Green's second identity on a compact bordered domain of a Riemann surface

Let $W$ be an open neighbourhood of ${\mathrm{\Omega }}^{\mathrm{\prime }}$ on which $u,v$ have $C^2$ chart expressions. By [F6] the boundary $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$ is a compact embedded $1$-submanifold of $X$, possibly empty, and every $x\in \mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$ has a half-slice chart $(V,\phi )$ with $\phi ({\mathrm{\Omega }}^{\mathrm{\prime }}\cap V)=\phi (V)\cap \{y\ge 0\}$ and $\phi (\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}\cap V)=\phi (V)\cap \{y=0\}$. For part 2 fix $j$ and a point $x\in \mathrm{\partial }{D}_j$, write ${\phi }_j(x)=w_j+r_j{e}^{i\alpha }$, and put $G(\zeta ):=(r_j-\mathrm{\mid }\zeta \mathrm{\mid },\mathrm{Im}\zeta )$ for $\zeta$ near $r_j$ and $\mathrm{\Psi }:=G\circ (\zeta \mapsto e^{-i\alpha }(\zeta -w_j))\circ {\phi }_j$. The rotation is a plane diffeomorphism and $G$ is smooth near $r_j$ with $DG(r_j)=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$ invertible, so $\mathrm{\Psi }$ restricts to a chart of the smooth structure of $X$ near $x$, with $\mathrm{\Psi }(x)=0$, in which $D_j=\{{\mathrm{\Psi }}_1\ge 0\}$ and $X\setminus D_j=\{{\mathrm{\Psi }}_1<0\}$; hence each $D_j$ is an embedded submanifold with boundary and its boundary points have half-slice charts with the disc on the side $\{{\mathrm{\Psi }}_1\ge 0\}$ [F7]. Since $D_j$ has positive distance from $\mathrm{\partial }\mathrm{\Omega }$ (as $\overline{D_j}$ is compact and lies in the open set $\mathrm{\Omega }$) and $\mathrm{\Omega }$ is open, $\mathrm{\Omega }$ agrees with $X$ near $\mathrm{\partial }{D}_j$, so ${\mathrm{\Omega }}_K$ is locally $\{{\mathrm{\Psi }}_1\le 0\}$ at every point of $\mathrm{\partial }{D}_j$, agrees with $\mathrm{\Omega }$ near $\mathrm{\partial }\mathrm{\Omega }$, where the half-slice charts of the bordered domain $\bar{\mathrm{\Omega }}$ from [F6] present it as a half-space, and is locally all of $X$ at its remaining points; moreover $\overline{{\mathrm{\Omega }}_K}=\bar{\mathrm{\Omega }}\setminus (D_1^{\circ }\cup \cdots \cup D_m^{\circ })$ is the closure of its interior, since $\mathrm{\Omega }=\mathrm{int}\bar{\mathrm{\Omega }}$ and each $D_j^{\circ }$ is an open disc meeting $\mathrm{\Omega }$. Therefore $\overline{{\mathrm{\Omega }}_K}$ is a compact bordered domain with $\mathrm{\partial }{\mathrm{\Omega }}_K=\mathrm{\partial }\mathrm{\Omega }\bigsqcup \mathrm{\partial }{D}_1\bigsqcup \cdots \bigsqcup \mathrm{\partial }{D}_m$; it is compact because it is closed in the compact space $\bar{\mathrm{\Omega }}$ [F8], and the hypothesis gives $C^2$ chart expressions for $u,v$ near it.

