Complex structure

As is well-known, the orientation and the conformal class $[g]$ of metrics on M determine the unique complex structure (biholomorphic sub-atlas of the smooth oriented atlas on M) on it, such that, a) in any holomorphic coordinate z on M, the metric g is of the form $ds^2_g(z)=\rho (z)|dz|^2$ , where $\rho (z)>0$ , or, equivalently, b) any holomorphic coordinate z on M obeys $(\star +i\,\mathrm {Id}) dz=0$ , where $\star $ is the Hodge operator on $(M,g)$ . Given this complex structure, a function w on M is holomorphic (resp., anti-holomorphic) if and only if the Cauchy–Riemann condition $d\Im w=\star d\Re w$ (resp., $d\Im w=-\star d\Re w$ ) holds. The space of functions holomorhic on $\mathrm {int}M$ and smooth up to the boundary $\Gamma $ is denoted by $\mathscr {A}(M)$ .

The operator

$$ \begin{align*}\Phi: \ A\mapsto (\star A^\flat)^\sharp\end{align*} $$

(here $\flat : \ TM\mapsto T^*M$ and $\sharp :=\flat ^{-1}$ are the musical isomorphisms defined by $A^\flat :=g(A,\cdot )$ ) acts as the counterclockwise right angle rotation in each tangent space $T_x M$ ( $x\in M$ ). Note that both $\star $ and $\Phi $ are independent of the choice of metric g from the conformal class $[g]$ . The Cauchy–Riemann condition can be rewritten as $\nabla \Im w=\Phi \nabla \Re w$ (in any metric from $[g]$ ).

Choose a unit tangent vector field $\gamma $ on $\Gamma $ . In the subsequent, we agree that the orientations of M and $\Gamma $ are related by

(2.1) $$ \begin{align} \Phi\nu=\gamma. \end{align} $$

Harmonic fields

Denote by $L_2(M;TM)$ the space of the real square integrable vector fields on M, endowed with the inner product $(A,B):=\int _M g(A,B)dS$ (we also denote the complexification of $L_2(M;TM)$ by $L^{\mathbb {C}}_2(M;TM)$ ). The harmonic fields constitute the (closed) subspace

$$ \begin{align*}\mathcal{H}:=\{A\in L_2(M;TM) \ | \ \mathrm{div}(\Phi A)=\mathrm{div}A=0 \text{ in } M\backslash\Gamma\}\end{align*} $$

in $L_2(M;TM)$ . By definition, the rotation $\Phi $ is an isometric automorphism of $L_2(M;TM)$ which preserves harmonicity and obeys $\Phi ^{-1}=\Phi ^*=-\Phi $ . Also, each harmonic field A on M can be represented as $A=\nabla u$ (with harmonic u) in any simply connected subdomain of M.

Introduce the subspace of potential fields $\mathcal {E}:=\{\nabla u\in \mathcal {H}\}$ and denote by $\mathcal {N}$ its orthogonal complement in $\mathcal {H}$ . Let $\mathcal {D}=\Phi \mathcal {N}$ . In view of the Stokes theorem, formula

$$ \begin{align*}(A,\nabla u)=\int\limits_M \mathrm{div}(uA)=\int\limits_\Gamma uA_\nu dl\end{align*} $$

holds for any $A\in \mathcal {H}$ and $u\in C^\infty (\overline {M})$ , where $A_\nu :=g(A,\nu )$ . Thus, a harmonic field belongs to $\mathcal {N}$ ( $\mathcal {D}$ ) if and only if it is tangent (normal) to $\Gamma $ . In particular, any $A\in \mathcal {N}$ ( $A\in \mathcal {D}$ ) is smooth up to the boundary due to the increasing smoothness theorems for solutions to elliptic boundary value problems. Note that

(2.2) $$ \begin{align} \mathrm{dim}\mathcal{N}=\mathrm{dim}\mathcal{D}=2\mathfrak{g}. \end{align} $$

Denote $A_\gamma :=g(A,\gamma )$ . Let $A\in \mathcal {N}$ ; then $\Phi A\in \mathcal {H}$ and the Stokes theorem yields $\int _{\Gamma } A_\gamma dl=-\int _M \mathrm {div}(\Phi A)dS=0$ . Thus, each $A\in \mathcal {N}\cup \mathcal {D}$ can be represented as $A=\nabla u$ in a tubular neighborhood of $\Gamma $ . In particular, the maps $\mathcal {N}\ni A\mapsto A_\gamma $ , $\mathcal {D}\ni B\mapsto B_\nu $ are injections due to the uniqueness of solution to the Cauchy problem for the Laplace equation.

Hilbert transform

Denote by P the orthogonal projection on $\mathcal {E}$ in $L_2(M;TM)$ and introduce the reduced rotation $\hat {\Phi }:=P\Phi P$ . Since $\Phi $ is anti-hermitian, so is $\hat {\Phi }$ . In what follows, we also consider the complexification $\hat {\Phi }(A+iB)=\hat {\Phi }A+i\hat {\Phi }B$ of $\hat {\Phi }$ acting in $L^{\mathbb {C}}_2(M;TM)$ .

Let $u=u^f$ be a harmonic function in M with trace f on $\Lambda $ . Then $\Phi \nabla u$ is a harmonic field. From the orthogonal decomposition $\mathcal {H}=\mathcal {E}\oplus \mathcal {N}$ , we have

(2.3) $$ \begin{align} \Phi\nabla u^f=\nabla v^h+A, \end{align} $$

where $A\in \mathcal {N}$ and $v^h$ is some harmonic function with the trace h. Hence, $\hat {\Phi }\nabla u^f=\nabla v^h$ and $-\hat {\Phi }\nabla v^h=\nabla u^f+P\Phi A$ , i.e.,

$$ \begin{align*}(\hat{\Phi}+iI)\nabla w=-P\Phi A, \text{ where } w=u^f+iv^h.\end{align*} $$

Note that $P\Phi A=0$ if and only if $A=0$ . Indeed, if $P\Phi A=0$ , then $(\Phi A,\nabla u)=(\Phi A,P\nabla u)=(P\Phi A,\nabla u)=0$ for any smooth u on M. Since $\Phi A$ is harmonic, it means that

$$ \begin{align*}0=\int_M\mathrm{div}(u\Phi A)dS=\int_\Gamma u(\Phi A)_\nu dl=-\int_\Gamma uA_\gamma dl,\end{align*} $$

whence $A_\gamma =0$ on $\Gamma $ and $A=0$ in M. In view of (2.3), we obtain the following criteria: w is holomorphic (resp., antiholomorphic) in $M\backslash \Gamma $ if and only if $\nabla w$ is an eigenvector of $\hat {\Phi }$ corresponding to the eigenvalue $-i$ (resp., $+i$ ).

In view of (2.3), the equality $\hat {\Phi }\nabla u^f=0$ implies $\nabla v^h=\hat {\Phi }\nabla u^f=0$ and $\partial _\gamma f=-(\Phi \nabla u^f)_\nu =-\partial _\nu v^h+0=0$ , whence $u^f=\mathrm {const}$ and $\nabla u^f=0$ . In addition, (2.3) implies that $(\hat {\Phi }-\Phi )\nabla u^f=0$ if and only if $A=0$ , i.e., if and only if $\nabla u^f\in \mathrm {Ker}(\hat {\Phi }-i) \oplus \mathrm { Ker}(\hat {\Phi }+i)$ .

Let us show that

(2.4) $$ \begin{align} (\Phi-\hat{\Phi})\mathcal{E}=\mathcal{N}. \end{align} $$

Indeed, since the left-hand side is equal to $(I-P)\Phi \mathcal {E}$ , it is contained in $\mathcal {N}$ . Next, suppose that the field $A\in \mathcal {N}$ is orthogonal to $(I-P)\Phi \mathcal {E}$ . Then $-(\Phi A,\nabla u)=(A,\Phi \nabla u)=(A,P\Phi \nabla u)=(PA,\Phi \nabla u)=0$ for any $u\in C^\infty (\overline {M})$ and $\Phi A\in \mathcal {N}$ . The last equality means that $A_\gamma =-(\Phi A)_\nu =0$ . Therefore, $A=0$ in M.

In view of the above, the eigenvalues of $\hat {\Phi }$ of infinite multiplicity are $0$ (the corresponding eigenspace is $L^{\mathbb {C}}_2(M;TM)\ominus \mathcal {E}^{\mathbb {C}}$ ), $-i$ and $+i$ (the corresponding eigenspaces consist of gradients of holomorphic and anti-holomorphic functions on M, respectively) while the remaining eigenvalues have the total multiplicity $\mathrm {dim}(\Phi -\hat {\Phi })\mathcal {E}=\mathrm { dim}\mathcal {N}=2\mathfrak {g}$ in view of (2.2). Since $\hat {\Phi }\nabla \overline {w}=\overline {\hat {\Phi }\nabla }w$ and $\hat {\Phi }$ is anti-hermitian, the remaining eigenvalues (counted with their multiplicities) can be represented as

(2.5) $$ \begin{align} \lambda_{\pm k}=i\mu_{\pm k}, \qquad \mu_{\pm k}=-\mu_{\mp k}\in\mathbb{R} \qquad (k=1,\dots,\mathfrak{g}). \end{align} $$

Denote by $(\cdot ,\cdot )_\Gamma $ the inner product in $L^{\mathbb {C}}_2(\Gamma ;dl)$ . Let $\langle f\rangle :=(f,1)_\Gamma /(1,1)_\Gamma $ denotes the mean value of f on $(\Gamma ,dl)$ . In view of the Green formula

(2.6) $$ \begin{align} (\nabla u^f,\nabla u^h)=(\Lambda f,h)_\Gamma=:(f,h)_\Lambda, \end{align} $$

the map $\mathfrak {E}: \ f\mapsto \nabla u^f$ is an isometry from the space $\partial _\gamma H^{3/2}(\Gamma ;\mathbb {C})=\{f\in H^{1/2}(\Gamma ;\mathbb {C}) \ | \ \langle f\rangle =0\}$ equipped with the inner product $(\cdot ,\cdot )_\Lambda $ onto $\mathcal {E}^{\mathbb {C}}$ .

