A New Golden Age of Minimal

Joaquín Pérez

Figure 1. A minimal , like this soap film, is characterized by the property that small pieces minimize for given boundary.

Joaquín Pérez is professor of geometry and topology at the Uni- versity of Granada and director of the Research Institute IEMath-GR. His e-mail address is [email protected]. Research partially supported by the Spanish MEC/FEDER grant No. MTM2014-52368-P. For permission to reprint this article, please contact: [email protected]. DOI: http://dx.doi.org/10.1090/noti1500

April 2017 Notices of the AMS 347 Introduction In this article we hope to convince the reader that as A minimal surface, like the soap film of Figure 1, has with the previous milestones, we are currently witnessing the property that small pieces minimize area for a given a new golden age of minimal surfaces, mostly favored boundary, even though the whole surface may be unstable. by a new tool discovered in 2004: the so-called Colding– At first, there were few explicit examples (see Figure 2): Minicozzi theory. This work, published in an impressive the , the of Euler (1741), and the series of four articles in the same issue of Annals of of Meusnier (1776). Mathematics [2], analyzes the convergence of sequences Many of the greatest mathematicians in history have of embedded minimal disks without imposing a priori been challenged by minimal surfaces; some of them made uniform bounds on area or . We will sketch spectacular advances in a relatively definite period, pro- how this theory has helped to solve open problems that ducing golden ages of this theory: The first one occurred were considered inaccessible until recently, and we will approximately in the period 1830–1890, when renowned venture, with all the reservations that predictions deserve, mathematicians such as Enneper, Scherk, Schwarz, Rie- to expose some of the most interesting open problems in mann, and Weierstrass made major advances on minimal this field. surfaces through the application of the newly created In order to de- field of by providing analytic formulas velop these objectives for a general minimal surface. Also in this period, fun- The theory of in a limited number damental research by Plateau on gave of pages, we must pay a physical interpretation to the problem of minimizing minimal surfaces the price of not going area with a given contour, which allowed the spread of into detail. There are this minimization problem beyond mathematics, to the is a confluence of many articles, books, point that since then it is customary to refer to it as the and chapters of books Plateau problem. A second golden age of minimal surfaces many branches of where interested read- took place from about 1914 to 1950, with the incipient mathematics. ers can satisfy their theory of partial differential equations: here, we highlight curiosity, such as the the contributions of Bernstein, Courant, Douglas (who in volume by Colding and 1936 won the first Fields Medal1 for his solution of the Minicozzi [1] or the survey by Meeks and me [3]. Plateau problem), Morrey, Morse, Radó, and Shiffman. A third golden age started in the 1960s, when giants of the Basic Results stature of Almgren, Alt, Calabi, do Carmo, Chern, Federer, The theory of minimal surfaces is a confluence of many Finn, Fleming, Gackstatter, Gulliver, Hardt, Hildebrandt, branches of mathematics. We can define minimality in at Jenkins, Lawson, Nitsche, Osserman, Serrin, Simon, and least eight different but equivalent ways, based on the Simons opened new routes through the use of multiple theory that we are most passionate about. Let 푋∶ 푀 → ℝ3 be an isometric of a Rie- mannian surface in three-dimensional Euclidean space, and let 푁∶ 푀 → 핊2(1) ⊂ ℝ3 be its unit normal or Gauss map (here 핊2(1) denotes the of radius 1 and center the origin of ℝ3). If we perturb 푋 in a relatively compact domain Ω ⊂ 푀 by a compactly supported dif- ∞ ferentiable function 푓 ∈ 퐶0 (Ω), then 푋 + 푡푓푁 is again an immersion for |푡| < 휀 and 휀 > 0 small enough. The 퐻 ∈ 퐶∞(푀) of 푋 (arithmetic mean of the principal ) is related to the area functional 퐴(푡) = Area((푋+푡푓푁)(Ω)) by means of the first variation Figure 2. The first explicit examples of minimal of area formula: surfaces were the plane, the catenoid (Euler, 1741), (1) 퐴′(0) = −2 ∫ 푓퐻푑퐴, and the helicoid (Meusnier, 1776). Ω where 푑퐴 is the area element of 푀. Now we can state the techniques, from Riemann surfaces to geometric measure first two equivalent definitions of minimality. theory, passing through integrable systems, conformal geometry, and functional analysis. The appearance of Definition 1. A surface 푀 ⊂ ℝ3 is minimal if it is a crit- computers was crucial for the discovery in the eighties of ical point of the area functional for all variations with new examples of complete minimal surfaces without self- compact support. intersections. The abundance of these newly discovered Definition 2. A surface 푀 ⊂ ℝ3 is minimal when its mean examples led to new problems and conjectures about the curvature vanishes identically. classification and structure of families of surfaces with 3 prescribed topology. More recent major contributors are Locally and after a rotation, every surface 푀 ⊂ ℝ too numerous to list here. can be written as the graph of a differentiable function 푢 = 푢(푥, 푦). In 1762, Lagrange wrote the foundations of 1Shared with Ahlfors for his work on Riemann surfaces. the by finding the PDE associated

