key: cord-0444490-9u4dx4dp authors: Forstneric, Franc title: Minimal surfaces in Euclidean spaces by way of complex analysis date: 2021-08-30 journal: nan DOI: nan sha: ee7ab85ef6c644828a266d167a85d15c45b678a7 doc_id: 444490 cord_uid: 9u4dx4dp This is an expanded version of my plenary lecture at the 8th European Congress of Mathematics in Portorov{z} on 23 June 2021. The main part of the paper is a survey of recent applications of complex-analytic techniques to the theory of conformal minimal surfaces in Euclidean spaces. New results concern approximation, interpolation, and general position properties of minimal surfaces, existence of minimal surfaces with a given Gauss map, and the Calabi-Yau problem for minimal surfaces. To be accessible to a wide audience, the article includes a self-contained elementary introduction to the theory of minimal surfaces in Euclidean spaces. Minimal surfaces are among the most beautiful and aesthetically pleasing geometric objects. These are surfaces in space which locally minimize area, in the sense that any small enough piece of the surface has the smallest area among surfaces with the same boundary. From the physical viewpoint, these are surfaces minimizing tension, hence in equilibrium position. They appear in a variety of applications to engineering, biology, architecture, and others. The subject has a luminous history, going back to 1744 when Leonhard Euler [32] showed that pieces of the surface now called catenoid (see Example 2.7) have smallest area among all surfaces of rotation in the 3-dimensional Euclidean space R 3 . The catenoid derives it name from catenary, the curve that an idealized hanging chain assumes under its own weight when supported only at its ends. The model catenary is the graph of the hyperbolic cosine function = cosh , and a catenoid is obtained by rotating this curve around the -axis in the ( , , )-space. Topologically, a catenoid is a cylinder, and as a conformal surface it is the puncture plane C * = C \ {0}. From mathematical viewpoint, the catenoid is one of the most paradigmatic examples of minimal surfaces, and it appears in several important classification results and in proofs of major theorems. The subject of minimal surfaces was put on solid footing by Joseph-Louis Lagrange who developed the calculus of variations during 1760-61, thereby reducing the problem of finding stationary points of functionals to a second order partial differential equation, now called Lagrange's equation. His work was published in 1762 by Accademia delle scienze di Torino [51, 50] and is available in his collected works [52] . In the second paper [50] , Lagrange applied his new method to a variety of problems in physics, dynamics, and geometry. In particular, he derived the equation of minimal graphs. The term minimal surface has since been used for a surface which is a stationary point of the area functional. The question whether a domain in a minimal surface truly minimizes the area among nearby surfaces with the same boundary can be analysed by considering the second variation of area. It was later shown that a minimal graph in R 3 over a compact convex domain in R 2 is an absolute area minimizer, and hence small enough pieces of any minimal surface are area minimizers. In 1776, Jean Baptiste Meusnier [65] discovered that domains in a surface in R 3 are minimal in the sense of Lagrange if and only if the surface has vanishing mean curvature at every point. He also described the second known minimal surface, the helicoid; see Example 2.8. It is obtained by a line in 3-space rotating at a constant rate as it moves at a constant speed along the axis of rotation, which is perpendicular to the rotating line. Helicoid is the geometric shape of a device known as Archimedes' screw (or the water screw, screw pump, or Egyptian screw), named after Greek philosopher and mathematician Archimedes who described it around 234 BC on the occasion of his visit to Egypt. There is evidence that this device had been used in ancient Egypt much earlier. The helicoid is sometimes called "double spiral staircase" -each of the two half-lines sweeps out a spiral staircase, and these two staircases only meet along the axis of rotation. Therefore, its physical model is a convenient device for letting people ascend and descend a staircase without the two crowds meeting in-between. From a different field, DNA molecules assume the shape of a helicoid. Topologically and conformally the helicoid is the plane. Its name derives from helix -for every point on the helicoid, there is a helix (a spiral curve) contained in the helicoid which passes through that point. The helicoid plays a major role in the classification of properly embedded minimal surfaces in R 3 ; see the survey paper [28] by Tobias H. Colding and William P. Minicozzi. Minimal surfaces appear naturally in the physical world. Laws of physics imply that a soap film spanned by a given frame (i.e., a closed Jordan curve) is a minimal surface. The reason is that this shape minimizes the surface tension and puts it in equilibrium position. Soap films, bubbles, and surface tension were studied by the Belgian physicist Joseph Plateau in the 19th century. Based on his experiments, Karl Weierstrass formulated in 1873 the Plateau problem, conjecturing that any closed Jordan curve in R 3 spans a minimal surface (in fact, a minimal disc). This was confirmed by Tibor Radó [70, 71] (1930) and Jesse Douglas [31] (1931) . For his work on the Plateau problem, Douglas received one of the first two Fields Medals at the International Congress of Mathematicians in Oslo in 1936. Half a century later, it was shown that the disc of smallest area with given boundary curve (the Douglas-Morrey solution of the Plateau problem) has no branch points; see the monograph by Anthony Tromba [76] . Furthermore, if the curve lies in the boundary of a convex domain in R 3 then the solution is embedded according to William H. Meeks and Shing Tung Yau [62, 63] . Minimal surfaces are also studied in more general Riemannian manifolds of dimension at least three. Holomorphic curves in complex Euclidean spaces C for > 1, or in any complex Kähler manifold of complex dimension at least two, are special but important examples of minimal surfaces. As pointed out by Colding and Minicozzi [28] , there are several fields where minimal surfaces are actively used in understanding physical phenomena. In particular, they come up in the study of compound polymers, protein folding, etc. They also play a prominent role in art, especially in architecture. The connection between minimal surfaces in Euclidean spaces and complex analysis has been known since mid-19th century. The basic fact is that a conformal immersion : → R from a Riemann surface parameterizes a minimal surface if and only if the map is harmonic (see Theorem 2.1); equivalently, the