Proof. Let $\dim W=2n+1\geqslant 3$ . By Darboux’s theorem (see Theorem A.2), every point of $W$ has a neighborhood $U\subset W$ and holomorphic coordinates $\unicode[STIX]{x1D6F7}=(x_{1},y_{1},\ldots ,x_{n},y_{n},z)$ on $U$ such that $\mathscr{L}|_{U}=\ker \unicode[STIX]{x1D6F7}^{\ast }(\unicode[STIX]{x1D702})$ , where $\unicode[STIX]{x1D702}$ is the standard contact form given by (1.1). Let $V\Subset U$ be a relatively compact domain. Theorem 1.2 provides a continuous injective map $F_{0}:M\rightarrow \unicode[STIX]{x1D6F7}(V)\subset \mathbb{C}^{2n+1}$ whose restriction to ${M\unicode[STIX]{x0030A}}$ is a holomorphic $\unicode[STIX]{x1D702}$ -Legendrian embedding which is complete with respect to the Euclidean metric $g_{0}:=|dz|^{2}+\sum _{j=1}^{n}(|dx_{j}|^{2}+|\,dy_{j}|^{2})$ on $\mathbb{C}^{2n+1}$ . As a consequence, $F:=\unicode[STIX]{x1D6F7}^{-1}\circ F_{0}:M\rightarrow W$ is $\mathscr{L}$ -Legendrian. Further, since $\overline{V}$ is compact, the restriction to $V$ of any Riemannian metric $g$ on $W$ is comparable to $\unicode[STIX]{x1D6F7}^{\ast }(g_{0}|_{\unicode[STIX]{x1D6F7}(V)})$ , and hence, $F|_{{M\unicode[STIX]{x0030A}}}$ is complete with respect to any such $g$ . This completes the proof.◻

Proof. We have $(\widetilde{F}^{\ast }\unicode[STIX]{x1D702})(\unicode[STIX]{x1D701})=z^{\prime }(\unicode[STIX]{x1D701})\,d\unicode[STIX]{x1D701}-(F^{\ast }\unicode[STIX]{x1D702})(\unicode[STIX]{x1D701})+\sum _{j=1}^{n}x_{j}(\unicode[STIX]{x1D701})y_{j}^{\prime }(\unicode[STIX]{x1D701})\,d\unicode[STIX]{x1D701}=0.$ ◻

Proof. For simplicity of notation we shall consider the case $n=1$ ; the same proof will apply also for $n>1$ .

Since we are looking for Legendrian disks $G:\overline{\mathbb{D}}\rightarrow \mathbb{C}^{3}$ satisfying certain approximate conditions in relationship to $f$ and $F$ , we may assume by approximation that all our holomorphic Legendrian disks are defined on a fixed open neighborhood of $\overline{\mathbb{D}}$ . Indeed, for $f$ just use Mergelyan’s approximation for $x$ and $y$ and define $z$ in a neighborhood of $\overline{\mathbb{D}}$ accordingly; for the boundary disks proceed likewise but using the parametric version of Megelyan’s theorem. For $(u,v)\in \mathbb{T}\times \overline{\mathbb{D}}$ we have

where the coefficients $a_{j}$ , $b_{k}$ , and $c_{n}$ are continuous functions of $u\in \mathbb{T}$ and we have

The Legendrian condition for the map $v\mapsto F(u,v)$ (with a fixed $u\in \mathbb{T}$ ) says that $Z_{v}+XY_{v}=0$ , where the subscript denotes the partial derivative with respect to the indicated variable. From the power series expansions for $X$ and $Y$ we obtain

Comparison with

gives the equations

By approximation we may assume that there are only finitely many nonzero coefficients $a_{j}$ , $b_{k}$ and hence $c_{n}$ , i.e., the Legendrian curves $v\mapsto F(u,v)$ are polynomial in $v\in \mathbb{C}$ of bounded degree independent of $u\in \mathbb{T}$ . Furthermore, we may approximate each of the coefficients $a_{j}$ and $b_{k}$ (which are continuous functions on $\mathbb{T}$ ) by a rational function with the only pole at $0$ . In view of the above formulas for the coefficients $c_{n}$ of $Z$ these also become rational functions on $\mathbb{C}$ with the only pole at $0$ . We denote the resulting functions and maps by the same letters. Note that this gives a family of polynomial Legendrian curves $F(u,\cdot )=(X(u,\cdot ),Y(u,\cdot ),Z(u,\cdot )):\mathbb{C}\rightarrow \mathbb{C}^{3}$ for $u\in \mathbb{C}\setminus \{0\}$ which are Laurent polynomials in the variable $u$ . In particular, we have that

Let $N_{0}$ be the biggest degree of pole of any of these coefficients at $0$ . For any $N\in \mathbb{N}$ with $N>N_{0}$ the map $F_{N}:\mathbb{C}\rightarrow \mathbb{C}^{3}$ given by

is a holomorphic polynomial with $F_{N}(0)=f(0)$ . It is well known that for sufficiently big $N\in \mathbb{N}$ the map $F_{N}$ satisfies properties (i)–(iv) in the lemma (see e.g. [Reference Drinovec Drnovšek and ForstneričDF12, Lemma 3.1]).

Although the maps $F_{N}$ obtained in this way need not be Legendrian, we shall now show that for all sufficiently big $N\in \mathbb{N}$ the map $F_{N}$ is as close as desired uniformly on $\overline{\mathbb{D}}$ to a Legendrian disk $G=G_{N}=(X_{N},Y_{N},\widetilde{Z}_{N}):\overline{\mathbb{D}}\rightarrow \mathbb{C}^{3}$ of the form given by Lemma 3.1; this will complete the proof. To this end we estimate the expression

We have that

By adding these two equations and taking into account the condition $Z_{v}+XY_{v}=0$ for the Legendrian disk $v\mapsto F(u,v)$ we obtain

From the power series expansions of $F=(X,Y,Z)$ we get

The term with $n=0$ in the above sum equals $c_{0}^{\prime }+a_{0}b_{0}^{\prime }=z^{\prime }+xy^{\prime }=0$ since the curve $f=(x,y,z)$ is Legendrian, and hence it drops out from the sum.

Recall that each of the coefficients $a_{j}$ , $b_{k}$ and $c_{n}$ is Laurent polynomial of the form $P(u,1/u)$ where $P$ is a holomorphic polynomial on $\mathbb{C}^{2}$ . The same is then true for their derivatives, and hence for the coefficients $c_{n}^{\prime }+\sum _{j+k=n}a_{j}b_{k}^{\prime }$ in the above expansion of $Z_{u}+XY_{u}$ . Let $N_{1}$ be the maximal power of $1/u$ that appears in any of these finitely many coefficients. If $N\geqslant N_{1}$ then the function $(Z_{u}+XY_{u})(u,u^{N})$ is a polynomial in $u$ whose lowest order term is $u^{N-N_{1}}$ or higher. Note that $\int _{0}^{\unicode[STIX]{x1D701}}u^{N-N_{1}}\,du=\unicode[STIX]{x1D701}^{N-N_{1}+1}/(N-N_{1}+1)$ which converges to zero uniformly on the disk $|\unicode[STIX]{x1D701}|\leqslant 1$ when $N\rightarrow \infty$ . Since we have finitely many such terms in the sum, it follows that the integral

converges to zero uniformly on $\overline{\mathbb{D}}$ as $N\rightarrow +\infty$ . Furthermore, if the last assumption in the lemma holds then we can perform the same construction on a somewhat bigger compact simply connected domain $D\subset \mathbb{C}$ containing the disk $\overline{\mathbb{D}}$ such that $I\subset bD$ and $\overline{\mathbb{D}}\setminus U\subset {D\unicode[STIX]{x0030A}}$ ; the $\mathscr{C}^{0}$ -estimate on $D\cong \overline{\mathbb{D}}$ then yields $\mathscr{C}^{1}$ estimate on $\overline{\mathbb{D}}\setminus U$ in view of the Cauchy estimates. Finally, setting $\widetilde{Z}_{N}(\unicode[STIX]{x1D701})=Z_{N}(\unicode[STIX]{x1D701})-\int _{0}^{\unicode[STIX]{x1D701}}F_{N}^{\ast }\unicode[STIX]{x1D702}$ (cf. Lemma 3.1) gives a sequence of holomorphic Legendrian maps $G_{N}=(X_{N},Y_{N},\widetilde{Z}_{N}):\mathbb{C}\rightarrow \mathbb{C}^{3}$ satisfying the conclusion of Lemma 3.2.◻

Proof. For simplicity of notation we consider the case $n=1$ ; the same proof applies in general by considering $x$ and $y$ as vector-valued functions and writing $x\,dy=\sum _{i=1}^{n}x_{i}\,dy_{i}$ .