348 Notices of the AMS Volume 64, Number 4 to a critical point of the area functional when the surface Definition 6. An isometric immersion 푋 = (푥1, 푥2, 푥3)∶ is a graph: 푀 → ℝ3 of a Riemannian surface in three-dimensional Euclidean space is said to be minimal if its coordinate Definition 3. A surface 푀 ⊂ ℝ3 is minimal if around functions are harmonic: Δ푥 = 0, 푖 = 1, 2, 3. any point it can be written as the graph of a function 푖 푢 = 푢(푥, 푦) that satisfies the second-order, quasi-linear From a physical point of view, the so-called Young’s elliptic partial differential equation equation shows that the mean curvature of a surface sep- arating two media expresses the difference of pressures (2) (1 + 푢2)푢 − 2푢 푢 푢 + (1 + 푢2 )푢 = 0. 푥 푦푦 푥 푦 푥푦 푦 푥푥 between the media. When both media are under the same The above PDE can also be written in divergence form: pressure, the surface that separates them is minimal. This happens after dipping a wire frame (mathematically, a ∇푢 (3) div ( ) = 0. nonnecessarily planar Jordan curve) in soapy water. How- √1 + |∇푢|2 ever, soap bubbles that we all have blown have nonzero Neglecting the gradient in the denominator of (3) leads constant mean curvature, because they enclose a volume to the celebrated Laplace equation. This means that on of air whose pressure is greater than the atmospheric a small scale (where 푢 is close to a constant), minimal pressure. surfaces inherit properties of harmonic functions, such as Definition 7. A surface 푀 ⊂ ℝ3 is minimal if each point the maximum principle, Harnack’s inequality, and others. 푝 ∈ 푀 has a neighborhood that matches the soap film On a large scale, dramatic changes appear in the way spanned by the boundary of this neighborhood. that global solutions to the Laplace and minimal surface equations behave; perhaps the paradigmatic example of To give the last definition of minimality, remember this dichotomy is Bernstein’s theorem: the only solutions that the differential 푑푁푝 at each point 푝 ∈ 푀 of the Gauss of (3) defined in the whole of ℝ2 are the affine functions, map 푁 is a self-adjoint endomorphism of the tangent while of course there are many global harmonic functions. plane 푇푝푀. Therefore, there exists an orthonormal basis A consequence of the second variation of area formula of 푇푝푀 where 푑푁푝 diagonalizes (principal directions at 푝), (i.e., the expression for 퐴′′(0)) shows that every minimal being the opposite of the eigenvalues of 푑푁푝, the so-called surface minimizes area locally. This property justifies the principal curvatures of 푀 at 푝. As the mean curvature 퐻 word minimal for these surfaces (not to be confused with is the arithmetic mean of the principal curvatures, the being a global area minimizer, which is a much more minimality of 푀 is equivalent to the vanishing of the trace restrictive property: the unique complete surfaces in ℝ3 of 푑푁푝 or, equivalently, to the property that the matrix of that minimize area globally are the affine planes). 푑푁푝 in any orthonormal basis of 푇푝푀 is of the form 푎 푏 Definition 4. A surface 푀 ⊂ ℝ3 is minimal if every 푑푁 = ( ). 푝 푏 −푎 point 푝 ∈ 푀 admits a neighborhood that minimizes area among all surfaces with the same boundary. After identifying 푁 with its onto the extended complex plane, the Cauchy–Riemann Definitions 1 and 4 place minimal surfaces as 2- equations allow us to enunciate the eighth equivalent dimensional analogues of in Riemannian geom- version of minimality. etry and connect them with the calculus of variations. Another functional of great importance is the energy, Definition 8. A surface 푀 ⊂ ℝ3 is minimal when its stere- ographically projected Gauss map 푔∶ 푀 → ℂ ∪ {∞} is a 2 퐸 = ∫ |∇푋| 푑퐴, meromorphic function. Ω 3 In fact, for a min- where again 푋∶ 푀 → ℝ is an isometric immersion and 3 Ω ⊂ 푀 is a relatively compact domain. Area and en- imal surface 푀 ⊂ ℝ , ergy are related by the inequality 퐸 ≥ 2퐴, with equality not only is the Gauss Minimal surfaces occurring exactly when 푋 is conformal. The fact that map meromorphic but every Riemannian surface admits local conformal (isother- also the whole immer- appear frequently mal) coordinates allows us to give two other equivalent sion can be expressed definitions of minimality. by means of holomor- in nature. phic data: as the third 3 Definition 5. A conformal immersion 푋∶ 푀 → ℝ is coordinate function 푥3 of 푀 is a , ∗ minimal if it is a critical point of the energy functional then it admits locally a conjugate harmonic function 푥3 . ∗ for every compactly supported variation or, equivalently, Thus, the height differential 푑ℎ ∶= 푑푥3 + 푖푑푥3 is a well- when every point on the surface admits a neighborhood defined holomorphic 1-form on 푀, and the surface can that minimizes energy among all surfaces with the same be conformally parameterized by the explicit formula boundary. (4) 푝 1 1 푖 1 The classical formula Δ푋 = 2퐻푁 that links the Lapla- 푋∶ 푀→ℝ3, 푋(푝) = ℜ ∫ ( ( − 푔) , ( + 푔), 1) 푑ℎ, 3 2 푔 2 푔 cian of an isometric immersion 푋∶ 푀 → ℝ with its mean 푝0 curvature function 퐻 and Gauss map 푁 leads us to the where ℜ stands for real part and 푝0 is the point of 푀 next definition. that we choose to be sent by 푋 to the origin in ℝ3 (i.e.,

April 2017 Notices of the AMS 349 Figure 3. (left) The (A. Schoen, 1970) has been observed in diblock copolymer systems; (right) nature often seeks optimal forms in terms of perimeter and area such as minimal surfaces.

purely mathematical viewpoint, minimal surfaces have been studied in other ambient spaces besides Euclidean space, giving rise to applications in such diverse prob- lems as the positive mass and the Penrose conjectures in , the Smith conjecture on diffeomor- phisms of finite order of the three-dimensional sphere, and Thurston’s geometrization conjecture in 3- theory.

