Proof. Denote by ${\mathcal{C}}(M,{\mathbb{R}}^n)$ the space of all continuous maps $M\to{\mathbb{R}}^n$ endowed with the compact-open topology. Since $M$ is hemicompactFootnote ³ and ${\mathbb{R}}^n$ is complete with the Euclidean metric, we have that ${\mathcal{C}}(M,{\mathbb{R}}^n)$ is completely metrizable (see Arens [Reference Arens6]; see also, e.g. [Reference McCoy and Ntantu11, p. 100]). Recall that a map $u=(u_1,\ldots,u_n):M\to{\mathbb{R}}^n$ lies in ${\rm CMI}(M,{\mathbb{R}}^n)$ if and only if u is a harmonic map and its complex derivative $\partial u$ (i.e. the $(1,0)$-part of the exterior derivative ${\rm d}u$ of u) satisfies $\sum_{i=1}^n (\partial u_i)^2=0$ and $\partial u\neq 0$ everywhere on M (see, e.g. [Reference Alarcón, Forstnerič and López5, Theorem 2.3.1]). Denote by ${\rm CHM}(M,{\mathbb{R}}^n)$ the subspace of ${\mathcal{C}}(M,{\mathbb{R}}^n)$ consisting of all harmonic maps $u:M\to{\mathbb{R}}^n$ satisfying $\sum_{i=1}^n (\partial u_i)^2=0$ everywhere on $M$. By Harnack’s theorem ${\rm CHM}(M,{\mathbb{R}}^n)$ is a closed subspace of ${\mathcal{C}}(M,{\mathbb{R}}^n)$, hence ${\rm CHM}(M,{\mathbb{R}}^n)$ is completely metrizable as well. Recall now that a subspace of a completely metrizable space is completely metrizable if and only if it is a G_δ set (see, e.g. [Reference Willard15, Theorem 24.12, p. 179] or [Reference Engelking8, Theorems 4.3.23 and 4.3.24, p. 274]). Since

\begin{equation*} {\rm CMI}(M,{\mathbb{R}}^n)=\big\{u\in{\rm CHM}(M,{\mathbb{R}}^n)\colon \partial u(p)\neq 0\ \text{for all } p\in M\big\} \end{equation*}

is a subspace of ${\rm CHM}(M,{\mathbb{R}}^n)$, to complete the proof it remains to see that ${\rm CMI}(M,{\mathbb{R}}^n)$ is a G_δ set in ${\rm CHM}(M,{\mathbb{R}}^n)$; i.e. it can be written as a countable intersection of open sets. For that, it is then clear that ${\rm CMI}(M,{\mathbb{R}}^n)=\bigcap_{j\in{\mathbb{N}}} U_j$, where

\begin{equation*} U_j=\big\{u\in {\rm CHM}(M,{\mathbb{R}}^n)\colon\partial u(p)\neq 0\ \text{for all } p\in K_j\big\},\quad j\in{\mathbb{N}}. \end{equation*}

(see (2.1).) To finish, just note that U_j is an open subset in ${\rm CHM}(M,{\mathbb{R}}^n)$ for all $j\in{\mathbb{N}}$ by Cauchy estimates.

Proof. Let ${\mathcal{C}}(K,{\mathbb{R}}^n)$ denote the space of all continuous maps $K\to{\mathbb{R}}^n$ endowed with the compact-open topology. Since $K$ is compact Hausdorff and ${\mathbb{R}}^n$ is a metric space, this topology coincides with the metric topology of the maximum norm (see, e.g. [Reference Bredon7, Theorem 2.12, p. 440]). Note that ${\mathcal{C}}(K,{\mathbb{R}}^n)$ is separable by Riesz’s Theorem; see, e.g. [Reference Royden and Fitzpatrick14, p. 251], and take into account that the product of finitely many separable metric spaces is separable. Since every subspace of a separable metric space is separable (see e.g. [Reference Royden and Fitzpatrick14, Proposition 26, p. 204] or [Reference Willard15, 16G.1]), we have that ${\mathrm{CMI}}(K,{\mathbb{R}}^n)\subset {\mathcal{C}}(K,{\mathbb{R}}^n)$ is separable too.

