Chartwise harmonic and subharmonic functions on a Riemann surface

Let $X$ be a Riemann surface with complex structure given by a holomorphic atlas $\mathcal{A}$ (Riemann surfaces and holomorphic atlases), and let $W\subseteq X$ be open. For a chart $\phi :U\to \phi (U)$ of $\mathcal{A}$ with $U\cap W\mathrm{\ne }\mathrm{\varnothing }$ and a function $u$ on $W$, the chart expression of $u$ in $\phi$ is $u_{\phi }:=u\circ (\phi {\mathrm{\mid }}_{U\cap W}{)}^{-1}\mathrm{.}$ Its value set is the value set of $u$, including $-\mathrm{\infty }$ when allowed below.

A function $u:W\to \mathbb{R}$ is harmonic on $W$ when $u$ is continuous and the chart expression $u_{\phi }$ is harmonic in the sense of Plane harmonic functions for every chart $\phi$ of $\mathcal{A}$ whose domain meets $W$. A function $u:W\to [-\mathrm{\infty },\mathrm{\infty })$ is subharmonic on $W$ when the restriction of $u_{\phi }$ to each connected component of $\phi (U\cap W)$ is subharmonic in the sense of Subharmonic functions on plane domains, for every chart whose domain meets $W$. Components of a plane open set are open, so these restrictions have plane domains as their domains. Both notions are nonvacuous only on nonempty open sets. Finite constants are harmonic and subharmonic; the constant $-\mathrm{\infty }$ is excluded from subharmonicity on any nonempty open set.

The definitions are independent of the atlas. On each connected component of a chart overlap inside $W$, the transition and its inverse are biholomorphisms between plane domains. The conformal invariance of harmonicity Plane harmonicity is preserved by holomorphic and antiholomorphic changes of coordinate and Plane subharmonicity is invariant under biholomorphic change of coordinate apply there. Restriction of a plane subharmonic function to a smaller domain preserves upper semicontinuity and the submean inequalities; it also preserves the nontriviality condition because local integrability implies finiteness almost everywhere (Plane subharmonic functions are locally integrable). Thus any compatible chart expression is locally harmonic or locally subharmonic. Harmonicity is local because the $C^2$ condition and the equation $\mathrm{\Delta }u=0$ are local.

For completeness, local plane subharmonicity gives subharmonicity on every component as follows. Upper semicontinuity is local, and local integrability excludes an identically $-\mathrm{\infty }$ component. On a closed disc $\overline{D(a,r)}$ inside the component, let $h$ be a continuous harmonic majorant of the boundary values. If $s-h$ were positive somewhere inside, then for a sufficiently small $\epsilon >0$ the upper semicontinuous function $q(z)=s(z)-h(z)+\epsilon (\mathrm{\mid }z-a{\mathrm{\mid }}^2-r^2)$ would have a positive maximum at an interior point $b$ of this compact disc. On a sufficiently small circle about $b$, local subharmonicity of $s$ and the harmonic mean-value property Plane harmonic functions satisfy the mean-value property give $q(b)\le \mathrm{avg}q-\epsilon t^2<\mathrm{avg}q$, contradicting maximality. Hence $s\le h$ inside, and Subharmonicity is equivalent to harmonic comparison on compactly contained discs proves subharmonicity. This proves that testing any one atlas gives the same notions.

Chartwise Laplacian. For $u$ of class $C^2$ in charts, write ${\mathrm{\Delta }}_{\phi }u:=(\mathrm{\Delta }(u_{\phi }))\circ \phi \text{on }U\cap W,\mathrm{\Delta }:={\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2\mathrm{.}$ This expression depends on the chart. Under the transition above one has $\mathrm{\Delta }(u_{\psi })=\mathrm{\mid }{\tau }^{\mathrm{\prime }}{\mathrm{\mid }}^2\cdot (\mathrm{\Delta }(u_{\phi }))\circ \tau$ on the overlap, by the chain rule for a holomorphic $\tau$. The factor $\mathrm{\mid }{\tau }^{\mathrm{\prime }}{\mathrm{\mid }}^2$ is positive, so vanishing is chart independent, and $u$ is harmonic exactly when every ${\mathrm{\Delta }}_{\phi }u$ vanishes. For a specified conformal metric ${\rho }^2\mathrm{\mid }dz{\mathrm{\mid }}^2$, the Laplace–Beltrami operator is ${\rho }^{-2}\mathrm{\Delta }$ in that chart: the metric determinant has square root ${\rho }^2$ and its inverse matrix is ${\rho }^{-2}I$, so these factors cancel inside the divergence. Thus the same vanishing criterion gives harmonicity for any conformal metric, without identifying the unweighted chart expressions as one global operator.

Harmonic conjugates. Let $u:W\to \mathbb{R}$ be harmonic on the open set $W$. A function $v:W\to \mathbb{R}$ is a harmonic conjugate of $u$ when $v$ is harmonic on $W$ and, for every chart $\phi$ of $\mathcal{A}$ with $U\cap W\mathrm{\ne }\mathrm{\varnothing }$, the function $(u+iv)\circ (\phi {\mathrm{\mid }}_{U\cap W}{)}^{-1}=u_{\phi }+i{v}_{\phi }$ is holomorphic on $\phi (U\cap W)$. In the chart this is exactly the pair of Cauchy–Riemann equations ${\mathrm{\partial }}_x{v}_{\phi }=-{\mathrm{\partial }}_y{u}_{\phi }$, ${\mathrm{\partial }}_y{v}_{\phi }={\mathrm{\partial }}_x{u}_{\phi }$, so the convention fixed here is that the conjugate is taken so that $u+iv$ is holomorphic; this convention fixes the sign of the conjugate once a chart is chosen. A harmonic conjugate $v$ is itself harmonic, because in each chart its expression is the imaginary part of a holomorphic function, and on a connected $W$ two harmonic conjugates of the same $u$ differ by a constant, because the difference has vanishing gradient in every chart. On an abstract surface such a $v$ need not exist globally even when $W$ is connected and $u$ has no singularities; the existence, and the resulting multivaluedness, is treated where it is needed later on this page. This chartwise notion of conjugate is the one used throughout the hyperbolic-surface arguments.