Let $\phi ,\psi$ be holomorphic charts with coordinates $z=x+iy$ and $w=u+iv$ on their overlap and transition $h=\phi \circ {\psi }^{-1}$, so that $z=h(w)$; then $h$ is holomorphic with $h^{\mathrm{\prime }}\mathrm{\ne }0$ on the overlap by [F1]. For a $C^2$ function $f$ one has $f_{\phi }\circ h=f_{\psi }$, so the chain rule in [F10] applied to the transformation law of [F2] gives $\mathrm{\Delta }(f_{\phi })(h(w))=\mathrm{\mid }{h}^{\mathrm{\prime }}(w){\mathrm{\mid }}^{-2}\mathrm{\Delta }(f_{\psi })(w)$; since $dz=h^{\mathrm{\prime }}(w)dw$ and $dx\wedge dy=\frac{i}{2}dz\wedge d\bar{z}$, one has $dx\wedge dy=\mathrm{\mid }{h}^{\mathrm{\prime }}(w){\mathrm{\mid }}^{2}du\wedge dv$. Multiplying the two identities, $u_{\phi }\mathrm{\Delta }(v_{\phi })dx\wedge dy=u_{\psi }\mathrm{\Delta }(v_{\psi })du\wedge dv$ on the overlap, so $(u\mathrm{\Delta }v-v\mathrm{\Delta }u)dA$ is a well-defined continuous $2$-form on the neighbourhood $W$ of ${\mathrm{\Omega }}^{\mathrm{\prime }}$.

For every $x\in \mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$ choose a holomorphic coordinate chart centred at $x$. The embedded-boundary condition of step 1.1 makes the boundary image a smooth curve. Rotate the coordinate so its tangent at $x$ is horizontal. By the inverse function theorem [F7], after shrinking, the boundary is a smooth graph $y=\gamma (x)$ and the interior of ${\mathrm{\Omega }}^{\mathrm{\prime }}$ lies on one side, say $y>\gamma (x)$. Choose a small coordinate rectangle $R_x$ with closure inside the holomorphic chart domain and the neighbourhood $W$ whose vertical sides meet that graph transversely and whose horizontal sides avoid it. Then the interior region $R_x\cap \mathrm{int}{\mathrm{\Omega }}^{\mathrm{\prime }}$ is a bounded piecewise $C^1$ planar domain with the graph as one boundary face. Compactness of $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$ gives finitely many such holomorphic rectangles $R_1,\dots ,R_{\mathrm{\ell }}$ covering it.

With $h$ as in step 2.1 write $h^{\mathrm{\prime }}(w)=\lambda e^{i\theta }$, $\lambda >0$; the real derivative of $h$ is multiplication by the complex number $h^{\mathrm{\prime }}(w)$, that is, $Dh(w)=\lambda R_{\theta }$ for the rotation $R_{\theta }$ by $\theta$, which is conformal and orientation preserving by [F1] and [F10]. Hence at corresponding boundary points the unit tangent vectors satisfy ${\tau }_z=R_{\theta }{\tau }_w$ and the outward unit conormals satisfy ${\nu }_z=R_{\theta }{\nu }_w$, while arclengths satisfy $d{s}_z=\lambda d{s}_w$. Transposing the chain rule $D{f}_{\psi }(w)=D{f}_{\phi }(h(w))Dh(w)$ gives ${\mathrm{\nabla }}_w{f}_{\psi }=\lambda R_{-\theta }{\mathrm{\nabla }}_z{f}_{\phi }$, hence ${\mathrm{\partial }}_{{\nu }_z}{v}_{\phi }={\lambda }^{-1}{\mathrm{\partial }}_{{\nu }_w}{v}_{\psi }$ and ${\mathrm{\partial }}_{{\nu }_z}{v}_{\phi }d{s}_z={\mathrm{\partial }}_{{\nu }_w}{v}_{\psi }d{s}_w$. With the chart-independent values of $u$, the pairing $u{\mathrm{\partial }}_{\nu }vds$ is therefore chart-independent along $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$.

The set $L:={\mathrm{\Omega }}^{\mathrm{\prime }}\setminus (R_1\cup \cdots \cup R_{\mathrm{\ell }})$ is compact and disjoint from $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$ by [F8] and step 2.2. Each $y\in L$ lies in the open complement of $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$, so there is a chart ball $B$ about $y$ with closure in a larger holomorphic chart domain inside $W$ and $\bar{B}\cap \mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}=\mathrm{\varnothing }$; these balls cover $L$, so by [F8] finitely many of them, say $B_{\mathrm{\ell }+1},\dots ,B_N$, cover $L$. Then $U_1:=R_1,\dots ,U_{\mathrm{\ell }}:=R_{\mathrm{\ell }},U_{\mathrm{\ell }+1}:=B_{\mathrm{\ell }+1},\dots ,U_N:=B_N$ are holomorphic coordinate domains covering ${\mathrm{\Omega }}^{\mathrm{\prime }}$.