For the second assertion, note that ${\mathrm{CMI}}(K_j,{\mathbb{R}}^n)$ is separable for all $j\in {\mathbb{N}}$ (see (2.1)). Since the Runge theorem for conformal minimal immersions in [Reference Alarcón, Forstnerič and López5, Theorem 3.6.1] (see Theorem 2.4 below) enables us to uniformly approximate every element in ${\mathrm{CMI}}(K_j,{\mathbb{R}}^n)$ by a sequence in ${\mathrm{CMI}}(M,{\mathbb{R}}^n)$, the latter space is separable as well.

Proof. By the continuity of ${\mathcal{F}}(\cdot,C)$, the set $\Lambda_i^C=\{u\in {\rm CMI}(M,{\mathbb{R}}^n) \colon {\mathcal{F}}(u,C) \gt i\}$ is open in ${\rm CMI}(M,{\mathbb{R}}^n) $ for all $C\in E$ and $i\in {\mathbb{N}}$, and so is $\Lambda_i=\bigcup_{C\in E} \Lambda_i^C$ for all $i\in {\mathbb{N}}$. Since ${\rm CMI}(M,{\mathbb{R}}^n)$ is completely metrizable by Claim 2.1 and $\Lambda_i$ is also dense in ${\rm CMI}(M,{\mathbb{R}}^n)$, the Baire Category Theorem ensures that $\bigcap _{i\in {\mathbb{N}}} \Lambda_i$ is a dense G_δ subset in ${\rm CMI}(M,{\mathbb{R}}^n)$.

Proof of (i)

Let ${\mathfrak{g}}$ be a Riemannian metric in ${\mathbb{R}}^n$. Fix a point $p_0\in\mathring K_1$, consider the family of compact sets $E=\{bK_j\colon j\in {\mathbb{N}}\}$ (see (2.1)), and set

\begin{equation*} {\mathcal{F}}(u,C)= {\mathrm{dist}}_u^{\mathfrak{g}}(p_0,C)\ \text{for every $u\in {\rm CMI}(M,{\mathbb{R}}^n)$ and $C\in E$,} \end{equation*}

where ${\mathrm{dist}}_u^{\mathfrak{g}}$ denotes the distance on M associated to the pull-back metric $u^*{\mathfrak{g}}$. Let $C=b K_j\in E$, $u\in {\rm CMI}(M,{\mathbb{R}}^n)$, and µ > 0. The Cauchy estimates (relying on the fact that the uniform convergence on a compact set of harmonic or holomorphic functions implies the one of all their derivatives) guarantee the existence of a number σ > 0 with the following property: for any $v\in {\rm CMI}(M,{\mathbb{R}}^n)$ with $\|v-u\|_{K_j} \lt \sigma$ the metric $v^*{\mathfrak{g}}$ is so close to $u^*{\mathfrak{g}}$ on K_j that $|{\mathrm{dist}}_v^{\mathfrak{g}}(p_0,C)- {\mathrm{dist}}_u^{\mathfrak{g}}(p_0,C)| \lt \mu$. This proves Claim 2.3 (A).

Let $i\in {\mathbb{N}}$ and fix $u$, $K$, and ϵ as in Claim 2.3 (B). Choose $C=bK_j\in E$, where $j\in{\mathbb{N}}$ is so large that $K\subset\mathring K_j$, and fix a number δ > 0 to be specified later. By [Reference Alarcón, Drnovšek, Forstnerič and López2, Lemma 4.1] (see also [Reference Alarcón, Forstnerič and López5, Lemma 7.3.1]) there is a conformal minimal immersion $\hat u:K_j\to{\mathbb{R}}^n$ such that $\|\hat u-u\|_{K_j} \lt \epsilon/2$ and ${\mathrm{dist}}_{\hat u}(p_0, C) \gt \delta$, where ${\mathrm{dist}}_{\hat u}$ denotes the distance function on M associated to the metric $\hat u^*{ds^2}$ induced on M via $\hat u$ by the standard Euclidean metric ds ² in ${\mathbb{R}}^n$. Since ds ² and the given metric ${\mathfrak{g}}$ are comparable in the compact set $\{x\in{\mathbb{R}}^n\colon {\mathrm{dist}}(x,u(K_j))\le \epsilon\}\subset{\mathbb{R}}^n$, we can choose δ > 0 so large that