We define the Hilbert transform as the isomorphic copy

$$ \begin{align*}H:=-\mathfrak{E}^{-1}\hat{\Phi}\mathfrak{E}\end{align*} $$

of the reduced rotation $\hat {\Phi }=P\Phi P$ (the minus sign is introduced to match the usual definition of the Hilbert transform on the circle). Then the Stokes theorem yields

$$ \begin{align*} (\Lambda Hf,h)_\Gamma=(Hf,h)_\Lambda=-(P\Phi P\nabla u^f,\nabla u^h)=&(\Phi \nabla u^f,\nabla u^h)\\ =&\int_M \mathrm{div}(\overline{u^h}\Phi\nabla u^f)dS=(-\partial_\gamma f,h)_\Gamma \end{align*} $$

for any $f,h\in \partial _\gamma H^{3/2}(\Gamma ;\mathbb {C})$ . Hence,

(2.7) $$ \begin{align} H=\Lambda^{-1}\partial_\gamma, \qquad H^{-1}=\partial^{-1}_\gamma\Lambda. \end{align} $$

Here $\partial _\gamma ^{-1}$ is the integration with respect to the length element along $\Gamma $ in the direction $\gamma $ . Since the images $\Lambda H^1(\Gamma ;\mathbb {C})$ and $\partial _\gamma H^1(\Gamma ;\mathbb {C})$ are orthogonal to constants in $L_2(\Gamma )$ , both operators (2.7) are well-defined on $H^1(\Gamma ;\mathbb {C})$ . In addition, the DN map $\Lambda $ is a pseudo-differential operator of the first order which coincides with $|\partial _\gamma |$ modulo smoothing operator [Reference Lee and Uhlmann14]. Thus $H=-|\partial _\gamma |^{-1}\partial _\gamma $ modulo smoothing operator and both operators (2.7) are well-defined on $L^{\mathbb {C}}_2(\Gamma ;dl)$ . Although the second operator $-\partial ^{-1}_\gamma \Lambda $ in (2.7) inverts H only on the orthogonal complement to constants, we keep the (slightly misleading) notation $H^{-1}$ for it.

Note that H coincides with the standard Hilbert transform on the circle if M is a closed unit disk $\overline {\mathbb {D}}$ . The extensions of the standard Hilbert transform on the circle were considered in [Reference Belishev and Korikov4, Reference Belishev and Sharafutdinov6], while the above definition (slightly different and based on the connection between H and the reduced rotation $\hat {\Phi }$ ) is proposed by Belishev.

Let $\mathscr {A}(M)$ be the algebra of smooth holomorphic functions on M. Denote by $\mathrm {Tr}$ the trace operator $w\mapsto w|_\Gamma $ . In view of the above, we arrive at the following statement.

Lemma 2.1 H is an anti-hermitian operator in the space $(\partial _\gamma H^{3/2}(\Gamma ;\mathbb {C}), \, (\cdot ,\cdot )_\Lambda )$ . The spectrum of H consists of $0$ (with $\mathrm {Ker}H=\mathbb {C}$ ), $\pm i$ (with the eigenspaces

$$ \begin{align*} \mathrm{Ker}&(H-i)=\mathrm{clos}_{H^{1/2}(\Gamma;\mathbb{C})}\big(\{\eta\in \mathrm{Tr}\mathscr{A}(M) \ | \ \langle\eta\rangle=0\}\big), \\ \mathrm{Ker}&(H+i)=\mathrm{clos}_{H^{1/2}(\Gamma;\mathbb{C})}\big(\{\eta\in \mathrm{Tr}\overline{\mathscr{A}(M)} \ | \ \langle\eta\rangle=0\}\big), \end{align*} $$

respectively; here $\overline {\cdot }$ denotes the complex conjugation), and eigenvalues (2.5). The eigenfunctions

(2.8) $$ \begin{align} \eta_{\pm k}=\overline{\eta_{\mp k}} \qquad (k=1,\dots,\mathfrak{g}) \end{align} $$

corresponding to $\lambda _{\pm k}$ are smooth. (In what follows, we assume that eigenfunctions (2.8) are normalized in $L_2^{\mathbb {C}}(\Gamma ;dl)$ and the eigenfunctions corresponding to the same eigenvalue are orthogonal in $L_2^{\mathbb {C}}(\Gamma ;dl)$ .)

Double cover

The double of $(M,g)$ is the surface $(\mathbb {M},\mathrm {g})$ obtained by gluing $(M,g)$ with its copy (endowed with the opposite orientation) along the boundary. In the subsequent, we consider M to be embedded into $\mathbb {M}$ . Introduce the involution $\tau $ on $\mathbb {M}$ which interchanges any point x of M with the same point on its copy. Then $\tau (x)=x$ if and only if $x\in \Gamma $ . The projection $\pi : \ \mathbb {M}\mapsto M$ defined by $\pi (x):=\pi (\tau (x)):=x$ for any $x\in M\subset \mathbb {M}$ is continuous, open and discrete, and the set of its ramification points coincides with $\Gamma $ .

The metric $\mathrm {g}$ , the rotation ${\boldsymbol {\Phi }}$ , and the Hodge operator $\boldsymbol {\star }$ on $\mathbb {M}$ are obtained by gluing together the corresponding metrics, rotations, et.c, on M and its copy. By construction, the metric is symmetric

$$ \begin{align*}\tau^*\mathrm{g}=\mathrm{g}=\pi^*g,\end{align*} $$

while the rotation and the Hodge operator are anti-symmetric

(2.9) $$ \begin{align} d\tau\circ{\boldsymbol{\Phi}}=-\boldsymbol{\Phi}\circ d\tau, \quad \tau^*\circ\boldsymbol{\star}=-\boldsymbol{\star}\circ\tau^* \end{align} $$

with respect to the involution $\tau $ . To check that ${\boldsymbol {\Phi }}$ and ${\boldsymbol {\star }}$ are correctly defined (i.e., they are continuous on the whole $\mathbb {M}$ , including $\Gamma $ ), it is sufficient to note that $d\tau (\gamma )=\gamma $ and $d\tau (\nu )=-\nu $ , whence $({\boldsymbol {\Phi }}\nu )|_{\Gamma _-}=-({\boldsymbol {\Phi }}\circ d\tau (\nu ))|_{\Gamma _-}=d\tau (({\boldsymbol {\Phi }}\nu )|_{\Gamma _+})=d\tau (\gamma )=\gamma =\Phi \nu =({\boldsymbol {\Phi }}\nu )|_{\Gamma _+}$ , where $\Gamma _+$ and $\Gamma _-$ denotes the sides of $\Gamma $ internal and external with respect to M, respectively. In particular, $\mathbb {M}$ is orientable.

Note that, although the metric $\mathrm {g}$ is, in general, only Lipschitz continuous on $\Gamma $ , the rotation and the Hodge operator are smooth on the whole $\mathbb {M}$ . Moreover, $\mathbb {M}$ is endowed with the complex structure compatible with the complex structures on M and its copy (these can be considered as complex submanifolds of $\mathbb {M}$ ). Indeed, it is sufficient to construct appropriate holomorphic charts in the neighborhood of $\Gamma $ . Let $x_0$ be an arbitrary point of $\Gamma $ and let u be a smooth harmonic function in M obeying $\partial _\nu u=0$ and $u(x_0)=0$ , $\partial _\gamma u(x_0)>0$ . In view of the Poincaré lemma, we have $\Phi \nabla u=\nabla v$ in some (simple connected) neighborhood U of $x_0$ in M. In particular $\partial _\gamma v=\partial _\nu u=0$ in $\Gamma \cap U$ and one can assume that $v=0$ on $\Gamma \cap U$ . Then $w=u+iv$ is holomorphic in U and real-valued in $\Gamma \cap U$ . Since $x_0$ is a simple zero of w and $\partial _\nu v(x_0)=-\partial _\gamma u(x_0)<0$ , one can assume, by making the U smaller, that $w: \ U\mapsto \mathbb {C}_+$ is an injection. Now we extend w on $U\cup \tau (U)$ by symmetry $w\circ \tau =\overline {w}$ ; then $w: \ U\mapsto \mathbb {C}$ is an injection and w is holomorphic on $\tau (U)$ in view of (2.9). Thus, $(U,w)$ is a holomorphic chart on $\mathbb {M}$ which is compatible with the complex atlases of M and its copy $\tau (M)$ .

Note that, if the function w is holomorphic in a domain $U\subset \mathbb {M}$ , then is holomorphic in $\tau (U)$ due to (2.9).

Abelian differentials

A (complex) 1-form $\omega $ on $\mathbb {M}$ is called an Abelian differential (of the first kind) if the equations

(2.10) $$ \begin{align} i{\boldsymbol{\star}}\omega=\omega, \quad d\omega=0 \end{align} $$

hold in $\mathbb {M}$ or, equivalently, if it can be locally represented as $\omega =dw$ , where w is a holomorphic function. The space $H^0(\mathbb {M};K)$ of Abelian differentials of the first kind has the complex dimension $\mathrm {dim}H^0(\mathbb {M};K)=\mathrm {gen}(\mathbb {M})=2\mathfrak {g}$ .