Classical Minimal Surface Theory Figure 4. Frei Otto modeled the Olympic stadium in By classical theory we will mean the study of connected, Munich on minimal surfaces. orientable, complete, embedded minimal surfaces in ℝ3. Let ℳ퐶 be the class of complete embedded minimal surfaces 푀 ⊂ ℝ3 with finite genus. In order to understand formula (4) defines 푋 up to an ambient translation). The this last word, recall that the maximum principle for pair (푔, 푑ℎ) is usually called the Weierstrass data of 푀. harmonic functions implies that there are no compact Minimal surfaces appear frequently in nature, not only minimal surfaces without boundary in ℝ3; therefore, in soap films or, more generally, interfaces separating complete minimal surfaces must have topological ends immiscible fluids at the same pressure but also for (roughly speaking, ways to go to infinity intrinsically on example in diblock copolymers, smectic liquid crystals the surface). After compactifying topologically a minimal (materials that have uniformly spaced layers with fluidlike surface 푀 by adding a point to each end, we define order within each layer), crystallography, semiconductor the genus of 푀 as the genus of its compactification. If technology,…even in the cuticular structure in the wing 푔 ∈ ℕ ∪ {0} ∪ {∞} and 푘 ∈ ℕ ∪ {∞}, we let ℳ퐶(푔, 푘) be scales of certain insects! See Figure 3. the subset of ℳ퐶 that consists of those surfaces with Minimization properties for this class of surfaces have genus 푔 and 푘 topological ends. When both 푔 and 푘 are motivated renowned architects such as Frei Otto to use finite, we will say that the surface has finite topology. them to design optimal structures such as the cover of the A surface 푀 ⊂ ℝ3 is called proper if every intrinsically Olympic stadium in Munich (Figure 4). The beauty of their divergent sequence of points of 푀 also diverges in ℝ3. balanced forms has awakened the interest of sculptors Roughly speaking, a complete surface is proper when its such as Robert Engman and Robert Longhurst. From a topological ends are placed at infinity in ℝ3 (an infinite

350 Notices of the AMS Volume 64, Number 4 roll of paper that wraps infinitely often and limits to an Regarding the relationship between ℳ퐶 and ℳ푃 in infinite cylinder from its inside or outside is an example the case of finite topology, we should highlight a deep of a complete surface that is not proper). We will denote result of Colding and Minicozzi which asserts that every by ℳ푃 the subset of ℳ퐶 formed by the proper minimal complete embedded minimal surface with finite topology is proper. Its proof is an application of the famous theory surfaces, and we let ℳ푃(푔, 푘) = ℳ푃 ∩ ℳ퐶(푔, 푘). Our goal in this section is to describe the main examples by Colding and Minicozzi, a topic which we will talk about of minimal surfaces in these families, attending to their a little later. topology, conformal structure, asymptotic behavior, and For our discussion of the case of finite topology, we will distinguish two subcases, depending on whether the the main results of classification. As we go through this number of ends of the minimal surface is one or more description, we will discuss some of the most interesting than one. open problems. Surfaces with finite genus and one end. In 2005, Meeks and Complete Minimal Surfaces with Finite Topology Rosenberg applied Colding–Minicozzi theory to show that the plane and the helicoid are the only possible examples The trivial example in this class is the plane. The first non- in ℳ (0, 1) (i.e., they gave the full classification of the trivial examples of minimal surfaces (Figure 2) belong to 푃 simply connected properly embedded minimal surfaces). this class: the catenoid discovered by Euler in 1741 (genus By the above properness result of Colding and Minicozzi, zero and two ends) and the helicoid found by Meusnier in the same uniqueness result holds in ℳ퐶(0, 1). As for the 1776 (genus zero, one end). Both surfaces support multi- asymptotic behavior of surfaces in ℳ푃(푔, 1) = ℳ퐶(푔, 1) ple characterizations; among the most classic ones we will with 1 ≤ 푔 < ∞, Bernstein and Breiner proved in 2011 mention that the catenoid is the unique minimal surface that every surface in ℳ푃(푔, 1) is asymptotic to a helicoid of revolution together with the plane (Euler) and that and conformally parabolic.2 For this reason, surfaces the helicoid and the plane are the unique ruled minimal in ℳ푃(푔, 1) are usually called of genus 푔. On surfaces (Catalan). A special mention among examples existence results in this line, it is worth mentioning that in this family is deserved by the Costa torus, the first Hoffman, Weber, and Wolf discovered in 2009 a helicoid complete minimal surface of finite topology discovered of genus one with the conformal structure of a rhombic after the aforementioned ones (after 206 years!), which torus minus one point and that Hoffman, Traizet, and has genus 1 and 3 ends, and its generalization to any White have recently proven the existence of examples in finite genus 푔 ≥ 1, discovered by Hoffman and Meeks, ℳ푃(푔, 1) for each finite 푔 ≥ 1 (arXiv 2015). An important also with three ends; see Figure 5. open problem about ℳ푃(푔, 1) is the possible uniqueness of examples with a given genus: this uniqueness is known in the case 푔 = 0, and it is conjectured that there exists a unique helicoid of genus 푔 for each 푔 ≥ 1, but even the local version of this result is not known. Surfaces with finite genus and 푘 ends, 2 ≤ 푘 < ∞. The main structural result in this case is due to Collin, who proved in 1997 that if 푀 ∈ ℳ푃(푔, 푘) has 푔, 푘 finite and 푘 ≥ 2, then 푀 lies in a particularly well-studied family: surfaces with finite total curvature, i.e., those where the Gaussian curvature 퐾 is integrable:

(5) ∫ 퐾푑퐴 = − ∫ |퐾|푑퐴 > −∞. 푀 푀 (Note that since the mean curvature, the sum of the principal curvatures, is zero, the Gauss curvature, the product of the principal curvatures, is nonpositive.) By previous work of Huber and Osserman, condition (5) implies that 푀 is conformally equivalent to a compact 필 of genus 푔 to which we have removed 푘 points (in particular, 푀 is conformally parabolic), and both the Gauss map 푔∶ 푀 → ℂ∪{∞} and the height differential 푑ℎ of 푀 extend to holomorphic objects defined on 필. This allows the application of powerful tools of complex analysis and algebraic geometry of compact Riemann surfaces; in some way, and given the lack of compactness of a complete minimal surface in ℝ3, those with finite total curvature are the closest ones to being compact. Figure 5. (top) The Costa torus; (bottom) a Hoffman– Meeks minimal surface. 2푀 is conformally parabolic if it does not admit any nonconstant, nonpositive subharmonic function.