complex derivative / in any local holomorphic coordinate on is holomorphic. Furthermore, the immersion is conformal if and only if / assumes values in the null quadric A ⊂ C , given by the equation 2 1 + 2 2 + · · · + 2 = 0 (see (2.23) ), and / ≠ 0 if is an immersion. This leads to the Enneper-Weierstrass representation of any conformally immersed minimal surface → R as the real part of the integral of a holomorphic map : Theorem 2.6) . The period vanishing conditions on along closed curves in ensure that the integral is welldefined. The formula is most concrete in dimension = 3 (see (2.25) ) due to an explicit 2-sheeted parameterization of the null quadric A ⊂ C 3 by C 2 . This connection between minimal surfaces and holomorphic maps was used by Bernhard Riemann around 1860 in his construction of properly embedded minimal surfaces in R 3 , now called Riemann's minimal examples [72] (see the paper [59] by William H. Meeks and Joaquín Pérez), and in numerous further works by other authors. It was popularized again in modern times by Robert Osserman [68] . Despite the long and illustrious history of the subject, the author in collaboration with Antonio Alarcón, Francisco J. López and others obtained in the last decade a string of new results by exploiting the Enneper-Weierstrass representation. The main point in our approach is that the punctured null quadric A * is a complex homogeneous manifold, hence an Oka manifold, a notion introduced in [35] and treated in [36, Chapter 5] . This implies that holomorphic maps from any open Riemann surface (and, more generally, from any Stein manifold, that is, a closed complex submanifold of a complex Euclidean space C ) to A * satisfy the Runge-Mergelyan approximation theorem and the Weierstrass interpolation theorem in the absence of topological obstructions. Together with methods of convexity theory, this gave rise to many new constructions of conformal minimal surfaces with interesting properties; see Theorem 3.1. By using parametric versions of these results, it was possible to determine the rough topological shape (i.e., the weak or strong homotopy type) of the space of nonflat conformal minimal immersions from any given open Riemann surface into R (see Theorem 3.2) . It was also shown that every natural candidate is the Gauss map of a conformal minimal surface in R (see Theorem 3.3) . Another complex analytic technique, which has recently had a major impact on the field, is an adaptation of the classical Riemann-Hilbert boundary value problem to conformal minimal surfaces and holomorphic null curves in Euclidean spaces. This led to an essentially optimal solution of the Calabi-Yau problem for minimal surfaces, originating in conjectures of Eugenio Calabi from 1965; see Theorems 3.5 and 3.6. This technique was also used in the construction of complete proper minimal surfaces in minimally convex domains of R (see [16, Chapter 8] ). The recent results presented in Section 3 are carefully explained in the monograph [16] published in March 2021. The corresponding developments on non-orientable minimal surfaces are described in the AMS Memoir [12] from 2020. It is needless to say that both these publication contain many other results not mentioned here. In 2021, David Kalaj and the author [34] obtained an optimal Schwarz-Pick lemma for conformal minimal discs in the ball of R and introduced the notion of hyperbolicity of domains in R , in analogy with Kobayashi hyperbolicity of complex manifolds. This new topic is currently being developed, and it is too early to include it here. To make the article accessible to a wide audience including advanced undergraduate students of Mathematics, we present in this section a self-contained introduction to the theory of minimal surfaces in Euclidean spaces. We assume familiarity with elementary calculus, topology, and rudiments of complex analysis; however, no a priori knowledge of differential geometry is expected. We shall use the fact that metric-related quantities such as length, area, and curvature of curves and surfaces in a Euclidean space R are invariant under translations and orthogonal maps of R ; these are the isometries of the Euclidean metric, also called rigid motions. For simplicity of presentation, we focus on minimal surfaces parameterized by plane domains, although the same methods apply on an arbitrary open Riemann surface. More complete treatment is available in a number of texts; see [54, 68, 20, 67, 26, 30, 57, 58, 16] , among others. For the theory of non-orientable minimal surfaces, see [12] . From the physical viewpoint, the most natural parameterization of a minimal surface is by a conformal map (from a plane domain, or a conformal surface). A conformal parameterization minimizes the total energy of the map and makes the tension uniformly spread over the surfaces. We give a brief introduction to the subject of conformal maps, referring to [16, Sections 1.8-1.9] for more details and further references. Let be a domain in R 2 with coordinates ( , ). A C 1 map : → R ( ≥ 2) is an immersion if the partial derivatives = / and = / are linearly independent at every point of . An immersion is said to be conformal if its differential at any point ∈ preserves angles. It is elementary to see (cf. [ Here, x · y denotes the Euclidean inner product between vectors x, y ∈ R and |x| = √ x · x is the Euclidean length of x. A smooth map : → R (of class C 1 , not necessarily an immersion) is called conformal if (2.1) holds at each point. It clearly follows that has rank zero at non-immersion points. Let be a topological surface. A conformal structure on is given by an atlas U = {( , )} ∈ with charts : → ⊂ R 2 whose transition maps are conformal diffeomorphisms of plane domains. Identifying R 2 with the complex plane C, each map , is biholomorphic or anti-biholomorphic. A surface endowed with a conformal structure (more precisely, with an equivalence class of conformal structures) is a conformal surface. If is orientable, then by choosing the charts in a conformal atlas to preserve orientation, the transition maps , are biholomorphic; hence, U is a complex atlas and ( , U) is a Riemann surface. A connected nonorientable conformal surface admits a two-sheeted conformal covering → by a Riemann surface . Assume now that is a Riemannian metric on a smooth surface , i.e., a smoothly varying family of scalar products on tangent spaces , ∈ . In any local coordinate ( , ) on , the metric has an expression where the coefficient functions , , satisfy − 2 > 0. A local chart ( , ) is said to be isothermal for if the above expression simplifies to for some positive function . An important result, first observed by Carl Friedrich Gauss, is that in a neighbourhood of any point of there exist smooth isothermal coordinates. One way to obtain such coordinates is from solutions of the classical Beltrami equation. We