(2.3)\begin{equation} {\mathrm{dist}}_{\hat u}^{\mathfrak{g}}(p_0,C) \gt i. \end{equation}

Next, by Theorem 2.4 we can approximate $\hat u$ uniformly on K_j by immersions $\tilde u\in{\mathrm{CMI}}(M,{\mathbb{R}}^n)$ with $\|\tilde u-\hat u\|_{K_j} \lt \epsilon/2$, and hence $\|\tilde u-u\|_K \lt \epsilon$. Moreover, if the approximation of $\hat u$ by $\tilde u$ on K_j is close enough then, by (2.3) and Cauchy estimates, we can also ensure that ${\mathrm{dist}}_{\tilde u}^{\mathfrak{g}}(p_0,C) \gt i$, and hence $\tilde u\in \Lambda_i$; see (2.2). This shows (B) and completes the proof in view of Claim 2.3; note that $\bigcap _{i\in {\mathbb{N}}} \Lambda_i$ equals the set of ${\mathfrak{g}}$-complete immersions in ${\rm CMI}(M,{\mathbb{R}}^n)$.

Proof of (ii)

Let $v:\overline{\mathbb D}\to {\mathbb{R}}^n$ be a conformal minimal immersion (on a neighborhood of $\overline{\mathbb D}$ in ${\mathbb{C}}$), consider the family $E=\{D\subset M\colon \text{$D$ smoothly bounded closed disc}\}$, and for each $D\in E$ denote by ${\mathcal{L}}_D$ the set of all biholomorphisms $\overline{\mathbb D}\to D$. Set

\begin{equation*} {\mathcal{F}}(u,D)=\sup_{\varphi\in {\mathcal{L}}_D} \frac1{\|u\circ \varphi-v\|_{\overline{\mathbb D}}} \in (0,+\infty] \quad \text{for $u\in {\rm CMI}(M,{\mathbb{R}}^n)$ and $D\in E$.} \end{equation*}

Let $i\in {\mathbb{N}}$ and fix $u$, $K$, and ϵ as in Claim 2.3 (B). We assume without loss of generality that $K\subset M$ is Runge, and pick $D\in E$ with $K\cap D=\varnothing$ and $\varphi\in {\mathcal{L}}_D$. Since $K\cup D$ is Runge, by Theorem 2.4 there is $\tilde u\in {\rm CMI}(M,{\mathbb{R}}^n)$ such that $|\tilde u-u| \lt \epsilon$ on K and $|\tilde u-v\circ \varphi^{-1}| \lt 1/i$ on D, hence $\tilde u\in \Lambda_i$ (see (2.2)), proving (B). Thus, Claim 2.3 ensures that the set $\bigcap _{i\in {\mathbb{N}}} \Lambda_i$, which equals the one of conformal minimal immersions $M\to {\mathbb{R}}^n$ rebuilding $v$, is a dense $G_\delta$ subset in ${\rm CMI}(M,{\mathbb{R}}^n)$.

Proof of (iii)–(vi)

Let $\varnothing \neq Z\subset {\mathbb{R}}^n$ be a closed set and $\rho\colon {\mathbb{R}}^n\to {\mathbb{R}}$ be a weight function. Let $E=\{K_j\colon j\in {\mathbb{N}}\}$ (see (2.1)), and for each $3\leq l\leq 6$ define the function ${\mathcal{F}}_l\colon {\rm CMI}(M,{\mathbb{R}}^n)\times E\to [0,+\infty]$ given by

\begin{equation*} {\mathcal{F}}_l(u,C)=\left\{ \begin{array}{ll} \displaystyle \frac1{{\mathrm{dist}}(u(C),Z)} & \text{if }l=3 \\ \displaystyle\sup_{C} |{\mathcal{K}}_u| (\rho\circ u) & \text{if }l=4 \\ \displaystyle \int_{C} |{\mathcal{K}}_u| (\rho\circ u)dA_u & \text{if }l=5 \\ \displaystyle \int_{C} (\rho\circ u) dA_u & \text{if }l=6\;. \end{array}\right. \end{equation*}