April 2017 Notices of the AMS 351 The asymptotic behavior of these minimal surfaces is structure to the space ℳ퐶(푔, 푘) around such a singular also well known: every end is asymptotic to a plane or surface (as an orbifold ?). half-catenoid. On uniqueness results, we highlight the following ones: Minimal Surfaces with Infinite Topology

1. Schoen proved in 1983 that if 푀 ∈ ℳ퐶(푔, 2) has Next we enter the world of classical minimal surfaces finite total curvature, then 푀 is a catenoid. This is with infinite topology, i.e., those that have either infinitely an application of the famous reflection method of many ends or infinite genus. The most basic examples in moving planes of Alexandrov, which is based on this family were discovered by Riemann in the nineteenth the maximum principle for the equation (2). century (and posthumously published by his disciple Hat- 2. In 1991, López and Ros characterized the catenoid tendorf) and consist of a 1-parametric family of properly as the only surface in ℳ퐶(0, 푘) with finite total embedded minimal surfaces, invariant by a translation, curvature together with the plane. Again the idea is with genus zero and infinitely many ends asymptotic to based on the maximum principle, but now applied equally spaced parallel planes. The Riemann minimal ex- to what has since been dubbed the López–Ros defor- amples admit the following fascinating characterization: mation, a 1-parameter family of minimal surfaces together with the plane, the helicoid and the catenoid, defined in terms of the Weierstrass data (푔, 푑ℎ) of they are the unique properly embedded minimal surfaces a given minimal surface 푀. The López–Ros defor- in ℝ3 that can be foliated by circles and lines in parallel mation only exists under a certain hypothesis on planes (indeed, Riemann discovered these examples by 3 the flux map of the original surface. imposing this property); see Figure 6. 3. In 1984, Costa classified the surfaces in ℳ퐶(1, 3) The Riemann minimal examples show how the period- with finite total curvature. These surfaces reduce icity of a surface can be regarded as a method to produce to the Costa torus and a 1-parametric family of examples of infinite topology: if the quotient surface by thrice-punctured tori discovered by Hoffman and the group of is not simply connected, then Meeks by deforming the Costa torus (and studied the lifted surface in ℝ3 has infinite topology. The same later by Hoffman and Karcher). thing happens with other examples of minimal surfaces The previous result by Costa was the first complete discovered in the nineteenth century, such as those shown description of a moduli space ℳ퐶(푔, 푘) that does not in Figure 6: reduce to a single surface: ℳ퐶(1, 3) has the structure of a 1. The singly periodic Scherk minimal surface (second noncompact 1-dimensional manifold, identifiable with an from the left in Figure 6) is invariant by a cyclic open interval. Generalizing this result, in 1996 Pérez and group of translations. The quotient surface by the Ros endowed the moduli spaces ℳ퐶(푔, 푘) (0 ≤ 푔 < ∞, cyclic group has genus zero and four ends (asymp- 2 ≤ 푘 < ∞) with a differentiable structure of dimension totic to half-planes; these ends are called Scherk 푘 − 2 around each minimal surface 푀 ∈ ℳ퐶(푔, 푘) with type ends); viewed in ℝ3, this surface has infinite an additional nondegeneracy assumption that affects the genus and one end. We can see this example as linear space of Jacobi functions on 푀, which are the a desingularization of two orthogonal planes by solutions 푢∶ 푀 → ℝ to the second-order, linear elliptic introducing infinitely many alternating holes form- PDE ing a 45∘ angle with the planes along their line of Δ푢 − 2퐾푢 = 0 on 푀, intersection. As in the case of the Riemann mini- where 퐾 is the Gaussian curvature of 푀. Until now, all mal examples, the singly periodic Scherk minimal known examples in ℳ퐶(푔, 푘) satisfy this nondegeneracy surface can be deformed by a 1-parametric family hypothesis. We highlight that the dimension of the space of singly periodic, properly embedded minimal sur- of nondegenerate surfaces in ℳ퐶(푔, 푘) does not depend faces, obtained by desingularization of planes that on the genus 푔, but only on the number of ends 푘. intersect with an angle 휃 ∈ (0, 휋). A major open problem is the Hoffman–Meeks conjec- 2. The doubly periodic Scherk minimal surface (third ture: If 푀 ∈ ℳ퐶(푔, 푘), then 푘 ≤ 푔 + 2. The best known from the left in Figure 6) is invariant by an infinite result to date in this regard is due to Meeks, Pérez, and group generated by two translations of linearly Ros (arXiv 2016), who proved the existence of an upper independent vectors. Again, the quotient surface bound for 푘 depending only on 푔, again by application of has genus zero and four ends (of Scherk type); the Colding–Minicozzi theory. viewed in ℝ3, this surface has infinite genus and one Another important open problem consists of deciding end. It can be considered as the desingularization of if there exist surfaces in some moduli space ℳ퐶(푔, 푘) that two infinite families of equally spaced vertical half- do not satisfy the nondegeneracy condition mentioned planes, one family inside {(푥, 푦, 푧) | 푧 > 0} and the above, and if they do exist, provide any “reasonable” other one in {(푥, 푦, 푧) | 푧 < 0}, in such a way that half- planes in different families cut at right angles. This 3The flux of a minimal surface 푀 ⊂ ℝ3 is the linear map surface also lies in a 1-parameter family of properly 퐹∶ 퐻 (푀) → ℝ3 that associates to each 1-dimensional homology 1 embedded, doubly periodic minimal surfaces, each class [푐] ∈ 퐻1(푀) the integral along a representative 푐 ∈ [푐] of the unit vector field along 푐 that is tangent to 푀 and orthogonal to of which desingularizes two infinite families of 푐. The condition for the López–Ros deformation to be well defined vertical half-planes in the open upper and lower on 푀 is that the range of 퐹 is at most 1. half-spaces of ℝ3, where the parameter is the angle

352 Notices of the AMS Volume 64, Number 4 Figure 6. From left to right: a Riemann minimal example, singly and doubly periodic Scherk surfaces, and the triply periodic Schwarz 푃-surface.