refer to [16, Secs. 1.8-1.9] for a more precise statement and references. Since the transition map between any pair of isothermal charts is a conformal diffeomorphism, we thus obtain a conformal atlas on consisting of isothermal charts. The upshot is that every Riemannian metric on a smooth surface determines a conformal structure. Furthermore, a pair of Riemannian metrics ,˜ on determine the same conformal structure if and only if˜ = for a smooth positive function on . Denote by x = ( 1 , . . . , ) the Euclidean coordinates on R and by 2 = 2 1 + · · · + 2 the Euclidean metric. If = ( 1 , . . . , ) : → R is a smooth immersion, then is a Riemannian metric on , called the first fundamental form. By the definition of , the map : ( , ) → (R , 2 ) is an isometric immersion. By what has been said, determines a conformal structure on (assuming now that is a surface), and in this structure the map is a conformal immersion. More precisely, ( , ) is conformal in any isothermal local coordinate ( , ) on . This shows that any immersion : → R from a smooth surface determines a unique conformal structure on which makes a conformal immersion. If in addition is oriented, we get the structure of a Riemann surface. Results of conformality theory imply that if is a domain in R 2 and : → R is an immersion, then there is a diffeomorphism : ′ → from another domain ′ ⊂ R 2 such that the immersion • : ′ → R is conformal. In particular, if is the disc then we may take ′ = . The same arguments and conclusions apply to immersions of a smooth surface into an arbitrary Riemannian manifold ( ,˜ ) in place of (R , 2 ). Assume that ⊂ R 2 ( , ) is a bounded domain with piecewise smooth boundary and : → R is a smooth immersion. Precomposing with a diffeomorphism from another such domain in R 2 , we may assume that is conformal; see (2.1). We consider the area functional and the Dirichlet energy functional We have elementary inequalities which are equalities if and only if x, y is a conformal frame, i.e., |x| = |y| and x · y = 0. Applying this to the vectors x = and y = gives Area( ) ≤ D ( ), with equality if and only if is conformal. Hence, these two functionals have the same critical points on the set of conformal immersions. It is elementary to find critical points of these functional. The calculation is simpler for the Dirichlet functional D, but the expression for the first variation is the same for both functionals at a conformal map . Assuming that : → R is a smooth map vanishing on , the first variation of D at in direction equals For completeness, we also calculate the first variation of area at a conformal immersion . Let : → R be as above. Consider the expression under the integral (2.2) for the map = + , ∈ R. Taking into account (2.1) we obtain and therefore (We integrated by parts and used that | = 0. The factor 2| | 2 also appears in the denominator when differentiating the expression for Area( + ) at = 0, so this term cancels.) Comparing with (2.4), we see that The same result holds on any compact domain with piecewise smooth boundary in a conformal surface . A conformal diffeomorphism changes the Laplacian by a multiplicative factor, so there is a well-defined notion of a harmonic function on . In this section, we prove a result due to Meusnier [65] which characterizes minimal surfaces in terms of vanishing mean curvature; see Theorem 2.3. To explain the notion of curvature of a smooth plane curve ⊂ R 2 at a point ∈ , we apply a rigid change of coordinates in R 2 taking to (0, 0) and the tangent line to the -axis, so locally near (0, 0) the curve is the graph = ( ) of a smooth function on an interval around 0 ∈ R, with (0) = ′ (0) = 0. Therefore, Let us find the circle which agrees with this graph to the second order at (0, 0). Clearly, such a circle has centre on the -axis, so it is of the form 2 + ( − ) 2 = 2 for some ∈ R \ {0}, unless ′′ (0) = 0 when the -axis (a circle of infinite radius) does the job. Solving the equation on near (0, 0) gives is the unique number for which the circle agrees with the curve (2.5) to the second order at (0, 0). This best fitting circle is called the osculating circle. The number is the signed curvature of the curve (2.5) at (0, 0), its absolute value | | = | ′′ (0)| ≥ 0 is the curvature, and | | = 1/| | = 1/| ′′ (0)| is the curvature radius. If ′′ (0) = 0 then the curvature is zero and the curvature radius is +∞. Consider now a smooth surface ⊂ R 3 . Let ( , , ) be coordinates on R 3 . Fix a point ∈ . A rigid change of coordinates gives = (0, 0, 0) and Then, is locally near the origin a graph of the form The symmetric matrix is called the Hessian matrix of at (0, 0). Given a unit vector = ( 1 , 2 ) in the ( , )-plane, let Σ be the 2-plane through 0 ∈ R 3 spanned by and the -axis. The intersection := ∩ Σ is then a planar curve contained in , given by for ∈ R near 0. Since | | = 1, the parameters ( , ) on Σ are Euclidean parameters, i.e., the Euclidean metric 2 on R 3 restricted to the plane Σ is given by 2 + 2 . From our discussion of curves and the formula (2.6), we infer that the number is the signed curvature of the curve at the point (0, 0). On the unit circle | | 2 = 2 1 + 2 2 = 1 the quadratic form ↦ → · reaches its maximum 1 and minimum 2 ; these are the principal curvatures of the surface (2.7) at (0, 0). Since is symmetric, 1 and 2 are its eigenvalues. The real numbers are, respectively, the mean curvature and the Gaussian curvature of at (0, 0, 0). Note that the trace of (2.8) equals the Laplacian Δ (0, 0). On the other hand, the trace of a matrix is the sum of its eigenvalues. This implies (2.11) Let be a domain in R 2 . If : → R is a smooth conformal immersion, then for every ∈ the vector Δ ( ) is orthogonal to the plane (R 2 ) ⊂ R . Equivalently, the following identities hold on : Proof. Recall from (2.1) that is conformal if and only if · = · and · = 0. Differentiating the first identity on and the second one on yields Likewise, differentiating the first identity on and the second one on gives Δ · = 0. We can now prove the following result due to Meusnier [65] . Proof. Fix a point 0 ∈ ; by a translation of coordinates we may assume that 0 = (0, 0) ∈ R 2 . Since the differential (0,0) : R 2 → R 3 is a conformal linear map, we may assume up to a rigid motion on R 3 that (0, 0) = (0, 0, 0) and 14) The implicit function theorem shows that there is a neighbourhood ⊂ of the origin such that the surface = ( ) is a graph = ( , ) with (0,0) = 0, so is of the form (2.7). Since the immersion is conformal, (2.12) shows that Δ is orthogonal to the ( , )-plane R 2 × {0} at the origin, which means that We now calculate Δ (0, 0). Differentiation of ( , ) = ( ( , ), ( , )) gives At the point (0, 0), taking into account (2.13) and = = 0 we get = 2 . A similar calculation gives = 2 at (0, 0), so we conclude that where is the mean curvature of at the origin (see (2.11)). Denoting by N = (0, 0, 1) the unit normal vector to at 0 ∈ R 3 , it follows from (2.14), (2.15) and (2.16) that holds at (0, 0) ∈ . In particular, Δ = 0 if and only if = 0. This formula is clearly independent of the choice of a Euclidean coordinate system. Although we used conformal parameterizations, neither curvature nor area depend on the choice of parameterization. This motivates the following definition. [54, 68, 16] or any other standard source for the details. In this section we explain the Enneper-Weierstrass formula, which provides a connection between holomorphic maps → C with special properties from domains ⊂ C and conformal minimal immersions → R for ≥ 3. The same connection holds more generally for maps from any open Riemann surface. Let = + be a complex coordinate on C. Let us recall the following basic operators of complex analysis, also called Wirtinger derivatives: The differential of a function ( ) can be written in the form Note that is the C-linear part and ¯ ¯ is the C-antilinear part of . In particular, / ¯ = 0 holds for holomorphic functions, and / = 0 holds for antiholomorphic ones. In terms of these operators, the Laplacian equals Hence, a function : → R is harmonic if and only if / is holomorphic. It follows that a smooth map / is holomorphic and the component functions have no common zero. Furthermore, conformality of is equivalent to the following nullity condition: Comparing with the conformality conditions (2.1) proves the claim. Since we know by Theorem 2.1 that a conformal immersion is harmonic if and only it parameterizes a minimal surface, this gives the following result. determines a conformal minimal immersion : → R given by for any base point 0 ∈ and vector ∈ R . Conditions (2.20) guarantee that the integral in (2.21) is well-defined, that is, independent of the path of integration. The imaginary components of the periods define the flux homomorphism : 1 ( , Z) → R on the first homology group of . Indeed, by Green's formula the period ∮ only depends on the homology class [ ] ∈ 1 ( , Z) of a closed path ⊂ . Remark (The first homology group). If is a domain in R 2 C then its first homology group 1 ( , Z) is a free abelian group Z ℓ (ℓ ∈ {0, 1, 2, . . . , ∞}) with finitely or countably many generators. If is bounded, connected, and its boundary consists of 1 Jordan curves Γ 1 , . . . , Γ 1 and 2 isolated points (punctures) 1 , . . . , 2 , then the group 1 ( , Z) has ℓ = 1 + 2 − 1 generators which are represented by loops in based at any given point 0 ∈ , each surrounding one of the holes of . (By a hole, we mean a compact connected component of the complement C \ . A hole which is an isolated point of C \ is called a puncture.) Indeed, if Γ 1 is the outer boundary curve of , then every other boundary curve Γ 2 , . . . , Γ 1 of is contained in the bounded component of C \ Γ 1 , so it bounds a hole of . Likewise, each of the points It is clear from Theorem 2.6 that the following quadric complex hypersurface in C plays a special role in the theory of minimal surfaces in R : This is called the null quadric in C , and A * = A \ {0} is the punctured null quadric. Note that A is a complex cone with the only singular point at 0. Theorem 2.6 says that we get all conformal minimal surfaces → R as integrals of holomorphic maps : → A * ⊂ C satisfying the period vanishing conditions (2.20). The Enneper-Weierstrass representation in R 3 . In dimension = 3, the null quadric A admits a 2-sheeted quadratic parameterization : C 2 → A given by This map is branched at 0 ∈ C 2 , and : C 2 \ {0} → A * is a 2-sheeted holomorphic covering map. It follows that every conformal minimal immersion = ( 1 , 2 , 3 ) : → R 3 can be written in the following form (see [68] or [16, pp. 107-108]): Here, = = ( 1 , 2 , 3 ), and is a holomorphic map to the Riemann sphere (a meromorphic function on ), called the complex Gauss map of . Identifying CP 1 with the unit 2-sphere 2 ⊂ R 3 by the stereographic projection from the point (0, 0, 1) ∈ 2 , corresponds to the classical Gauss map N = × /| × | : → 2 of . Many important quantities and properties of a minimal surface are determined by its Gauss map. In particular, we have that Here, is the Gauss curvature function (2.10) of the metric * 2 and 2 CP 1 is the spherical metric on CP 1 . It follows that the total Gaussian curvature (see (2. ( , ) = (cos · cosh , sin · cosh , ) . (2.28) It is 2 -periodic in the variable, hence infinitely-sheeted. Introducing the variable = e − + ∈ C * , we pass to the quotient C/(2 Z) C * and obtain a single-sheeted parameterization : C * → R 3 having the Enneper-Weierstrass representation (2.30) Its Weierstrass representation in the complex coordinate = + ∈ C is [27] . Remark (Branch points). Our definition of a conformal map : → R of class C 1 ( ) requires that equations (2.1) hold. We have already observed that such a map has rank zero at non-immersion points. Assuming that is harmonic at immersion points, it follows that = / : → C is a continuous map with values in the null quadric A (2.23) which is holomorphic at immersion points of and vanishes at non-immersion points. By a theorem of T. Radó [69] There is a family of holomorphic curves in C which are close relatives of conformal minimal surfaces in R . A holomorphic map = ( 1 , . . . , ) : → C for ≥ 3 from a domain ⊂ C satisfying the nullity condition integrates to a holomorphic null curve where 0 ∈ is any given base point and ∈ C . Indeed, conditions (2.31) guarantee that the integral in (2.32) is independent of the choice of a path of integration. These period conditions are trivial on a simply connected domain . If = + : → C is an immersed holomorphic null curve, then its real part = ℜ : → R and imaginary part = ℑ : → R are conformal minimal surfaces which are harmonic conjugates of each other. Indeed, denoting the complex variable in C by = + , the Cauchy-Riemann equations imply satisfies the nullity condition (2.19), is a conformal minimal immersion. In the same way we find that = ′ = + = 2 , so is a conformal minimal immersion. Being harmonic conjugates, and are called conjugate minimal surfaces. Conformal minimal surfaces in the 1-parameter family = ℜ(e ) : → R , ∈ R are called associated minimal surfaces of the holomorphic null curve . Conversely, if : → R is a conformal minimal surface and the holomorphic map = 2 : → A −1 satisfies period vanishing conditions (2.31), then integrates to a holomorphic null curve : → C (2.32) with ℜ = . In general, the imaginary parts of the periods (2.32) determine the flux homomorphism 1 ( , Z) → R of the minimal surface (see (2.22) ); hence, is the real part of a holomorphic null curve if and only if it has vanishing flux. The periods (2.31) always vanish on a simply connected domain , and hence every conformal minimal immersion → R is the real part of a holomorphic null curve → C . The relationship