Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 24G / Solution

The identity theorem says that if two holomorphic functions on a connected Riemann surface agree on a set having an accumulation point, then they agree everywhere.

Apply local coordinates to their difference $F$. The power-series proof in the plane shows that near any point, either $F$ vanishes identically or its zeros are isolated: if the first nonzero Taylor coefficient has order $m$, then $F(z)=(z-z_0{)}^{m}G(z)$ with $G(z_0)\ne 0$. An accumulation point of the zero set therefore has a neighbourhood on which $F=0$. The set of points having such a neighbourhood is nonempty and open. It is also closed: near a limit point, a coordinate neighbourhood contains an open set on which $F=0$, and the planar identity theorem makes $F$ vanish throughout that coordinate neighbourhood. Connectedness now makes this set the whole surface.

For a nonempty connected open set $U\subset {\mathbb{R}}^2$, a function $h:U\to \mathbb{R}$ is harmonic when $h\in C^2(U)$ andOn a sufficiently small disc, the one-formis closed and hence equals $dk$ for some $k$. The Cauchy--Riemann equations then make $h+ik$ holomorphic. Holomorphic functions are smooth, so $h\in C^{\infty }(U)$.

A real-valued function $H$ on a Riemann surface is harmonic when, in every holomorphic chart $z$, its coordinate expression is harmonic. This definition is chart-independent. If $w=w(z)$ is a holomorphic transition map, the chain rule and Cauchy--Riemann equations giveThus vanishing of the Laplacian is preserved by every change of holomorphic coordinate.

Solved by gpt-5.6-sol high.