휃 ∈ (0, 휋) that the half-planes in the two families merged is the Costa torus (i.e., 푔 = 1). Also by gluing tech- form. There is a direct relationship between the niques, but using Riemann surfaces with nodes, Traizet Scherk singly and doubly minimal surfaces, which was able to prove in 2012 the existence of a complete, reflects the fact that every harmonic function admits nonperiodic minimal surface with infinite genus and in- (locally) a conjugate harmonic function. finitely many ends asymptotic to half- (Figure 7 3. The triply periodic Schwarz 푃-surface (Figure 6, right). In summary, there are lots of examples in the case right) is invariant by the group generated by three of infinite topology. translations of linearly independent vectors. The As for uniqueness results for minimal surfaces of quotient surface by this lattice of translations is infinite topology, it is clear in light of the previous compact with genus three and lives in a three- paragraph that we must distinguish in some way the dimensional cubic torus. Viewed in ℝ3, this surface families that we have found: a reasonable starting point has infinite genus and one end. The Schwarz 푃- could be imposing some kind of periodicity. Here, it surface is one of the most famous triply periodic is worth mentioning the following classification results minimal surfaces, a class of surfaces with multi- for moduli spaces of periodic minimal surfaces with ple applications to crystallography and material prescribed topology: science: the group of each triply periodic 1. The Riemann minimal examples are the unique minimal surface 푀 ⊂ ℝ3 is a crystallographic group, properly embedded minimal tori with finitely many 3 and the quotient surface 푀 over the lattice Γ of planar ends in a quotient of ℝ by a translation translations of rank 3 that leaves 푀 invariant di- (Meeks, Pérez, and Ros, 1998). The number of vides the three-dimensional torus ℝ3/Γ into two ends must be even, and when we fix this number regions of equal volume called labyrinths. The gy- the corresponding moduli space is a noncompact roid (Figure 3, left) is another famous triply periodic manifold of dimension 1. minimal surface with compact quotient of genus 2. The Scherk doubly periodic minimal surfaces are the three. The classification of triply periodic embedded unique properly embedded minimal surfaces with genus zero and finitely many ends in a quotient of minimal surfaces with quotient of genus three (the 3 lowest possible nontrivial value) is another major ℝ by two independent translations (Lazard–Holly and Meeks, 2001). Again the number of ends is open problem. necessarily even, and for a fixed number of ends, In view of the above examples, we could ask ourselves the moduli space is diffeomorphic to an open if the only method of producing minimal surfaces with interval. infinite topology is by imposing periodicity. The answer is 3. The moduli spaces of properly embedded minimal no, as shown in 2007 by Hauswirth and Pacard, who used tori with any fixed finite number of parallel planar 4 gluing techniques to merge a Hoffman–Meeks minimal ends in a quotient of ℝ3 by two linearly independent surface (we mentioned these surfaces when describing translations were described in 2005 by Pérez, Ro- examples in ℳ퐶(푔, 3) in the subsection “Complete Min- dríguez, and Traizet. Each of these moduli spaces imal Surfaces with Finite Topology”) with two halves of (for any fixed even number of ends) is a noncom- a Riemann minimal surface ℛ. In Figure 7 (left) we can pact manifold of dimension 3 whose surfaces are see a schematic representation of one of the examples called KMR examples (in honor of Karcher, Meeks, by Hauswirth and Pacard, when the central surface to be and Rosenberg, who previously found 1-parameter families of these surfaces in this moduli space). 4 This technique consists of a sophisticated application of the im- 4. The moduli spaces of properly embedded minimal plicit function theorem to the mean curvature operator defined surfaces with genus zero and finitely many ends of between certain Banach spaces. Scherk type in a quotient of ℝ3 by a translation were

April 2017 Notices of the AMS 353 Figure 7. From left to right: schematic representations of a Hauswirth–Pacard minimal surface with genus 1 and of the example of infinite genus by Traizet: two nonperiodic, complete embedded minimal surfaces with infinite topology.