between conformal minimal surfaces and holomorphic null curves extends to maps having (isolated) branch points. Hence, is a holomorphic null curve. Consider the 1-parameter family of its associated minimal surfaces in R 3 for ∈ [0, 2 ]: This section is a survey of recent results in the theory of minimal surfaces in Euclidean spaces, which were discussed in my lecture at 8 ECM. A detailed presentation is available in the monograph [16] and, for non-orientable surfaces, in the AMS Memoir [12] by Alarcón, López and myself. Holomorphic approximation is a central topic in complex analysis. Holomorphic functions and maps with interesting properties are often constructed inductively, exhausting the manifold by an increasing sequence of compact sets such that one can approximate holomorphic functions uniformly on each one by holomorphic functions on . The quintessential example is Runge's theorem from 1885 [74] on approximation of holomorphic functions on a compact set ⊂ C with connected complement by holomorphic polynomials. A major extension is Mergelyan's theorem [64] from 1951. In order to generalize Runge's theorem, we need the following concept. Denote by O( ) the algebra of holomorphic functions on a complex manifold . Given a compact set in , its O( )-convex hull (or holomorphic hull) is the set If = then is said to be holomorphically convex, or O( )-convex, or a Runge compact. If is the complex plane or, more generally, an open Riemann surface, then the hull is the union of and all relatively compact connected components of \ (the holes of in ). There is no topological characterization of the hull in higher dimensional complex manifolds. Holomorphically convex sets are the natural sets for holomorphic approximation. Runge's theorem was extended to open Riemann surfaces by H. Behnke and K. Stein [21] in 1949, who proved that any holomorphic function on a neighborhood of a Runge compact in open Riemann surface can be approximated uniformly on by holomorphic functions on . A related result on higher dimensional complex manifolds is the Oka-Weil theorem which pertains to Runge compacts in C and, more generally, in any Stein manifold (a closed complex submanifold of a Euclidean space C ). A recent survey of holomorphic approximation theory can be found in [33] . We have seen in Subsection 2.4 that every conformal minimal immersion → R from an open Riemann surface is the integral of a holomorphic map : → A * ⊂ C into the punctured null quadric A * ; furthermore, must satisfy the period vanishing conditions (2.20) . Hence, a Runge-type approximation theorem for conformal minimal surfaces in R (or holomorphic null curves in C ) reduces to the approximation problem for holomorphic maps : → A * satisfying the period vanishing conditions (2.20) (or (2.31) when considering null curves). This is a nonlinear approximation problem. The first part, ignoring the period conditions, fits within Oka theory. In particular, the manifold A * is easily seen to be a homogeneous space of the complex orthogonal group (C). Runge-type approximation theorems for holomorphic maps from Stein manifolds to complex homogeneous manifolds were proved by Hans Grauert [40] (1957) and Grauert and Kerner [41] (1963). More generally, a complex manifold is said to be an Oka manifold if and only if approximation results of this type hold for holomorphic maps → from any Stein manifold in the absence of topological obstructions. Oka theory also includes interpolation theorems for holomorphic maps, generalizing classical theorems of K. Weierstrass [77] and H. Cartan [22] . For the theory of Oka manifolds, see [36] . The second part of the problem, ensuring the period vanishing conditions (2.20) or (2.31) for holomorphic maps to A * , can be treated by using sprays of holomorphic maps together with elements of convexity theory. More precisely, Gromov's onedimensional convex integration lemma from [42] is useful in this regard. The main techniques underlying all subsequent developments were established in [6] (2014). Their application led to the following result, which is a summary of several individual theorems. Parts (i), (ii) and (iv) are due to Alarcón, López, and myself [6, 13, 12] (the special case of (i) for = 3 was obtained beforehand in [19] ), while (iii) was proved by Alarcón and Castro-Infantes [2, 3] . Related results for conformal minimal surfaces of finite total curvature were given by Alarcón and López [18] . (iii) In addition, one can prescribe the values of on any closed discrete subset of (Weierstrass-type interpolation). (iv) The analogous results hold for non-orientable minimal surfaces in R and for holomorphic null curves in C , ≥ 3. The proof of Theorem 3.1 is fairly complex, and we shall only outline the main idea. Fix a nowhere vanishing holomorphic 1-form on the open Riemann surface . (Such a 1-form always exists; see [43] .) By Enneper-Weierstrass (Theorem 2.6), it suffices to prove the Runge approximation theorem for holomorphic maps : → A * satisfying the period vanishing conditions (2.20) . Consider an inductive step. Assume that ⊂ are connected Runge compacts with piecewise smooth boundaries in , : → R is a conformal minimal surface, and = 2 / : → A * . We wish to approximate by a conformal minimal immersion : → R . We may assume that ( ) is not contained in a complex ray C * z of the null quadric A * , for otherwise the result is trivial. There are two main cases to consider, the noncritical case and the critical case. The noncritical case: there is no change of topology from to . It is well known that there are closed curves 1 , . . . , ℓ in forming a basis of 1 ( , Z) whose union = ℓ =1 is a Runge compact. Let B denote the unit ball of C . By using flows of holomorphic vector fields on C tangent to A, we construct a smooth map which is holomorphic on˚ × B , such that the associated period map is biholomorphic onto its image. Such period dominating spray can be found of the form where each is the flow of a holomorphic vector field tangent to A and ∈ O( ). We first construct smooth functions on which give a period dominating spray; this can be done since the convex hull of A equals C . As is Runge in , we can approximate the 's by holomorphic functions on , thereby obtaining a holomorphic period dominating spray as above. In the next key step, we use that A * is an Oka manifold, so we can approximate by a holomorphic map : × B ℓ → A * . (There is no topological obstruction since A * is connected.) If the approximation is close enough, the implicit function theorem furnishes a parameter value˜ ∈ B ℓ close to 0 such that the map˜ = (· ,˜ ) : → A * has vanishing real periods on the curves 1 , . . . , ℓ . Hence, fixing a point 0 ∈ , the map : → R given by is a conformal minimal immersion which approximates : → R on . The critical case. Assume now that is an embedded smooth arc in \˚ attached