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Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere

Abstract

We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere \(\hat {\mathbb {C}}\). Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

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Acknowledgements

The authors especially thank Stefan Sechelmann for writing software and creating examples for numerical experiments.

Funding

Open Access funding enabled and organized by Projekt DEAL. This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.

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Correspondence to Ulrike Bücking.

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Communicated by: F W Nijhoff

Appendices

Appendix A.1: Interpolation Function on Boundary Triangles and Estimates on Corresponding Edge Weights

In the following, we expose the calculations for the energy of the interpolation function and the corresponding edge weights.

Let Δ[x,y,z] ∈ Fρ be a boundary triangle. Without loss of generality, we assume that the vertices are labelled such that xBρ(0) and \(y,z\in \mathbb {C}\setminus B_{\rho }(0)\). Therefore, the triangle Δ[x,y,z] is bounded by two straight edges [x,y] and [x,z] and by the trace of the curve \(s:[0,1]\to \mathbb {C}\), \(s(t)=\frac {yz}{z+t(y-z)}\), connecting y and z which is in general a circular arc. We parametrize this triangle by

$$ p:[0,1]\times[0,1]\to{{\varDelta}}[x,y,z],\quad p(\tau,\sigma)= x+\sigma(s(\tau)-x). $$

Note that p is bijective for σ≠ 0. In this parametrization, the interpolation function is uΔ(τ,σ) := ITu(p(τ,σ)) = ux + σ(uyux + τ(uzuy)) as explained in Section 2.1. Here we use the notation uv = u(v) for the values of the given smooth function u at the vertices x,y,z. Therefore, we obtain

$$ \begin{array}{@{}rcl@{}} \underset{{{\varDelta}}[x,y,z]}{\int}|\nabla I_{T}u|^{2} \!&=&\! {{\int}_{0}^{1}}{{\int}_{0}^{1}} Du_{{{\varDelta}}}(\tau,\sigma) Dp^{-1}(p(\tau,\sigma))(Dp^{-1}(p(\tau,\sigma)))^{T} (Du_{{{\varDelta}}}(\tau,\sigma))^{T} |\det Dp(\tau,\sigma)| d\tau d\sigma \\ \!&=&\! {{\int}_{0}^{1}}{{\int}_{0}^{1}} \left( (u_{z}-u_{y})^{2} \frac{|s(\tau)-x|^{2} -2\tau\text{Re}((s(\tau)-x) \overline{s^{\prime}(\tau)}) +\tau^{2}|s^{\prime}(\tau)|^{2}}{|\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})|^{2}}\right. \\&&\qquad\qquad+ (u_{y}-u_{x})^{2} \frac{|s^{\prime}(\tau)|^{2}}{|\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})|^{2}} \\ &&\qquad\qquad \left.+ (u_{y}v-\!u_{x})(u_{z} - u_{y}) \frac{2\tau|s^{\prime}(\tau)|^{2} - 2\text{Re}((s(\tau) - x) \overline{s^{\prime}(\tau)})}{|\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})|^{2}}\right) |\det \mathit{Dp}(\tau,\sigma)| \mathit{d}\tau \mathit{d}\sigma \\ \!&=&\!\frac{1}{2} {{\int}_{0}^{1}} \left( (u_{z}-u_{y})^{2} \frac{|s(\tau)-x|^{2} -2\tau\text{Re}((s(\tau)-x) \overline{s^{\prime}(\tau)}) +\tau^{2}|s^{\prime}(\tau)|^{2}}{|\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})|} \right. \\&&\qquad\qquad\quad+ (u_{y}-u_{x})^{2} \frac{|s^{\prime}(\tau)|^{2}}{|\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})|} \\ &&\qquad\qquad\quad \left.+ (u_{y}-u_{x})(u_{z}-u_{y}) \frac{2\tau|s^{\prime}(\tau)|^{2} -2\text{Re}((s(\tau)-x) \overline{s^{\prime}(\tau)})}{|\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})|}\right) d\tau,\end{array} $$
(12)

