Suppose that we have a Kähler manifold $(M, \omega_0)$ and another $(1,1)$ form $\eta$ on $M$. Let $\varphi$ be a smooth function such that $\omega_{\varphi} = \omega_0 + i \partial \bar \partial \varphi$ is positive. Denote the metric tensor of $\omega_0$ by $g_{i \bar j}$ and $g_0^{i \bar j} = \delta_{i j}$, first derivative of $g_0$ vanishes and $\partial_{i} \bar \partial_{j} \varphi = \varphi_{i \bar j}= \varphi_{i \bar i}$ if $i = j$ and $0$ if $ i \neq j$.
My question is about how to properly estimate tensorial quantity in local coordinates:
For instance if we look at and $\sum_{i}\frac{R_{i \bar i j \bar k} \varphi_{j} \varphi_{\bar k}}{1+\varphi_{i \bar i}}$, one can estimate:
$\sum_{i}\frac{R_{i \bar i j \bar k} \varphi_{j} \varphi_{\bar k}}{1+\varphi_{i \bar i}} \geq -C |\partial \varphi|^2 \sum_{i} \frac{1}{1+\varphi_{i \bar i}}$
where $R_{i \bar i j \bar k} = -\partial_{j} \partial_{\bar k} g_{i \bar i} + g^{p \bar q}\partial_{j} g_{i \bar q} \partial_{\bar k} g_{p \bar i}$
1). How to make sense of this $C$ here? Is it coming from the bound on $|R_{i \bar i j \bar k}|$? I do not think it make sense to give bounds to individual component function of tensors which is not invariant under coordinate change.
2). In normal coordinates $R_{i \bar i j \bar k}$ only depends on mixed second derivative of the metric tensor, but in the usual coordinates it also depends on the inverse of the metric tensor and the first derivative of the metric tensor. What is really the dependence on $\omega_0$?
The bounty is about to expire, anyone interested in answering?