We may assume that $M$ is connected. Choose a smoothly bounded simply connected domain $D\subset U$ (a disk) such that $D$ is a neighborhood of the closed arc $I$ . By denting $bM$ slightly inward along a neighborhood of $I$ we can find a smoothly bounded compact domain $M^{\prime }\subset M$ such that $M=M^{\prime }\cup D$ and the following separation condition holds:

Thus, $(M^{\prime },D)$ is a Cartan pair (cf. [Reference ForstneričFor11, Definition 5.7.1]).

Let $C_{1},\ldots ,C_{\ell }\subset {M\unicode[STIX]{x0030A}}^{\prime }$ be closed curves forming a basis of the homology group $H_{1}(M^{\prime };\mathbb{Z})\cong H_{1}(M;\mathbb{Z})=\mathbb{Z}^{\ell }$ such that the union $\bigcup _{j=1}^{\ell }C_{j}$ is Runge in $M$ . Consider the period map

whose $j\text{th}$ component equals

Note that ${\mathcal{P}}(x,y)=0$ if and only if the 1-form $x\,dy$ is exact, and this holds if and only if $(x,y)$ is the Lagrange projection of a Legendrian curve $f=(x,y,z):M\rightarrow \mathbb{C}^{3}$ .

We shall first assume that the second component $y$ of $f$ is not constant; then $y|_{C_{j}}$ is not constant for any $j=1,\ldots ,\ell$ by the identity principle. By the Runge property of $\bigcup _{j=1}^{\ell }C_{j}$ there exist holomorphic functions $g_{1},\ldots ,g_{\ell }$ on $M$ such that for every $j,k=1,\ldots ,\ell$ the number $\int _{C_{j}}g_{k}\,dy\approx \unicode[STIX]{x1D6FF}_{j,k}$ is close to $1$ if $j=k$ and to $0$ if $j\neq k$ . (Here, $\unicode[STIX]{x1D6FF}_{j,k}$ is the Kronecker symbol. We first construct smooth functions $g_{k}$ on $\bigcup _{j=1}^{\ell }C_{j}$ such that $\int _{C_{j}}g_{k}\,dy=\unicode[STIX]{x1D6FF}_{j,k}$ and then use Mergelyan’s theorem to approximate them by holomorphic functions on $M$ .) Let $\unicode[STIX]{x1D701}=(\unicode[STIX]{x1D701}_{1},\ldots ,\unicode[STIX]{x1D701}_{\ell })\in \mathbb{C}^{\ell }$ . Consider the function $\widetilde{x}:M\times \mathbb{C}^{\ell }\rightarrow \mathbb{C}$ given by

Note that for all $j,k\in \{1,\ldots ,\ell \}$ we have

If the above approximations are close enough then

If this holds, then the map (3.5) is called a period dominating holomorphic spray with the core $\widetilde{x}(\cdot ,0)=x$ .

Assume now that $(x,y)\in \mathscr{A}^{1}(M)^{2}$ is the Lagrange projection of the given Legendrian map $f=(x,y,z):M\rightarrow \mathbb{C}^{3}$ ; hence ${\mathcal{P}}(x,y)=0$ . By the inverse function theorem there is a ball $r\mathbb{B}\subset \mathbb{C}^{\ell }$ around the origin such that the map $r\mathbb{B}\ni \unicode[STIX]{x1D701}\mapsto {\mathcal{P}}(\widetilde{x}(\cdot ,\unicode[STIX]{x1D701}),y)\in \mathbb{C}^{\ell }$ is biholomorphic onto its image (a neighborhood of $0\in \mathbb{C}^{\ell }$ ). Fix a point $u_{0}\in D$ and consider the function $\widetilde{z}:D\times \mathbb{C}^{\ell }\rightarrow \mathbb{C}$ of class $\mathscr{A}^{1}(D\times \mathbb{C}^{\ell })$ given by

Recall that $z$ is the third component of the Legendrian map $f=(x,y,z):M\rightarrow \mathbb{C}^{3}$ and the integral is taken over any path in the disk $D$ . Note that $\widetilde{z}(\cdot ,0)=z|_{D}$ since $\widetilde{x}(\cdot ,0)=x$ and $dz=-x\,dy$ . Let $\widetilde{f}:D\times \mathbb{C}^{\ell }\rightarrow \mathbb{C}^{3}$ be the family of Legendrian disks

depending holomorphically on $\unicode[STIX]{x1D701}\in \mathbb{C}^{\ell }$ . (The second component $y$ is independent of the parameter  $\unicode[STIX]{x1D701}$ .) Note that $\widetilde{f}(u,0)=f(u)$ for $u\in D$ .

For each point $u\in bD\cap bM$ and for every $\unicode[STIX]{x1D701}\in \mathbb{C}^{\ell }$ we let

be the Legendrian disk of class $\mathscr{A}^{1}(\mathbb{D})$ given by

When $\unicode[STIX]{x1D701}=0$ , we have $\widetilde{X}(u,v,0)=X(u,v)$ , $\widetilde{Y}(u,v,0)=Y(u,v)$ and hence

so we see that $\widetilde{F}(u,v,0)=F(u,v)$ is the given Legendrian disk in the theorem. Furthermore, setting $v=0$ we have

Finally, for every point $u\in bD\cap bM\setminus I$ and for all $\unicode[STIX]{x1D701}\in \mathbb{C}^{\ell }$ we have

so $\widetilde{F}(u,\cdot ,\unicode[STIX]{x1D701})$ is the constant disk. We extend $\widetilde{F}$ to all points $u\in bD$ by setting

Note that $\widetilde{f}(\cdot ,\unicode[STIX]{x1D701}):D\rightarrow \mathbb{C}^{3}$ and $\widetilde{F}(u,\cdot ,\unicode[STIX]{x1D701}):\overline{\mathbb{D}}\rightarrow \mathbb{C}^{3}$ are families of holomorphic Legendrian maps, depending continuously on $u\in bD$ (this only pertains to $\widetilde{F}$ ) and holomorphically on $\unicode[STIX]{x1D701}\in \mathbb{C}^{\ell }$ , which satisfy the assumptions of Lemma 3.2 on the disk $D\cong \overline{\mathbb{D}}$ . Hence there is a family of Legendrian disks $\widetilde{G}(\cdot ,\unicode[STIX]{x1D701}):D\rightarrow \mathbb{C}^{3}$ satisfying the approximation conditions in Lemma 3.2 with respect to the central Legendrian disk $\widetilde{f}(\cdot ,\unicode[STIX]{x1D701})$ and the family of Legendrian disks $\widetilde{F}(u,\cdot ,\unicode[STIX]{x1D701})$ over the boundary point $u\in bD$ . It is easily seen from the proof of Lemma 3.2 that the family $\widetilde{G}(\cdot ,\unicode[STIX]{x1D701})$ may be chosen to depend holomorphically on $\unicode[STIX]{x1D701}\in \mathbb{C}^{\ell }$ , and the estimates in the lemma can be made uniform for all points $\unicode[STIX]{x1D701}$ in any given bounded domain in $\mathbb{C}^{\ell }$ , in particular, on the ball $r\mathbb{B}\subset \mathbb{C}^{\ell }$ .

Let $V\subset D\setminus M^{\prime }$ be a small neighborhood of the arc $I\subset bD$ . By condition (iv) in Lemma 3.2 we may assume that $\widetilde{G}(\cdot ,\unicode[STIX]{x1D701})$ is as close as desired to $\widetilde{f}(\cdot ,\unicode[STIX]{x1D701})$ in the $\mathscr{C}^{1}$ norm on the set $D\setminus V$ , and hence on $M^{\prime }\cap D\subset D\setminus V$ . In particular, given $\unicode[STIX]{x1D6FF}>0$ we may assume that

We shall write $\widetilde{G}=(\widetilde{G}_{1},\widetilde{G}_{2},\widetilde{G}_{3})$ .

