Riemann extension across a holomorphic hypersurface zero set

Uniqueness is immediate from [L4]: if two holomorphic extensions agree with $f$ on $U\setminus Z(g)$, then their difference vanishes on the nonempty open set $U\setminus Z(g)$ and hence vanishes identically on the domain $U$.

Fix $a\in Z(g)$. The germ of $g$ at $a$ is nonzero, else [L4] would make $g$ vanish identically on $U$. To prove local extendability at $a$, we may translate $a$ to $0$ and compose with the invertible complex-linear coordinate change from [L2], because holomorphicity, local boundedness, and the existence of a local extension are preserved under such coordinate changes. After that change, [L2] makes the germ of $g$ regular in $z_m$ and then yields a smaller product neighbourhood $V\times \{\mathrm{\mid }\zeta \mathrm{\mid }<r\}$ on which $g=uW$ with $W$ Weierstrass, $u$ nowhere zero, and $W(z^{\mathrm{\prime }},\zeta )\mathrm{\ne }0$ for $\mathrm{\mid }\zeta \mathrm{\mid }=r$. Shrinking once more if needed, $u$ stays nonzero on this neighbourhood, so $Z(g)=Z(W)$ there. The compact boundary cylinder $V\times \{\mathrm{\mid }\zeta \mathrm{\mid }=r\}$ is therefore disjoint from $Z(g)$, so $f$ is holomorphic on a neighbourhood of it and hence bounded there.

Fix $z^{\mathrm{\prime }}\in V$. The slice $\zeta \mapsto f(z^{\mathrm{\prime }},\zeta )$ is holomorphic on the disc $\mathrm{\mid }\zeta \mathrm{\mid }<r$ with the finitely many zeros of $W(z^{\mathrm{\prime }},\cdot )$ removed. By step 1.2 it is bounded near each removed point, so [L1] extends that slice holomorphically across all of them. Call the extended slice $F_{z^{\mathrm{\prime }}}$.

Define $F(z^{\mathrm{\prime }},z_m):=\frac{1}{2\pi i}{\int }_{\mathrm{\mid }\zeta \mathrm{\mid }=r}\frac{f(z^{\mathrm{\prime }},\zeta )}{\zeta -z_m}d\zeta \mathrm{.}$ Fixing all variables except one coordinate, [L4] makes the integrand holomorphic in that coordinate and [L3] makes the corresponding slice of $F$ holomorphic. The boundedness from step 1.2 and the ML estimate make $F$ locally bounded. Hence [L3] makes $F$ holomorphic on $V\times \{\mathrm{\mid }{z}_m\mathrm{\mid }<r\}$.

For fixed $z^{\mathrm{\prime }}$, the one-variable Cauchy formula from [L3] applied to the holomorphic slice extension $F_{z^{\mathrm{\prime }}}$ from step 2.1 shows that the integral in step 2.2 equals $F_{z^{\mathrm{\prime }}}(z_m)$ for every $\mathrm{\mid }{z}_m\mathrm{\mid }<r$. In particular, when $W(z^{\mathrm{\prime }},z_m)\mathrm{\ne }0$ this value is the original $f(z^{\mathrm{\prime }},z_m)$. So step 2.2 gives a local holomorphic extension in the chosen coordinates, and undoing the coordinate change extends $f$ across the original point $a$. By step 1.1 these local extensions agree on overlaps, and therefore glue to a unique global holomorphic extension on $U$.