# Berenstein rigidity EP-1-H2-to-H4

## Setting

You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.

## Problem

H2: Let \(D=\{z\in\mathbb C:|z|<1\}\). Let \(\phi\) be holomorphic on a neighborhood of \(\overline D\), with
\[
\phi(0)=0,\qquad \phi'(0)=1,\qquad
\operatorname{Re}\phi'(z)>0\quad(z\in\overline D).
\]
Suppose \(k>0\) and a real-valued function \(U\in C^2(\overline D)\) satisfy
\[
\Delta U+k^2|\phi'|^2U=0\quad\text{in }D,
\]
\[
U=0,\qquad \partial_r U=|\phi'|\quad\text{on }\partial D,
\]
where \(\partial_r\) is the outward radial derivative on the unit circle.

Then \(\phi\) is injective on a neighborhood of \(\overline D\). The set \(\Omega=\phi(D)\) is a bounded simply connected domain with real-analytic Jordan boundary, and
\[
u=U\circ\phi^{-1}
\]
is a nonzero real-valued function in \(C^2(\overline\Omega)\) satisfying
\[
\Delta u+k^2u=0\quad\text{in }\Omega,
\qquad
u=0\quad\text{on }\partial\Omega,
\qquad
\partial_\nu u=1\quad\text{on }\partial\Omega,
\]
where \(\nu\) is the outward unit normal.

Prove H4: Let \(D=\{z\in\mathbb C:|z|<1\}\). There exist an integer \(m\ge2\), a real number \(k>0\), a holomorphic function \(\phi\) on a neighborhood of \(\overline D\), and a real-valued function \(U\in C^2(\overline D)\), such that
\[
\phi(0)=0,\qquad \phi'(0)=1,\qquad
\operatorname{Re}\phi'(z)>0\quad(z\in\overline D),
\]
\[
\phi(e^{2\pi i/m}z)=e^{2\pi i/m}\phi(z)\quad(z\in D),
\qquad \phi\not\equiv z,
\]
and
\[
\Delta U+k^2|\phi'|^2U=0\quad\text{in }D,
\]
\[
U=0,\qquad \partial_r U=|\phi'|\quad\text{on }\partial D,
\]
where \(\partial_r\) is the outward radial derivative on the unit circle.

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.