Recall that the first component of $\widetilde{f}$ (see (3.8)) is the function $\widetilde{x}$ (cf. (3.5)) which is defined on all of $M$ . By solving a Cousin-I problem with bounds on the Cartan pair $(M^{\prime },D)$ we glue $\widetilde{x}$ and $\widetilde{G}_{1}$ and obtain a function $H_{1}(\cdot ,\unicode[STIX]{x1D701}):M\rightarrow \mathbb{C}$ of class $\mathscr{A}^{1}(M)$ , holomorphic in $\unicode[STIX]{x1D701}$ , such that for all $\unicode[STIX]{x1D701}\in r\mathbb{B}$ we have

where the constant $C$ only depends on the Cartan pair $(M^{\prime },D)$ . This is accomplished by first patching the two functions smoothly, using the separation condition (3.3), and then correcting the $\mathscr{C}^{1}$ -small error by solving the $\overline{\unicode[STIX]{x2202}}$ -equation with $\mathscr{C}^{1}$ -estimates on the bordered Riemann surface $M$ . The variable $\unicode[STIX]{x1D701}$ is treated as a parameter. This is standard; see e.g. [Reference ForstneričFor11, proof of Lemma 8.5.2] and use $\mathscr{C}^{1}$ estimates instead of $\mathscr{C}^{0}$ estimates.

Likewise, we can glue the second component $\widetilde{y}=y\in \mathscr{A}^{1}(M)$ of $\widetilde{f}$ with the function $\widetilde{G}_{2}(\cdot ,\unicode[STIX]{x1D701})$ into a function $H_{2}(\cdot ,\unicode[STIX]{x1D701}):M\rightarrow \mathbb{C}$ of class $\mathscr{A}^{1}(M)$ , holomorphic in $\unicode[STIX]{x1D701}$ , such that for all $\unicode[STIX]{x1D701}\in r\mathbb{B}$ we have the estimates

From the above estimates on $M^{\prime }$ and the fact that $\bigcup _{j=1}^{\ell }C_{j}\subset M^{\prime }$ it follows that the period map $\unicode[STIX]{x1D701}\mapsto {\mathcal{P}}(H_{1}(\cdot ,\unicode[STIX]{x1D701}),H_{2}(\cdot ,\unicode[STIX]{x1D701}))$ (cf. (3.4)) approximates the biholomorphic period map $\unicode[STIX]{x1D701}\mapsto {\mathcal{P}}(\widetilde{x}(\cdot ,\unicode[STIX]{x1D701}),\widetilde{y}(\cdot ,\unicode[STIX]{x1D701}))$ uniformly on the ball $\unicode[STIX]{x1D701}\in r\mathbb{B}$ . Assuming that $\unicode[STIX]{x1D6FF}>0$ is chosen small enough, it follows that there is a point $\unicode[STIX]{x1D701}^{\prime }\in r\mathbb{B}$ near the origin such that

For this value of $\unicode[STIX]{x1D701}^{\prime }$ we obtain a Legendrian curve

whose third component equals

The integral is independent of the choice of the path in $M$ since by (3.9) all the periods over closed curves in $M$ vanish. (Note however that we do not get a Legendrian curve for parameter values $\unicode[STIX]{x1D701}\neq \unicode[STIX]{x1D701}^{\prime }$ since the period condition (3.9) fails.) It follows from the construction that $H$ satisfies the conclusion of Theorem 3.3 provided that the approximations made in the proof were close enough. This completes the proof under the assumption that the second component $y$ of $f$ is nonconstant.

Assume now that $y=y_{0}$ is constant. If the first component $x$ is nonconstant, we consider a spray of the form (3.5) over the second component:

where the functions $g_{1},\ldots ,g_{\ell }\in \mathscr{O}(M)$ are chosen such that

This ensures that the period map $\unicode[STIX]{x1D701}\mapsto {\mathcal{P}}(x,\widetilde{y}(\cdot ,\unicode[STIX]{x1D701}))$ has maximal rank at $\unicode[STIX]{x1D701}=0$ , so we can proceed as before, keeping the component $x$ fixed during the proof.

Finally, if $x=x_{0}$ and $y=y_{0}$ are both constant, then any perturbation of either $x$ or $y$ integrates to a Legendrian curve which brings us back to the second case considered above. Of course we must also adjust the Legendrian disks $F(u,\cdot )$ $(u\in bM)$ accordingly so that the condition $F(u,0)=f(u)$ is satisfied.◻

Proof. Since $S$ is a deformation retract of $R$ , we have that $R$ is of finite topological type and $S$ is connected and Runge in $R$ . Without loss of generality, we can assume that $K\neq \emptyset$ . Indeed, otherwise $S=\unicode[STIX]{x1D6E4}$ consists of a single closed curve or Jordan arc. Then we choose a small smoothly bounded close disk $K$ in $R$ such that $S^{\prime }=K\cup \unicode[STIX]{x1D6E4}$ is admissible and connected, and $K\cap S$ is a single Jordan arc. Approximating $f$ by a generalized Legendrian curve $S^{\prime }\rightarrow \mathbb{C}^{2n+1}$ reduces the proof to the case when $K\neq \emptyset$ .

Assume as we may that $K\neq \emptyset$ . It follows that every component of $\unicode[STIX]{x1D6E4}$ (hence of $S$ ) meets  $K$ . Let $C_{1},\ldots ,C_{\ell }\subset S$ be closed curves forming a basis of the homology group $H_{1}(S;\mathbb{Z})\,=\,\mathbb{Z}^{\ell }$ $(\ell \in \mathbb{Z}_{+})$ such that the union $\bigcup _{k=1}^{\ell }C_{k}$ is Runge in $R$ . By our assumptions on $S$ , we may ensure that each curve $C_{k}$ , $k=1,\ldots ,\ell$ , contains a subarc $\widetilde{C}_{k}$ lying in ${K\unicode[STIX]{x0030A}}$ .

The argument at the end of the proof of Theorem 3.3 and Remark 2.1 enable us to assume without loss of generality that $y_{1}$ is not constant on any component of $K$ . Let

be the period map (3.4) with the components

Let us construct a spray $\widetilde{x}_{1}(\cdot ,\unicode[STIX]{x1D701}):S\rightarrow \mathbb{C}$ of the form

(cf. (3.5)) with the core $\widetilde{x}_{1}(\cdot ,0)=x_{1}$ (the first component of $f$ ), depending holomorphically on $\unicode[STIX]{x1D701}\in \mathbb{C}^{\ell }$ , such that the $\unicode[STIX]{x1D701}$ -derivative of the period map ${\mathcal{P}}(\widetilde{x}_{1}(\cdot ,\unicode[STIX]{x1D701}),y_{1},\ldots ,x_{n},y_{n})$ at $\unicode[STIX]{x1D701}=0$ is an isomorphism (cf. (3.7)). We proceed as follows. Since $y_{1}$ is holomorphic and nonconstant on each component of $K$ , we may first construct smooth functions $g_{k}$ on $C$ , with support on $\widetilde{C}_{k}\subset {K\unicode[STIX]{x0030A}}$ , defining the spray $\widetilde{x}_{1}(\cdot ,\unicode[STIX]{x1D701})$ on $C$ and then, by Mergelyan theorem, assume that each $g_{k}$ is holomorphic on $R$ (recall that $C$ is Runge in $R$ ). This ensures the existence of the desired spray on $S$ , the domain of definition of $x_{1}$ .

Next, we approximate $(x_{1},y_{1},\ldots ,x_{n},y_{n})\in \mathscr{A}^{1}(S)^{2n}$ in the $\mathscr{C}^{1}(S)$ -norm by a holomorphic map $(x_{1}^{\prime },y_{1}^{\prime },\ldots ,x_{n}^{\prime },y_{n}^{\prime })\in \mathscr{O}(R)^{2n}$ such that $\sum _{j=1}^{n}x_{j}^{\prime }\,dy_{j}^{\prime }$ does not vanish everywhere on $R$ and all the component functions $x_{1}^{\prime },y_{1}^{\prime },\ldots ,x_{n}^{\prime },y_{n}^{\prime }$ are nonconstant. For that, recall that $S$ is Runge in $R$ and apply Mergelyan approximation with jet-interpolation. Let $\widetilde{x}_{1}^{\prime }:R\times \mathbb{C}^{\ell }\rightarrow \mathbb{C}$ be the spray (4.3) obtained by replacing the core $x_{1}$ by $x_{1}^{\prime }$ :

If the approximations are close enough, then the period map ${\mathcal{P}}(\widetilde{x}_{1}^{\prime }(\cdot ,\unicode[STIX]{x1D701}),y_{1}^{\prime },\ldots ,x_{n}^{\prime },y_{n}^{\prime })$ is so close to ${\mathcal{P}}(\widetilde{x}_{1}(\cdot ,\unicode[STIX]{x1D701}),y_{1},\ldots ,x_{n},y_{n})$ that there is a point $\unicode[STIX]{x1D701}^{\prime }\in \mathbb{C}^{\ell }$ close to $0$ for which