with its endpoints to such that ∪ is a deformation retract of . (Thus, has the same topology as ∪ . This situation arises when passing a critical point of index 1 of a strongly subharmonic Morse exhaustion function on .) Let , ∈ denote the endpoints of . We extend smoothly across to a map : ∪ → A * such that This is possible since the convex hull of A * equals C . We then proceed as in the noncritical case: embed into a period dominating spray of smooth maps ∪ → A * which are holomorphic on˚ = \ , approximate it by a holomorphic spray on by Mergelyan's theorem, and pick a parameter value for which the map in the spray has vanishing real periods on ∪ , and hence on . The Enneper-Weierstrass formula gives a conformal minimal surface : → R approximating on . The proof of the basic approximation theorem (i) (without the properness condition) is then completed by induction on a suitable exhaustion of by Runge compacts, alternatively using the above two cases. Critical points of index 2 do not arise. Interpolation (part (iii)) is easily built into the same inductive construction. Indeed, in each of the two cases considered above, we can arrange that none of the points ∈ at which we wish to interpolate lies on the boundary of or . By choosing the functions in the spray (3.1) to vanish at those points which lie in the interior of , we ensure that the spray is fixed at these points (independent of the parameter ), and hence the approximating map will agree with at these points. For each of the finitely many points ∈˚ \ we choose a smooth embedded arc ⊂ \˚ with one endpoint and the other endpoint ∈ such that \ { } ⊂ \ and these arcs are pairwise disjoint. The set = ∪ is then a Runge compact. We extend the map : → A * smoothly to such that for each , ∫ has the correct value which ensures that the integral assumes the prescribed value at . It remains to apply the same method as above with a spray which is period dominating also on each of the arcs and to use Mergelyan approximation on the set . Properness of the approximating conformal minimal immersion : → R (part (ii) of the theorem) requires considerable additional work. The main point is to prove a relative version of the approximation theorem in part (i) in which all but two components of the given map extend to harmonic functions on all of . One can keep these components fixed while approximating the remaining two components such that the resulting map is a conformal minimal immersion. This requires a more precise version of the Oka principle. This result is then used in an inductive scheme which is designed so that | ( )| tends to infinity as the point ∈ goes to the ideal boundary of (i.e., it exists any compact subset). Finally, the general position theorem in part (ii) uses the same technique together with the transversality theorem. The details of proof are considerably more involved from the technical viewpoint, and we shall not deal with this subject here. is an open Riemann surface. Fix a nowhere vanishing holomorphic 1-form on . Let ≥ 3. An immersion → R is said to be nonflat if its image is not contained in an affine 2-plane. We introduce the following notation: • O( , A * ) and C( , A * ) denote spaces of holomorphic and continuous maps → A * , respectively. • CMI( , R ) denotes the space of conformal minimal immersions → R . • CMI nf ( , R ) is the subspace of CMI( , R ) consisting of nonflat immersions. • NC( , C ) is the space of holomorphic null immersions → C . • NC nf ( , C ) is the subspace of NC( , C ) consisting of nonflat immersions. Consider the commutative diagram • the maps : NC nf ( , C ) → O( , A * ) and : CMI nf ( , C ) → O( , A * ) are given by ↦ → / and ↦ → 2 / , respectively; • the map NC nf ( , C ) → ℜNC nf ( , C ) is the projection = + ↦ → ; • the maps : ℜNC nf ( , C ) ↩→ CMI nf ( , R ) and : O( , A * ) ↩→ C( , A * ) are the natural inclusions. Recall that a continuous map : → between topological spaces is said to be a weak homotopy equivalence if it induces a bijection of path components of the two spaces and, for each integer ∈ N, an isomorphism ( ) : ( ) → ( ) of their -th homotopy groups. The map is a homotopy equivalence if there is a continuous map : → such that • : → is homotopic to the identity on and • : → is homotopic to the identity on . These notions indicate that the spaces and have the same rough topological shape. Since A * is an Oka manifold, the inclusion : O( , A * ) ↩→ C( , A * ) is a weak homotopy equivalence by the Oka-Grauert principle (see [36, Corollary 5.5.6] ), and by Lárusson [53] it is a homotopy equivalence if is of finite topological type, i.e., if the homology group 1 ( , Z) is a finitely generated abelian group. The real-part projection map ℜ : NC nf ( , C ) → ℜNC nf ( , C ) is evidently a homotopy equivalence. It turns out that all other maps in the above diagram are also weak homotopy equivalences. The first part of the following theorem was proved by Lárusson and myself in [38] , and the second part was proved by Alarcón, López and myself in [15] . Validity of statement (a) for CMI( , R ) and NC( , C ) remains an open problem. (a) Each of the maps , , in the above diagram is a weak homotopy equivalence, and a homotopy equivalence if is of finite topological type. (b) The map • : CMI( , R ) → C( , A * ) induces a bijection of path components of the two spaces. Hence, It follows that each of the spaces NC nf ( , C ) and CMI nf ( , C ) is weakly homotopy equivalent to the space C( , A * ) of continuous maps → A * , and is homotopy equivalent to C( , A * ) if the surface has finite topological type. The group Z 2 = {0, 1}, which appears in part (b), is the fundamental group of the punctured null quadric A * ⊂ C 3 ; see (2.24) and note that C 2 \ {0} is simply connected. If ∈ CMI( , R 3 ) then / : → A * maps every generator of the homology group 1 ( , Z) either to the generator of 1 (A * ) or to the trivial element. This gives 2 ℓ choices, each one determining a connected component of CMI( , R 3 ). The null quadric A * ⊂ C for > 3 is simply connected. These results are proved by using the parametric versions of techniques discussed in Subsection 3.1. Each of the maps in question satisfies the parametric h-principle, which implies that it is a weak homotopy equivalence. The Gauss map is of major importance in the theory of minimal surfaces. We have already seen that the Gauss map of a conformal minimal immersion : → R 3 is a holomorphic map : → CP 1 (2.26), which coincides with the classical Gauss map → 2 under the stereographic projection from 2 onto CP 1 . In general for any dimension ≥ 3 one defines the generalized Gauss map of a conformal minimal immersion = ( 1 , 2 , . . . , ) : → R as the Kodaira-type holomorphic map where is the projectivization of the punctured null quadric A * , a smooth quadric complex hypersurface in CP −1 . A recent discovery is the following