as DuΔ(τ,σ) = (σ(uzuy),uyux + τ(uzuy)), \(Dp(\tau ,\sigma )= \left (\begin {array}{ll} \sigma \text {Re}(s^{\prime }(\tau )) & \text {Re}(s(\tau )-x)\\ \sigma \text {Im}(s^{\prime }(\tau )) & \text {Im}(s(\tau )-x) \end {array}\right )\), and \(\det (Dp(\tau ,\sigma ))= \sigma \text {Im}((s(\tau )-x) \overline {s^{\prime }(\tau )})\). Thus we deduce that

$$ \begin{array}{@{}rcl@{}} \underset{{{\varDelta}}[x,y,z]}{\int} |\nabla I_{T}u|^{2} &=& C_{[x,y]}(u(x)-u(y))^{2} + C_{[y,z]}(u(y)-u(z))^{2} +C_{[z,x]}(u(z)-u(x))^{2}, \\ \text{where}\qquad C_{[x,y]}&=&\frac{1}{2} {{\int}_{0}^{1}} \frac{(1-\tau)|s^{\prime}(\tau)|^{2} +\text{Re}((s(\tau)-x)) \overline{s^{\prime}(\tau)})}{|\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})|} d\tau, \end{array} $$
(13)
$$ \begin{array}{@{}rcl@{}} C_{[y,z]} &=& \frac{1}{2} {{\int}_{0}^{1}} \frac{|s(\tau)-x|^{2}+ \tau(\tau-1)|s^{\prime}(\tau)|^{2} +\text{Re}((s(\tau)-x) \overline{s^{\prime}(\tau)})}{|\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})|} d\tau, \end{array} $$
(14)
$$ \begin{array}{@{}rcl@{}} C_{[z,x]}&=&\frac{1}{2} {{\int}_{0}^{1}} \frac{\tau |s^{\prime}(\tau)|^{2} -\text{Re}((s(\tau)-x) \overline{s^{\prime}(\tau)})}{|\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})|} d\tau. \end{array} $$
(15)

This gives an explicit way to calculate the edge weights. Note that by the same method we can obtain the usual cotan-weights on the Euclidean triangle with vertices x,y,z, if we use sE(t) = y + t(zy) instead of s(t) for t ∈ [0, 1]. The function sE is the usual linear parametrization of the straight edge from y to z.

An important observation is that these seemingly ‘complicated’ weights are in fact only small perturbations of the usual cotan-weights if the edge length is small enough. To see this, we will estimate the quantities in the above integrals compared to the corresponding quantities for sE.

Proof Proof of the estimate in Remark 1

We assume that there is some δ > 0 such that all angles in the triangle Δ[x,y,z] are in [δ,πδ] and also all angles of the Euclidean triangle with vertices x,y,z are in [δ,πδ]. In the following, we will always assume that τ ∈ [0, 1] as in the integral terms above. Also, we are not interested in the best possible estimates, any constant, depending only on the indicated parameters, will suffice.

First note that \(|\text {Im}((s(\tau )-x) \overline {s^{\prime }(\tau )})|\geqslant |s(\tau )-x||s^{\prime }(\tau )|\sin \limits \delta \) and \(|\text {Im}((s_{E}(\tau )-x) \overline {s_{E}^{\prime }(\tau )})|\geqslant |s_{E}(\tau )-x||z-y|\sin \limits \delta \). Denote by h the maximal edge length of Δ[x,y,z], so \(h\geqslant \max \limits \{|y-x|,|z-x|,\text {length}(s)\}\). As Δ[x,y,z] is a boundary triangle, we have

$$ \frac{\rho^{2}}{(\rho+h)^{2}}\leqslant \left|\frac{s^{\prime}(\tau)}{s_{E}(\tau)}\right|= \frac{|s^{\prime}(\tau)|}{|y-z|} \leqslant \frac{(\rho+h)^{2}}{\rho^{2}}. $$

