Isometric Deformations of Minimal Surfaces Darin Mohr · 2004-04-07 · Isometric Deformations of...
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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 Minimal surfaces are found in nature when a closed loop is submersed in a soapy liquid, causing the formation of a soap film. It is found that amongst all possible surfaces that fill the loop, nature selects that surface having minimal area. 10 5 0 -5 -10 -10 -5 0 5 14 10 12 10 8 6 4 2 0 10 5 0 -5 -10 -10 -5 0 5 14 10 12 10 8 6 4 2 0 10 5 0 -5 -10 -10 -5 0 5 14 10 12 10 8 6 4 2 0 Figure 1: A curve in space, a filling-surface whose area can be reduced, and the filling-surface with minimal area. There are many surfaces that fill the curve, in fact there are infinitely many degrees of freedom in specifying such a surface. Determining the surface of minimal area that spans a given curve is an old problem known as the Plateau Problem [2]. The development of string theory by physicists has catalyzed a renewed interest in the theory of minimal surfaces [7]. 2. Derivation of the Partial Differential Equation Given a parametric surface X(u, v )= 〈x(u, v ),y (u, v ),z (u, v )〉 with parameter domain D, the area is given by A(X)= D ∂ X ∂u × ∂ X ∂v du dv. Mathematicians study the critical points of the function A. Using the calculus of vari- ations, we find [6] that critical points are solutions to the partial differential equation x uu (y u z v − z u y v )+ y uu (z u x v − x u z v )+ z uu (x u y v − y u x v ) ||X v || 2 − (2 x uv (y u z v − z u y v )+2 y uv (z u x v − x u z v )+2 z uv (x u y v − y u x v )) X v · X v + x vv (y u z v − z u y v )+ y vv (z u x v − x u z v )+ z vv (x u y v − y u x v ) ||X u || 2 =0. In the language of differential geometry, solutions to this equation are characterized as those surfaces that have zero mean curvature. Differential geometers [3] use the shape operator S to describe the curvature of a surface at a point. The shape operator is a symmetric two-by-two matrix. The eigenvalues of S are called the principal curvatures. The trace of S is the mean curvature, and the determinant is the Gaussian curvature. For a minimal surface, the eigenvalues of the matrix S are opposites of one another, and thus the Gaussian curvature is always negative. 3. Some Classical Solutions Surfaces with vanishing mean curvature exist in abundance. Here are some of the most renowned solutions. Euler showed the catenoid was minimal in 1744, and Meusnier showed the helicoid was minimal in 1776. No more minimal surfaces were found until Scherk discovered one in 1835, and Enneper in 1864. 20 10 0 -10 -20 -20 -10 0 10 20 50 40 30 20 10 0 1.5 1 0.5 0 1 -0.5 1.5 0.5 1 0.5 0 -1 0 -0.5 -0.5 -1 -1.5 -1 -1.5 6 4 2 0 -2 4 -4 2 -6 -4 0 -2 0 -2 -6 2 4 -4 6 Figure 2: Helicoid, Catenoid, Enneper and Scherk’s surfaces. Helicoid: Catenoid: X(u, v )= 〈a sinh (v ) cos (u) , X(u, v )= 〈a cos (u) cosh (v ) , a sinh (v ) sin (u) , bu〉 a sin (u) cosh (v ) , bv 〉 Enneper’s surface: Scherk’s surface: X(u, v )= 〈u − 1/3 u 3 + uv 2 , X(u, v )= 〈u, v, ln cos(v ) − ln cos(u)〉 − v +1/3 v 3 − vu 2 ,u 2 − v 2 〉 4. The Gauss Map A central tool in analyzing minimal surfaces is the Gauss Map. This is a map G : X → S 2 , where S 2 is the unit sphere, and is defined by associating with a point p on the surface X the unit normal vector to the tangent plane at p, which in turn can be identified with a point on S 2 . Although G can be defined for any surface in space, it is a remarkable theorem that the Gauss map of a minimal surface is conformal, i.e., it does not distort angles. By another theorem [1], conformal maps of two-dimensional surfaces can be identified with complex analytic functions, so we can investigate minimal surfaces using the tools from analysis of functions of a single complex variable. Figure 3: The Gauss Map of the Helicoid, of Enneper’s Surface and of the Catenoid. Coloring computed to correspond to mapping correspondences. 