This means that $\widetilde{x}_{1}^{\prime }(\cdot ,\unicode[STIX]{x1D701}^{\prime })\,dy_{1}^{\prime }+\sum _{j=2}^{n}x_{j}^{\prime }\,dy_{j}^{\prime }$ is an exact holomorphic $1$ -form on $R$ . Pick an initial point $u_{0}\in {K\unicode[STIX]{x0030A}}$ and define the holomorphic function $z^{\prime }\in \mathscr{O}(R)$ by

Clearly, the map $\widetilde{f}=(\widetilde{x}_{1}^{\prime }(\cdot ,\unicode[STIX]{x1D701}^{\prime }),y_{1}^{\prime },\ldots ,x_{n}^{\prime },y_{n}^{\prime },z^{\prime }):R\rightarrow \mathbb{C}^{2n+1}$ is then a holomorphic Legendrian curve approximating $f$ in the $\mathscr{C}^{1}(S)$ -topology and having no constant component function provided that $\unicode[STIX]{x1D701}^{\prime }$ is close enough to $0\in \mathbb{C}^{\ell }$ . This proves the first part of the lemma.

For the second part, let $\unicode[STIX]{x1D70E}\in \{1,\ldots ,2n+1\}$ and assume that conditions (i) and (ii) hold. By Remark 2.1 we may assume that $\unicode[STIX]{x1D70E}\in \{2,2n+1\}$ .

Case 1: Assume that  $\unicode[STIX]{x1D70E}=2$ . In this case, the argument above works with the only difference that when approximating $(x_{1},y_{1},\ldots ,x_{n},y_{n})\in \mathscr{A}^{1}(S)^{2n}$ in the $\mathscr{C}^{1}(S)$ -norm by a holomorphic map $(x_{1}^{\prime },y_{1}^{\prime },\ldots ,x_{n}^{\prime },y_{n}^{\prime })\in \mathscr{O}(R)^{2n}$ , we choose $y_{1}^{\prime }=y_{1}$ . Take into account that, in this case, $y_{1}$ is holomorphic and nonconstant on $R$ by assumption (i), and so neither the argument at the end of the proof of Theorem 3.3 nor Remark 2.1 are required.

Case 2: Assume that  $\unicode[STIX]{x1D70E}=2n+1$ . By condition (ii) and Remark 2.1 we may assume that $x_{1}$ and $y_{1}$ are not constant on any component of $K$ , $x_{1}$ has no zeros in $\unicode[STIX]{x1D6E4}$ , and $y_{1}$ has no critical points in $\unicode[STIX]{x1D6E4}$ . Denote by ${\mho}$ the subset of $\mathscr{A}^{1}(S)^{2n}$ consisting of those maps $(a_{1},a_{2},b_{2},\ldots ,a_{n},b_{n},w):S\rightarrow \mathbb{C}^{2n}$ (note that $b_{1}$ is omitted) such that, for some (and hence for any) holomorphic $1$ -form $\unicode[STIX]{x1D703}$ vanishing nowhere on $R$ , the map

is well defined and of class $\mathscr{A}^{0}(S)$ (see (4.1) and (4.2)). The fact that $f$ is a generalized Legendrian curve implies that $(x_{1},x_{2},y_{2},\ldots ,x_{n},y_{n},z)\in {\mho}$ . Consider the period map

with the components

We shall now construct a spray $\widetilde{x}_{1}(\cdot ,\unicode[STIX]{x1D701}):S\rightarrow \mathbb{C}$ of the form

where $g_{l}\in \mathscr{O}(R)$ for all $l\in \{1,\ldots ,\ell \}$ , with the core $\widetilde{x}_{1}(\cdot ,0)=x_{1}$ , depending holomorphically on $\unicode[STIX]{x1D701}\in \mathbb{C}^{\ell }$ , such that the $\unicode[STIX]{x1D701}$ -derivative of the period map ${\mathcal{P}}(\widetilde{x}_{1}(\cdot ,\unicode[STIX]{x1D701}),x_{2},y_{2},\ldots ,x_{n},y_{n},z)$ at $\unicode[STIX]{x1D701}=0$ is an isomorphism. (Note that $(\widetilde{x}_{1}(\cdot ,\unicode[STIX]{x1D701}),x_{2},y_{2},\ldots ,x_{n},y_{n},z)\in {\mho}$ for all $\unicode[STIX]{x1D701}\in \mathbb{C}^{\ell }$ since $(x_{1},x_{2},y_{2},\ldots ,x_{n},y_{n},z)\in {\mho}$ and $e^{\sum _{l=1}^{\ell }\unicode[STIX]{x1D701}_{l}g_{l}}$ has no zeros.) We proceed as follows. Note that, for such a spray,

As above, since $\widetilde{C}_{k}\subset {K\unicode[STIX]{x0030A}}$ and $y_{1}$ is holomorphic and nonconstant on every component of $K$ , we may first construct smooth functions $g_{l}$ on $C$ , with support on $\widetilde{C}_{k}$ , defining the spray $\widetilde{x}_{1}(\cdot ,\unicode[STIX]{x1D701})$ on $C$ , such that

(Recall that $\unicode[STIX]{x1D6FF}_{l,k}$ is the Kronecker symbol.) By the Mergelyan theorem, we may assume that each $g_{l}$ is holomorphic on $R$ . As above, this guarantees the existence of the desired spray $\widetilde{x}_{1}(\cdot ,\unicode[STIX]{x1D701})$ on $S$ . We next approximate $(x_{1},x_{2},y_{2},\ldots ,x_{n},y_{n})\in \mathscr{A}^{1}(S)^{2n-1}$ in the $\mathscr{C}^{1}(S)$ -norm by a holomorphic map $(x_{1}^{\prime },x_{2}^{\prime },y_{2}^{\prime },\ldots ,x_{n}^{\prime },y_{n}^{\prime })\in \mathscr{O}(R)^{2n-1}$ such that the following hold.

1. (a) Each component function $x_{1}^{\prime },x_{2}^{\prime },y_{2}^{\prime },\ldots ,x_{n}^{\prime },y_{n}^{\prime }$ is nonconstant.

2. (b) The zeros of $x_{1}^{\prime }$ in $R$ are those of $x_{1}$ in $S$ (which lie in $K$ , see assumption (ii)), with the same order; in particular, $x_{1}^{\prime }$ has no zeros in $R\setminus K$ .

3. (c) The $1$ -form $dz+\sum _{j=2}^{n}x_{j}^{\prime }\,dy_{j}^{\prime }$ does not vanish identically on $R$ , but it does vanish at the zeros of $dz+\sum _{j=2}^{n}x_{j}\,dy_{j}=-x_{1}\,dy_{1}$ in $K$ , with the same order. (Observe that, by assumption (ii), $x_{1}\,dy_{1}$ does not vanish identically on $K$ and has no zeros in $\unicode[STIX]{x1D6E4}$ .)

To ensure these conditions we make use of Mergelyan approximation with jet interpolation.

Let $\widetilde{x}_{1}^{\prime }:R\times \mathbb{C}^{\ell }\rightarrow \mathbb{C}$ be the spray (4.5) obtained by replacing the core $x_{1}$ by $x_{1}^{\prime }$ :

Since $(x_{1},x_{2},y_{2},\ldots ,x_{n},y_{n},z)\in {\mho}$ , conditions (b) and (c) above guarantee that

As above, if the approximations are close enough, there is a point $\unicode[STIX]{x1D701}^{\prime }\in \mathbb{C}^{\ell }$ close to $0$ for which ${\mathcal{P}}(\widetilde{x}_{1}^{\prime }(\cdot ,\unicode[STIX]{x1D701}^{\prime }),x_{2}^{\prime },y_{2}^{\prime },\ldots ,x_{n}^{\prime },y_{n}^{\prime },z)=0$ . Thus, choosing an initial point $u_{0}\in {K\unicode[STIX]{x0030A}}$ and defining

it follows from (4.7) that $y_{1}^{\prime }:R\rightarrow \mathbb{C}$ is holomorphic and, in view of conditions (a) and (c), the map

is a holomorphic Legendrian curve satisfying the conclusion of the lemma, provided that the approximations are sufficiently close. ◻

Proof. We assume that $n=1$ ; the same proof applies in general. We may also assume without loss of generality that $M$ is connected.

