The Calabi–Yau problem for Riemann surfaces with finite genus and countably many ends

Abstract

In this paper, we show that if R is a compact Riemann surface and M=R\setminus\bigcup_i D_i is a domain in R whose complement is a union of countably many pairwise disjoint smoothly bounded closed discs, D_i , then there is a complete conformal minimal immersion X\colon M\to\mathbb{R}^3 , extending to a continuous map X\colon \overline M\to\mathbb{R}^3 , such that X(bM)=\bigcup_i X(bD_i) is a union of pairwise disjoint Jordan curves. In particular, M is the complex structure of a complete bounded minimal surface in \mathbb{R}^3 . This extends a recent result for finite bordered Riemann surfaces.

Schwarz-Pick lemma for harmonic maps which are conformal at a point

“…However, assuming that |f | ≤ c for some 0 < c < 1, it is optimal up to a constant. In this connection, we wish to mention that for any n ≥ 3 there exist bounded conformal minimal immersions M → R n from any bordered Riemann surface (even with infinitely many boundary components) which are complete, in the sense that the intrinsic distance to any boundary point is infinite, which also implies that the area of the image is infinite; see [1,2] and the references therein. This exposes the interesting question whether the asymptotic rate of growth of the area, given by the right hand side of (2.4) in the case when M is the disc, can be achieved.…”

“…If the map f is harmonic then ω is a holomorphic function (see (7.2)). This is not the case for the Beltrami coefficient µ from the Beltrami equation 2 , this implies that f preserves the orientation at every point where f z = 0; such maps are called sense preserving. A fundamental result of quasiconformal theory says that for every measurable function µ on D with µ ∞ < 1 there exists a quasiconformal homeomorphism f : D → Ω onto a given simply connected domain Ω C satisfying the Beltrami equation f z = µf z .…”

Minimal surfaces in Euclidean spaces by way of complex analysis

“…By this method, the Calabi-Yau property has been established in several geometries: for holomorphic curves in complex manifolds [5], holomorphic null curves in C and conformal minimal surfaces in R for ≥ 3 [7,4,10], holomorphic Legendrian curves in complex contact manifolds [14,8], and superminimal surfaces in self-dual or antiself-dual Einstein 4-manifolds [37]. For a survey and further references, see [16,Sect.…”

“…In this case, we actually showed that any conformal minimal immersion → R can be approximated uniformly on by a map as in the theorem. The general case for countably many ends was obtained in [10]; an approximation theorem also holds in that case.…”

A strong parametric h-principle for complete minimal surfaces

“…In view of (C j−1 ) there is a number ǫ j > 0 satisfying (D j ); use the Cauchy estimates and see [7,Section 2]. By (4.2), (4.3), and (A j−1 ), Lemma 3.1 applies with K j and K j−1 and furnishes us with a homotopy ũt :…”