5. The Weierstrass Representation Using complex analysis, Weierstrass discovered in 1866 [5] that given any complex analytic function f and any meromorphic function g , the parameterized surface given by X(u, v ) = Re z a f (ζ )(1 − g (ζ ) 2 ) dζ,i z a f (ζ )(1 + g (ζ ) 2 ) dζ, 2 z a f (ζ )g (ζ ) dζ is a minimal surface, where z = u + iv and i = √ −1. Careful inspection reveals that if we identify S 2 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 e it f , 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 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. For contrast, we know that we cannot isometrically deform a flat sheet of paper into a portion of a sphere−any flat map of the earth must necessarily distort distances. 6. An Example of an Isometric Deformation The Weierstrass representations of the catenoid and helicoid both have g (z )= z . For the catenoid f c (z )=1/z 2 while f h (z )= i/z 2 for the helicoid. Notice that f c = e πi/2 f h , and thus the helicoid arises as an isometric deformation of the catenoid. Figure 5: Isometric Deformation of a Helicoid to a Catenoid. The penultimate frame suggests why we must work in the category of immersed surfaces, i.e., surfaces with self-intersection. References [1] L. Ahlfors. Complex Analysis. McGraw-Hill, third edition, 1979. [2] J. Douglas. Solution of the problem of Plateau. Trans. Amer. Math. Soc., 1931. [3] J. Gray. Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press, second edition, 1998. [4] D. Hoffman. The computer-aided discovery of new embedded minimal surfaces. Math. Intell., pages 8–21, 1987. [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. Acknowledgments • Philip S. Zivnuska for financial support of student-faculty collaborations in the UWEC Mathematics Department. • Graphics computed and rendered with Maple9. • UW-Eau Claire Center of Excellence for Faculty and Undergraduate Student Research Collaboration.
Transcript of Isometric Deformations of Minimal Surfaces Darin Mohr · 2004-04-07 · Isometric Deformations of...
Isometric Deformations of Minimal Surfaces
Darin MohrFaculty Mentor: Alex Smith
UWEC Mathematics Department Zivnuska Scholarship ProjectUniversity of Wisconsin-Eau Claire
April 2004
1. Minimal Surfaces
Minimal surfaces are found in nature when a closed loop is submersed in a soapy liquid,causing the formation of a soap film. It is found that amongst all possible surfaces that fillthe loop, nature selects that surface having minimal area.
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Figure 1: A curve in space, a filling-surface whose area can be reduced, and the filling-surface with
minimal area. There are many surfaces that fill the curve, in fact there are infinitely many degrees of
freedom in specifying such a surface.
Determining the surface of minimal area that spans a given curve is an old problem knownas the Plateau Problem [2]. The development of string theory by physicists has catalyzeda renewed interest in the theory of minimal surfaces [7].
2. Derivation of the PartialDifferential Equation
Given a parametric surface X(u, v) = 〈x(u, v), y(u, v), z(u, v)〉 with parameter domain D,the area is given by
A(X) =
∫∫
D
∣
∣
∣
∣
∣
∣
∣
∣
∂X
∂u× ∂X
∂v
∣
∣
∣
∣
∣
∣
∣
∣
du dv.
Mathematicians study the critical points of the function A. Using the calculus of vari-ations, we find [6] that critical points are solutions to the partial differential equation
xuu (yuzv − zuyv) + yuu (zuxv − xuzv) + zuu (xuyv − yuxv) ||Xv||2− (2 xuv (yuzv − zuyv) + 2 yuv (zuxv − xuzv) + 2 zuv (xuyv − yuxv))Xv · Xv
+ xvv (yuzv − zuyv) + yvv (zuxv − xuzv) + zvv (xuyv − yuxv) ||Xu||2 = 0.