Let $R$ be an open Riemann surface containing $M$ as a smoothly bounded compact domain which is a deformation retract of $R$ . By Lemma 4.3 we may assume that $f$ is a holomorphic Legendrian curve on $R$ having no constant component function. Write $f=(x,y,z):R\rightarrow \mathbb{C}^{3}$ .

The proof will proceed in two steps. In the first step we shall approximate $f$ in the $\mathscr{C}^{1}(M)$ -topology by a Legendrian immersion $R\rightarrow \mathbb{C}^{3}$ whose $(x,y)$ -projection is an immersion $R\rightarrow \mathbb{C}^{2}$ . In the second step we shall show how to remove double points on $M$ of the new Legendrian immersion and hence obtain a Legendrian embedding $M{\hookrightarrow}\mathbb{C}^{3}$ . If the approximation in both steps is close enough then the resulting Legendrian embedding will have no constant component function.

Let $C_{1},\ldots ,C_{\ell }\subset {M\unicode[STIX]{x0030A}}$ be closed curves forming a basis of the homology group $H_{1}(M;\mathbb{Z})=\mathbb{Z}^{\ell }$ such that the union $\bigcup _{k=1}^{\ell }C_{k}$ is Runge in ${M\unicode[STIX]{x0030A}}$ , hence in $R$ . Let

be the period map (3.4) with the components

For the first step of the proof, let $u_{1},u_{2},\ldots$ be the zeros of $dy$ in $R$ . We shall approximate $x$ in the $\mathscr{C}^{1}(M)$ -norm by a function $x^{\prime }\in \mathscr{O}(R)$ such that

The first condition ensures that the map $(x^{\prime },y):R\rightarrow \mathbb{C}^{2}$ is an immersion, and the second one that $x^{\prime }\,dy$ is an exact $1$ -form on $R$ . Choosing an initial point $u_{0}\in M$ and setting

we shall obtain a desired Legendrian immersion $f^{\prime }=(x^{\prime },y,z^{\prime }):R\rightarrow \mathbb{C}^{3}$ .

We now explain how to find the function $x^{\prime }\in \mathscr{O}(R)$ satisfying (4.8). Choose a function $h\in \mathscr{O}(R)$ such that for every $j=1,2,\ldots$ we have $dh(u_{j})=0$ if $dx(u_{j})\neq 0$ and $dh(u_{j})\neq 0$ if $dx(u_{j})=0$ ; the existence of such a function is ensured, for instance, by the theorem of Gunning and Narasimhan (see [Reference Gunning and NarasimhanGN67]). Then, for every $\unicode[STIX]{x1D6FF}\in \mathbb{C}\setminus \{0\}$ the function $x_{\unicode[STIX]{x1D6FF}}:=x+\unicode[STIX]{x1D6FF}h\in \mathscr{O}(R)$ satisfies

Hence, $(x_{\unicode[STIX]{x1D6FF}},y):R\rightarrow \mathbb{C}^{2}$ is an immersion. We need to correct $x_{\unicode[STIX]{x1D6FF}}$ in order to achieve the period vanishing condition in (4.8). To this end, choose the curves $C_{1},\ldots ,C_{\ell }$ above such that $C:=\bigcup _{k=1}^{\ell }C_{k}$ does not contain any of the critical points $u_{1},u_{2},\ldots$ of $y$ . Let $\widetilde{x}(\cdot ,\unicode[STIX]{x1D701}):R\rightarrow \mathbb{C}(\unicode[STIX]{x1D701}\in \mathbb{C}^{\ell })$ be the spray (3.5) with the core $\widetilde{x}(\cdot ,0)=x$ , where the functions $g_{k}\in \mathscr{O}(R)$ are chosen such that the derivative of the period map ${\mathcal{P}}(\widetilde{x}(\cdot ,\unicode[STIX]{x1D701}),y)$ at $\unicode[STIX]{x1D701}=0$ (see (3.7)) is an isomorphism and $dg_{k}(u_{j})=0$ for all $k=1,2,\ldots ,\ell$ and $j=1,2,\ldots .$ Let $\widetilde{x}_{\unicode[STIX]{x1D6FF}}(\cdot ,\unicode[STIX]{x1D701}):R\rightarrow \mathbb{C}$ be the holomorphic spray

If $x_{\unicode[STIX]{x1D6FF}}$ is sufficiently close to $x$ on the compact set $C$ (which can be achieved by choosing $|\unicode[STIX]{x1D6FF}|>0$ small enough), then the period map ${\mathcal{P}}(\widetilde{x}_{\unicode[STIX]{x1D6FF}}(\cdot ,\unicode[STIX]{x1D701}),y)$ is close to ${\mathcal{P}}(\widetilde{x}(\cdot ,\unicode[STIX]{x1D701}),y)$ . If the approximation is close enough, the implicit function theorem furnishes a point $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FF}}\in \mathbb{C}^{\ell }$ near $0$ such that ${\mathcal{P}}(\widetilde{x}_{\unicode[STIX]{x1D6FF}}(\cdot ,\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FF}}),y)=0$ and $\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FF}}=0$ . Fix such $\unicode[STIX]{x1D6FF}$ and set $x^{\prime }=\widetilde{x}_{\unicode[STIX]{x1D6FF}}(\cdot ,\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FF}})\in \mathscr{O}(R)$ ; hence the $1$ -form $x^{\prime }\,dy$ is exact on $R$ . Since $dg_{k}(u_{j})=0$ , we also have that $dx^{\prime }(u_{j})=dx_{\unicode[STIX]{x1D6FF}}(u_{j})\neq 0$ for all $j=1,2,\ldots$ (cf. (4.10)), so $(x^{\prime },y):R\rightarrow \mathbb{C}^{2}$ is an immersion. Finally, defining the function $z^{\prime }\in \mathscr{O}(R)$ by (4.9) we obtain a Legendrian immersion $f^{\prime }=(x^{\prime },y,z^{\prime }):R\rightarrow \mathbb{C}^{3}$ which approximates $f$ in the $\mathscr{C}^{1}(M)$ -norm. This concludes the first step of the proof.

It remains to show that a holomorphic Legendrian immersion $f:M\rightarrow \mathbb{C}^{3}$ can be approximated in the $\mathscr{C}^{1}(M)$ -norm by holomorphic Legendrian embeddings $\widetilde{f}:M\rightarrow \mathbb{C}^{3}$ . We shall follow the idea in [Reference Alarcón, Forstnerič and LópezAFL16c, proof of Theorem 4.1] where the analogous result was shown for holomorphic null curves in $\mathbb{C}^{n}$ for $n\geqslant 3$ ; a similar idea was also used in [Reference Alarcón and ForstneričAF14, proof of Theorem 2.4] to find conformal minimal embeddings of open Riemann surfaces into $\mathbb{R}^{n}$ for any $n\geqslant 5$ .

To a map $f:M\rightarrow \mathbb{C}^{3}$ we associate the difference map $\unicode[STIX]{x1D6FF}f:M\times M\rightarrow \mathbb{C}^{3}$ defined by

Clearly, $f$ is injective if and only if

Assuming that $f$ is an immersion, there is an open neighborhood $U\subset M\times M$ of the diagonal $D_{M}$ such that $\unicode[STIX]{x1D6FF}f$ does not assume the value $0\in \mathbb{C}^{3}$ on $\overline{U}\setminus D_{M}$ .

Assume that $f:M\rightarrow \mathbb{C}^{3}$ is a holomorphic Legendrian immersion. We shall approximate $f$ in the $\mathscr{C}^{1}(M)$ -norm by a holomorphic Legendrian immersion $\widetilde{f}:M\rightarrow \mathbb{C}^{3}$ whose difference map  $\unicode[STIX]{x1D6FF}\widetilde{f}$ is transverse to the origin $0\in \mathbb{C}^{3}$ on $M\times M\setminus U$ . Since $\dim M\times M=2<3=\dim \mathbb{C}^{3}$ , this will imply that $\unicode[STIX]{x1D6FF}\widetilde{f}$ does not assume the value zero on $M\times M\setminus U$ , so $\widetilde{f}(u)\neq \widetilde{f}(v)$ if $(u,v)\in M\times M\setminus U$ . If on the other hand $(u,v)\in U\setminus D_{M}$ , then $\widetilde{f}(u)\neq \widetilde{f}(v)$ provided that $\widetilde{f}$ is close enough to $f$ ; hence the map $\widetilde{f}$ is an embedding.

