In the present paper, the authors prove Runge’s theorem (Theorem 1.2, a special case of Theorem 6.1) for complete minimal surfaces with finite total curvature, which is an analogue of the Behnke-Stein and Royden theorem on approximation and interpolation for functions on compact Riemann surfaces.
Theorem 1.2. Let \(\Sigma\) be a compact Riemann surface and \(\emptyset \ne E \subset \Sigma\) be a finite subset. Also let \(K \subset \Sigma\setminus E\) be a smoothly bounded Runge compact domain and let \(E_0\) and \(\Lambda\) be a pair of disjoint (possibly empty) finite sets in the interior of \(K.\) If \(X: K \setminus E_0 \rightarrow \mathbb R^n\), \(n \geq 3\), is a complete conformal minimal immersion with finite total curvature, then for any \(\varepsilon > 0\) and any integer \(r \geq 0\) there is a conformal minimal immersion \(Y : \Sigma \setminus (E \cup E_0) \rightarrow \mathbb R^n\) satisfying the following conditions:

(i)

\(Y\) is complete and has finite total curvature.

(ii)

\(Y - X\) extends harmonically to \(K\) and \(|Y - X|< \varepsilon\) on \(K.\)

(iii)

\(Y - X\) vanishes at least to order \(r\) at every point of \(E_0 \cup \Lambda.\)

Theorem 1.2 has already been proved for \(n =3\) in the special case \(\Lambda = \emptyset\) by F. J. López [Trans. Am. Math. Soc. 366, No. 12, 6201–6227 (2014; Zbl 1304.53007)] and the case \(E_0 = \emptyset \) by A. Alarcón et al. [Calc. Var. Partial Differ. Equ. 58, No. 1, Paper No. 21, 20 p. (2019; Zbl 1407.53007)]. The methods in both these papers rely on the spinor representation formula for minimal surfaces in \(\mathbb R^3,\) while the proof of Theorem 1.2 uses sprays generated by the flows of complete vector fields on the null-quadric \(\{(z_1,\ldots, z_n) \in \mathbb C^n\setminus 0: \, \sum_{j=1}^n z_j^2 = 0 \},\) and can therefore be applied to dimensions \(n \geq 3.\)
As an application of Theorem 1.2, the authors also obtain an analogue of the Mittag-Leffler theorem for minimal surfaces (Theorem 1.3).
An interested reader may find more on this topic in the book [A. Alarcón et al., Minimal surfaces from a complex analytic viewpoint. Cham: Springer (2021; Zbl 1520.53001)].