{w}_k} \right) u^{\bar{w}_k \alpha _w}\\&\qquad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u^{\bar{w}_p \alpha _w} \left( - u^{\bar{z}_b z_a} u^{\bar{z}_d z_c} u_{z_a \bar{z}_d w_l} u_{z_c \bar{z}_b \bar{w}_p} + u^{\bar{w}_b w_a} u^{\bar{w}_d w_c} u_{w_a \bar{w}_d w_l} u_{w_c \bar{w}_b \bar{w}_p} \right) \\&\qquad - u^{\bar{z}_b z_a} \left[ - u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_b} u^{\bar{w}_q \alpha _w} - u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q \alpha _w} \right. \\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q \alpha _w} \\&\qquad \left. + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s \alpha _w} \right] \\&\qquad + u^{\bar{w}_b w_a} \left[ - u_{\alpha _z \bar{w}_k w_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{w}_b} u^{\bar{w}_q \alpha _w} - u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q \alpha _w} \right. \\&\qquad \left. + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q \alpha _w} \right. \\&\qquad \left. + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s \alpha _w} \right] \\&\quad =: \sum _{i=1}^{12} A_i. \end{aligned}$$
Observing that W is Hermitian, and that the operator \(\frac{\partial }{\partial t}- \mathcal L\) is Hermitian we obtain (6.5). \(\square \)
Lemma 6.2
With the setup above,
$$\begin{aligned} Q \le 0. \end{aligned}$$
Proof
Using Lemma 6.1 we compute
$$\begin{aligned}&Q(\alpha ,\bar{\alpha }) = Q_{\alpha _z \bar{\alpha }_z} + Q_{\alpha _z \bar{\alpha }_w} + Q_{\alpha _w \bar{\alpha }_z} + Q_{\alpha _w \bar{\alpha }_w}\\&\quad = - u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q \alpha _z} u_{z_r \bar{z}_s \bar{\alpha }_z} + u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q \alpha _z} u_{z_r \bar{z}_s \bar{w}_k} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z}\\&\qquad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_p} u_{z_r \bar{z}_s \bar{w}_q} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\\ \end{aligned}$$
$$\begin{aligned}&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_l} u_{z_r \bar{z}_s \bar{\alpha }_z}\\&\qquad + u^{\bar{z}_b z_a} \left[ - u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_b} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} + u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z \bar{z}_b} \right. \\&\qquad - u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} \\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z} \\&\qquad - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z \bar{z}_b}\\ \end{aligned}$$
$$\begin{aligned}&\qquad \left. + u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z z_a} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z z_a} \right] \\&\qquad - u^{\bar{w}_b w_a} \left[ - u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z}\right. \\&\qquad + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_l} u_{w_l \bar{\alpha }_z} \\&\qquad \left. + u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_l} u_{w_l \bar{\alpha }_z w_a} - u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_l} u_{w_l \bar{\alpha }_z w_a} \right] \\&\qquad - u^{\bar{\alpha }_w w_k} u^{\bar{w}_l \alpha _w} u^{\bar{z}_q z_r} u^{\bar{z}_s z_p} u_{z_p \bar{z}_q w_k} u_{z_r \bar{z}_s \bar{w}_l} \\ \end{aligned}$$
$$\begin{aligned}&\qquad + u^{\bar{z}_l z_k} u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \bar{z}_l} u^{\bar{w}_q w_j} u_{w_j \bar{w}_k z_k} u^{\bar{w}_k \alpha _w}\\&\qquad + u^{\bar{z}_l z_k} u^{\bar{\alpha }_w w_j} u_{w_j \bar{w}_k z_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_l} u^{\bar{w}_q \alpha _w}\\&\qquad - u^{\bar{w}_l w_k} u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \bar{w}_l} u^{\bar{w}_q w_j} u_{w_j \bar{w}_r w_k} u^{\bar{w}_r \alpha _w}\\&\qquad - u^{\bar{z}_b z_a} u^{\bar{z}_d z_c} u_{z_a \bar{z}_d \alpha _z} u_{z_c \bar{z}_b \bar{w}_k} u^{\bar{w}_k \alpha _w} + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_l} u^{\bar{w}_p \alpha _w} u^{\bar{z}_b z_a} u^{\bar{z}_d z_c} u_{z_a \bar{z}_d w_l} u_{z_c \bar{z}_b \bar{w}_p}\\&\qquad - u^{\bar{z}_b z_a} \left[ - u_{\alpha _z \bar{w}_k z_a} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q \bar{z}_b} u^{\bar{w}_q \alpha _w} - u_{\alpha _z \bar{w}_k \bar{z}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q \alpha _w} \right. \\&\qquad \left. + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q \alpha _w} \right. \\&\qquad \left. + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s \alpha _w} \right] \\&\qquad + u^{\bar{w}_b w_a} \left[ - u_{\alpha _z \bar{w}_k \bar{w}_b} u^{\bar{w}_k w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q \alpha _w}\right. \\ \end{aligned}$$