Let $i\in {\mathbb{N}}$ and fix $u$, $K$, and ϵ as in Claim 2.3 (B). We assume without loss of generality that $K\subset M$ is Runge. Let $j\in{\mathbb{N}}$ be so large that $K\subset K_j\in E$ and take a smoothly bounded closed disc $D\subset K_{j+1}\setminus K_j$. Let $B\subset {\mathbb{R}}^n$ be an open ball and δ > 0 such that $\rho|_{\overline B} \gt \delta$. By Theorem 2.4 applied to a suitable extension of $u|_{K_j}$ to $K_j\cup D$, for each $l\in \{3,4,5,6\}$ there is $\tilde u^l\in {\rm CMI}(M,{\mathbb{R}}^n)$ such that $|\tilde u^l-u| \lt \epsilon$ on K, $\tilde u^l(D)\subset B$ provided that $4\leq l\leq 6$, and

\begin{equation*} {\mathrm{dist}}(\tilde u^3(D),Z) \lt 1/i,\quad \sup_D |{\mathcal{K}}(\tilde u^4)| \gt i/\delta, \end{equation*}

\begin{equation*} \int_{D} |{\mathcal{K}}(\tilde u^5)|dA_{\tilde u^5} \gt i/\delta, \quad {\rm Area}(\tilde u^6(D))=\int_{D}dA_{\tilde u^6} \gt i/\delta. \end{equation*}

For cases $l=5,6$, use that the open ball B contains complete bounded minimal discs [Reference Nadirashvili13], having infinite total curvature and area. Since $\rho|_{\overline B} \gt \delta$ and $\tilde u^l(D)\subset B$, $4\leq l\leq 6$, this shows that $\tilde u^l\in\Lambda_i^l=\{u\in {\rm CMI}(M,{\mathbb{R}}^n)\colon \sup_E {\mathcal{F}}_l(u,\cdot) \gt i\}$ and hence $\Lambda_i^l$ is dense in ${\rm CMI}(M,{\mathbb{R}}^n)$ for all $i\in {\mathbb{N}}$ and $3\leq l \leq 6$. Thus, $\bigcap_{i\in {\mathbb{N}}} \Lambda_i^l$ is a dense G_δ in ${\rm CMI}(M,{\mathbb{R}}^n)$ for each l by Claim 2.3. Since these are the sets in (iii)–(vi), the proof is done.

Proof of (vii)

Let $A\subset{\mathbb{R}}^n$ be a properly immersed submanifold of codimension $\ge 2$ and $S\subset A$ be a compact subset with empty interior. By Claim 2.1 and the Baire category theorem, it suffices to prove that the set

\begin{equation*} \Theta_i=\Big\{u\in{\rm CMI}(M,{\mathbb{R}}^n)\colon u(K_i)\cap S=\varnothing\Big\}, \end{equation*}

(see (2.1)) is open and dense in ${\rm CMI}(M,{\mathbb{R}}^n)$ for every $i\in {\mathbb{N}}$; observe that $\Theta=\bigcap_{i\in{\mathbb{N}}}\Theta_i$ equals the set of immersions in ${\rm CMI}(M,{\mathbb{R}}^n)$ with range in ${\mathbb{R}}^n\setminus S$. Indeed, the openness is clear by compactness of $K_i$ and $S$. For the density, fix $i\in{\mathbb{N}}$ and choose an immersion $u\in{\mathrm{CMI}}(M,{\mathbb{R}}^n)$, a compact set $K\subset M$, and a number ϵ > 0. Since $\Theta_i\supset \Theta_j$ for all $j\geq i$, we can assume that $i\in{\mathbb{N}}$ is so large that $K\subset K_i$. An inspection of the proof of the general position theorem in [Reference Alarcón, Forstnerič and López5, Theorem 3.4.1] shows that there is a conformal minimal immersion $\hat u:K_i\to{\mathbb{R}}^n$ such that $\|\hat u-u\|_{K_i} \lt \epsilon/3$, $\hat u$ intersects $A\subset{\mathbb{R}}^n$ transversely, and $\hat u(bK_i)\cap A=\varnothing$; see [Reference Alarcón, Forstnerič and López5, p. 145] and recall that $\dim A\le n-2$. In particular, the set $\hat u(K_i)\cap A$ is finite, and hence so is $\hat u(K_i)\cap S$. Since $S\subset A$ is compact and has empty relative interior, post-composing $\hat u$ with a suitable small translation in ${\mathbb{R}}^n$ we can obtain a conformal minimal immersion $\hat u':K_i\to{\mathbb{R}}^n$ such that $\|\hat u'-\hat u\|_{K_i} \lt \epsilon/3$ and $\hat u'(K_i)\cap S=\varnothing$. Finally, Theorem 2.4 allows to approximate $\hat u'$ uniformly on $K_i$ by an immersion $\tilde u\in{\mathrm{CMI}}(M,{\mathbb{R}}^n)$ with $\|\tilde u-\hat u'\|_{K_i} \lt \epsilon/3$, and hence $\|\tilde u-u\|_{K_i} \lt \epsilon$. Taking into account that both $K_i$ and $S$ are compact, if the approximation of $\hat u'$ by $\tilde u$ on K_i is close enough then $\tilde u(K_i)\cap S=\varnothing$, and hence $\tilde u\in \Theta_i$. This completes the proof.