In view of (2.9), the map

preserves equations (2.10) and therefore it is an involution on $H^0(\mathbb {M};K)$ . We call $\omega \in H^0(\mathbb {M};K)$ symmetric and write $w\in H^1_{sym}(\mathbb {M};K)$ if . Then $H^0_{sym}(\mathbb {M};K)$ is a real linear space of dimension $\mathfrak {g}$ and any $\omega \in H^0(\mathbb {M};K)$ admits the decomposition $\omega =\omega _++i\omega _-$ , where and belong to $H^0_{sym}(\mathbb {M};K)$ .

The important observation used in the article is the following connection between the tangent harmonic fields on M and the symmetric Abelian differentials on its double $\mathbb {M}$ .

Homology groups

Let $(X,g)$ be an oriented surface (possibly with non-empty boundary) of genus m and let l be a finite (possibly empty) collection of closed oriented curves in M. By definition, the following operations preserve homology class (“cycle”) $[l]$ of l: a) a homotopic deformation of each curve in l, b) cutting the curves into a finite number of segments and gluing them together in a different order in such a way that the resulting curves are closed and the orientation of each segment is preserved, c) adding or excluding an oriented boundary of some (arbitrarily oriented) domain in X. The set of cycles endowed with the addition $[l]+[l^{\prime }]=[l\cup l^{\prime }]$ is an Abelian group $H_1(X,\mathbb {Z})$ called the first homology group. Note $-[l]=[-l]$ , where $-l$ is obtained from l by reversing the orientation of all curves. It is well known that $H_1(X,\mathbb {Z})\simeq \pi _1(X^\circ )/[\pi _1(X^\circ ),\pi _1(X^\circ )]\simeq \mathbb {Z}^{2m}$ , where $X^\circ $ is obtained from X by attaching disks to all connected components of $\partial X$ .

Let $l,l^{\prime }$ be closed oriented curves in X; by homotopic deformation one can assume that they are smooth, oriented by unit tangent vectors $\gamma ,\gamma ^{\prime }$ , respectively, l intersects $l^{\prime }$ a finite number of times, and each intersection is transversal. The intersection is positive if $(\gamma ,\gamma ^{\prime })$ is positively oriented with respect to the orientation of X, and negative otherwise. By definition, the intersection number $[l]\sharp [l^{\prime }]$ is the difference between numbers of positive and negative intersections of l and $l^{\prime }$ (if $l,l^{\prime }$ are collections of the curves, then $[l]\sharp [l^{\prime }]$ is obtained by the summation of the intersection numbers of all pairs from $l\times l^{\prime }$ ). It can be shown that $[l]\sharp [l^{\prime }]$ is invariant with respect to operations (a)–(c) and thereby is well defined on homology classes. Moreover, $\sharp : \ H_1(X,\mathbb {Z})\times H_1(X,\mathbb {Z})\times H_1(X,\mathbb {Z})\mapsto \mathbb {Z}$ is an alternating bilinear form.

We say that $[l_\cdot ]=\{[l_1],\dots ,[l_{2m}]\}$ form a homology basis on X if they generate $H_1(X,\mathbb {Z})$ . Introduce the intersection matrix J of the basis $[l_\cdot ]$ by $J_{ij}:=[l_i]\sharp \,[l_j]$ . The homology basis is called canonical if its intersection matrix coincides with the standard symplectic matrix

$$ \begin{align*}\Omega_{(m)}=\left(\begin{array}{cc} 0 & I_{m}\\ -I_{m} & 0\end{array}\right).\end{align*} $$

In this case we call that $a_1=[l_1],\dots ,a_m=[l_m]$ are a-cycles and $b_1=[l_{m+1}],\dots ,b_m=[l_{2m}]$ are b-cycles. The canonical bases always exist. Two homology bases $[l_\cdot ]$ , $[l^{\prime }_\cdot ]$ are simultaneously (non-)canonical if and only if $[l^{\prime }_i]=\sum _j M_{ij}[l_j]$ ( $i=1,\dots ,m$ ), where $M\in \mathrm {Sp}(m,\mathbb {Z})$ (here $\mathrm {Sp}(m,\mathbb {Z})$ is the group of symplectic $m\times m$ -matrices with integer entries). A compact Riemann surface X with empty boundary endowed with a choice of a canonical homology basis $[l_\cdot ]$ is called a Torelli marked surface.

Let $\partial X$ be empty or diffeomorphic to a circle, let $\omega $ be a harmonic 1-form on X, and let $A=\omega ^\sharp $ . The integral

$$ \begin{align*}T(\omega|[l])\equiv T(A|[l]):=\int_{l}\omega=\int_{l}g(A,\gamma)dl\end{align*} $$

(where $\gamma $ and $dl$ are tangent unit vector and the length element on l, respectively) depends only on $[l]$ ; this integral is called the period of $\omega $ (or of A) along the cycle $[l]$ .

We say that $\omega $ is normal (tangent) to $\partial X$ if $\omega (\gamma )=0$ ( $\omega (\nu )=0$ ) on $\partial X$ , or, equivalently, if $A=\omega ^\sharp $ is normal (tangent) to $\partial X$ . Then harmonic 1-forms $\omega $ normal (tangent) to $\partial X$ are determined by their period vectors

$$ \begin{align*}\mathrm{T}(\omega|[l_\cdot])\equiv\mathrm{T}(\omega|[l_\cdot]):=(T(\omega|[l_1],\dots,T(\omega|[l_m])^T\end{align*} $$

with respect to a given homology basis $[l_\cdot ]$ .

Lemma 2.3 Let $(X,g)$ be an orientable surface of genus m (possibly with non-empty boundary), let $\Phi $ be a rotation on X and let $[l_\cdot ]$ be a homology basis on X. Suppose that $A,B\in C^\infty (X;TX)$ satisfy $\mathrm {div}A=\mathrm {div}(\Phi B)=0$ in X. In addition, suppose that A is tangent and/or B is normal to $\partial X$ . Then their inner product in $L_2(X;TX)$ admits the representation

(2.11) $$ \begin{align} (B,A)=\mathrm{T}(B|\,[l_\cdot])^T J^{-1}\mathrm{T}(-\Phi A|\,[l_\cdot]), \end{align} $$

where J is the intersection matrix of $[l_\cdot ]$ .

Proof Let $\star $ be the Hodge operator on X (then $\star \omega =(\Phi \omega ^\sharp )^\flat $ for any 1-form $\omega $ ). Denote $\omega =B^\flat $ and $\eta ={\star }A^\flat $ , then $d\omega =d\eta =0$ and at least one of $A,B$ is normal to $\partial X$ . The left-hand side of (2.11) can be rewritten as $(B,A)=\int _X \omega ^\wedge \eta $ while the right-hand side is given by $\mathrm {T}(\omega |\,[l_\cdot ])^T(-J^{-1})\mathrm {T}(\eta |\,[l_\cdot ])$ . Let us show that the right-hand side is independent of the choice of a homology basis. Let $[l^{\prime }_\cdot ]$ be a new homology basis connected with ${l_\cdot }$ via $[l^{\prime }_i]=\sum _{ij}M_{ij}[l_j]$ (i.e., $M,M^{-1}$ have integer entries). Then the period vectors and the intersection matrices obey the transformation rules $\mathrm {T}(\cdot |[l^{\prime }_\cdot ])=M\mathrm { T}(\cdot |[l_\cdot ])$ and

(2.12) $$ \begin{align} J^{\prime}=MJM^T, \end{align} $$

whence $\mathrm {T}(\omega |\,[l^{\prime }_\cdot ])^T(-J^{^{\prime }-1})\mathrm {T}(\eta |\,[l^{\prime }_\cdot ])=\mathrm {T}(\omega |\,[l_\cdot ])^T(-J^{-1})\mathrm {T}(\eta |\,[l_\cdot ])$ . Thus, one can check (2.11) assuming that $[l_\cdot ]=\{a_1,\dots ,a_m,b_1,\dots ,b_m\}$ is canonical. Then (2.11) takes the familiar form

(2.13) $$ \begin{align} \int_X \omega\wedge\eta=\sum_{j=1}^m\Big(\!\int_{a_j}\omega\int_{b_j}\eta-\int_{b_j}\omega\int_{a_j}\eta\Big), \end{align} $$

which is just the Riemann bilinear identity if $\partial X=\varnothing $ . It remains to prove that (2.13) is valid if $\partial X\ne \varnothing $ and one of $\omega ,\eta $ is normal to $\partial X$ . Let $X^\circ $ be the Riemann surface obtained by attaching a disk $D=\{z\in \mathbb {C} \ | \ |z|\le 1\}$ to each connected component of $\partial X$ (to construct the complex charts near $\partial X\subset X^\circ $ , one can use the procedure described after (2.9)). Let $\omega _\circ ,\eta _\circ $ be the smooth extensions to $X^\circ $ of $\omega ,\eta $ , respectively, given by $\omega ^\circ =du_1$ , $\eta ^\circ =du_2$ , where $u_k$ are smooth on $\tilde {X}=X^\circ \backslash \mathrm {int}X$ . Denote by $\chi $ a smooth function with compact support on $[0,+\infty )$ equal to $1$ in the neighborhood of zero. Introduce the function $\chi _\varepsilon $ given by $\chi _\varepsilon (x)=\chi (\varepsilon ^{-1}(|z(x)|-1))$ on each disk in $\tilde {X}$ . Suppose that $\omega $ is normal to $\partial X$ ; then one can chose $u_1$ in such a way that $u_1=0$ on $\partial X$ . Let $\omega _\varepsilon $ be the smooth closed extension of $\omega $ given by $\omega _\varepsilon =d(\chi _\varepsilon u_1)$ on $\tilde {X}$ . Since $u(z)=O(1-|z|)=O(\varepsilon )$ on the support of $\chi _\varepsilon $ , we have $\|\omega _\varepsilon \|_{L_2(\tilde {X}\,;T^*\tilde {X})}=O(\varepsilon ^{1/2})$ , whence

(2.14) $$ \begin{align} \int_{X_\circ} \omega_\varepsilon\wedge\eta_\circ\to \int_X \omega\wedge\eta \qquad (\varepsilon\to 0). \end{align} $$

Since each closed curve in $X^\circ $ is homotopic to a curve in $X\subset X^\circ $ , we have $H_1(X^\circ ,\mathbb {Z})=H_1(X,\mathbb {Z})$ and each homology class on $X^\circ $ is an extension of a homology class on X. Thus, formula (2.13) is valid with the left-hand side replaced by the left-hand side of (2.13). Now formula (2.14) is obtained by passing to the limit as $\varepsilon \to 0$ .