classified in 2007 by Pérez and Traizet. In this case, The strategy sketched in the preceding paragraph fails these moduli spaces are noncompact of badly if we seek classification results for minimal surfaces dimension 2푘 − 3 (here 2푘 is the number of ends), with infinite topology without imposing periodicity, but whose surfaces were discovered by Karcher in 1988 in this case the Colding–Minicozzi theory is of great help, as a generalization of the Scherk simply periodic as we will explain next. minimal surfaces. The four uniqueness results listed above have a com- Colding–Minicozzi Theory mon flavor. First, periodicity is used in a strong way, Consider the following question: since it allows working in the quotient of ℝ3 by the Problem 9. What are the properly embedded minimal sur- corresponding group of isometries, and the quotient min- faces in ℝ3 with genus zero? imal surfaces always have finite total curvature in the sense of equation (5); in this setting, one can control Suppose that 푀 ⊂ ℝ3 is a surface that meets the the asymptotic geometry and the conformal representa- conditions of Problem 9. As explained above, in the case tion of the minimal surfaces under study. Second, the that 푀 has only one end we know that 푀 is a plane or desired uniqueness follows from a continuity argument: a helicoid (Meeks and Rosenberg). If 푀 has 푘 ends with one starts by proving that any surface 푀 in each of 2 ≤ 푘 < ∞, then 푀 is a catenoid by the theorems of these moduli spaces can be deformed within the moduli Collin and López–Ros. It remains to study the case that space (openness part) until arriving at a point 푀∞ in 푀 has infinitely many ends. If we knew that such an 푀 the boundary of the moduli space, which turns out to were invariant by a translation 푇, then it would not be be a properly embedded minimal surface with simpler difficult to check that the quotient surface of 푀 by the topology or periodicity than those in the original moduli cyclic group generated by 푇 is a torus with finitely many space (compactness part); this compactness requires a ends, and hence 푀 is a Riemann minimal example by the rather complete understanding of the possible limits of 1998 result of Meeks, Pérez, and Ros. Therefore, a way of sequences of minimal surfaces in the original moduli solving Problem 9 is to prove that if the number of ends space. Once we have arrived at 푀∞, the desired global of 푀 is infinite, then 푀 is periodic. uniqueness follows from an inverse function theorem Often we face the problem of understanding the possi- argument (local uniqueness around 푀∞) that needs the ble limits of a sequence of embedded minimal surfaces. previous classification of the moduli space of minimal As a trivial example, let us think of a surface 푀 ⊂ ℝ3 that 3 surfaces to which 푀∞ belongs. This last aspect reveals is invariant by a translation of vector 푣 ∈ ℝ − {0}. The a stratified structure in the moduli spaces of embedded constant sequence {푀푛 ∶= 푀 − 푛푣 = 푀}푛∈ℕ has as trivial minimal surfaces with prescribed topology and periodic- limit 푀 itself. This naive example suggests a possible way ity: the boundary of a given moduli space is the union to solve Problem 9: suppose that a properly embedded of other moduli spaces of minimal surfaces with simpler minimal surface 푀 ⊂ ℝ3 with infinitely many ends is a topology or periodicity. For instance, the description of solution to this problem. As the number of ends of 푀 is the moduli space in item 3 above requires solving the infinite, one can deduce that 푀 has infinite total curvature, classification problem in item 4. whence we can find a divergent sequence of points 푝푛 ∈ 푀

354 Notices of the AMS Volume 64, Number 4 where the unit normal to 푀 takes the same value. It is 1 U β L reasonable to try to conclude that {푀 = 푀 − 푝 } has (at 푛 푛 푛 Cβ least) a convergent subsequence as a step to demonstrate the desired periodicity of 푀. We have thus transformed 0 ϕ Problem 9 into another one, perhaps more ambitious: D β Problem 10. Under what conditions can we extract a con- Figure 8. The open set 퐴 is covered by images 푈 of vergent subsequence from a given sequence of embedded 훽 local charts 휑 , each one transforming a collection of minimal surfaces? 훽 disks at heights lying in a closed subset 퐶훽 of [0, 1] Suppose that {푀푛}푛 is a sequence of embedded minimal into portions of the leaves of ℒ. 3 surfaces in an open set 퐴 ⊂ ℝ . Also assume that {푀푛}푛 has at least one accumulation point, since we do not want provided any examples of a nontrivial minimal lamination the 푀 to completely escape and have nothing to analyze 푛 of ℝ3 that does not consist of a single embedded minimal in the limit. Each surface 푀 can be locally written as 푛 surface other than a collection of planes ℒ as above. the graph of a function 푢 defined in an open subset of 푍 푛 Coming back to our Problem 10, what can we say the tangent plane of 푀 at a given point, and the size 푛 about the limit of the 푀 if these embedded minimal of the domain of 푢 can be uniformly controlled if we 푛 푛 surfaces do not have uniform local bounds for their have uniform local bounds for the Gaussian curvatures second fundamental forms? Here is where the theory (equivalently, for the second fundamental forms) of the of Colding–Minicozzi comes to our aid. Following the surfaces. If in addition we have uniform local area bounds previous notation, the lack of uniform local curvature for the 푀 , then we will control the number of graphs that 푛 bounds implies that the Gaussian curvature of the 푀 lie in a given region of 퐴. Therefore, working in a very small 푛 blows up at some point of 퐴; i.e., the following set is (but uniform) scale, we will deduce that every surface 푀 푛 nonempty: gives rise to a single graphing function 푢푛. Thus we have transformed Problem 10 about convergence of surfaces ̂ (6) 풮 = {푥 ∈ 퐴 | sup |퐾푀푛∩픹(푥,푟)| → ∞, ∀푟 > 0} , into another problem, the convergence of graphs. In this where 퐾 denotes the Gaussian curvature of a surface Σ setting, the uniform local curvature bounds for the 푀 Σ 푛 and 픹(푥, 푟) is the closed ball centered at 푥 ∈ ℝ3 with radius produce equicontinuity of the 푢 , and the fact that we 푛 푟 > 0. The Colding–Minicozzi theory describes the limit are working locally produces uniform boundedness for of (a subsequence of) the 푀 in the above scenario under the 푢 . Therefore, the Arzelá–Ascoli Theorem insures 푛 푛 an additional hypothesis: each 푀 must be topologically that a subsequence of the 푢 converges uniformly to a 푛 푛 a compact disk that is contained in a ball of radius 푅 > 0, limit function 푢 that can be proven to satisfy the same 푛 ∞ say centered at the origin, with boundary 휕푀 contained PDE (2) as the 푢 . A prolongation argument now implies 푛 푛 in the boundary sphere of that ball. The description of that a subsequence of {푀 } converges to an embedded 푛 푛 this limit is very different depending upon whether the minimal surface in 퐴, and thus our Problem 10 is solved sequence of radii 푅 diverges or stays bounded. in this case. 푛 If we do not have local area bounds for the 푀푛 but still Theorem 11 (Colding–Minicozzi). Given 푛 ∈ ℕ, let 푀푛 assume local uniform curvature bounds, reasoning similar be an embedded minimal disk in a closed ball 픹(푅푛) = to the above leads to the conclusion that a subsequence 픹(⃗0, 푅푛) with 휕푀푛 ⊂ 휕픹(푅푛). If 푅푛 → ∞ and 풮̂ ∩ 픹(1) ≠ Ø, 3 of {푀푛}푛 converges to a natural generalization of the then a subsequence of the 푀푛 converges to a foliation of ℝ notion of minimal surface: a lamination whose leaves are by parallel planes, away from a straight line5 (called the minimal surfaces. Without going into detail, a lamination singular set of convergence), along which the curvature of ℒ of 퐴 is a closed union (in the induced topology on 퐴) of 푀푛 blows up when 푛 → ∞. surfaces embedded in 퐴, called leaves of ℒ, with a certain To better understand the last result, we will use the local product structure. This means that we can take following example. Consider the standard vertical helicoid local coordinates in 퐴 that transform the leaves into the 퐻 = {(푥, 푦, 푧) | 푥 sin 푧 = 푦 cos 푧}. Take a sequence of product of a two-dimensional disk with a closed subset positive numbers 휆 tending to zero, and consider for of ℝ, which we can think of as the heights of disjoint 푛 each 푛 ∈ ℕ the homothetic copy 푀 = 휆 퐻 of 퐻 by ratio copies of that disk placed horizontally (see Figure 8). 푛 푛 휆 that is again minimal and simply connected. As 푛 This local product structure endows the leaves of ℒ with 푛 increases, 푀 can be thought of as a new view of 퐻 from the structure of smooth, pairwise disjoint surfaces. A 푛 a viewpoint that becomes further and further away, as lamination is said to be minimal if its leaves are all in Figure 9. The farther away we look at the helicoid 퐻, minimal surfaces. For example, if 푍 is a nonempty closed the more it looks like a collection of horizontal planes subset of ℝ, then the collection of horizontal planes 2 separated by smaller and smaller distances, and so in the ℒ푍 = {푃푧 = ℝ × {푧} | 푧 ∈ 푍} is a minimal lamination of 3 퐴 = ℝ whose leaves are the planes 푃 . In the case that a 5 푧 In fact, Colding and Minicozzi proved that the singular set of lamination ℒ of 퐴 does not leave any empty spaces in 퐴, we convergence is a Lipschitz arc transverse to the limit foliation. Us- 3 call it a foliation of 퐴 (ℒ푍 is a foliation of ℝ when 푍 = ℝ). ing the uniqueness of the helicoid as the only nonflat surface in The theory of minimal laminations is a natural extension ℳ푃(0, 1), Meeks deduced in 2004 that this Lipschitz arc is indeed of the one of minimal surfaces. However, we still have not a line.