Proof of (viii)

For each $i\in {\mathbb{N}}$ choose an open neighborhood $\Delta_i$ of the diagonal $D_M$ of $M\times M$ such that $\Delta_i\supset \Delta_{i+1}$, $i\in {\mathbb{N}}$, and $\bigcap_{i\in {\mathbb{N}}} \Delta_i=D_M$. For every $i\in {\mathbb{N}}$ denote by $\Theta_i$ the set of those immersions $u\in{\rm CMI}(M,{\mathbb{R}}^n)$ such that $d(\sigma_u)_{(p,q)}:T_{(p,q)}M\times M\to {\mathbb{R}}^n$ is surjective for all $(p,q)\in \sigma_u^{-1}(0)\cap (K_i\times K_i)\setminus \Delta_i$; see Definition 1.1 (e) and (2.1). By Claim 2.1 and the Baire category theorem, it suffices to prove that the set $\Theta_i$ is open and dense in ${\rm CMI}(M,{\mathbb{R}}^n)$ for every $i\in {\mathbb{N}}$; observe that $\Theta=\bigcap_{i\in{\mathbb{N}}}\Theta_i$ equals the set of immersions in ${\rm CMI}(M,{\mathbb{R}}^n)$ that self-intersect nicely. To check that $\Theta_i$ is open, just observe that for a given $u\in \Theta_i$ the Cauchy estimates give a number ϵ > 0 such that $d(\sigma_{\hat u})_{(p,q)}$ is surjective for all $(p,q)\in \sigma_{\hat u}^{-1}(0)\cap (K_i\times K_i)\setminus \Delta_i$ and all $\hat u \in {\rm CMI}(M,{\mathbb{R}}^n)$ with $|\hat u-u| \lt \epsilon$ on $K_{i+1}$. For the density, given $u\in {\rm CMI}(M,{\mathbb{R}}^n)$, a Runge compact set $K\subset M$ with $K_i\subset K$, and a number ϵ > 0, an inspection of the proof of [Reference Alarcón, Forstnerič and López5, Theorem 3.4.1] provides a conformal minimal immersion $\hat u\colon K\to {\mathbb{R}}^n$ such that $|\hat u-u| \lt \epsilon$ on $K$ and $d(\sigma_{\hat u})_{(p,q)}$ is surjective for all $(p,q)\in \sigma_{\hat u}^{-1}(0)\cap (K_i\times K_i)\setminus \Delta_i$. By Theorem 2.4 we may assume that $\hat u\in{\mathrm{CMI}}(M,{\mathbb{R}}^n)$, hence $\Theta_i$ is dense.