Steps 2.1 and 3.1 show that both integrands of part 1 are intrinsic: the left-hand side is the integral over the compact set ${\mathrm{\Omega }}^{\mathrm{\prime }}$ of the continuous $2$-form $(u\mathrm{\Delta }v-v\mathrm{\Delta }u)dA$, and the right-hand side is the arclength integral of the continuous density $u{\mathrm{\partial }}_{\nu }v-v{\mathrm{\partial }}_{\nu }u$ over the compact boundary, evaluated on any finite chart cover of $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$, the finitely many corner points contributing zero arclength by [F5].

For each $j$ put $W_j:=U_j\cap \mathrm{int}{\mathrm{\Omega }}^{\mathrm{\prime }}$. For $j\le \mathrm{\ell }$, step 2.2 makes its chart image a bounded piecewise $C^1$ domain with graph and rectangle faces. For $j>\mathrm{\ell }$, the ball $B_j$ is connected and disjoint from $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$, and it meets $L\subseteq \mathrm{int}{\mathrm{\Omega }}^{\mathrm{\prime }}$, so it lies wholly in the interior and $W_j=B_j$ is a bounded smooth planar domain. Thus every $W_j$ is an open domain admissible for the planar Green identity.

The finite family $\{U_1,\dots ,U_N\}$ is an open cover of the compact set ${\mathrm{\Omega }}^{\mathrm{\prime }}$; by local compactness [F8] and compactness there are compact sets $K_j\subseteq U_j$ covering ${\mathrm{\Omega }}^{\mathrm{\prime }}$. By [F9] choose smooth bumps $b_j$ with $b_j=1$ on $K_j$ and $\mathrm{supp}{b}_j⋐U_j$; then $\mathrm{\Sigma }:={\sum }_j{b}_j$ is positive on a neighbourhood of ${\mathrm{\Omega }}^{\mathrm{\prime }}$, and another application of [F9] gives a smooth $\psi$ equal to $1$ near ${\mathrm{\Omega }}^{\mathrm{\prime }}$ with $\mathrm{supp}\psi \subseteq \{\mathrm{\Sigma }>0\}$. The functions ${\rho }_j:=\psi b_j\mathrm{/}\mathrm{\Sigma }$, extended by zero, are smooth, satisfy $\mathrm{supp}{\rho }_j⋐U_j$, and obey ${\sum }_j{\rho }_j=1$ on a neighbourhood of ${\mathrm{\Omega }}^{\mathrm{\prime }}$.

Fix $j$ and let $\theta$ be the larger holomorphic chart containing $\overline{U_j}$ chosen in steps 2.2 and 3.2. On a neighbourhood of $\overline{W_j}$ the functions $f_j:={\rho }_{j}u$ and $g:=v$ have $C^2$ chart expressions: $u$ and $v$ do by the hypothesis, ${\rho }_j$ is smooth, and products and scalar multiples of $C^2$ functions are $C^2$ by [F10].

Fix $j$. The plane domain $\theta (W_j)$ is admissible for [F3] by step 4.2, and the functions $f_j,g$ are $C^2$ up to its closure by step 5.1, so the plane second Green identity gives ${\int }_{W_j}(g\mathrm{\Delta }{f}_j-f_j\mathrm{\Delta }g)dA={\int }_{\mathrm{\partial }{W}_j}(g{\mathrm{\partial }}_{\nu }{f}_j-f_j{\mathrm{\partial }}_{\nu }g)ds$, with every normal outward and with the faces of the piecewise presentation counted once off the edge set.

Every face of $W_j$ contained in $\mathrm{\partial }{U}_j$ lies in the complement of $\mathrm{supp}{\rho }_j$ by step 4.3, so there ${\rho }_j=0$ and $\mathrm{\partial }{\rho }_j=0$; hence $f_j={\rho }_{j}u$ and its conormal derivative vanish on such a face, which therefore contributes zero to the boundary integral of step 6.1. The remaining faces lie in $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$, and on their interior points $W_j$ coincides with ${\mathrm{\Omega }}^{\mathrm{\prime }}$, so the outward normal of $W_j$ is the outward conormal of ${\mathrm{\Omega }}^{\mathrm{\prime }}$ there.