As easily follows from Lemma 2.3 and (2.10), Abelian differentials are determined by their a-periods.

Let $X=\mathbb {M}$ . The involution acting on curves in $\mathbb {M}$ by the rule induces the involution on $H_1(\mathbb {M},\mathbb {Z})$ obeying . Since the involution $\tau $ is orientation reversing, we have

(2.15)

Note that

(2.16)

Since $\partial M=\Gamma $ consists of one connected component, each homology class $[l]$ in $\mathbb {M}$ admits the decomposition , where $l_\pm $ are collections of the curves in M. In particular, we have . Due to this facts and (2.15), any homology basis $a_1,\dots ,a_{\mathfrak {g}},b_1,\dots ,b_{\mathfrak {g}}$ in $H_1(M,\mathbb {Z})$ defines the canonical homology basis

(2.17)

in $H_1(\mathbb {M},\mathbb {Z})$ . In what follows, a homology basis of the form (2.17) is called symmetric.

Period matrices

Consider a Torelli marked Riemann surface $(X,[l_\cdot ])$ of genus m (here $[l_\cdot ]=\{a_1,\dots ,a_{m},b_1,\dots ,b_{m}\}$ ). For a basis $\omega _\cdot =\{\omega _1,\dots ,\omega _{\mathrm {g}}\}$ in $H^0(X;K)$ , we introduce its period matrix $\mathbb {T}([l_\cdot ],\omega _\cdot )$ with entries $\mathbb {T}_{ij}([l_\cdot ],\omega _\cdot ):=T(\omega _i|[l_j])$ . There is the unique basis $\omega _\cdot $ whose period matrix is of the form

$$ \begin{align*}\mathbb{T}([l_\cdot],\omega_\cdot)=(I_{m}|\mathbb{B});\end{align*} $$

this basis is called dual to $[l_\cdot ]$ and the matrix $\mathbb {B}$ is called the b-period matrix of $(X,[l_\cdot ])$ . We say that a basis $\omega _\cdot $ in $H^0(X;K)$ is canonical if it is dual to some Torelli marking on X. Also we say that a matrix $\mathbb {B}$ is b-period matrix of X if there is a Torelli marking $[l_\cdot ]$ on X such that $\mathbb {B}$ is a b-period matrix of $(X,[l_\cdot ])$ .

Let $X=\mathbb {M}$ and let $\omega _\cdot =\{\omega _1,\dots ,\omega _{2\mathrm {g}}\}$ be a basis in $H^0(\mathbb {M};K)$ . The basis $\omega _\cdot $ is called symmetric canonical if it is dual to some symmetric canonical homology basis $[l_\cdot ]$ on $\mathbb {M}$ . In this case, the b-period matrix $\mathbb {B}$ of $(X,[l_\cdot ])$ is called symmetric.

Now, let $[l_\cdot ]=\{[l_1],\dots ,[l_{2\mathfrak {g}}]\}$ be a homology basis on M and let $B_\cdot =\{B_1,\dots ,B_{2\mathfrak {g}}\}$ be a basis in $\mathcal {D}$ . We say that $B_\cdot $ is dual to $[l_\cdot ]$ if $T(B_i|[l_j])=\delta _{ij}$ ; in this case, the matrix $\mathfrak {B}$ with the entries

$$ \begin{align*}\mathfrak{B}_{ji}=T(\Phi B_i|[l_j])\end{align*} $$

is called the auxiliary period matrix corresponding to the homology basis $[l_\cdot ]$ . The basis dual to $[l_\cdot ]$ exists and is unique. Indeed, if $B^{\prime }_\cdot =\{B^{\prime }_1,\dots ,B^{\prime }_{2\mathfrak {g}}\}$ is a basis in $\mathcal {D}$ , then the matrix M with entries $M_{ij}=T(B_i|[l_j])$ is invertible (otherwise, there is the vector $0\ne B=\sum _i c_i B_i\in \mathcal {N}$ which is harmonic, normal to $\Gamma $ and has periods $T(B_i|[l_j])=\sum _i c_iM_{ij}=0$ , a contradiction). Thus, the dual basis to $[l_\cdot ]$ consists of the vectors $B_i=\sum _j (M^{-1})_{ij}B^{\prime }_j$ .

We say that the basis $B_\cdot $ in $\mathcal {D}$ is dual if it is dual to some homology basis $[l_\cdot ]$ on M; if, in addition, $[l_\cdot ]$ is canonical, then we say that $B_\cdot $ is canonical. As follows from (2.12), two dual bases $B_\cdot $ and $B^{\prime }_\cdot $ are simultaneously (non-)canonical if and only if $B^{\prime }_i=\sum _j M_{ji}B_j$ , where $M\in \mathrm {Sp}(\mathfrak {g},\mathbb {Z})$ is arbitrary; the corresponding homology bases are related via $[l^{\prime }_i]=\sum _j(M^{-1})_{ij}[l_j]$ . Similarly, any two auxiliary period matrices $\mathfrak {B}$ and $\mathfrak {B}^{\prime }$ (corresponding to different homology bases) are (corresponding to different homology bases) are related via

(2.18) $$ \begin{align} \mathfrak{B}^{\prime}=M^{-1}\mathfrak{B}M \qquad (M\in \mathrm{Sp}(\mathfrak{g},\mathbb{Z})). \end{align} $$

The following lemma provides the criterion of the canonicity of the dual basis. Also, it provides the expression for the auxiliary period matrix of a canonical homology basis in terms of inner products of elements of its dual basis.

Proof a) Since $B_k$ are normal and $T(B_k|[l_s])=\delta _{ks}$ , Lemma 2.3 and the equality $\Phi ^2=-\mathrm {Id}$ imply

$$ \begin{align*}(B_i,\Phi B_j)=\mathrm{T}(B_i|\,[l_\cdot])^T J^{-1}\mathrm{T}(-\Phi^2 B_j|\,[l_\cdot])=\delta_{ik}(J^{-1})_{ks}\delta_{sj}=(J^{-1})_{ij}.\end{align*} $$

Thus, we have proved (2.19). Now (b) easily follows from (a). (c) In view of Lemma 2.3 and the equality $J^{-1}=\Omega _{(\mathfrak {g})}^{-1}=-\Omega _{(\mathfrak {g})}$ , we have

$$ \begin{align*}(B_i,B_j)=\mathrm{T}(B_i|\,[l_\cdot])^T\Omega_{(\mathfrak{g})}\mathrm{T}(\Phi B_j|\,[l_\cdot])=\delta_{ik}(\Omega_{(\mathfrak{g})})_{ks}\mathfrak{B}_{sj}=(\Omega_{(\mathfrak{g})}\mathfrak{B})_{ij}.\\[-34pt] \end{align*} $$

Let $[l_\cdot ]=\{a_1,\dots ,a_{\mathfrak {g}},b_1,\dots ,b_{\mathfrak {g}}\}$ be a canonical homology basis on M, let $B_\cdot $ be the corresponding dual basis in $\mathcal {D}$ and let $\mathfrak {B}$ be the corresponding auxiliary period matrix. Let us establish the connection between $\mathfrak {B}$ and the b-period matrix $\mathbb {B}$ of the cover $\mathbb {M}$ corresponding to the symmetric canonical basis $[\tilde {l}_\cdot ]$ related to $[l_\cdot ]$ via (2.17). Denote

$$ \begin{align*}\omega_i=(iB_i-\Phi B_i)^\flat \qquad (i=1,\dots,2\mathfrak{g}).\end{align*} $$