Proof of (ix)

Let $E=\{bK_j\colon j\in {\mathbb{N}}\}$ (see (2.1)) and set

\begin{equation*} {\mathcal{F}}(u,C)=\inf_C |u| \in [0,+\infty) \quad\text{for every $u\in {\rm CMI}(M,{\mathbb{R}}^n)$ and $C\in E$.} \end{equation*}

The function ${\mathcal{F}}(\cdot,C): {\rm CMI}(M,{\mathbb{R}}^n)\to [0,+\infty)$ is continuous for each $C\in E$. Let $i\in {\mathbb{N}}$ and fix u, K, and ϵ as in Claim 2.3 (B). Let $j\in {\mathbb{N}}$ be so large that $K\subset \mathring K_j$ and choose a smoothly bounded compact domain $K'\subset M$ such that $K\subset K'\subset\mathring K_j$ and Kʹ is a strong deformation retract of K_j. Up to post-composing u with a small translation in ${\mathbb{R}}^n$, assume without loss of generality that $|u| \gt 0$ everywhere on bKʹ. By [Reference Alarcón, Forstnerič and López5, Lemma 3.11.1] (see also [Reference Alarcón, Forstnerič and López3, Lemma 7.2]), there is a conformal minimal immersion $\tilde u :K_j\to{\mathbb{R}}^n$ such that $\|\tilde u-u\|_{K'} \lt \epsilon$ and $\inf_{bK_j}|\tilde u| \gt i$. Further, by Theorem 2.4 we may assume that $\tilde u \in {\rm CMI}(M,{\mathbb{R}}^n)$, and hence $\tilde u$ lies in the set $\Lambda_i$ given in (2.2). Thus, $\Lambda_i$ is dense in ${\rm CMI}(M,{\mathbb{R}}^n)$ and Claim 2.3 ensures that

\begin{equation*} \bigcap _{i\in {\mathbb{N}}} \Lambda_i=\big\{u\in {\rm CMI}(M,{\mathbb{R}}^n)\colon \forall i\in {\mathbb{N}} \;\exists j\in {\mathbb{N}}\; \text{such that}\; \inf_{bK_j} |u| \gt i\big\}, \end{equation*}

is a dense $G_\delta$ subset in ${\rm CMI}(M,{\mathbb{R}}^n)$. To complete the proof, note that every immersion $u\in \bigcap _{i\in {\mathbb{N}}} \Lambda_i$ is almost proper. Indeed, for any such $u$ there is a strictly increasing sequence $\{n_k\}_{k\in{\mathbb{N}}}\subset{\mathbb{N}}$ such that $\lim_{k\to+\infty} \inf_{bK_{n_k}}|u|=+\infty$. Thus, given a number r > 0, we have that $u^{-1}(\{x\in{\mathbb{R}}^n\colon |x|\le r\})\cap bK_{n_k}=\varnothing$ for every large enough $k\in {\mathbb{N}}$. Since $K_{n_1}\Subset K_{n_2}\Subset\cdots\subset \bigcup_{k\in{\mathbb{N}}} K_{n_k}=M$ is an exhaustion of M by compact domains, it turns out that every connected component of $u^{-1}(\{x\in{\mathbb{R}}^n\colon |x|\le r\})$ is compact, proving that $u:M\to{\mathbb{R}}^n$ is an almost proper map and completing the proof.

Proof of Corollary 1.3

Statements (i), (iii), and (vii) follow straightforwardly from Theorem 1.2 (i), (iii), and (vii), the Baire category theorem, and the fact that a countable intersection of $G_\delta$ sets is again a $G_\delta$. Indeed, for (iii) use that a subset $W\subset {\mathbb{R}}^n$ is dense if and only if ${\rm dist}(W,\{q\})=0$ for all $q\in {\mathbb{Q}}^n$, and for (vii) that every meagre set in $A_j\subset {\mathbb{R}}^n$ is contained in a countable union of compact sets with empty interior.

To prove (ii) recall that the space ${\mathrm{CMI}}(\overline{\mathbb D},{\mathbb{R}}^n)$ of all conformal minimal immersions ${\mathbb{C}}\supset \overline{\mathbb D}\to{\mathbb{R}}^n$, endowed with the compact-open topology, is separable by Claim 2.2. Let ${\mathcal{D}}=\{f_i\in {\mathrm{CMI}}(\overline{\mathbb D},{\mathbb{R}}^n)\colon i\in{\mathbb{N}}\}$ be a dense countable subset of ${\mathrm{CMI}}(\overline{\mathbb D},{\mathbb{R}}^n)$, and note that the set $\Sigma=\{u\in\mathrm{CMI}(M,\mathbb{R}^n):u\text{ rebuilds }f_i\text{ for all }i\in\mathbb{N}\}$ equals the set of immersions in ${\mathrm{CMI}}(M,{\mathbb{R}}^n)$ rebuilding every conformal minimal disc, while Theorem 1.2(ii) shows that $\Sigma$ is a dense $G_\delta$ set in ${\mathrm{CMI}}(M,{\mathbb{R}}^n)$.

