Second reformulation of the ODE [0230]

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Second reformulation of the ODE

We write for $0<t<1$ ,

s=\frac{1}{1-t}\in(1,\infty),\quad\mathfrak{w}(s)=\frac{w(t)}{1-t}.

(13)

Then

\frac{d\mathfrak{w}}{ds}=(1-t)^{2}\frac{d\mathfrak{w}}{dt}=(1-t)^{2}\frac{d}{dt}(\frac{w}{1-t})=(1-t)w^{\prime}+w.

\frac{d^{2}\mathfrak{w}}{ds^{2}}=(1-t)^{2}\frac{d}{dt}((1-t)w^{\prime}+w)=(1-t)^{3}w^{\prime\prime}.

Thus the ODE (12) can be reformulated as

\frac{d}{ds}(\frac{d\mathfrak{w}}{ds})^{n-1}=(n-1)\frac{d^{2}\mathfrak{w}}{ds^{2}}(\frac{d\mathfrak{w}}{ds})^{n-2}=\mathfrak{w}^{-3}.

(14)

The Kähler condition (cf. Remark 3.2) translates into

\mathfrak{w}>0,\quad\frac{d\mathfrak{w}}{ds}>0,\quad\frac{d^{2}\mathfrak{w}}{ds^{2}}>0.

(15)

The initial condition at $s=1$ is

\mathfrak{w}=w_{0}+O(t),\quad\frac{d\mathfrak{w}}{ds}=b_{1}t^{1/(n-1)}+O(t),\quad t=1-s^{-1}.

(16)