As follows from Lemma 2.2 and the equality $\Phi \mathcal {D}=\mathcal {N}$ , $\omega _i$ admit analytic continuation to symmetric abelian differentials on the double $\mathbb {M}$ (still denoted by ). Note that $\omega _1,\dots ,\omega _{2\mathfrak {g}}$ constitute a basis in $H^1(\mathbb {M};K)$ due to the linear independence of $B_1,\dots ,B_{2\mathfrak {g}}$ . In addition,

$$ \begin{align*}T(\omega_i|a_j)=i\delta_{ij}-\mathfrak{B}_{ji}, \qquad T(\omega_i|b_j)=i\delta_{i,j+\mathfrak{g}}-\mathfrak{B}_{j+\mathfrak{g},i}\end{align*} $$

for $j\le \mathfrak {g}$ . In view of (2.16) and (2.17), we have

for $j>\mathfrak {g}$ . Then the period matrix of the basis $\omega _\cdot $ is given by

$$ \begin{align*}\mathbb{T}([\tilde{l}_\cdot]|\omega_\cdot)=\big(i\chi^{+-}-\mathfrak{B}^T\chi^{++}|i\chi_{++}-\mathfrak{B}^T\chi_{+-}\big),\end{align*} $$

where

(2.20) $$ \begin{align} \chi^{\mathfrak{s},\mathfrak{s}^{\prime}}=\left(\begin{array}{cc} \mathfrak{s}I_{\mathfrak{g}} & \mathfrak{s}^{\prime}I_{\mathfrak{g}}\\ 0 & 0 \end{array}\right), \qquad \chi_{\mathfrak{s},\mathfrak{s}^{\prime}}=\left(\begin{array}{cc} 0 & 0 \\ \mathfrak{s}I_{\mathfrak{g}} & \mathfrak{s}^{\prime}I_{\mathfrak{g}} \end{array}\right) \qquad (\mathfrak{s},\mathfrak{s}^{\prime}=\pm) \end{align} $$

and $I_{m}$ is the unit $m\times m$ -matrix. Introduce the new basis $\tilde {\omega }_\cdot $ in $H^1(\mathbb {M};K)$ by $\tilde {\omega }_i=\sum _j M_{ij}\omega _j$ , where $M=(i\chi ^{+-}-\mathfrak {B}^T\chi ^{++})^{-1}$ . Then its period matrix is equal to

$$ \begin{align*}\mathbb{T}([\tilde{l}_\cdot]|\tilde{\omega}_\cdot)=(I_{2\mathfrak{g}} \ | \ (i\chi^{+-}-\mathfrak{B}^T\chi^{++})^{-1}(i\chi_{++}-\mathfrak{B}^T\chi_{+-})\big).\end{align*} $$

Hence, the basis $\tilde {\omega }_\cdot $ is dual to $[\tilde {l}_\cdot ]$ . In particular, the b-period matrix $\mathbb {B}$ of $\mathbb {M}$ corresponding to $[\tilde {l}_\cdot ]$ is related to $\mathfrak {B}$ via

(2.21) $$ \begin{align} \mathbb{B}=(i\chi^{+-}-\mathfrak{B}^T\chi^{++})^{-1}(i\chi_{++}-\mathfrak{B}^T\chi_{+-}). \end{align} $$

Symmetry (2.17) of the canonical homology basis leads to the symmetries of the dual basis $\omega _\cdot $ and the b-period matrix. Indeed, (2.17) and (2.16) imply

for $j\le \mathfrak {g}$ . Since the Abelian differentials are determined by their a-periods, we have for $i=1,\dots ,\mathfrak {g}$ . As a corollary, we obtain

(2.22)

Step 1. Determination of boundary data of harmonic normal vectors on M

Let $u=u^f$ be a harmonic function in M with trace f on $\Gamma $ . Then $\Phi \nabla u$ is a harmonic field and the decomposition $\mathcal {H}=\mathcal {E}\oplus \mathcal {N}$ yields

(3.1) $$ \begin{align} \Phi\nabla u^f=\nabla u^h+A, \end{align} $$

where $u^h$ is a harmonic function in M with trace h on $\Gamma $ and $A\in \mathcal {N}$ . Note that $A=(\Phi -\hat {\Phi })\nabla u^f$ and $\nabla u^h=\hat {\Phi }\nabla u^f$ .

The vector field

(3.2) $$ \begin{align} B=\Phi A \end{align} $$

is an element of $\mathcal {D}$ . Restricting equations (3.1) and (3.2) to the boundary and taking into account (2.1), we obtain

(3.3) $$ \begin{align} -\partial_\gamma f=\Lambda h, \qquad \Lambda f=\partial_\gamma h+A_\gamma, \qquad B_\nu=-A_\gamma. \end{align} $$

In particular, we have

(3.4) $$ \begin{align} B_\nu=\partial_\gamma h-\Lambda f=-(\partial_\gamma\Lambda^{-1}\partial_\gamma+\Lambda)f=-\partial_\gamma(H+H^{-1})f. \end{align} $$

As is easily seen from (3.4), f is determined by B up to an element of $\Re \mathrm {Ker}(H+H^{-1})=\mathrm {clos}_{H^{1/2}(\Gamma ;\mathbb {R})}\big (\Re \mathrm {Tr}\mathscr {A}(M)\big )$ . This is related to the fact that the fields $A,B$ do not change after adding the term $\Phi \nabla u^{\tilde {f}}=\nabla u^{\tilde {h}}$ to both sides of (3.1), where $u^{\tilde {f}}+iu^{\tilde {h}}=\tilde {w}\in \mathrm { clos}_{H^1(M,g)}\big (\mathscr {A}(M)\big )$ (i.e., the equation $\Phi \nabla u^{\tilde {f}}=\nabla u^{\tilde {h}}$ is exactly the Cauchy–Riemann condition for $\tilde {w}$ ). One can fix f by the additional condition

$$ \begin{align*}(f,\tilde{f})_\Lambda=0 \qquad \forall \ \tilde{f}\in\mathrm{Ker}(H+H^{-1});\end{align*} $$

then f is uniquely defined by $B_\nu $ and admits the representation

(3.5) $$ \begin{align} f=\sum_{\pm k=1}^{\mathfrak{g}}\hat{f}_k\eta_k \qquad (\hat{f}_{-k}=\overline{\hat{f}_k}\in\mathbb{C}), \end{align} $$

where $\eta _k$ are given by (2.8). In particular, one can assume that f is smooth.

In what follows, we say that the pair $\{B_\nu ,f\}$ is a boundary data for $B\in \mathcal {D}$ and denote $\{B_\nu ,f\}=\mathfrak {T}(B)$ . Note that each $B\in \mathcal {D}$ admits boundary data. Indeed, formula (2.4) implies that each $A=-\Phi B$ admits representation (3.1). The space $\mathfrak {T}(\mathcal {D})$ of all boundary data is denoted by $\mathcal {D}_\Gamma $ . Since each $B\in \mathcal {D}$ is determined by the normal component of its boundary trace, the linear map $\mathfrak {T}: \ \mathcal {D}\to \mathcal {D}_\Gamma $ is a bijection.

From (3.1), (3.2), (2.6), and (3.3) it follows that

(3.6) $$ \begin{align} \begin{aligned} \|B\|^2=&\|A\|^2=\|\Phi\nabla u^f\|^2-\|\nabla u^h\|^2=\|\nabla u^f\|^2-\|\nabla u^h\|^2=\\ =&(\Lambda f,f)_\Gamma-(\Lambda h,h)_\Gamma=(\Lambda f,f)_\Gamma-(\partial_\gamma f,\Lambda^{-1}\partial_\gamma f)_\Gamma=-(B_\nu,f)_\Gamma. \end{aligned} \end{align} $$

Due to (3.6) and the polarization identity, the inner products of elements of $\mathcal {D}$ can be found from their boundary data. Namely, we have

$$ \begin{align*}(B,B^{\prime})=-(B_\nu,f^{\prime})_\Gamma=-(f,B^{\prime}_\nu)_\Gamma,\end{align*} $$

where $\{B^{\prime }_\nu ,f^{\prime }\}$ is the boundary data of $B^{\prime }\in \mathcal {D}$ . Similarly, since the subspaces $\mathcal {N}$ and $\mathcal {D}=\Phi \mathcal {N}$ are $L_2(M;TM)$ -orthogonal to $\mathcal {E}$ and $\Phi \mathcal {E}$ , respectively, we have

$$ \begin{align*} (\Phi B,B^{\prime})&=-(B,\Phi B^{\prime})=-(\Phi A,\Phi B^{\prime})=-(A,B^{\prime})=(\nabla u^h-\Phi\nabla u^f,B^{\prime})=\\ &=(\Phi\nabla u^f,B^{\prime})+(\nabla v^h,B^{\prime})=0+(\nabla u^h,B^{\prime})=\\ &=\int_M \mathrm{div}(u^hB^{\prime})dS-\int_M u^h\mathrm{div}(B^{\prime})dS=\\&= \int_\Gamma h B^{\prime}_\nu dl+0=-(\Lambda^{-1}\partial_\gamma f,B^{\prime}_\nu)_\Gamma=-(Hf,B^{\prime}_\nu)_\Gamma. \end{align*} $$

We arrive at the following statement.

Proposition 3.1 Using the DN map $\Lambda $ of M, one can construct the isometric copy $\mathcal {D}_\Gamma $ of $\mathcal {D}$ in the following way:

• • The space $\mathcal {D}_\Gamma $ is defined by

  $$ \begin{align*} \mathcal{D}_\Gamma:=\Big\{\{B_\nu,f\}&\in C^\infty(\Gamma;\mathbb{C})\times C^\infty(\Gamma;\mathbb{C}) \quad \big| \\ \big| \quad f&=\sum_{\pm k=1}^{\mathfrak{g}}\hat{f}_k\eta_k, \quad \hat{f}_{-k}=\overline{\hat{f}_k}\in\mathbb{C}, \quad B_\nu=-\partial_\gamma(H+H^{-1})f\Big\}, \end{align*} $$
  where $H=\Lambda ^{-1}\partial _\gamma $ is the Hilbert map and $\eta _{\pm 1},\dots ,\eta _{\pm \mathfrak {g}}$ are eigenfunctions (2.8) corresponding to the eigenvalues $\lambda _{\pm 1},\dots ,\lambda _{\pm \mathfrak {g}}$ different from $\pm i$ .

