A New Golden Age of Minimal Surfaces Joaquín Pérez Figure 1. A minimal surface, like this soap film, is characterized by the property that small pieces minimize area for given boundary. Joaquín Pérez is professor of geometry and topology at the Uni- versity of Granada and director of the Research Mathematics 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 plane, the catenoid of Euler (1741), and the helicoid 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 curvature. 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 complex analysis 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 surface tension 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 immersion 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 sphere 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 mean curvature 퐻 ∈ 퐶∞(푀) of 푋 (arithmetic mean of the principal curvatures) 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 calculus of variations 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 stereographic projection 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 geodesics 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.