April 2017 Notices of the AMS 355 푛 Figure 9. Homothetic images of the same vertical helicoid 퐻, with ratio 휆푛 = 1/2 .

픻∗ = {(푥, 푦, 0) | 0 < 푥2 + 푦2 < 1}, and the other two are nonproper minimal surfaces 퐿+, 퐿− that rotate infinitely many times from above and below 픻∗, accumulating on 픻∗ as in Figure 10. In this case, the Gaussian curvature

of the 푀푛 blows up at the origin, but this time ⃗0 is a genuine singularity of the limit lamination ℒ, which does not admit a smooth extension across ⃗0. The theoretical description by Colding and Minicozzi for the limit of a sequence of compact, embedded minimal disks 푀푛 ⊂ 픹(푅푛) with 휕푀푛 ⊂ 휕픹(푅푛) and bounded radii 푅푛 is very technical, and we will omit it here. Instead, we will simply mention that after extracting a subsequence, the 푀푛 converge to a minimal lamination with singularities. The singularities of such a limit singular minimal lamination form a closed set, and each singularity is of one of the following two types: Figure 10. A minimal lamination of the punctured unit (a) Isolated singularities, in which case Figure 10 shows ball, with three leaves and a singularity at the origin. essentially the behavior of the limit object: there is a leaf 퐷∗ of the lamination that limits to the ∗ 3 singularity 푝 (in fact, 퐷 extends smoothly across limit we obtain the foliation of ℝ by horizontal planes. 푝 to an embedded minimal disk 퐷) and one or Observe that each leaf of this limit foliation is flat (its two nonproper leaves, which rotate infinitely many Gaussian curvature is identically zero) and the Gaussian times and accumulate at 퐷∗. Furthermore, portions curvatures of the 푀푛 converge to zero away from the ⟂ of the 푀푛 outside a solid cone of axis 푝 + (푇푝퐷) 푧-axis. However, since the Gaussian curvature of 퐻 along can be written as multivalued graphs over annular the 푧-axis is constant −1, the Gaussian curvature of 푀푛 regions of 푇 퐷, and as 푛 → ∞, these annular regions 2 푝 along the same axis is −1/휆푛, which tends to infinity. converge to a punctured disk, at the same time that In other words, the singular set 풮̂ defined in (6) is the the number of turns of the multivalued graphs 푧-axis in this example. Also note that the limit foliation inside the 푀푛 become arbitrarily large and the is perfectly regular along 풮̂; it is only the convergence of multivalued graphs collapse into 퐷∗ as in Figure 11 the 푀푛 to the limit that fails along 풮̂. This limit object (left). is known as a limiting parking garage structure with one (b) Nonisolated singularities, each of which is the limit column: away from the 푧-axis, the structure becomes of at least one sequence of isolated singularities, as arbitrarily flat and horizontal (this is where cars park), in Figure 11 (right). and to travel from one parking floor to another one, cars It should also be noted that in the above description, have to go up the ramp (around the column at the 푧-axis). the set 풮̂ where the Gaussian curvatures of the disks 푀푛 Well, Theorem 11 tells us that the general behavior of the blow up consists of not only the singularities of the limit limit of the embedded minimal disks 푀푛 when the radii lamination ℒ but also possibly embedded arcs of class 1,1 푅푛 tend to infinity is essentially the same as this example. 퐶 around which ℒ is a local foliation, as in Figure 11 The description when the radii 푅푛 remain bounded (left). In particular, these arcs are not singularities of ℒ can also be visualized with an example. In 2003, Colding except for their end points, as in the convergence of the and Minicozzi produced a sequence of minimal disks 푀푛 푀푛 to ℒ in Theorem 11. embedded in the closed unit ball 픹(1) and of helicoidal The previous description leads us directly to the study appearance, such that the number of turns that the of the singularities of a minimal lamination of an open 3 boundary curve 휕푀푛 makes around the 푧-axis tends subset of ℝ . Does this set have any reasonable structure? to infinity as 푛 → ∞, and the limit of the 푀푛 is a This question is another open central problem in minimal minimal lamination ℒ of 픹(1) − {⃗0} that consists of surface theory. Along this line, it is worth mentioning two three leaves: one is the horizontal punctured unit disk recent results of Meeks, Pérez, and Ros (2016):