• • $\mathcal {D}_\Gamma $ is endowed with the inner product

  (3.7) $$ \begin{align} (\{B_\nu,f\},\{B^{\prime}_\nu,f^{\prime}\}):=-(B_\nu,f^{\prime})_\Gamma=-(f,B^{\prime}_\nu)_\Gamma \end{align} $$
  and the alternating bilinear form
  (3.8) $$ \begin{align} {\boldsymbol{\langle}}\{B_\nu,f\},\{B^{\prime}_\nu,f^{\prime}\}{\boldsymbol{\rangle}}:=(B_\nu,Hf^{\prime})_\Gamma=-(Hf,B^{\prime}_\nu)_\Gamma. \end{align} $$

Then the map $\mathfrak {T}: \ \mathcal {D}\to \mathcal {D}_\Gamma $ introduced after (3.5) is an isometry obeying

(3.9) $$ \begin{align} (\Phi B,B^{\prime})={\boldsymbol{\langle}}\mathfrak{T}(B),\mathfrak{T}(B^{\prime}){\boldsymbol{\rangle}} \qquad (B,B^{\prime}\in\mathcal{D}). \end{align} $$

Step 2. Determination of boundary data of harmonic normal vectors with integer periods on M

Let us rewrite (3.1) and (3.2) as follows

(3.10) $$ \begin{align} \Phi\nabla u^h=B-\nabla u^f. \end{align} $$

Let U be an arbitrary simple connected neighborhood in M. Since $u^h$ is harmonic in U, the Poincaré lemma implies that there is a harmonic function V in U obeying $\nabla V=\Phi \nabla u^h$ . Hence, the function

(3.11) $$ \begin{align} x\mapsto W(x):=u^h(x)+iV(x)=u^h(x)+i\int_\cdot^x(\Phi\nabla u^h)^\flat+i\mathrm{const} \end{align} $$

is holomorphic in U. However W is not in general globally defined on M: after analytic continuation along the loop l from any non-trivial cycle $[l]\in H_1(M,\mathbb {Z})$ in M its value acquires the shift

$$ \begin{align*}T(\Phi\nabla u^h|[l])=T(B|[l])\end{align*} $$

(the equality follows from (3.10)). Note that one can chose a single-valued branch of W in a tubular neighborhood of $\Gamma $ due to $\int _\Gamma B^\flat =\int _\Gamma B_\gamma dl=0$ . In view of (3.10) and (3.3), the boundary trace of W obeys

$$ \begin{align*} \begin{aligned} \partial_\gamma W|_\Gamma &=\partial_\gamma h+ig(\nabla V,\gamma)=\partial_\gamma h+ig(\Phi\nabla u^h,\gamma)=\partial_\gamma h+i\partial_\nu u^h\\ & =(\partial_\gamma+i\Lambda) h=-(\partial_\gamma+i\Lambda)\Lambda^{-1}\partial_\gamma f=-\partial_\gamma(H+i)f. \end{aligned} \end{align*} $$

Hence,

$$ \begin{align*}W|_\Gamma=-(H+i)f+iC\end{align*} $$

Here $C\in \mathbb {R}$ is a constant on $\Gamma $ which depends on the choices of the constant in (3.11) and branch of W near $\Gamma $ . From now on, we assume that (the branch of) W is chosen in such a way that $C=0$ .

The multivalued function $e^{2\pi W}$ acquires the multiplier $e^{2\pi iT(B|[l])}$ after analytic continuation along each closed loop l in M. Therefore, $e^{2\pi W}$ is single-valued if and only if all the periods $T(B|[l])$ ( $[l]\in H_1(M,\mathbb {Z})$ ) of B are integer. In the last case, $e^{2\pi W}|_\Gamma $ is an element of $\mathrm {Tr}\mathscr {A}(M)\equiv (\mathrm {Ker}(H-i)\dotplus \mathbb {C})\cap C^{\infty }(\Gamma ;\mathbb {C})$ due to Lemma 2.1. So, B has integer periods only if the equation

(3.12) $$ \begin{align} \partial_\gamma(H-i)\big[e^{-2\pi (H+i)f}\big]=0 \end{align} $$

holds on $\Gamma $ . Note that (3.12) is invariant under the shift $f\mapsto f+q$ , where q is a smooth element of $\mathrm {Ker}(H+H^{-1})=\mathrm {Ker}(H-i)\dotplus \mathrm {Ker}(H+i)\dotplus \mathbb {C}$ . Indeed, if $q\in \mathrm {Ker}(H+i)\dotplus \mathbb {C}$ , then q is a boundary trace of some holomorphic function w and $e^{-2\pi (H+i)q}=e^{-4\pi i q}$ is the trace of $e^{-4\pi i w}$ . Then the condition $e^{-2\pi (H+i)f}\in \mathrm { Tr}\mathscr {A}(M)$ (equivalent to (3.12)) implies $e^{-2\pi (H+i)(f+q)}=e^{-2\pi (H+i)f}e^{-4\pi i w}\in \mathrm {Tr}\mathscr {A}(M)$ and vice versa.

Now, suppose that (3.12) holds on $\Gamma $ . Then there is a holomorphic function w on M whose boundary trace is equal to $e^{-2\pi (H+i)f}=e^{2\pi W}|_\Gamma $ . Since $e^{2\pi W}$ and w are holomorphic and $e^{2\pi W}=w$ on $\Gamma $ , they coincide everywhere where one of them can be analytically continued. Thus, $e^{2\pi W}=w$ on M and $e^{2\pi W}$ is single-valued. The latter means that B has integer periods $T(B|[l])$ ( $[l]\in H_1(M,\mathbb {Z})$ ). Thus, we arrive at the following statement.

Proposition 3.2 Introduce be the additive group

$$ \begin{align*}\mathscr{G}=\{B\in\mathcal{D} \ \ | \ T(B|[l])\in\mathbb{Z} \quad \forall [l]\in H_1(M,\mathbb{Z})\}\end{align*} $$

of vector fields with integer periods in $\mathcal {D}$ and denote by $\mathscr {G}_{\Gamma }=\mathfrak {T}(\mathscr {G})$ the corresponding group in $\mathcal {D}_\Gamma $ . Then $\mathscr {G}_{\Gamma }$ can be determined from the DN map $\Lambda $ via the formula

$$ \begin{align*}\mathscr{G}_{\Gamma}=\{\{B_\nu,f\}\in\mathcal{D}_\Gamma \ \ | \ f \text{ is a solution to }(3.12)\}.\end{align*} $$

Using representation (3.5) for f, one can rewrite equation (3.12) in more convenient form. Let $\hat {f}_k=\overline {\hat {f}_{-k}}=\alpha _k+i\beta _k$ , where $\alpha _k,\beta _k\in \mathbb {R}$ ( $k=1,\dots ,\mathfrak {g}$ ). Introduce the functions

(3.13) $$ \begin{align} \begin{aligned} p_k:&=\mathrm{exp}\big(-2\pi i[\eta_k(1+\mu_k)+\overline{\eta_k}(1-\mu_k)]\big), \\ q_k:&=\mathrm{exp}\big(2\pi[\eta_k(\mu_k+1)+\overline{\eta_k}(\mu_k-1)]\big), \end{aligned} \end{align} $$

where $\eta _k,\mu _k$ are given by (2.8) and (2.5). Then

$$ \begin{align*}-2\pi i(H+i)f=2\pi\sum_k [ c_k\eta_k(\mu_k+1)-\overline{2\pi c_k\eta_k}(\mu_k-1)]\end{align*} $$

and (3.12) is equivalent to (1.4). As easily seen from (1.4), condition (3.12) is actually an equation on $2\mathfrak {g}$ real variables $\alpha _k,\beta _k$ . Thus, $\varkappa:=(\alpha _1,\dots ,\alpha _{\mathfrak {g}},\beta _1\dots ,\beta _{\mathfrak {g}})$ is a solution to (3.12) if and only if

(3.14) $$ \begin{align} \begin{aligned} \{B_\nu(\varkappa),f(\varkappa)\}= \Big(i\sum_{k=1}^{\mathfrak{g}}[(\mu_k^{-1}-\mu_k)(\alpha_k+i\beta_k)\partial_\gamma\eta_k & +(\mu_k^{-1}+\mu_k)(\alpha_k-i\beta_k)\partial_\gamma\overline{\eta}_k],\\ \sum_{k=1}^{\mathfrak{g}}[(\alpha_k & +i\beta_k)\eta_k+(\alpha_k-i\beta_k)\overline{\eta}_k]\Big) \end{aligned} \end{align} $$

is the boundary data of an element of $\mathscr {G}$ .

Step 3. Determination of boundary data of canonical bases in $\mathcal {D}$

As shown in Proposition 3.2, the solutions to (3.12) (or to (1.4)) provide the boundary data of vectors with integer periods in $\mathcal {D}$ . The next step is to find among them the boundary data $\mathfrak {T}(B_1),\dots ,\mathfrak {T}(B_{2\mathfrak {g}})$ corresponding to some canonical basis $B_1,\dots ,B_{2\mathfrak {g}}$ in $\mathcal {D}$ . To this end, we apply the following statement.

Proof (a) The necessity follows from Lemma 2.4, (b). Let us prove the sufficiency. Let $Q_1,\dots ,Q_{2\mathfrak {g}}$ be a canonical basis in $\mathcal {D}$ and let $[l_\cdot ]$ be the corresponding canonical homology basis. Since $B_1,\dots ,B_{2\mathfrak {g}}$ are linearly independent, we have $B_i=M_{ij}Q_j$ , where M is an invertible matrix. Then

$$ \begin{align*}T(B_i|[l_k])=\sum_{j}M_{ij}T(Q_j|[l_k])=\sum_{j}M_{ij}\delta_{jk}=M_{ik}\end{align*} $$

and, since each $B_i$ has integer periods, the entries of M are integer.