converse result from [15] (see also [16, Theorem 5.4.1] ), which shows that every natural candidate is the Gauss map of a conformal minimal surfaces. Main Theorem 3.3. Assume that ≥ 3. Here, A −1 ( , −2 ) denotes the space of maps → −2 of class C −1 which are holomorphic in the interior \ of . Furthermore, the following assertions hold true in both cases in the above theorem. −→ CP 1 . This is the complex Gauss map (2.26) of , which appears in the Enneper-Weierstrass representation (2.25). The generalized Gauss map G : 2) of is then expressed by G = • , where : CP 1 → 1 is given by (3.3) . Let us say a few words about the proof of Theorem 3.3. The first step is to lift the given map G : → to a holomorphic map : → A * . Note that the natural projection A * → sending ( 1 , . . . , ) to [ 1 : · · · : ] is a holomorphic fibre bundle with fibre C * = C \ {0}. The existence of a continuous lifting follows by noting that the homotopy type of is a wedge of circles, and every oriented C * -bundle over a circle is trivial. Further, since C * is an Oka manifold, every continuous lifting is homotopic to a holomorphic lifting according to the Oka principle [36, Corollary 5.5.11] . In the second and main step of the proof, the holomorphic map : → A * is multiplied by a nowhere vanishing holomorphic function ℎ : → C * such that the product = ℎ : → A * has vanishing periods along closed curves in (see (2.31)), and hence it integrates to a holomorphic null immersion : → C . Its real part = ℜ : → R is then a conformal minimal immersion having the Gauss map G. The construction of such a multiplier ℎ follows the idea of proof of Theorem 3.1, but the details are fairly nontrivial and we refer to the cited works. There are many results in the literature relating the behaviour of a minimal surface to properties of its Gauss map. A particularly interesting question is how many hyperplanes in a general position in CP −1 can be omitted by the Gauss map of a complete conformal minimal surface of finite total curvature. A discussion this topic can be found in [16, Chapter 5] and in several other sources. A smooth immersion : → R is said to be complete if * 2 is a complete metric on . Equivalent, for every divergent path : [0, 1) → (i.e., such that ( ) leaves every compact set in as → 1) the image path • : [0, 1) → R has infinite Euclidean length. Clearly, if is proper then it is complete since any such path • ( ) diverges to infinity as → 1. The converse is not true; it is easy to construct complete immersions (and embeddings if ≥ 3) with bounded image ( ) ⊂ R . It is however not so easy to find complete bounded immersions with additional properties, such as conformal minimal or, in case when the target is a complex Euclidean space C , holomorphic. The following conjecture was posed by Eugenio Calabi in 1965, [49, p. 170 ]. Calabi's conjecture was also promoted by S. S. Chern [24, p. 212] . A particular reason which may have led Calabi to propose these conjectures was the theorem of S. S. Chern and R. Osserman [25] from that time. Their result says in particular that if : → R ( ≥ 3) is a complete conformal minimal surface of finite total Gaussian curvature TC( ) > −∞, then is the complement of finitely many points 1 , . . . , in a compact Riemann surface , the holomorphic 1-form has an effective pole at each point , and is proper. (The first statement holds even without the completeness assumption on , due to a result of Huber [46] from 1957.) The Chern-Osserman theorem says that such is complete if and only if has an effective pole at each puncture . The asymptotic behaviour of at the punctures was described by M. Jorge and W. Meeks [47] in 1983. It turns out that, at least in dimension = 3, Calabi's conjecture is both right and wrong, depending on whether the minimal surface is embedded or merely immersed. (This point was not specified in the original question.) In dimension = 3, the answer is radically different for these two cases, as we now explain. The first counterexample to Calabi's conjecture in the immersed case was given by L. P. de M. Jorge and F. Xavier in 1980 [48] , who constructed a complete nonflat conformal minimal immersion D → R 3 from the disc with the range contained in a slab between two parallel planes. In The problem was resolved for immersed surfaces by N. Nadirashvili [66] who in 1996 constructed a complete conformal minimal immersion D → R 3 with the image contained in a ball. Many subsequent results followed, showing similar results for topologically more general surfaces; see [16, Section 7.1] for a survey and references. However, the conformal type of the examples could not be controlled by the methods developed in those papers, except for the disc. The reason is that the increase of the intrinsic radius of a surface was achieved by applying Runge's theorem on pieces of a suitable labyrinth in the surface, chosen such that any divergent path avoiding most pieces has infinite length, while crossing a piece of the labyrinth increases the length by a prescribed amount. However, Runge's theorem does not allow to control the map everywhere, and hence small pieces of the surface had to be cut away in order to keep the image bounded. This surgery changes the conformal type of the surface, and only its topological type can be controlled by this method. After Nadirashvili's paper, Yau revisited the Calabi-Yau conjectures in his 2000 millenium lecture and proposed several new questions (see [80, p. 360] where is a compact Riemann surface without boundary and { } is a finite or countable family of pairwise disjoint compact geometric discs or points in . (A geometric disc in is a compact subset whose preimage in the universal holomorphic covering space of , which is one of the surfaces CP 1 , C, or D, is a family of pairwise disjoint round discs or points.) Such is called a circled domain in . Hence, Theorem 3.5 is a corollary to the following more precise result, which includes information about the boundary behaviour of surfaces. This means that the image ( ) is a complete immersed minimal surface whose boundary ( ) consists of pairwise disjoint Jordan curves. The control of conformal structures on complete minimal surfaces in Theorems 3.5 and 3.6 is one of the main new aspect of these results; the other one is that the surfaces in Theorem 3.6 have Jordan boundaries. These answer the aforementioned questions by Yau. For surfaces of type (3.4) with finitely many boundary components, Theorem 3.6 was proved in [4] . This covers all finite bordered Riemann surfaces in view of the uniformization theorem [75, Theorem 8.1] due to E. L. Stout. 