To find such $\widetilde{f}$ , it suffices to construct a holomorphic map $H:M\times \mathbb{C}^{N}\rightarrow \mathbb{C}^{3}$ for some big integer $N\in \mathbb{N}$ satisfying the following properties for some $r>0$ .

1. (a) $H(\cdot ,0)=f$ .

2. (b) The map $H(\cdot ,\unicode[STIX]{x1D709}):M\rightarrow \mathbb{C}^{3}$ is a holomorphic Legendrian immersion for every $\unicode[STIX]{x1D709}\in r\mathbb{B}$ , where $\mathbb{B}$ is the unit ball in $\mathbb{C}^{N}$ .

3. (c) The difference map $\unicode[STIX]{x1D6FF}H:M\times M\times r\mathbb{B}\rightarrow \mathbb{C}^{3}$ , defined by

   $$\begin{eqnarray}\unicode[STIX]{x1D6FF}H(u,v,\unicode[STIX]{x1D709})=H(v,\unicode[STIX]{x1D709})-H(u,\unicode[STIX]{x1D709}),\quad u,\,v\in M,~\unicode[STIX]{x1D709}\in r\mathbb{B},\end{eqnarray}$$
   is a submersive family of maps on $M\times M\setminus U$ , in the sense that
   (4.11) $$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}|_{\unicode[STIX]{x1D709}=0}\,\unicode[STIX]{x1D6FF}H(u,v,\unicode[STIX]{x1D709}):\mathbb{C}^{N}\rightarrow \mathbb{C}^{3}~\text{is surjective for every }(u,v)\in M\times M\setminus U.\end{eqnarray}$$

Assume for a moment that such $H$ exists. By compactness of $M\times M\setminus U$ it follows from (4.11) that the partial differential $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D6FF}H)$ is surjective on $(M\times M\setminus U)\times r^{\prime }\mathbb{B}$ for some $0<r^{\prime }\leqslant r$ . Hence the map $\unicode[STIX]{x1D6FF}H:(M\times M\setminus U)\times r^{\prime }\mathbb{B}\rightarrow \mathbb{C}^{3}$ is transverse to any submanifold of $\mathbb{C}^{3}$ , in particular, to the origin $0\in \mathbb{C}^{3}$ . By Abraham’s reduction to Sard’s theorem [Reference AbrahamAbr63] (see also [Reference ForstneričFor11, § 7.8] and the references therein for the holomorphic case) it follows that for a generic choice of $\unicode[STIX]{x1D709}\in r^{\prime }\mathbb{B}$ the difference map $\unicode[STIX]{x1D6FF}H(\cdot ,\cdot ,\unicode[STIX]{x1D709})$ is transverse to $0\in \mathbb{C}^{3}$ on $M\times M\setminus U$ , and hence it omits the value $0$ by dimension reasons. Choosing $\unicode[STIX]{x1D709}$ sufficiently close to $0\in \mathbb{C}^{N}$ we obtain a holomorphic Legendrian embedding $\widetilde{f}=H(\cdot ,\unicode[STIX]{x1D709}):M\rightarrow \mathbb{C}^{3}$ close to $f$ in the $\mathscr{C}^{1}(M)$ -norm.

The main point is to find for any given point $(p,q)\in M\times M\setminus U$ a spray $H$ as above, with $N=3$ , such that (4.11) holds at $(u,v)=(p,q)$ . Since the submersivity of the differential is an open condition and $M\times M\setminus U$ is compact, we obtain a spray $H$ that is submersive at all points of $M\times M\setminus U$ by composing finitely many such sprays as explained in [Reference Alarcón, Forstnerič and LópezAFL16c, proof of Theorem 4.1] or [Reference Alarcón and ForstneričAF14, proof of Theorem 2.4].

Fix a pair of distinct points $p\neq q$ in $M$ . Choose a smooth embedded arc $E\subset M$ connecting $p$ to $q$ . As above, there exist smooth closed curves $C_{1},\ldots ,C_{\ell }\subset M$ forming a basis of the homology group $H_{1}(M;\mathbb{Z})=\mathbb{Z}^{\ell }$ such that $(\bigcup _{k=1}^{\ell }C_{k})\cap E=\emptyset$ and $(\bigcup _{k=1}^{\ell }C_{k})\cup E$ is Runge in $M$ . Given a number $\unicode[STIX]{x1D707}>0$ , we choose holomorphic functions $g_{1},\ldots ,g_{\ell },h_{1},h_{2}\in \mathscr{O}(M)$ satisfying the following conditions:

1. (i) $\int _{C_{j}}g_{k}\,dy\approx \unicode[STIX]{x1D6FF}_{j,k}~\text{for all}~j,k=1,\ldots ,\ell$ ;

2. (ii) $|g_{k}(u)|<1$ for all $u\in E$ and $k=1,\ldots ,\ell$ ;

3. (iii) $h_{1}(p)=0$ , $h_{1}(q)=1$ , $h_{2}(p)=h_{2}(q)=0$ ;

4. (iv) $\int _{E}h_{2}\,dy=-1$ ;

5. (v) $|h_{j}(u)|<\unicode[STIX]{x1D707}$ and $|dh_{j}(u)|<\unicode[STIX]{x1D707}$ for all $u\in \bigcup _{k=1}^{\ell }C_{k}$ and $j=1,2$ .

Functions with these properties are easily found by first constructing suitable smooth functions on the curves $(\bigcup _{k=1}^{\ell }C_{k})\cup E$ and applying Mergelyan’s approximation theorem.

Let $\unicode[STIX]{x1D709}=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2},\unicode[STIX]{x1D709}_{3})\in \mathbb{C}^{3}$ and $\unicode[STIX]{x1D701}=(\unicode[STIX]{x1D701}_{1},\ldots ,\unicode[STIX]{x1D701}_{\ell })\in \mathbb{C}^{\ell }$ . Consider the following sprays of holomorphic functions of $u\in M$ :

Note that $\widetilde{x}(u,0,0)=x(u)$ and $\widetilde{y}(u,0)=y(u)$ . Condition (i) ensures that

Therefore, by the implicit function theorem, the period vanishing equation

(which holds at $\unicode[STIX]{x1D709}=0$ and $\unicode[STIX]{x1D701}=0$ ) can be solved in the form $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D709})$ , where $\unicode[STIX]{x1D70C}$ is a holomorphic map from a neighborhood of $0\in \mathbb{C}^{3}$ to a neighborhood of $0\in \mathbb{C}^{\ell }$ with $\unicode[STIX]{x1D70C}(0)=0$ . We must estimate the differential $d\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D709})$ at $\unicode[STIX]{x1D709}=0$ . Differentiating (4.14) gives in view of $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D709})$ and the chain rule that

From (4.12), (4.13) and condition (v) on $h_{1}$ and $h_{2}$ we see that the right-hand side of the above equation is of size $O(\unicode[STIX]{x1D707})$ . Indeed, its components are integrals over the curves $C_{1},\ldots ,C_{\ell }$ of terms which involve at least one of the functions $h_{1},h_{2}$ or $dh_{1}$ ; these are of size $O(\unicode[STIX]{x1D707})$ by condition (v). Since the first term on the left-hand side is close to the identity, we get

Define the holomorphic spray $\widetilde{z}(\cdot ,\unicode[STIX]{x1D709})$ with the core $\widetilde{z}(\cdot ,0)=z$ on $M$ by

Note that the integral is independent of the choice of the path from $p$ to $u$ . It follows that the spray $H(\cdot ,\unicode[STIX]{x1D709}):M\rightarrow \mathbb{C}^{3}$ defined by

consists of Legendrian maps and is holomorphic with respect to $\unicode[STIX]{x1D709}\in \mathbb{C}^{3}$ near the origin. Obviously, $H$ satisfies properties (a) and (b). It remains to see that it also satisfies property (c), i.e.,

Condition (iii) on $h_{1}$ and $h_{2}$ implies that

We claim that the following estimates hold:

The first two follow directly from the definitions of $\widetilde{x}$ and $\widetilde{y}$ (see (4.12) and (4.13)) and taking into account (4.15) and (4.18). To get (4.21), note that (4.16) implies

where the integral is taken over the arc $E$ . From conditions (ii), (iv), (4.12), (4.13) and (4.15) we infer that

which establishes (4.21). Choosing the constant $\unicode[STIX]{x1D707}>0$ small enough, we see from (4.19), (4.20) and (4.21) that (4.17) holds. By what has been said before, this shows that $f$ can be approximated on $M$ by Legendrian embeddings.◻