$$\begin{aligned}&\qquad \left. + u_{\alpha _z \bar{w}_k} u^{\bar{w}_k w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q \alpha _w}\right] \\&\qquad + u^{\bar{\alpha }_w w_p} u^{\bar{w}_q w_k} u_{w_k \bar{\alpha }_z} u^{\bar{z}_b z_a} u^{\bar{z}_d z_c} u_{z_a \bar{z}_d w_p} u_{z_c \bar{z}_b \bar{w}_q} - u^{\bar{\alpha }_w w_k} u^{\bar{z}_b z_a} u^{\bar{z}_d z_c} u_{z_a \bar{z}_d w_k} u_{z_c \bar{z}_b \bar{\alpha }_z}\\&\qquad - u^{\bar{z}_b z_a} \left[ u^{\bar{\alpha }_w w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_k} u_{w_k \bar{\alpha }_z} \right. \\&\qquad + u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_r} u_{w_r \bar{w}_s \bar{z}_b} u^{\bar{w}_s w_k} u_{w_k \bar{\alpha }_z}\\&\qquad \left. - u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q z_a} u^{\bar{w}_q w_k} u_{w_k \bar{\alpha }_z \bar{z}_b} - u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \bar{z}_b} u^{\bar{w}_q w_k} u_{w_k \bar{\alpha }_z z_a}\right] \\&\qquad + u^{\bar{w}_b w_a} \left[ u^{\bar{\alpha }_w w_r} u_{w_r \bar{w}_s \bar{w}_b} u^{\bar{w}_s w_p} u_{w_p \bar{w}_q w_a} u^{\bar{w}_q w_k} u_{w_k \bar{\alpha }_z} \right. \\&\qquad \left. - u^{\bar{\alpha }_w w_p} u_{w_p \bar{w}_q \bar{w}_b} u^{\bar{w}_q w_k} u_{w_k \bar{\alpha }_z w_a} \right] \\&\quad =: \sum _{i=1}^{36} A_i. \end{aligned}$$
We observe:
$$\begin{aligned} A_{21} + A_{22}+A_{29} + A_{30}&= \mathfrak {R}\left[ \ u^{\bar{z}_b z_a} u^{\bar{z}_d z_c} \left( u_{z_c \bar{z}_b \bar{w}_p} u^{\bar{w}_p \alpha _w} \right) \right. \\&\quad \times \left. \left( u_{z_a \bar{z}_d w_l} u^{\bar{w}_k w_l} u_{\alpha _z \bar{w}_k} - u_{\alpha _z z_a \bar{z}_d} \right) \right] \\&\le - \left( A_1 + A_2 + A_3 + A_{4} + A_{17} \right) . \end{aligned}$$
using Cauchy–Schwarz.
Next
$$\begin{aligned} A_{23} + A_{26} + A_{32}+ A_{33}&=Re \left[ \ u^{\bar{z}_b z_a} u^{\bar{w}_k w_p} \left( u_{w_p \bar{w}_q \bar{z}_b} u^{\bar{w}_q \alpha _w} \right) \right. \nonumber \\&\quad \times \left. \left( u_{\alpha _z z_a \bar{w}_k} - u_{z_a w_p \bar{w}_k} u^{\bar{w}_q w_p} u_{\alpha _z \bar{w}_q} \right) \right] \\&\le - \left( A_5 + A_6 + A_9 + A_{10} + A_{18} \right) . \end{aligned}$$
Next
$$\begin{aligned} A_{24} + A_{25} + A_{31}+ A_{34}&= \mathfrak {R}\left[ \ u^{\bar{z}_b z_a} u^{\bar{w}_k w_p} \left( u_{w_p \bar{w}_q z_a} u^{\bar{w}_q \alpha _w} \right) \right. \nonumber \\&\quad \times \left. \left( u_{\alpha _z \bar{w}_k \bar{z}_b} - u_{\alpha _z \bar{w}_s} u^{\bar{w}_s w_r} u_{\bar{z}_b w_r \bar{w}_k} \right) \right] \\&\le \ - \left( A_7 + A_8 + A_{11} + A_{12} + A_{19} \right) . \end{aligned}$$
Next
$$\begin{aligned} A_{27} + A_{28} + A_{35}+ A_{36}&= \mathfrak {R}\left[ \ u^{\bar{w}_b w_a} u^{\bar{w}_k w_p} \left( u_{w_p \bar{w}_q w_a} u^{\bar{w}_q \alpha _w} \right) \right. \nonumber \\&\quad \times \left. \left( u_{\alpha _z \bar{w}_s} u^{\bar{w}_s w_r} u_{w_r \bar{w}_k \bar{w}_b} - u_{\alpha _z \bar{w}_k\bar{w}_b} \right) \right] \\&\le - \left( A_{13}+A_{14}+A_{15}+A_{16} + A_{20} \right) . \end{aligned}$$
\(\square \)
Lemma 6.3
Let \(u_t\) be a solution to (1.2) such that \(u_t \in \mathcal E\) for all t. Then
$$\begin{aligned} \left( \frac{\partial }{\partial t}- \mathcal L\right) \frac{\partial u}{\partial t}= 0. \end{aligned}$$
Also,
$$\begin{aligned} \left( \frac{\partial }{\partial t}- \mathcal L\right) W \le 0. \end{aligned}$$
Proof
Let \(u_t\) be as in the statement. Define \(v_t : \mathbb C^n \rightarrow \mathbb R\), by
$$\begin{aligned} v_t(z_1,\dots ,z_n) = u_t({{\mathrm{Re}}}z_1,\dots , {{\mathrm{Re}}}z_n). \end{aligned}$$
Elementary calculations show that \(v_t \in \mathcal E\) and that \(v_t\) is a solution to (1.4). Moreover the matrix W associated to \(v_t\) via (3.2) agrees with the matrix \(\nabla ^2 w\) as in (2.6). The result follows from Lemmas 6.1 and 6.2. \(\square \)
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Streets, J., Warren, M. Evans–Krylov Estimates for a nonconvex Monge–Ampère equation. Math. Ann. 365, 805–834 (2016). https://doi.org/10.1007/s00208-015-1293-x
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DOI: https://doi.org/10.1007/s00208-015-1293-x