Let us check (iv); the proofs of (v) and (vi) follow analogously and we leave the details out. Let $\{B_j\subset {\mathbb{R}}^n\colon j\in {\mathbb{N}}\}$ be a sequence of open balls being a countable basis of the Euclidean topology in ${\mathbb{R}}^n$. For each $j\in {\mathbb{N}}$ let $\rho_j:{\mathbb{R}}^n\to {\mathbb{R}}$ be a weight function on ${\mathbb{R}}^n$ with the support ${\rm supp}(\rho_j)\Subset B_j$, $0\leq \rho_j\leq 1$, and $\rho_j|_{B_j'}=1$ for some ball $B_j'\Subset B_j$. By Theorem 1.2(iv) and the Baire category theorem, the set $X$ of all immersions $u\in{\mathrm{CMI}}(M,{\mathbb{R}}^n)$ such that $\sup_M |K_u| (\rho_j\circ u)=+\infty$ for all $j\in {\mathbb{N}}$ is a dense $G_\delta$ subset. To finish, let us show that $X$ equals the set $Y$ of all immersions $u\in {\rm CMI}(M,{\mathbb{R}}^n)$ having dense image and unbounded curvature on $u^{-1}(\Omega)$ for every open set $\Omega\subset {\mathbb{R}}^n$. Indeed, if $u\in X$ and $\Omega\subset {\mathbb{R}}^n$ is open, there is $j\in {\mathbb{N}}$ such that $B_j\subset \Omega$. Since $\sup_M |K_u| (\rho_j\circ u)=+\infty$ and ${\rm supp}(\rho_j)\subset B_j$, we have that $\varnothing\neq u^{-1}(B_j)\subset u^{-1}(\Omega)$. This implies that $u$ has dense image. Furthermore, since $0\leq \rho_j\leq 1$ it turns out that

\begin{equation*} \sup_{u^{-1}(\Omega)} |K_u|\geq\sup_{u^{-1}(\Omega)} |K_u|(\rho_j\circ u)\geq \sup_{u^{-1}(B_j)} |K_u|(\rho_j\circ u)= \sup_M |K_u| (\rho_j\circ u)=+\infty, \end{equation*}

which implies that $u\in Y$. On the other hand, if $u\in Y$ and $j\in {\mathbb{N}}$ then $u^{-1}(B_j')\neq \varnothing$, and since $\rho_j|_{B_j'}=1$ we infer that

\begin{equation*} \sup_M |K_u| (\rho_j\circ u)=\sup_{u^{-1}(B_j)} |K_u|(\rho_j\circ u)\geq \sup_{u^{-1}(B_j')} |K_u| (\rho_j\circ u)=\sup_{u^{-1}(B_j')} |K_u|=+\infty, \end{equation*}

where the last equality holds since $u\in Y$ and $B_j'$ is open in ${\mathbb{R}}^n$. This proves that $u\in X$ and concludes the proof.

Proof of Theorem 1.4

Let $M$ be an open Riemann surface and $B\subset {\mathbb{R}}^3$ be an open ball. Reasoning as in the proof of Corollary 1.3 (iv), it suffices to show that the set $X$ of all immersions $u\in {\rm CMI}(M,{\mathbb{R}}^3)$ with $u^{-1}(B)\neq \varnothing$ and having infinite index of stability on $u^{-1}(B)$ is a dense $G_\delta$ subset. For this, take an exhaustion $K_j$, $j\in {\mathbb{N}}$, of $M$ as in (2.1), and for each $i\in {\mathbb{N}}$ set

\begin{equation*} \Lambda_i=\{u\in {\rm CMI}(M,{\mathbb{R}}^3)\colon \sup_{j\in {\mathbb{N}}} {\mathcal{I}}_B(u,j) \gt i\}, \end{equation*}