In the language of differential geometry, solutions to this equation are characterized as thosesurfaces that have zero mean curvature. Differential geometers [3] use the shape operator
S to describe the curvature of a surface at a point. The shape operator is a symmetrictwo-by-two matrix. The eigenvalues of S are called the principal curvatures. The trace ofS is the mean curvature, and the determinant is the Gaussian curvature.For a minimal surface, the eigenvalues of the matrix S are opposites of one another, and thusthe Gaussian curvature is always negative.
3. Some Classical Solutions
Surfaces with vanishing mean curvature exist in abundance. Here are some of the mostrenowned solutions. Euler showed the catenoid was minimal in 1744, and Meusnier showed thehelicoid was minimal in 1776. No more minimal surfaces were found until Scherk discoveredone in 1835, and Enneper in 1864.
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Figure 2: Helicoid, Catenoid, Enneper and Scherk’s surfaces.
Helicoid: Catenoid:X(u, v) = 〈a sinh (v) cos (u) , X(u, v) = 〈a cos (u) cosh (v) ,
a sinh (v) sin (u) , bu〉 a sin (u) cosh (v) , bv〉
Enneper’s surface: Scherk’s surface:
X(u, v) = 〈u − 1/3 u3 + uv2, X(u, v) = 〈u, v, ln cos(v) − ln cos(u)〉− v + 1/3 v3 − vu2, u2 − v2〉
4. The Gauss Map
A central tool in analyzing minimal surfaces is the Gauss Map. This is a map G : X → S2,where S2 is the unit sphere, and is defined by associating with a point p on the surface X theunit normal vector to the tangent plane at p, which in turn can be identified with a point onS2. Although G can be defined for any surface in space, it is a remarkable theorem that theGauss map of a minimal surface is conformal, i.e., it does not distort angles. By anothertheorem [1], conformal maps of two-dimensional surfaces can be identified with complexanalytic functions, so we can investigate minimal surfaces using the tools from analysis of
functions of a single complex variable.
Figure 3: The Gauss Map of the Helicoid, of Enneper’s Surface and of the Catenoid.
Coloring computed to correspond to mapping correspondences.
5. The Weierstrass Representation
Using complex analysis, Weierstrass discovered in 1866 [5] that given any complex analyticfunction f and any meromorphic function g, the parameterized surface given by
X(u, v) = Re
⟨∫ z
af (ζ)(1 − g(ζ)2) dζ, i
∫ z
af (ζ)(1 + g(ζ)2) dζ, 2
∫ z
af (ζ)g(ζ) dζ
⟩
is a minimal surface, where z = u + iv and i =√−1. Careful inspection reveals that if we
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 belocally isometric to the original. This means that locally, any minimal surface
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.For contrast, we know that we cannot isometrically deform a flat sheet of paper into a portionof a sphere−any flat map of the earth must necessarily distort distances.
6. An Example of an IsometricDeformation
The Weierstrass representations of the catenoid and helicoid both have g(z) = z. For the
catenoid fc(z) = 1/z2 while fh(z) = i/z2 for the helicoid. Notice that fc = eπi/2fh, andthus the helicoid arises as an isometric deformation of the catenoid.
Figure 5: Isometric Deformation of a Helicoid to a Catenoid. The penultimate frame suggests why
we must work in the category of immersed surfaces, i.e., surfaces with self-intersection.
References[1] L. Ahlfors. Complex Analysis. McGraw-Hill, third edition, 1979.
[2] J. Douglas. Solution of the problem of Plateau. Trans. Amer. Math. Soc., 1931.
[3] J. Gray. Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press, second edition, 1998.
[4] D. Hoffman. The computer-aided discovery of new embedded minimal surfaces. Math. Intell., pages 8–21, 1987.
[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.
Acknowledgments
• Philip S. Zivnuska for financial support of student-faculty collaborations in the UWEC Mathematics Department.
• Graphics computed and rendered with Maple9.
• UW-Eau Claire Center of Excellence for Faculty and Undergraduate Student Research Collaboration.
