Proper superminimal surfaces of given conformal types in the hyperbolic four-space (Q6173849)

Let \(M\) be an oriented immersed surface inside an oriented Riemannian four-manifold and consider any \(x\in M\). Any normal tangent vector \(n\in N_x\) acts on \(v\) through the associated shape operator \(S_n:T_xM\to T_xM\), so we may consider the curve \(I_x(v)=\{S_x(n)v: n\in N_x, |n|=1\}\subset T_xM\). Now \(M\) is called superminimal of positive (negative) spin if, for every \(x\in M\) and every \(v\in T_xM\), \(I_x(v)\) is a circle (potentially of radius \(0\)) inside \(T_xM\) and the map \(n\mapsto S_n(v)\) is orientation-preserving (-reversing). This paper concerns immersed superminimal surfaces inside a very special Riemannian four-manifold, namely hyperbolic four-space \(H\). This is a quaternionic Kähler manifold, hence admits a twistor bundle \(\pi: Z\to H\), where \(Z\) is a complex three-fold. The Riemannian geometry of \(H\) is encoded in the complex geometry of \(Z\), a fact which is exploited in this paper. Concretely, the author relies on a result of \textit{R. L. Bryant} [J. Differ. Geom. 17, 455--473 (1982; Zbl 0498.53046)] which transforms the study of superminimal surfaces in \(H\) into that of holomorphic Legendrian curves in \(Z\) (which admits a holomorphic contact structure). Moreover, both \(H\) and its twistor space \(Z\) admit very concrete and simple descriptions by taking a quaternionic viewpoint, which the author clearly explains and subsequently exploits. If \(R\) is a compact Riemann surface and \(\{\Delta_i\}\) a finite set of pairwise disjoint compact disks in \(R\), the surface \(M=R\setminus\bigcup_i \Delta_i\) is known as a bordered Riemann surface. Its closure \(\bar M\) is called a compact bordered Riemann surface. With these definitions, the main result of the paper is that every conformal, superminimal immersion of a compact bordered Riemann surface \(\bar M\) into \(H\) can be approximated uniformly on compacts in \(M\) by \textit{proper} conformal superminimal immersions \(M\to H\). These approximations can moreover be chosen such that they agree with the original immersion to a given finite order at finitely many points in \(M\). As a corollary, every bordered Riemann surface is the conformal structure of a properly immersed superminimal surface inside \(H\). The novelty of this result lies in the fact that not only the topology but also the conformal type of the surfaces is controlled, and the main technical contribution consists of the development of the so-called Riemann-Hilbert modification technique for holomorphic Legendrian curves in \(\mathbb{C}\mathrm{P}^3\). The author points out that it is not known whether the approximating maps \(M\to H\) can be chosen so they extend to \(\bar M\), or whether similar results can be proven if \(H\) is replaced by a different quaternionic Kähler structure on the 4-ball (which admits an infinite-dimensional space of them).