Summing the identities of step 6.1 over $j$ and inserting step 7.1 gives the equality of ${\sum }_j{\int }_{W_j}(v\mathrm{\Delta }({\rho }_{j}u)-{\rho }_{j}u\mathrm{\Delta }v)dA$ with ${\sum }_j{\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}\cap U_j}(v{\mathrm{\partial }}_{\nu }({\rho }_{j}u)-{\rho }_{j}u{\mathrm{\partial }}_{\nu }v)ds$.

For the volume sum, each integrand $v\mathrm{\Delta }({\rho }_{j}u)-{\rho }_{j}u\mathrm{\Delta }v$ is supported in $\overline{W_j}$ by step 4.3, so summing over $j$ and applying the linearity of $\mathrm{\Delta }$ from [F2] and [F10] together with ${\sum }_j{\rho }_j=1$ near ${\mathrm{\Omega }}^{\mathrm{\prime }}$ gives ${\sum }_j(v\mathrm{\Delta }({\rho }_{j}u)-{\rho }_{j}u\mathrm{\Delta }v)=v\mathrm{\Delta }u-u\mathrm{\Delta }v$ on a neighbourhood of ${\mathrm{\Omega }}^{\mathrm{\prime }}$; and the boundary has area zero, hence the volume sum equals ${\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}(v\mathrm{\Delta }u-u\mathrm{\Delta }v)dA$.

For the boundary sum, ${\sum }_j(v{\mathrm{\partial }}_{\nu }({\rho }_{j}u)-{\rho }_{j}u{\mathrm{\partial }}_{\nu }v)=v{\mathrm{\partial }}_{\nu }u-u{\mathrm{\partial }}_{\nu }v$ on $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$: the sum is finite, ${\sum }_j{\rho }_j=1$ near $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$ by step 4.3, and the same conormal and arclength density are used for every $j$ by the chart independence of step 3.1; the finitely many corner points carry no arclength. Hence the boundary sum equals ${\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}}(v{\mathrm{\partial }}_{\nu }u-u{\mathrm{\partial }}_{\nu }v)ds$.

Combining steps 8.1, 9.1 and 9.2 gives ${\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}(v\mathrm{\Delta }u-u\mathrm{\Delta }v)dA={\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}}(v{\mathrm{\partial }}_{\nu }u-u{\mathrm{\partial }}_{\nu }v)ds$ and hence, after multiplying both sides by $-1$, the displayed identity of part 1.

Part 2. Apply part 1 to the compact bordered domain $\overline{{\mathrm{\Omega }}_K}$ of step 1.1. The volume integrand $u\mathrm{\Delta }v-v\mathrm{\Delta }u$ vanishes identically, because $u$ and $v$ are harmonic on $\mathrm{int}{\mathrm{\Omega }}_K$ and hence have vanishing chartwise Laplacian there by [F2]; continuity of their second derivatives extends this vanishing to $\overline{{\mathrm{\Omega }}_K}$. On the boundary component $\mathrm{\partial }\mathrm{\Omega }$ both functions vanish identically, and their conormal derivatives are finite there because $u,v$ have $C^2$ chart expressions near $\overline{{\mathrm{\Omega }}_K}$; hence the integrand $u{\mathrm{\partial }}_{\nu }v-v{\mathrm{\partial }}_{\nu }u$ is pointwise zero on $\mathrm{\partial }\mathrm{\Omega }$. Since $\mathrm{\partial }{\mathrm{\Omega }}_K$ is the disjoint union of $\mathrm{\partial }\mathrm{\Omega }$ and the circles $\mathrm{\partial }{D}_1,\dots ,\mathrm{\partial }{D}_m$ by step 1.1, the identity of part 1 reduces exactly to ${\sum }_{j=1}^m{\int }_{\mathrm{\partial }{D}_j}(u{\mathrm{\partial }}_{\nu }v-v{\mathrm{\partial }}_{\nu }u)ds=0$. The case $m=0$ gives the empty sum, and if $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$ is empty in part 1 both sides vanish because the empty boundary contributes zero.