Furthermore, we deduce that \(1-\mathsf {Const}_{\delta ,\rho }\cdot h\leqslant \left |\frac {s(\tau )-x}{s_{E}(\tau )-x}\right |\leqslant 1+\mathsf {Const}_{\delta ,\rho }\cdot h\) as

$$\left|\frac{s(\tau)-s_{E}(\tau)}{s_{E}(\tau)-x}\right| = \left|\frac{\tau(\tau-1)(y-z)^{2}}{z+\tau(y-z)}\cdot \frac{1}{(y-x)+\tau(z-y)}\right| \leqslant \frac{h}{\rho\sin^{2}\delta}.$$

Further, note that using the sine law

$$ \sin^{2}\delta \leqslant \frac{|s_{E}(\tau)-x|}{|s_{E}^{\prime}(\tau)|} =\frac{|(y-x)+\tau(z-y)|}{|y-z|} \leqslant 1+\frac{1}{\sin\delta}. $$

Combining these estimates, we have

$$ \begin{array}{@{}rcl@{}} \frac{|s(\tau)-s_{E}(\tau)| |s^{\prime}(\tau)|}{|\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})|} &\leqslant& \frac{|s(\tau)-s_{E}(\tau)|}{|s_{E}(\tau)-x|} \cdot \left|\frac{s^{\prime}(\tau)}{s_{E}(\tau)}\right| \cdot\frac{1}{\sin\delta} \leqslant \mathsf{Const}_{\delta,\rho}\cdot h,\\ \frac{|s^{\prime}(\tau)|^{2}}{|\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})|} &\leqslant& \left|\frac{s^{\prime}(\tau)}{s_{E}(\tau)}\right| \cdot \frac{|s_{E}^{\prime}(\tau)|^{2}}{|s_{E}^{\prime}(\tau)| |s_{E}(\tau)-x|\sin\delta} \leqslant \mathsf{Const}_{\delta,\rho}\cdot h,\\ \frac{|s(\tau)-x|^{2}}{|\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})|} &\leqslant& \frac{|s(\tau)-x|^{2}}{|s_{E}(\tau)-x|^{2}}\cdot \frac{|s_{E}(\tau)-x|^{2}}{|s_{E}^{\prime}(\tau)| |s_{E}(\tau)-x|\sin\delta} \leqslant \mathsf{Const}_{\delta,\rho}\cdot h. \end{array} $$

Furthermore, as \(h\leqslant \rho /2\),

$$ \frac{|s_{E}(\tau)-x| |s^{\prime}(\tau)-s_{E}^{\prime}(\tau)|}{|\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})|} \leqslant \frac{|s^{\prime}(\tau)-s_{E}^{\prime}(\tau)|}{|y-z|\sin\delta} = \frac{|y-z||\tau^{2}(y-z)^{2}+2\tau z-z|}{|z+\tau(y-z)|^{2}\sin\delta} \leqslant \frac{6}{\rho\sin\delta}\cdot h. $$

Therefore, we obtain

$$ \begin{array}{@{}rcl@{}} &&\left| \frac{\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})}{\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})} -1\right| = \left| \frac{\text{Im}((s(\tau)-s_{E}(\tau)) \overline{s^{\prime}(\tau)})}{\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})} \right.\\&&\left.+ \frac{\text{Im}((s(\tau)-x) \overline{(s^{\prime}(\tau)-s_{E}^{\prime}(\tau))})}{\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})}\right| \leqslant \mathsf{Const}_{\rho,\delta}\cdot h\\[2ex] && \left| \frac{\text{Re}((s(\tau)-x) \overline{s^{\prime}(\tau)})- \text{Re}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})}{\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})}\right| =\left| \frac{\text{Re}((s(\tau)-s_{E}(\tau)) \overline{s^{\prime}(\tau)})}{\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})} \right.\\&&\left.+ \frac{\text{Re}((s_{E}(\tau)-x) \overline{(s^{\prime}(\tau)-s_{E}^{\prime}(\tau))})}{\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})}\right| \leqslant \mathsf{Const}_{\rho,\delta}^{\prime}\cdot h \end{array} $$

