Proof of Theorem A. [0287]

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Proof of Theorem A. We consider first the case $M={\mathbb{C}}^{n}$ . Fix $c>1$ . We define inductively a sequence $\{m_{j}\}$ with the following properties: $m_{-1}=m_{0}=0<m_{1}$ , $\{u<m_{1}\}\neq\emptyset$ , and for $j\geq 1$ , $m_{j}>m_{j-1}$ is chosen large enough so that

a_{j}=\frac{m_{j-1}-m_{j-2}+1}{m_{j}-m_{j-1}}\leq\frac{\log c}{2^{j}}\;.

Since $\gamma_{j}\geq\gamma_{0}=1$ we have by (1),

\gamma_{j}(m_{j}-m_{j-1})\leq\gamma_{j-1}(m_{j}-m_{j-2}+1)\Longrightarrow\gamma_{j}\leq\gamma_{j-1}(1+a_{j}).

Thus

\gamma_{j}<\gamma=\prod_{j=1}^{\infty}(1+a_{j})\;,\;\;\log\gamma\leq\sum_{j=1}^{\infty}a_{j}\leq\log c.

Let $\psi=\psi_{c}$ be the psh extension of $\varphi$ provided by Proposition 1.3 for this sequence $\{m_{j}\}$ . Then for every $z\in{\mathbb{C}}^{n}$ we have

\psi(z)<\gamma\max\{u(z),0\}\leq c\max\{u(z),0\}.

Assume now that $M$ is a Stein manifold of dimension $n$ . Then $M$ can be properly embedded in ${\mathbb{C}}^{2n+1}$ , hence we may assume that $M$ is a complex submanifold of ${\mathbb{C}}^{2n+1}$ (see e.g. [Ho, Theorem 5.3.9]). Proposition 1.1 implies the existence of a continuous psh exhaustion function $\widetilde{u}$ on ${\mathbb{C}}^{2n+1}$ so that $\widetilde{u}=u$ on $M$ . By what we already proved, given $c>1$ there exists a psh function $\widetilde{\psi}$ on ${\mathbb{C}}^{2n+1}$ which extends $\varphi$ and such that $\widetilde{\psi}<c\max\{\widetilde{u},0\}$ on ${\mathbb{C}}^{2n+1}$ . We let $\psi=\widetilde{\psi}\,|_{{}_{M}}$ . $\Box$