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. The situation regarding point ends remains elusive and does not have a clear-cut answer. On the one hand, a bounded conformal minimal surface cannot be complete at an isolated point end (a puncture) since a bounded harmonic function extends across a puncture. On the other hand, it was shown in [10, Theorem 5.1] that an analogue of Theorem 3.6 holds for connected domains of the form where is a compact set in a compact Riemann surface and ⊂ \ are pairwise disjoint geometric discs such that the distance to is infinite within . In particular, there are complete bounded conformal minimal surfaces in R 3 with point ends which are limits of disc ends. Our construction uses an adaptation of the Riemann-Hilbert boundary value problem to holomorphic null curves and conformal minimal surfaces, together with a method of exposing boundary points of such surfaces. This technique is explained in detail in [16, Chapter 6] . The modifications which we use provide a good control of the position of the whole surface in the ambient space, thereby keeping it bounded. The main technical lemma of independent interest (see [16, Lemma 7.3.1] ) enables one to make the intrinsic radius of a conformal bordered minimal surface in R as large as desired by a deformation of the surface which is uniformly as small as desired. One uses this lemma in an inductive process which converges to a bounded complete limit surface. This lemma also allows the construction of complete minimal surfaces with other interesting geometric properties. In particular, every bordered Riemann surface admits a complete proper conformal minimal immersion into any convex domain in R (embedding if ≥ 5) and, more generally, into any minimally convex domain (see [16, Section 8.3] ). A smoothly bounded domain in R 3 is minimally convex if and only if the boundary has nonnegative mean curvature at each point. We give a brief description of the modifications which lead to proof of the above results. A complete presentation of this technique is given in [16, Chapter 6] , and Theorem 3.6 is proved in [16, Chapter 7] . Illustrations can be found in my lecture at https://8ecm.si/system/admin/abstracts/presentations/000/000/663/original/8EC Each step consists of two substeps. In the first substep, we choose a large but finite number of roughly equidistributed points on the boundary of the surface and change the surface so that it grows long spikes (tentacles) at these points, which however remain uniformly close to the attachment points. (Imagine the picture of a corona virus.) The effect of this modification is that curves in the surface which terminate near one of the exposed boundary points get elongated by a prescribed amount. (See [16, Sect. 6.7] .) In the second substep, we perform a Riemann-Hilbert type modification which increases the intrinsic radius along each of the boundary arcs between a pair of exposed points, without destroying the effect of substep one. To each boundary arc between a pair of exposed points we attach a 3-dimensional cylinder, consisting of a 1-parameter family of conformal minimal discs centred at points of the given arc. The boundaries of these discs form a 2-dimensional cylinder, a product of the arc with a circle, and their radii shrink to zero near the exposed endpoints of the arc. Is then possible to modify the surface by pushing each arc very near the corresponding 2-dimensional cylinder, with the modification tempering out near the exposed endpoints and away from the arcs. So, the modification in substep 2 is big very close to the boundary (except near the exposed points), and it is arbitrarily small outside a given neighbourhood of the boundary. The new conformal minimal surface is contained in an arbitrarily small neighbourhood of the union of the surface from substep 1 and the 3-dimensional cylinders that have been attached to the arcs in substep 2. The metric effect of the modification in substep 2 is that the length of any path in the surface terminating at an interior point of one of the boundary arcs increases almost by the radius of the disc that was attached at this point. (For curves terminating near the exposed points a desired elongation was already achieved in substep 1.) For technical reasons, we actually work with -derivatives of these conformal minimal surfaces, including the boundary discs, so the entire picture concerns families of holomorphic maps with values in the punctured null quadric A * . In order to control the period conditions, we work with sprays of such configurations, like in the proof of Theorem 3.1. Special attention is paid to avoid introducing branch points to our surfaces in the process. As said before, this provides the main modification lemma, and its inductive application leads to the proof of Theorem 3.6. 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. 7.4 ]. An axiomatic approach to the Calabi-Yau problem was proposed in [11] . The analogue of the Calabi-Yau problem for complex submanifolds in C , which is known as Paul Yang's problem who raised it in 1977 [78] , has also received a lot of recent attention. In particular, J. Globevnik showed [39] that for any pair of integers 1 ≤ < , the ball of C admits holomorphic foliations by complete -dimensional proper complex subvarieties, most of which are without singularities (submanifolds). Another construction using a different technique was given by Alarcón et al. [17] , and it was also shown that there are nonsingular holomorphic foliations of the ball having complete leaves (Alarcón [1] ). Furthermore, there are nonsingular holomorphic foliations of the ball whose leaves are complete properly embedded discs [9] . The techniques in these papers do not apply to more general minimal surfaces, and they do not provide control of complex structures of examples. In conclusion, I propose the following conjecture. Although I am fully aware of the lack of technical tools to solve it in this generality, I believe that it is true. Conjecture 3.7. The Calabi-Yau property holds for bordered minimal surfaces in any smooth Riemannian manifold ( , ) with dim ≥ 3. Explicity, for every bordered Riemann surface, , and conformal minimal immersion : → it is possible to approximate uniformly on by complete conformal minimal immersions → . 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Hattendorf Function theory in the unit ball of C Zur Theorie der Eindeutigen Analytischen Functionen Bounded holomorphic functions on finite Riemann surfaces A theory of branched minimal surfaces Ueber die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen Curvatures of complex submanifolds of C Problem section Review of geometry and analysis Review of geometry and analysis Acknowledgments. My research is supported by Program P1-0291 from ARRS, Republic of Slovenia. I wish to thank Antonio Alarcón and Francisco J. López for having introduced me to this beautiful subject back in 2011. I also thank the mentioned colleagues, as well as Barbara Drinovec Drnovšek and Finnur Lárusson, for the continuing collaboration on this subject.