This also implies \(\left | \frac {\text {Re}((s(\tau )-x) \overline {s^{\prime }(\tau )})}{\text {Im}((s_{E}(\tau )-x) \overline {s_{E}^{\prime }(\tau )})}\right | \leqslant \left | \frac {\text {Re}((s_{E}(\tau )-x) \overline {s_{E}^{\prime }(\tau )})}{\text {Im}((s_{E}(\tau )-x) \overline {s_{E}^{\prime }(\tau )})}\right | + \mathsf {Const}_{\rho ,\delta }^{\prime }\cdot h \leqslant \mathsf {Const}_{\rho ,\delta }^{\prime \prime }\) Denote by \({\alpha ^{E}_{z}}\in [\delta ,\pi -\delta ]\) the angle in the Euclidean triangle with vertices x,y,z. Then the previous estimates imply

$$ \begin{array}{@{}rcl@{}} |C_{[x,y]}-\frac{1}{2}{\cot\alpha_{z}^{E}}| &\leqslant& \frac{1}{2} {{\int}_{0}^{1}} \left( \frac{(1-\tau)(|s^{\prime}(\tau)|^{2}\left|\frac{\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})}{\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})} -1\right|+\left|\frac{|s^{\prime}(\tau)|^{2}}{|s_{E}^{\prime}(\tau)|^{2}} -1\right| |s_{E}^{\prime}(\tau)|^{2})}{|\text{Im} ((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})|}\right. \\&&\qquad\qquad\quad+ \frac{|\text{Re}((s(\tau)-x) \overline{s^{\prime}(\tau)})| \left|\frac{\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})}{\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})} -1\right|}{|\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})|}\\ &&\qquad\qquad\quad \left.+ \frac{|\text{Re}((s(\tau)-x) \overline{s^{\prime}(\tau)})- \text{Re}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})|}{|\text{Im}((s_{E}(\tau)-x) \overline{s_{E}^{\prime}(\tau)})|} \right) d\tau \\ &\leqslant& \mathsf{Const}_{\rho,\delta}^{\prime\prime\prime}\cdot h. \end{array} $$

Similarly, we can deduce that \(|C_{[y,z]}-\frac {1}{2}{\cot \alpha _{x}^{E}}| \leqslant \mathsf {Const}_{\rho ,\delta }^{\prime \prime \prime }\cdot h\) and \(|C_{[z,x]}-\frac {1}{2}{\cot \alpha _{y}^{E}}| \leqslant \mathsf {Const}_{\rho ,\delta }^{\prime \prime \prime }\cdot h\). □

Appendix : A.2: Proof of Lemma 11

Note that as u is smooth, there exist constants Constu,ρ,constu,ρ > 0 such that for 0 < h < constu,ρ we have \(|u(x)-u(y)|^{2}/|x-y|^{2}\leqslant \mathsf {Const}_{\delta ,\rho }\cdot \underset {w\in {{\varDelta }}}{\max \limits }\|D^{1}u(w)\|^{2} \leqslant \mathsf {Const}_{u,\rho }\) for all edges e = [x,y] of boundary triangles in Fρ with edge lengths smaller than h.

Using our estimates of Appendix A we deduce that there exists a constant Constδ,ρ such that under our assumptions on angles and edge lengths we have the following estimates: For every boundary triangle Δ[x,y,z] ∈ Fρ and with the notation of Appendix A

$$ \begin{array}{@{}rcl@{}} |z-y|^{2} \frac{|s(\tau)-x|^{2}}{|\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})|^{2}}&\leqslant& \mathsf{Const}_{\delta,\rho},\quad |y-x|^{2} \frac{|s^{\prime}(\tau)|^{2}}{|\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})|^{2}} \\&\leqslant& \mathsf{Const}_{\delta,\rho} , |y-x||z-y| \frac{|\text{Re}((s(\tau)-x) \overline{s^{\prime}(\tau)})|}{|\text{Im}((s(\tau)-x) \overline{s^{\prime}(\tau)})|^{2}}\\&\leqslant& \mathsf{Const}_{\delta,\rho}. \end{array} $$