Proof. Let $\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{k}$ $(k\in \mathbb{N})$ be pairwise disjoint smooth Jordan curves which are the connected components of $bM_{1}$ . Likewise, let $\unicode[STIX]{x1D6FD}_{1},\ldots ,\unicode[STIX]{x1D6FD}_{k}$ be the pairwise disjoint smooth Jordan curves in $bM_{2}$ , labeled so that $M_{2}\setminus {M\unicode[STIX]{x0030A}}_{1}$ consists of $k$ closed annuli $A_{1},\ldots ,A_{k}$ with boundaries $bA_{l}=\unicode[STIX]{x1D6FC}_{l}\cup \unicode[STIX]{x1D6FD}_{l}$ , $l=1,\ldots ,k$ ; take into account that $M_{1}$ is a strong deformation retract of $M_{2}$ . Since $\max \{|\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D70E}}\circ f|,|\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D70D}}\circ f|\}>\unicode[STIX]{x1D707}$ on $bM_{1}$ (cf. (5.1)), there exist an integer $m\geqslant 2$ and compact connected subarcs $\{\unicode[STIX]{x1D6FC}_{l,a}:a\in \mathbb{Z}_{m}=\mathbb{Z}/m\mathbb{Z}\}$ of $\unicode[STIX]{x1D6FC}_{l}$ for $l=1,\ldots ,k$ such that the following conditions hold.

1. (A1) $\bigcup _{a\in \mathbb{Z}_{m}}\unicode[STIX]{x1D6FC}_{l,a}=\unicode[STIX]{x1D6FC}_{l}$ .

2. (A2) The arcs $\unicode[STIX]{x1D6FC}_{l,a-1}$ and $\unicode[STIX]{x1D6FC}_{l,a}$ meet at a point $p_{l,a}$ and are otherwise disjoint for all $a\in \mathbb{Z}_{m}$ , whereas $\unicode[STIX]{x1D6FC}_{l,a}\cap \unicode[STIX]{x1D6FC}_{l,a^{\prime }}=\emptyset$ provided that $a^{\prime }\notin \{a-1,a,a+1\}\subset \mathbb{Z}_{m}$ .

3. (A3) The set $I:=\{1,\ldots ,k\}\times \mathbb{Z}_{m}$ splits into two disjoint subsets $I_{\unicode[STIX]{x1D70E}}$ and $I_{\unicode[STIX]{x1D70D}}$ such that

   $$\begin{eqnarray}|\unicode[STIX]{x1D70B}_{c}\circ f|>\unicode[STIX]{x1D707}\text{ on }\unicode[STIX]{x1D6FC}_{l,a}\text{ for all }(l,a)\in I_{c},c\in \{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70D}\}.\end{eqnarray}$$
   (Either of the sets $I_{\unicode[STIX]{x1D70E}}$ or $I_{\unicode[STIX]{x1D70D}}$ could be empty.)

Choose a family of pairwise disjoint smooth Jordan arcs $\unicode[STIX]{x1D6FE}_{l,a}\subset M_{2}\setminus {M\unicode[STIX]{x0030A}}_{1}$ , $(l,a)\in I$ , such that $p_{l,a}\in bM_{1}$ is an endpoint of $\unicode[STIX]{x1D6FE}_{l,a}$ , the other endpoint of $\unicode[STIX]{x1D6FE}_{l,a}$ , which will be called $q_{l,a}$ , lies in $bM_{2}$ , and $\unicode[STIX]{x1D6FE}_{l,a}\setminus \{p_{l,a},q_{l,a}\}\subset {M\unicode[STIX]{x0030A}}_{2}\setminus M_{1}$ . Moreover, we choose the arcs $\unicode[STIX]{x1D6FE}_{l,a}\subset M_{2}\setminus {M\unicode[STIX]{x0030A}}_{1}$ for $(l,a)\in I$ such that the set

is admissible in $R$ (see Definition 4.1). Note that the set $A_{l}\setminus (\unicode[STIX]{x1D6FC}_{l}\cup \unicode[STIX]{x1D6FD}_{l}\cup (\bigcup _{a\in \mathbb{Z}_{m}}\unicode[STIX]{x1D6FE}_{l,a}))$ consists of $m$ pairwise disjoint open disks; we denote by $\unicode[STIX]{x1D6FA}_{l,a}$ the one whose closure contains $\unicode[STIX]{x1D6FC}_{l,a}$ , $(l,a)\in I$ . We also set $\unicode[STIX]{x1D6FD}_{l,a}:=\unicode[STIX]{x1D6FD}_{l}\cap \overline{\unicode[STIX]{x1D6FA}}_{l,a}$ for all $(l,a)\in I$ , and hence $\unicode[STIX]{x1D6FA}_{l,a}$ is bounded by the arcs $\unicode[STIX]{x1D6FC}_{l,a}$ , $\unicode[STIX]{x1D6FE}_{l,a}$ , $\unicode[STIX]{x1D6FE}_{l,a+1}$ , and $\unicode[STIX]{x1D6FD}_{l,a}$ . (See Figure 5.1.)

Figure 5.1. Sets in the proof of Lemma 5.2.

In the first step we extend the map $f:M_{1}\rightarrow \mathbb{C}^{2n+1}$ to a generalized Legendrian curve $\widehat{g}:S\rightarrow \mathbb{C}^{2n+1}$ (see Definition 4.2) such that:

1. – $|\unicode[STIX]{x1D70B}_{c}\circ \widehat{g}|>\unicode[STIX]{x1D707}$ on $\unicode[STIX]{x1D6FE}_{l,a}\cup \unicode[STIX]{x1D6FC}_{l,a}\cup \unicode[STIX]{x1D6FE}_{l,a+1}$ for all $(l,a)\in I_{c}$ , $c\in \{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70D}\}$ ;

2. – $\min \{|\unicode[STIX]{x1D70B}_{c}(\widehat{g}(q_{l,a}))|,|\unicode[STIX]{x1D70B}_{c}(\widehat{g}(q_{l,a+1}))|\}>\unicode[STIX]{x1D707}+1$ for all $(l,a)\in I_{c}$ , $c\in \{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70D}\}$ .

To construct such extensions we just use property (A3) and the fact that every compact path in $\mathbb{C}^{2n+1}$ may be uniformly approximated by Legendrian paths (see Theorem A.6). Lemma 4.3 then provides a holomorphic Legendrian curve $g:M_{2}\rightarrow \mathbb{C}^{2n+1}$ with no constant component function and satisfying the following conditions:

1. (B1) the map $g$ approximates $f$ in the $\mathscr{C}^{1}(M_{1})$ -topology;

2. (B2) $|\unicode[STIX]{x1D70B}_{c}\circ g|>\unicode[STIX]{x1D707}$ on $\unicode[STIX]{x1D6FE}_{l,a}\cup \unicode[STIX]{x1D6FC}_{l,a}\cup \unicode[STIX]{x1D6FE}_{l,a+1}$ for all $(l,a)\in I_{c}$ , $c\in \{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70D}\}$ ;

3. (B3) $\min \{|\unicode[STIX]{x1D70B}_{c}(g(q_{l,a}))|,|\unicode[STIX]{x1D70B}_{c}(g(q_{l,a+1}))|\}>\unicode[STIX]{x1D707}+1$ for all $(l,a)\in I_{c}$ , $c\in \{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70D}\}$ .

Let $\unicode[STIX]{x1D6FA}\subset M_{2}$ be a compact domain meeting the following requirements.

1. – $S\subset \unicode[STIX]{x1D6FA}$ and $S\cap b\unicode[STIX]{x1D6FA}=\{q_{l,a}:(l,a)\in I\}$ .

2. – The set $b\unicode[STIX]{x1D6FA}\cap \unicode[STIX]{x1D6FD}_{l}$ consists of $m$ pairwise disjoint compact arcs $\{\unicode[STIX]{x1D706}_{l,a}:a\in \mathbb{Z}_{m}\}$ such that $\unicode[STIX]{x1D706}_{l,a}\subset \unicode[STIX]{x1D6FD}_{l,a-1}\cup \unicode[STIX]{x1D6FD}_{l,a}$ and $q_{l,a}$ is in the relative interior of $\unicode[STIX]{x1D706}_{l,a}$ for all $a\in \mathbb{Z}_{m}$ , $l=1,\ldots ,k$ .

