3.3 Global matching problem [0236]

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3.3 Global matching problem

We have solved the ODE with the prescribed initial condition, on the interval $0<t<1$ . For the application to the generalized Calabi ansatz, we need solutions over $0<t<\infty$ , and for this purpose the ODE (14) is inadequate. From the ODE (12), it is however a priori clear that $t=1$ is not a singularity.

Lemma 3.6.

For any given $w_{0}>0$ , the solution to the ODE (12) exists smoothly on $0<t<1+\epsilon(w_{0})$ for some $\epsilon(w_{0})>0$ .

Proof.

The ODE (12) can be smoothly extended as long as $w$ remains bounded positively below and $(1-t)w^{\prime}+w$ remains bounded (which imply boundedness of $w^{\prime\prime}$ , and in particular the boundedness of $w,w^{\prime}$ ). Notice

\frac{d}{dt}((1-t)w^{\prime}+w)=(1-t)w^{\prime\prime}>0,

so $(1-t)w^{\prime}+w$ is monotone increasing, and in particular positive. By

\frac{d}{dt}((1-t)w^{\prime}+w)^{n-1}=\frac{1-t}{w^{3}},

we see $(1-t)w^{\prime}+w$ will be bounded as long as $w$ is bounded positively below.

The convexity of $w$ is ensured whenever the solution is smooth. Thus for some small $\epsilon>0$ ,

w(t)\geq w(\epsilon)-w^{\prime}(\epsilon)(t-\epsilon),\quad t\geq\epsilon.

For small $\epsilon$ , we have $w_{0}-w_{0}\epsilon<w(\epsilon)<w_{0}$ and $-w_{0}<w^{\prime}(\epsilon)<0$ , so $w(\epsilon)/w^{\prime}(\epsilon)>1-\epsilon$ , whence $w(t)$ has an a priori lower bound slightly beyond $t=1$ . ∎

Recall the ODE has a symmetry under $t\to t^{-1}$ (cf. Remark 3.3). Our strategy to achieve both the $t=0$ and the $t=\infty$ boundary conditions, is to look for symmetric solutions:

w(t)=t\tilde{w}(1/t),\quad w(t)=\tilde{w}(t).

This is a functional equation on $w(t)$ , and it amounts to a matching condition at $t=1$ :

w(1)=\tilde{w}(1),\quad w^{\prime}(1)=\tilde{w}^{\prime}(1).

This is equivalent to

w^{\prime}(1)=\frac{1}{2}w(1).

(20)

The problem is then to look for $w_{0}>0$ to solve (20). The key is to extract $w^{\prime}(1)$ from the asymptote of $\mathfrak{w}$ as $s\to+\infty$ , namely $t\to 1$ . The starting point is the identity

w^{\prime}(t)=s\frac{d\mathfrak{w}}{ds}-\mathfrak{w}.

(21)

which means the value of $w^{\prime}$ is related to the Legendre transform of $\mathfrak{w}$ .