Now formula (1) leads to the estimate

$$ \underset{{{\varDelta}}[x,y,z]}{\int}|\nabla I_{T}u|^{2} \leqslant \mathsf{Const}_{\delta,\rho,u}\cdot \text{Area}({{\varDelta}}[x,y,z]), $$
(16)

where \(\mathsf {Const}_{\delta ,\rho ,u} \leqslant \mathsf {Const}_{\delta ,\rho }\cdot \max \limits _{w\in {{\varDelta }}}\|D^{1}u(w)\|^{2}\). Summing up these energies, we obtain

$$ \begin{array}{@{}rcl@{}} E_{F_{\rho}}(u) &=&\underset{{{\varDelta}}\in F_{\rho}}{\sum} E_{T_{{{\varDelta}}}}(u) =\underset{{{\varDelta}}\in F_{\rho}}{\sum} \underset{{{\varDelta}}[x,y,z]}{\int}|\nabla I_{T}u|^{2} \leqslant \underset{{{\varDelta}}\in F_{\rho}}{\sum} \mathsf{Const}_{\delta,\rho,u} \cdot \text{Area}({{\varDelta}}[x,y,z]) \\ &\leqslant& \mathsf{Const}_{\delta,\rho,u}\cdot \text{Area}(B_{\rho+h}(0)\setminus B_{\rho-h}(0)) \leqslant \mathsf{Const}_{\delta,\rho,u} \cdot 4h\rho \cdot d \leqslant \mathsf{Const}_{u,\delta,\rho,{\mathscr{R}}}\cdot h, \end{array} $$

where d denotes the degree of the covering map for \({\mathscr{R}}\).

Appendix : A.3: Proof of Equicontinuity Lemma 15

First note that condition (D) from Section 2.4 implies that for every path in a non-degenerate uniform adapted triangulation T with consecutive vertices \(v_{0}v_{1}{\dots } v_{m}\) we have

$$ \begin{array}{@{}rcl@{}} E_{v_{0}v_{1}{\dots} v_{m}}(u)\!:=\! \sum\limits_{k=1}^{m} c([v_{k-1},v_{k}])\cdot (u(v_{k}) - u(v_{k-1}))^{2} &\geqslant& \sum\limits_{k=1}^{m} \frac{\mathsf{Const}}{e}\cdot (u(v_{k})-u(v_{k-1}))^{2}\\ &\geqslant& \frac{\mathsf{Const}}{e}\cdot \frac{1}{m} \left( \sum\limits_{k=1}^{m} u(v_{k}) - u(v_{k-1})\right)^{2}\\&=&\frac{\mathsf{Const}}{e\cdot m} (u(v_{m})-u(v_{0}))^{2}. \end{array} $$
(17)

Here e denotes the eccentricity as defined in Section 4.1 and for the last estimate we have used Schwarz’s inequality.

Now consider a simply connected triangulation \(T^{\prime }\) with boundary contained in an open disc \(B_{r}(v)\subset \mathbb {C}\). This is the assumption for part (i) of the lemma. In the case of part (ii), we consider the image triangulation \(T_{g}^{\prime }\) by the chart \(g_{O}(z)=(z-O)^{\gamma _{O}}\). By abuse of notation, we still denote this image triangulation by \(T^{\prime }\). Also, we denote the vertices of \(T^{\prime }\) by Z and W, which are the actual vertices z, w in the first case and the images Z = gO(z), W = gO(w) in the case of a branch point. For simplicity, we assume that the edges between vertices are straight line segments, as we do not need the actual, possibly curved edges.

Let \(u:V^{\prime }\to \mathbb {R}\) be any function which assumes its maximum and its minimum on the boundary for any subgraph of \(T^{\prime }\). Let Z, W be two distinct interior vertices of \(T^{\prime }\). Denote by ZW the straight line segment joining these points. Let \(dist(ZW,\partial T^{\prime })\) be the Euclidean distance of this straight line segment to the curve of boundary edges. We assume that \(|Z-W|<r/3<dist(ZW,\partial T^{\prime })/3\) for some r > 0. Let \(h^{\prime }\) denote twice the maximum circumradius of the triangles of \(T^{\prime }\). Let \(m=\lfloor \frac {r-|Z-W|}{2h^{\prime }}\rfloor \) be the largest integer smaller than \(\frac {r-|Z-W|}{2h^{\prime }}\). We consider auxiliary rectangles Rk, \(k=1,\dots ,m\), which are centered at (Z + W)/2 with one pair of sides parallel to ZW with length \(ZW+2k\cdot h^{\prime }\) and other pair of sides orthogonal to ZW of length \(2k\cdot h^{\prime }\). Then the interior of Rk, \(k=1,\dots ,m\), is covered by triangles of \(T^{\prime }\). Denote by \(V_{k}^{\prime }\) the set of vertices contained in RkRk− 1, where R0 = ZW. Then any two vertices \(v_{A},v_{B}\in V^{\prime }_{k}\) may be connected by a path \(v_{0}v_{1}{\dots } v_{N}\) with v0 = vA, vN = vB and all vertices \(v_{j}\in V^{\prime }_{k}\) as \(h^{\prime }\) is larger than any edge length.