where ${\mathcal{I}}_B(u,j)$ is the index of stability of $u$ on $\mathring K_j\cap u^{-1}(B)$, that is to say, the supremum of the stability index of $u|_K$ among all the smoothly bounded compact domains $K\subset \mathring K_j\cap u^{-1}(B)$. Since the function ${\mathcal{I}}_B(\cdot ,j): {\rm CMI}(M,{\mathbb{R}}^3)\to {\mathbb{R}}$ takes its values in ${\mathbb{Z}}_+$ and is lower semicontinuous by Cauchy estimates, we have that $\Lambda_i$ is open for all $i\in {\mathbb{N}}$. Observe there are conformal minimal discs ${\mathbb D}\to B$ with index of stability as big as desired (e.g. pieces of suitable helicoids with axes intersecting $B$), and hence an analogous argument to that in the proof of Theorem 1.2(iii)–(vi) shows that $\Lambda_i$ is dense for all $i\in {\mathbb{N}}$. Since $X=\bigcap_{i\in {\mathbb{N}}}\Lambda_i$, the Baire category theorem completes the proof in view of Claim 2.1.

Proof of Corollary 1.6

Item (i) follows trivially from Theorem 1.2 (viii), since a conformal minimal immersion $M\to {\mathbb{R}}^n$, $n\geq 5$, self-intersects nicely if and only if it is injective. The first part of (ii) follows from [Reference Alarcón, Forstnerič and López5, Theorem 3.4.1(c)] and the ideas in the proof of Theorem 1.2 (viii). Note that if $u\in {\rm CMI}(M,{\mathbb{R}}^4)$ has simple double points and $K\subset M$ is compact then the set $\{p\in K\colon u^{-1}(u(p))\cap K\neq \varnothing\}$ is finite, hence ${\rm DP}(u)$ is countable. Likewise, if $u\in {\rm CMI}(M,{\mathbb{R}}^3)$ self-intersects nicely then the set $\{p\in K\colon u^{-1}(u(p))\cap K\neq \varnothing\}$ is a countable union of regular curves and boundary points. The proof of items (ii) and (iii) is then completed by Corollary 1.3 (ii) and the following claim; see Definition 1.1.

To prove the claim, assume that $u\in{\mathrm{CMI}}(M,{\mathbb{R}}^n)$ rebuilds every conformal minimal disc and choose a smoothly bounded, relatively compact, open disc $V\subset M$. We need to prove that $V\cap{\rm DP}(u)\neq\varnothing$; see (1.1). So, pick $p\in V$ and let $f:\overline{\mathbb D}\to{\mathbb{R}}^n$ be a conformal minimal immersion such that $f(\overline{\mathbb D})$ is a planar round disc in an affine 2-plane in ${\mathbb{R}}^n$,

(3.1)\begin{equation} f(0)=u(p), \end{equation}

and $du_p(T_pM)+df_0(T_0{\mathbb D})={\mathbb{R}}^n$; recall that $n\le 4$. Up to replacing $V$ by a smaller neighborhood of $p$, we can assume that

(3.2)\begin{equation} du_q(T_qM)+df_z(T_z{\mathbb D})={\mathbb{R}}^n\quad \text{for all }q\in V\text{and }z\in{\mathbb D}, \end{equation}

take into account that $df_z(T_z{\mathbb D})\equiv df_0(T_0{\mathbb D})$ for all $z\in{\mathbb D}$. Since $u$ approaches every conformal minimal disc, it turns out that for any ϵ > 0 there are a smoothly bounded closed disc $D$ in $M$ and a biholomorphism $\varphi:\overline{\mathbb D}\to D$ such that $\|u\circ\varphi-f\|_{\overline{\mathbb D}} \lt \epsilon$, and thus $\|u-f\circ\varphi^{-1}\|_D \lt \epsilon$ as well. Since $n\in\{3,4\}$, conditions (3.1) and (3.2) ensure that $u(V)\cap u(D)\neq\varnothing$ and $V\cap D=\varnothing$ whenever that ϵ > 0 is sufficiently small, and hence $V\cap {\rm DP}(u)\neq\varnothing$. This shows that ${\rm DP}(u)$ is dense in $M$.