356 Notices of the AMS Volume 64, Number 4 L+ pm D(pm) p D p D(p)

Γ

Figure 11. Left: schematic representation of an isolated singularity 푝 of a singular minimal lamination ℒ + obtained as a limit of embedded minimal disks 푀푛, with a nonproper leaf 퐿 at one side of the disk leaf 퐷 that ⟂ + passes through 푝. Locally around 푝 and outside a solid cone of vertex 푝 and axis 푝 + (푇푝퐷) , 퐿 is a multivalued graph. The other side of 퐷 is foliated by leaves of ℒ. The convergence is singular along a 퐶1,1 arc Γ. Right: at a nonisolated singularity 푝, the disk leaf 퐷 = 퐷(푝) passing through 푝 is also the limit of the corresponding disk leaves 퐷(푝푚) associated to isolated singularities 푝푚 that converge to 푝.

Theorem 12 (Local Removable Singularity Theorem). genus zero. In the first paragraph of the section “Colding– Let ℒ ⊂ 픹(1) − {⃗0} be a minimal lamination. Then ℒ ex- Minicozzi Theory” we explained that the problem reduces tends to a minimal lamination of 픹(1) (i.e., the singularity to proving that if 푀 has infinitely many ends, then 푀 at ⃗0 is removable) if and only if the Gaussian curvature is periodic. This strategy, which uses Colding–Minicozzi theory as we have mentioned above, was the one used by function 퐾ℒ of the lamination does not blow up at the origin faster than the square of the extrinsic distance to ⃗0; Meeks, Pérez, and Ros [5] to prove the following result: 2 i.e., |퐾ℒ|(푥) ⋅ ‖푥‖ is bounded in ℒ. Theorem 14. Every properly embedded minimal surface 푀 ⊂ ℝ3 with genus zero is either a plane, a helicoid, a It follows from Theorem 12 that if the function |퐾 |(푥)⋅ ℒ catenoid, or one of the Riemann minimal examples. In par- ‖푥‖2 is bounded in a minimal lamination ℒ ⊂ 픹(1) − {⃗0}, ticular, 푀 is foliated by circles or straight lines in parallel then |퐾 |(푥) ⋅ ‖푥‖2 extends across the origin with value ℒ planes. zero. Another consequence of this theorem is that in the example of Figure 10, the Gaussian curvature of the disks A final remark about the proof of Theorem 14 is in 푀푛 blows up faster than the square of the distance to the order. The theory of Colding and Minicozzi yields only origin as 푛 → ∞. the quasi-periodicity of 푀 (this means that if {푝푛}푛 is a Another result about singularities of minimal lamina- divergent sequence of points in 푀, then a subsequence tions is a global version of Theorem 12 that classifies the of {푀 − 푝푛}푛 converges to a properly embedded minimal 3 minimal laminations of ℝ3 − {⃗0} with quadratic decay of surface in ℝ with genus zero and infinitely many ends). curvature: The key to proving the desired periodicity of 푀 once we know it is quasi-periodic is a fascinating application of Theorem 13. Let ℒ ⊂ ℝ3 − {⃗0} be a nonflat minimal lami- the theory of integrable systems and more precisely of the nation such that |퐾|(푥) ⋅ ‖푥‖2 is bounded. Then ℒ extends holomorphic Korteweg-de Vries equation, a third-order across the origin to a minimal lamination of ℝ3 that con- PDE that models mathematically the behavior of waves sists of a single leaf 푀, which is a properly embedded min- on shallow water surfaces. imal surface with finite total curvature. In particular, |퐾| decays much faster than quadratically with the distance References to the origin: |퐾|(푥) ⋅ ‖푥‖4 is bounded in 푀. [1] T. H. Colding and W. P. Minicozzi II, Minimal Surfaces, vol- ume 4 of Courant Lecture Notes in Mathematics, New York We have said that in order to apply the theory of University, Courant Institute of Mathematical Sciences, New Colding–Minicozzi to a sequence 푀푛 of embedded minimal York, 1999. MR1683966, Zbl 0987.49025. surfaces, we need to assume that the 푀푛 are compact [2] , The space of embedded minimal surfaces of fixed disks with boundaries in ambient . This condition genus in a 3-manifold, Parts I–IV, Ann. of Math. (2) 160 (2004). is not really a restriction, as it can be naturally obtained MR2119718, MR2123932, MR2123933. by a rescaling argument so that the injectivity radius [3] W. H. Meeks III and J. Pérez, The classical theory of mini- mal surfaces, Bulletin of the AMS, (N.S.) 48 (2011), 325–407. function of the rescaled minimal surfaces is uniformly MR2801776, Zbl 1232.53003. bounded away from zero (Meeks, Pérez, and Ros [4]). [4] W. H. Meeks III, J. Pérez, and A. Ros, The local picture theo- rem on the scale of topology, to appear in J. Differential Ge- Classification of the Properly Embedded Minimal ometry, preprint at https://arxiv.org/abs/1505.06761. Surfaces in ℝ3 with Genus Zero [5] , Properly embedded minimal planar domains, Ann. of To finish our brief tour through the current state of the Math. (2) 181 (2015), 473–546.MR3275845, Zbl 06399442. classical minimal surface theory, we return to Problem 9 on the properly embedded minimal surfaces 푀 in ℝ3 with

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