In view to Lemma 2.3, condition (3.15) implies

$$ \begin{align*} (\Omega_{\mathfrak{g}})_{ij}=(\Phi B_i,B_j)=-(B_i,\Phi B_j)=\sum_{ks}T(B_i|[l_k])(-\Omega_{\mathfrak{g}}^{-1})_{ks}T(-\Phi^2 B_j|[l_s])=\\ =\sum_{ks}T(B_i|[l_k])(\Omega_{\mathfrak{g}})_{ks}T(B_j|[l_s])=\sum_{ks}M_{ik}(\Omega_{\mathfrak{g}})_{ks}M^T_{sj}=(M\Omega_{\mathfrak{g}}M^T)_{ij}. \end{align*} $$

Thus, we have $\Omega _{\mathfrak {g}}=M\Omega _{\mathfrak {g}} M^T$ and $0\ne \mathrm {det}(\Omega _{\mathfrak {g}})=\mathrm {det}(\Omega _{\mathfrak {g}})(\mathrm {det}(M))^2$ . Since the entries of M are integer, we have $\mathrm { det}(M)=\pm 1$ . Thus, entries of $M^{-1}$ are also integer and $B_1,\dots ,B_{2\mathfrak {g}}$ is a basis in $\mathcal {D}$ dual to the homology basis

$$ \begin{align*}[\tilde{l}_i]=(M^{-1})_{ij}[l_j].\end{align*} $$

Moreover, the new homology basis $[\tilde {l}_\cdot ]$ is canonical due to Lemma 2.4, (b).

(b) Denote $B_k=\mathfrak {T}^{-1}(\kappa _k)$ , then conditions (3.15) and (3.16) are equivalent due to (3.9). Therefore, (b) follows from (a).

Step 4. Determination of period matrices of M and $\mathbb {M}$

Let $\kappa _1=\mathfrak {T}(B_1),\dots ,\kappa _{2\mathfrak {g}}=\mathfrak {T}(B_{2\mathfrak {g}})$ be elements of $\mathcal {G}_\Gamma $ obeying condition (3.16). In view of Proposition 3.3, (b), vectors $B_1,\dots ,B_{2\mathfrak {g}}$ constitute a basis in $\mathcal {D}$ dual to some canonical homology basis $[l_\cdot ]$ on M. In view of Lemma 2.4, (c) and Proposition 3.1, the auxiliary period matrix $\mathbb {B}$ corresponding to $[l_\cdot ]$ obeys

(3.17) $$ \begin{align} (\Omega_{(\mathfrak{g})}\mathfrak{B})_{ij}=(B_i,B_j)={\boldsymbol{(}\kappa_i,\kappa_j\boldsymbol{)}} \end{align} $$

(the inner product on $\mathcal {D}_\Gamma $ is defined by (3.7)).

So, using the previous steps and formula (3.17), one determines the auxiliary period matrix $\mathfrak {B}$ of M corresponding to some canonical homology basis $[l_\cdot ]=\{a_1,\dots ,a_{\mathbb {g}},b_1,\dots ,b_{\mathbb {g}}\}$ on it. Then the b-period matrix $\mathbb {B}$ of $\mathbb {M}$ , corresponding to symmetric canonical basis (2.17) is derived from $\mathfrak {B}$ by applying formulas (2.21) and (2.20).

It remains to note that, although one cannot control the choice of $[l_\cdot ]$ , it is still possible to find all other auxiliary period matrices (corresponding to all possible canonical homology bases on M) by applying transformations (2.18) to $\mathfrak {B}$ . Then the substitution of these matrices into (2.21) and (2.20) provides all symmetric b-period matrices of $\mathbb {M}$ .

3.1 On the stability of the algorithm under small noise in the boundary data

Let $\Lambda $ be a DN map of some (unknown) surface $(M,g)$ (we assume that the boundary $\Gamma $ of $(M,g)$ is given). We now explain the implementation of Steps 1–4 for the case in which only some approximation $\Lambda ^{\prime }$ of $\Lambda $ is known. Namely, we assume that $\Lambda ^{\prime }$ is a continuous operator acting from $H^1(\Gamma ;\mathbb {C})$ to $L_2^{\mathbb {C}}(\Gamma ;dl)$ and obeying

(3.18) $$ \begin{align} \|\Lambda-\Lambda^{\prime}\|_{H^1(\Gamma;\mathbb{C})\to L_2^{\mathbb{C}}(\Gamma;dl)}\le \varepsilon, \end{align} $$

where $\varepsilon $ is a small parameter called the noise bound. In what follows, we suppose that the noise bound is known to the one who applies Steps 1–4.

Now, we describe the implementation of Steps 1–4 to obtain the approximation of some b-period matrix $\mathbb {B}$ of the double of $(M,g)$ via $\Lambda ^{\prime }$ .

Step 1 (implementation)

Introduce the approximate Hilbert transform $H^{\prime }=\Lambda ^{^{\prime }-1}\partial _\gamma $ . In view of (3.18), the operator $H^{^{\prime }-1}=\partial _\gamma ^{-1}\Lambda ^{\prime }$ obeys

$$ \begin{align*}\|H^{^{\prime}-1}-H^{-1}\|_{H^{1}(\Gamma;\mathbb{C})\mapsto H^{1}(\Gamma;\mathbb{C})}=O(\varepsilon).\end{align*} $$

Here and in the subsequent, all estimates are assumed to be uniform in $\Lambda ^{\prime }$ (but not uniform in $\Lambda $ ). In particular, the spectrum $\mathrm {Sp}(H^{^{\prime }-1})$ of $H^{^{\prime }-1}$ is contained in the $O(\varepsilon )$ -neighborhood of the spectrum $\mathrm {Sp}(H^{-1})$ of $H^{-1}$ . The essential spectrum of $H^{-1}$ is $\{i\}\cup \{-i\}$ . Since the set of Fredholm operators is open in the operator norm (see [Reference Murphy17, Theorem 1.4.17]), the essential spectrum of $H^{^{\prime }-1}$ is contained in the $O(\varepsilon )$ -neighborhoods of $\pm i$ . The same estimates are valid for the spectra of $H^{\prime }$ ,H.

To find the approximations for eigenvalues (2.5) and eigenfunctions (2.8), we apply the following simple lemma.

Lemma 3.4 Suppose that $\lambda $ is a regular eigenvalueFootnote ¹ of a continuous operator A (acting in some Banach space E) of finite multiplicity and there is a punctured $c_0$ -neighborhood of $\lambda $ which does not intersect the spectrum of A. Let $A^{\prime }$ be an arbitrary continuous operator in E obeying $\|A^{\prime }-A\|<\varepsilon $ for sufficiently small $\varepsilon $ . Let $(\lambda ^{\prime },f^{\prime })$ be any eigenpair of $A^{\prime }$ obeying $|\lambda ^{\prime }-\lambda |<\varepsilon $ and $\|f^{\prime }\|=1$ . Then there is $f\in \mathrm {Ker}(A-\lambda )$ such that $\|f-f^{\prime }\|=O(\varepsilon )$ .

Proof Consider the decomposition $E=\mathrm {Ker}(A-\lambda )\dot {+}\tilde {E}$ , where $\tilde {E}$ is a closed subspace in E (such $\tilde {E}$ exists since $\mathrm {Ker}(A-\lambda )$ is finite-dimensional). Since A is continuous, $(A-\lambda )\tilde {E}$ is closed and the operator $\tilde {A}=((A-\lambda )|_{\tilde {E}})^{-1}: \ (A-\lambda )\tilde {E}\to \tilde {E}$ is continuous due to the closed graph theorem. Decompose $f^{\prime }$ as $f^{\prime }=f+\tilde {f}$ , where $f\in \mathrm {Ker}(A-\lambda )$ and $\tilde {f}\in \tilde {E}$ . Since

$$ \begin{align*}\tilde{f}=\tilde{A}(A-\lambda)f^{\prime}=\tilde{A}(A-A^{\prime})f^{\prime}-(\lambda-\lambda^{\prime})\tilde{A}f^{\prime},\end{align*} $$

we have $\|\tilde {f}\|=O(\varepsilon )$ .

Thus, to construct approximations of the eigenfunctions of H corresponding to the unknown eigenvalue $\lambda =\lambda _i$ , we find all (normalized in $L_2^{\mathbb {C}}(\Gamma ;dl)$ ) eigenfunctions of $H^{\prime }$ corresponding to the (nonzero) eigenvalues $\lambda ^{\prime }_k$ obeying $|\lambda ^{\prime }\pm i|>\sqrt {\varepsilon }$ and $|\lambda ^{\prime }_k-\lambda ^{\prime }_l|<\sqrt {\varepsilon }$ and then chose among them the maximal collection of pairwise orthogonal (in $L_2^{\mathbb {C}}(\Gamma ;dl)$ ) eigenfunctions $\eta ^{\prime }_i,\dots ,\eta ^{\prime }_{i+m(i)}$ . As a result, for sufficiently small $\varepsilon $ , we obtain the approximations $(\lambda ^{\prime }_{\pm k},\eta ^{\prime }_{\pm k})$ of the eigenpairs $(\lambda _{\pm k},\eta _{\pm k})$ obeying

(3.19) $$ \begin{align} |\lambda^{\prime}_{\pm k}-\lambda_{\pm k}|+\|\eta^{\prime}_{\pm k}-\eta_{\pm k}\|_{H^{1}(\Gamma;\mathbb{C})}=O(\varepsilon) \qquad (k=1,\dots,\mathfrak{g}). \end{align} $$