3. – The set $\overline{A_{l}\setminus \unicode[STIX]{x1D6FA}}$ consists of $m$ pairwise disjoint, smoothly bounded, simply connected compact domains $\unicode[STIX]{x1D6F6}_{l,a}$ , $a\in \mathbb{Z}_{m}$ , $l=1,\ldots ,k$ . Up to relabeling, we assume that $\unicode[STIX]{x1D6F6}_{l,a}\subset \overline{\unicode[STIX]{x1D6FA}}_{l,a}$ for all $(l,a)\in I$ . Note that $\unicode[STIX]{x1D6F6}_{l,a}\cap (\unicode[STIX]{x1D6FE}_{l,a}\cup \unicode[STIX]{x1D6FC}_{l,a}\cup \unicode[STIX]{x1D6FE}_{l,a+1})=\emptyset$ .

Furthermore, in view of properties (B2) and (B3), continuity of $g$ enables us to choose $\unicode[STIX]{x1D6FA}$ sufficiently close to $S$ such that the following properties hold:

1. (C1) $|\unicode[STIX]{x1D70B}_{c}\circ g|>\unicode[STIX]{x1D707}$ on $\overline{\unicode[STIX]{x1D6FA}_{l,a}\setminus \unicode[STIX]{x1D6F6}_{l,a}}$ for all $(l,a)\in I_{c}$ , $c\in \{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70D}\}$ ;

2. (C2) $|\unicode[STIX]{x1D70B}_{c}\circ g|>\unicode[STIX]{x1D707}+1$ on $(\unicode[STIX]{x1D706}_{l,a}\cup \unicode[STIX]{x1D706}_{l,a+1})\cap \unicode[STIX]{x1D6FD}_{l,a}$ for all $(l,a)\in I_{c}$ , $c\in \{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70D}\}$ .

Assume that $I_{\unicode[STIX]{x1D70E}}\neq \emptyset$ ; otherwise $I_{\unicode[STIX]{x1D70D}}=I\neq \emptyset$ and we reason in a symmetric way. For each $(l,a)\in I_{\unicode[STIX]{x1D70E}}$ pick an arc $\unicode[STIX]{x1D6FF}_{l,a}\subset \overline{\unicode[STIX]{x1D6FA}_{l,a}\setminus \unicode[STIX]{x1D6F6}_{l,a}}$ with an endpoint in the relative interior of $\unicode[STIX]{x1D6FC}_{l,a}$ , the other endpoint in $\unicode[STIX]{x1D6F6}_{l,a}\setminus \unicode[STIX]{x1D6FD}_{l,a}$ , and otherwise disjoint from $\unicode[STIX]{x1D6F6}_{l,a}\cup b\overline{\unicode[STIX]{x1D6FA}}_{l,a}$ . (See Figure 5.1.) Moreover, we choose the family of arcs $\unicode[STIX]{x1D6FF}_{l,a}$ , $(l,a)\in I_{\unicode[STIX]{x1D70E}}$ , such that the set

(see (5.2)) and

The latter condition is easy to fulfil since $g$ is holomorphic and has no constant component function. Notice that $S_{\unicode[STIX]{x1D70E}}\subset M_{2}$ is a deformation retract of $M_{2}$ . Set

(cf. Definition 4.1). Let us prove the following claim.

With Claim 5.3 in hand, and since $g$ has no constant component function, we may apply Lemmas 4.3 and 4.4 to obtain a holomorphic Legendrian embedding $h_{\unicode[STIX]{x1D70E}}:M_{2}{\hookrightarrow}\mathbb{C}^{2n+1}$ with the following properties:

1. (E1) $h_{\unicode[STIX]{x1D70E}}$ approximates $\widetilde{h}_{\unicode[STIX]{x1D70E}}$ in the $\mathscr{C}^{1}(S_{\unicode[STIX]{x1D70E}})$ -topology;

2. (E2) $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D70E}}\circ h_{\unicode[STIX]{x1D70E}}=\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D70E}}\circ g$ .

Thus, in view of properties (D3) and (D1), if the approximation in property (E1) is close enough, we have that:

1. (E3) $|\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D70D}}\circ h_{\unicode[STIX]{x1D70E}}|>\unicode[STIX]{x1D707}+1$ on $\unicode[STIX]{x1D6F6}_{l,a}$ for all $(l,a)\in I_{\unicode[STIX]{x1D70E}}$ ;

1. (E4) $h_{\unicode[STIX]{x1D70E}}$ has no constant component function.

We claim that if $I_{\unicode[STIX]{x1D70D}}=\emptyset$ , then $\widetilde{f}:=h_{\unicode[STIX]{x1D70E}}$ satisfies the conclusion of Lemma 5.2. Indeed, properties (B1) and (E1) ensure that $\widetilde{f}$ approximates $f$ in the $\mathscr{C}^{1}(M_{1})$ -topology. If on the other hand $I_{\unicode[STIX]{x1D70D}}=\emptyset$ , then $I_{\unicode[STIX]{x1D70E}}=I$ and hence

and

Thus, properties (C2), (E2), and (E3) give condition (i), whereas properties (C1), (E2), and (E3) guarantee condition (ii).

Assume now that $I_{\unicode[STIX]{x1D70D}}\neq \emptyset$ . As above, for each $(l,a)\in I_{\unicode[STIX]{x1D70D}}$ , we take an arc $\unicode[STIX]{x1D6FF}_{l,a}\subset \overline{\unicode[STIX]{x1D6FA}_{l,a}\setminus \unicode[STIX]{x1D6F6}_{l,a}}$ with an endpoint in the relative interior of $\unicode[STIX]{x1D6FC}_{l,a}$ , the other endpoint in $\unicode[STIX]{x1D6F6}_{l,a}\setminus \unicode[STIX]{x1D6FD}_{l,a}$ , and otherwise disjoint from $\unicode[STIX]{x1D6F6}_{l,a}\cup b\overline{\unicode[STIX]{x1D6FA}}_{l,a}$ , such that the set

is admissible in $R$ and no component function of $h_{\unicode[STIX]{x1D70E}}$ has a zero or a critical point in $\bigcup _{(l,a)\in I_{\unicode[STIX]{x1D70E}}}\unicode[STIX]{x1D6FF}_{l,a}$ . Reasoning as above, in a symmetric way, we may construct a holomorphic Legendrian embedding $h_{\unicode[STIX]{x1D70D}}:M_{2}\rightarrow \mathbb{C}^{2n+1}$ enjoying the following properties:

1. (F1) $h_{\unicode[STIX]{x1D70D}}$ approximates $h_{\unicode[STIX]{x1D70E}}$ in the $\mathscr{C}^{1}(S_{\unicode[STIX]{x1D70D}})$ -topology;

2. (F2) $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D70D}}\circ h_{\unicode[STIX]{x1D70D}}=\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D70D}}\circ h_{\unicode[STIX]{x1D70E}}$ ;

3. (F3) $|\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D70E}}\circ h_{\unicode[STIX]{x1D70D}}|>\unicode[STIX]{x1D707}+1$ on $\unicode[STIX]{x1D6F6}_{l,a}$ for all $(l,a)\in I_{\unicode[STIX]{x1D70D}}$ .

We claim that, if the approximations in properties (B1), (D1), (E1), and (F1) are close enough, the Legendrian curve $\widetilde{f}:=h_{\unicode[STIX]{x1D70D}}:M_{2}\rightarrow \mathbb{C}^{2n+1}$ satisfies the conclusion of the lemma. Indeed, the mentioned properties ensure that $\widetilde{f}$ approximates $f$ in the $\mathscr{C}^{1}(M_{1})$ -topology. It remains to check conditions (i) and (ii). Let $p\in M_{2}\setminus {M\unicode[STIX]{x0030A}}_{1}$ . If $p\in \unicode[STIX]{x1D6F6}_{l,a}$ for some $(l,a)\in I_{\unicode[STIX]{x1D70E}}$ , we have

and together with property (F3) we infer that

On the other hand, if $p\in \overline{\unicode[STIX]{x1D6FA}_{l,a}\setminus \unicode[STIX]{x1D6F6}_{l,a}}$ for some $(l,a)\in I_{\unicode[STIX]{x1D70E}}$ , we have

whereas if $p\in \overline{\unicode[STIX]{x1D6FA}_{l,a}\setminus \unicode[STIX]{x1D6F6}_{l,a}}$ for some $(l,a)\in I_{\unicode[STIX]{x1D70D}}$ , then

So, the estimates (5.6) and (5.7) together with the inequalities (5.5), (C1), and (C2), and the facts that

guarantee conditions (i) and (ii). This concludes the proof. ◻