Without loss of generality, assume that \(u(Z)\geqslant u(W)\). As u assumes its maximum and minimum on the boundary, there exists \(Z_{k},W_{k}\in V^{\prime }_{k}\) such that \(u(Z_{k})\geqslant u(Z)\geqslant u(W)\geqslant u(W_{k})\). The length of the path joining Zk and Wk is at most the number of vertices in \(V^{\prime }_{k}\). The set of these vertices can be covered by at most \(\mathsf {Const}\cdot (|Z-W|+4kh^{\prime })/h^{\prime }\) discs of radius \(h^{\prime }/2\). Therefore, by condition (U) of Section 2.4 the number of vertices in \(V^{\prime }_{k}\) is less than \(\mathsf {Const}\cdot e\cdot (|Z-W|/h^{\prime }+4k)\). Therefore, we can estimate the energy for a path \(v_{0}v_{1}{\dots } v_{N}\) in \(V_{k}^{\prime }\) from v0 = Zk, vN = Wk using (6)

$$ \begin{array}{@{}rcl@{}} E_{Z_{k}{\dots} W_{k}}(u)\geqslant &\frac{\mathsf{Const}}{e\cdot (\mathsf{Const}\cdot e\cdot (|Z-W|/h^{\prime}+4k))}\cdot (u(Z_{k})-u(W_{k}))^{2} = \mathsf{Const}\cdot \frac{(u(Z)-u(W))^{2}}{e^{2}}\cdot \frac{h^{\prime}}{kh^{\prime}+ |Z-W|/4}. \end{array} $$

Summing these estimates and estimating \({\sum }_{k=1}^{m} \frac {h^{\prime }}{kh^{\prime }+ |Z-W|/4} \geqslant \mathsf {Const}\) \( {\int \limits }_{h^{\prime }}^{\frac {r-|Z-W|}{2}}\) \(\frac {dt}{t+ |Z-W|/4}\) we get

$$ \begin{array}{@{}rcl@{}} E^{\prime}(u)&\geqslant& \sum\limits_{k=1}^{m} E_{Z_{k}{\dots} W_{k}}(u) \geqslant \mathsf{Const}\cdot \frac{(u(Z)-u(W))^{2}}{e^{2}} \cdot {\int}_{h^{\prime}}^{\frac{r-|Z-W|}{2}}\frac{dt}{t+ |Z-W|/4} \\&\geqslant& \mathsf{Const}\cdot \frac{(u(Z)-u(W))^{2}}{e^{2}}\cdot \log \frac{2r-|Z-W|}{4h^{\prime}+|Z-W|} \\ &\geqslant& \mathsf{Const}\cdot \frac{(u(Z)-u(W))^{2}}{e^{2}}\cdot \log \frac{r}{3\max\{|Z-W|,h^{\prime}\}}. \end{array} $$

This implies the desired inequalities (8) and (9).

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Bobenko, A.I., Bücking, U. Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere. Math Phys Anal Geom 24, 23 (2021). https://doi.org/10.1007/s11040-021-09394-2

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Keywords

  • Discrete analytic function
  • Riemann surface
  • Period matrix
  • Discrete holomorphic integral
  • Dirichlet energy
  • Approximation

Mathematics Subject Classification (2010)

  • Primary 39A12
  • 65M60
  • 30E10; Secondary 14K20
  • 30F30