Now, we introduce the space $\mathcal {D}^{\prime }_\Gamma $ and the bilinear forms $(\cdot ,\cdot )^{\prime }$ , $\langle \cdot ,\cdot \rangle $ in the same way as in Proposition 3.1, where $\eta _{\pm k}$ are replaced by $\eta ^{\prime }_{\pm k}$ . Denote

$$ \begin{align*}\kappa^{\prime}_{\pm k}:=\{-\partial_\gamma(H^{\prime}+H^{^{\prime}-1})\eta^{\prime}_{\pm k},\eta^{\prime}_{\pm k}\},\end{align*} $$

then formulas (3.18) and (3.19) imply the closeness between the structures on $\mathcal {D}^{\prime }_\Gamma $ and $\mathcal {D}_\Gamma $ ,

(3.20) $$ \begin{align} \begin{aligned} (\kappa^{\prime}_{\pm i},\kappa^{\prime}_{(\pm) j})^{\prime}& -(\kappa_{\pm i},\kappa_{(\pm) j})=O(\varepsilon), \\ \langle\kappa^{\prime}_{\pm i},\kappa^{\prime}_{(\pm) j}\rangle^{\prime} & -\langle\kappa_{\pm i},\kappa_{(\pm) j}\rangle=O(\varepsilon) \qquad (k=1,\dots,\mathfrak{g}). \end{aligned} \end{align} $$

Step 2 (implementation)

Instead of (1.4), we consider the (approximate) equation

$$ \begin{align*}\partial_\gamma(H^{\prime}-i)\big[(p^{\prime}_1)^{\alpha^{\prime}_1}\dots (p^{\prime}_{\mathfrak{g}})^{\alpha^{\prime}_{\mathfrak{g}}}(q^{\prime}_1)^{\beta^{\prime}_1}\dots (q^{\prime}_{\mathfrak{g}})^{\beta^{\prime}_{\mathfrak{g}}}\big]=0,\end{align*} $$

where $p^{\prime }_k$ , $q^{\prime }_k$ are given by formula (3.13) with $\eta _k$ replaced by $\eta ^{\prime }_k$ . Introduce the functions

$$ \begin{align*} \mathscr{E}(\kappa)&:=\|\partial_\gamma(H-i)\big[p_1^{\alpha_1}\cdots p_{\mathfrak{g}}^{\alpha_{\mathfrak{g}}}q_1^{\beta_1}\cdots q_{\mathfrak{g}}^{\beta_{\mathfrak{g}}}\big]\|_{L^{\mathbb{C}}_2(\Gamma;dl)},\\ \mathscr{E}^{\prime}(\kappa^{\prime})&:=\|\partial_\gamma(H^{\prime}-i)\big[(p^{\prime}_1)^{\alpha^{\prime}_1}\cdots (p^{\prime}_{\mathfrak{g}})^{\alpha^{\prime}_{\mathfrak{g}}}(q^{\prime}_1)^{\beta^{\prime}_1}\cdots (q^{\prime}_{\mathfrak{g}})^{\beta^{\prime}_{\mathfrak{g}}}\big]\|_{L^{\mathbb{C}}_2(\Gamma;dl)}, \end{align*} $$

where $\varkappa:=(\alpha _1,\dots ,\alpha _{\mathfrak {g}},\beta _1\dots ,\beta _{\mathfrak {g}})$ and $\varkappa^{\prime }:=(\alpha ^{\prime }_1,\dots ,\alpha ^{\prime }_{\mathfrak {g}},\beta ^{\prime }_1\dots ,\beta ^{\prime }_{\mathfrak {g}})$ .

Recall that the global minima (i.e., zeroes) of $\mathscr {E}$ correspond to the boundary data of elements of $\mathcal {D}$ with integer periods via (3.14). Let $\mathscr {B}$ be a sufficiently large closed ball in the parameter space $\mathbb {R}^{2\mathfrak {g}}$ of $\varkappa$ whose interior contains the zeroes of $\mathscr {E}$ corresponding to the elements of some canonical dual basis in $\mathcal {D}$ . Then estimates (3.19) and (3.18), formula (3.13), and the continuity of the embedding $H^{1}(\Gamma ;\mathbb {C})\subset C(\Gamma ;\mathbb {C})$ yield

(3.21) $$ \begin{align} \|\mathscr{E}^{\prime}-\mathscr{E}\|_{C(\mathbb{R}^{2\mathfrak{g}})}=O(\varepsilon). \end{align} $$

Let $\varkappa\in \mathscr {B}$ be a zero of $\mathscr {E}$ , then $\Pi =p_1^{\alpha _1}\cdots p_{\mathfrak {g}}^{\alpha _{\mathfrak {g}}}q_1^{\beta _1}\cdots q_{\mathfrak {g}}^{\beta _{\mathfrak {g}}}$ is a trace on $\Gamma $ of holomorphic invertible function on $(M,g)$ . For small variations $\delta \varkappa:=(\delta \alpha _1,\dots ,\delta \alpha _{\mathfrak {g}},\delta \beta _1,\dots ,\delta \beta _{\mathfrak {g}})$ , we have

$$ \begin{align*} \partial_\gamma(H-i)\big[p_1^{\alpha_1+\delta\alpha_1}\cdots &p_{\mathfrak{g}}^{\alpha_{\mathfrak{g}}+\delta\alpha_{\mathfrak{g}}}q_1^{\beta_1+\delta\beta_1}\cdots q_{\mathfrak{g}}^{\beta_{\mathfrak{g}}+\delta\beta_{\mathfrak{g}}}\big]\\ =&\partial_\gamma(H-i)\sum_k\big[\mathrm{log}p_k\delta \alpha_k+\mathrm{log}q_k\delta \beta_k\big]\Pi+O(|\delta\varkappa|^2). \end{align*} $$

Note that the first term vanishes only if $\delta \varkappa=0$ . Indeed, otherwise, there is the nonzero linear combination of $\Pi \mathrm {log}p_k$ and $\Pi \mathrm {log}q_k$ which is a trace of a holomorphic function. Since $\Pi ^{-1}$ is the trace of holomorphic function and $\mathrm {log}p_k,\mathrm {log}q_k$ admit representations (3.13), we conclude that there is a nonzero linear combination of $\eta _{\pm k}$ which is the trace of holomorphic function (i.e., an element of $\mathrm {Ker}(H-i)\dot {+}\mathbb {C}$ ), this gives a contradiction. Thus, we obtain the non-degeneracy of all minima $\kappa $ of $\mathscr {E}$ ,

$$ \begin{align*}0<c_0<\frac{\mathscr{E}(\varkappa+\delta\varkappa)}{|\delta\varkappa|}<c_1<+\infty \qquad (|\delta\varkappa|<c_3),\end{align*} $$

where the constants $c_1,c_2,c_3$ depend on $\Lambda $ and $\mathscr {B}$ . In particular, the inequality $|\mathscr {E}(\varkappa^{\prime })|<\epsilon \ll 1$ implies that $\varkappa^{\prime }$ lies in $O(\epsilon )$ -neighborhood of some solution $\varkappa$ to (1.4).

Let us find the minimum $\varkappa^{\prime }\in \mathscr {B}$ of $\mathscr {E}^{\prime }$ . Then $|\mathscr {E}^{\prime }(\varkappa^{\prime })|<\varepsilon $ and (3.21) yields $|\mathscr {E}(\varkappa)|=O(\varepsilon )$ . Thus, $|\varkappa^{\prime }-\varkappa|=O(\varepsilon )$ , where $\varkappa$ is a solution to (1.4). Now, remove from $\mathscr {B}$ the $\sqrt {\varepsilon }$ -neighborhood of $\varkappa^{\prime }$ and repeat the procedure, etc. As a result, we find all the approximations of the solutions to (1.4) in $\mathscr {B}$ . For each approximation $\varkappa^{\prime }$ , we construct the approximate boundary data $\kappa ^{\prime }:=(B^{\prime }_\nu (\varkappa^{\prime }),f^{\prime }(\varkappa^{\prime }))$ via formula (3.14) with $\alpha _k,\beta _k,\eta _k,\mu _k$ replaced by $\alpha ^{\prime }_k,\beta ^{\prime }_k,\eta ^{\prime }_k,\mu ^{\prime }_k$ , respectively. As a result, we obtain approximations $\kappa ^{\prime }$ of all boundary data $\kappa $ (with parameters $\varkappa$ in $\mathscr {B}$ ) obeying

(3.22) $$ \begin{align} \|\kappa^{\prime}-\kappa\|_{L^{\mathbb{C}}_2(\Gamma;dl)\times H^{1}(\Gamma;\mathbb{C})}=O(\varepsilon). \end{align} $$

Since the radius of $\mathscr {B}$ is unknown, we actually start with some ball $\mathscr {B}^{\prime }$ , then enlarge it and repeat the above procedure, etc., until we obtain the sufficiently large number of solutions $\kappa ^{\prime }$ to successfully perform the next step (finding the approximation of the boundary data of some canonical dual basis).

Step 3 (implementation)

Let us find a collection of approximations $\kappa ^{\prime }_1,\dots ,\kappa ^{\prime }_{2\mathfrak {g}}$ obeying

$$ \begin{align*}|{\boldsymbol{\langle}}\kappa^{\prime}_i,\kappa^{\prime}_j{\boldsymbol{\rangle}}^{\prime}-(\Omega_{\mathfrak{g}})_{ij}|<\sqrt{\varepsilon}.\end{align*} $$

In view of (3.22) and (3.20), the corresponding exact boundary data $\kappa _1,\dots ,\kappa _{2\mathfrak {g}}$ obey condition (3.16) up to the discrepancy $O(\sqrt {\varepsilon })$ . Since the left-hand side of (3.16) is integer, this means that $\kappa _1,\dots ,\kappa _{2\mathfrak {g}}$ constitute the boundary data of some canonical dual basis.