Isometric Deformations of Minimal Surfaces Darin Mohr Faculty Mentor: Alex Smith UWEC Mathematics Department Zivnuska Scholarship Project University of Wisconsin-Eau Claire April 2004

1. Minimal Surfaces 2. Derivation of the Partial 3. Some Classical Solutions Differential Equation

Minimal surfaces are found in nature when a closed loop is submersed in a soapy liquid, Surfaces with vanishing mean curvature exist in abundance. Here are some of the most causing the formation of a soap film. It is found that amongst all possible surfaces that fill Given a parametric surface X(u,v) = x(u,v),y(u,v),z(u,v) with parameter domain D, renowned solutions. Euler showed the catenoid was minimal in 1744, and Meusnier showed the the loop, nature selects that surface having minimal area. the area is given by h i helicoid was minimal in 1776. No more minimal surfaces were found until Scherk discovered ∂X ∂X one in 1835, and Enneper in 1864. (X)= du dv. A ZZ ¯¯ ∂u × ∂v ¯¯ D ¯¯ ¯¯ ¯¯ ¯¯ Mathematicians study the critical points of¯¯ the function¯¯ . Using the calculus of vari- A -1 0 -4 10 10 10 ations, we find [6] that critical points are solutions to the partial differential equation -0.5 -2 0 0 0 10 5 5 5 2 2 2 20 0 1.5 0 6 20 4 10 1 4 4 4 2 0.5 2 2 0 0 0 0 0.5 0 6 6 6 X 30 0 4 -2 xuu (yuzv zuyv)+ yuu (zuxv xuzv)+ zuu (xuyv yuxv) v -10 1 -4 8 -5 8 -5 8 -5 -20 -0.5 -6 -4 -6 40 1.5 -2 0 2 10 10 10 − − − || || 1 0.5 -1 4 6 0 -0.5 -1.5 12 -10 12 -10 12 -10 X X -20 -1 (2 xuv (yuzv zuyv)+2 yuv (zuxv xuzv)+2 zuv (xuyv yuxv)) v v 50 -10 -1.5 -10 -10 -10 0 14 -5 14 -5 14 -5 10 0 0 0 − − − − · 20 10 5 10 5 10 5 2 + xvv (y z z y )+ yvv (z x x z )+ zvv (x y y x ) X =0. u v − u v u v − u v u v − u v || u|| In the language of differential geometry, solutions to this equation are characterized as those Figure 2: Helicoid, Catenoid, Enneper and Scherk’s surfaces. Figure 1: A curve in space, a filling-surface whose area can be reduced, and the filling-surface with surfaces that have zero mean curvature. Differential geometers [3] use the shape operator minimal area. There are many surfaces that fill the curve, in fact there are infinitely many degrees of S to describe the curvature of a surface at a point. The shape operator is a symmetric Helicoid: Catenoid: freedom in specifying such a surface. two-by-two matrix. The eigenvalues of S are called the principal curvatures. The trace of X(u,v)= a sinh (v) cos (u) , X(u,v)= a cos (u) cosh (v) , h a sinh (v)sin(u) ,bu h a sin(u) cosh (v) ,bv S is the mean curvature, and the determinant is the Gaussian curvature. i i Determining the surface of minimal area that spans a given curve is an old problem known For a minimal surface, the eigenvalues of the matrix S are opposites of one another, and thus Enneper’s surface: Scherk’s surface: as the Plateau Problem [2]. The development of string theory by physicists has catalyzed the Gaussian curvature is always negative. X(u,v)= u 1/3 u3 + uv2, X(u,v)= u,v, ln cos(v) ln cos(u) a renewed interest in the theory of minimal surfaces [7]. h − h − i v + 1/3 v3 vu2,u2 v2 − − − i

4. The Gauss Map 5. The Weierstrass Representation 6. An Example of an Isometric Deformation

A central tool in analyzing minimal surfaces is the Gauss Map. This is a map : X S2, Using complex analysis, Weierstrass discovered in 1866 [5] that given any complex analytic where S2 is the unit sphere, and is defined by associating with a point p on the surfaceG →X the function f and any meromorphic function g, the parameterized surface given by The Weierstrass representations of the catenoid and helicoid both have g(z) = z. For the 2 2 πi/2 unit normal vector to the tangent plane at p, which in turn can be identified with a point on catenoid fc(z) = 1/z while fh(z) = i/z for the helicoid. Notice that fc = e fh, and 2 S . Although can be defined for any surface in space, it is a remarkable theorem that the z z z thus the helicoid arises as an isometric deformation of the catenoid. Gauss map ofG a minimal surface is conformal, i.e., it does not distort angles. By another X(u,v) = Re f(ζ)(1 g(ζ)2) dζ, i f(ζ)(1 + g(ζ)2) dζ, 2 f(ζ)g(ζ) dζ ¿Z − Z Z À theorem [1], conformal maps of two-dimensional surfaces can be identified with complex a a a analytic functions, so we can investigate minimal surfaces using the tools from analysis of is a minimal surface, where z = u + iv and i = √ 1. Careful inspection reveals that if we functions of a single complex variable. identify S2 with the Riemann sphere C , then− g is the Gauss map ! ∪ {∞} G Figure 4: Costa’s Surface, recently discovered in 1984, has f(ζ) = ℘(ζ) and

g(ζ)= A/℘′(ζ), where ℘ is the Weierstrass P function. It is notoriously difficult to com- pute and render [4]. Now consider what happens if we change f to eitf, and leave g unchanged. By the Theorema Egregium of Gauss [3], the new surface, having the same Gauss map as the original, will be locally isometric to the original. This means that locally, any minimal surface Figure 3: The Gauss Map of the Helicoid, of Enneper’s Surface and of the Catenoid. can be isometrically deformed in such a way that the deformations remain minimal. An intuitive example of this is the deformation of sheet of paper into a cylinder. Coloring computed to correspond to mapping correspondences. Figure 5: Isometric Deformation of a Helicoid to a Catenoid. The penultimate frame suggests why For contrast, we know that we cannot isometrically deform a flat sheet of paper into a portion we must work in the category of immersed surfaces, i.e., surfaces with self-intersection. of a sphere any flat map of the earth must necessarily distort distances. − References Acknowledgments [1] L. Ahlfors. Complex Analysis. McGraw-Hill, third edition, 1979. [2] J. Douglas. Solution of the problem of Plateau. Trans. Amer. Math. Soc., 1931. Philip S. Zivnuska for financial support of student-faculty collaborations in the UWEC Mathematics Department. • [3] J. Gray. Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press, second edition, 1998. Graphics computed and rendered with Maple9. • [4] D. Hoffman. The computer-aided discovery of new embedded minimal surfaces. Math. Intell., pages 8–21, 1987. UW-Eau Claire Center of Excellence for Faculty and Undergraduate Student Research Collaboration. • [5] J. Oprea. The Mathematics of Soap Films: Explorations with Maple. Student Mathematical Library. Amer. Math. Soc., 2000. [6] R. Osserman. A Survey of Minimal Surfaces. Dover, 1986. [7] S. Paycha S. Scarlatti S. Albeverio, J. Jost. A Mathematical Introduction to String Theory. Cambridge University Press, 1997.