Rami Mohieddine
Math 401 Presentation
April 24, 2007
Minimal Surface
I.
History
The surface with the least area, for a given boundary curve, is called a minimal surface.
Minimal surfaces originated from research that began in the earlier part of the eighteenth century
but, at the time, mathematicians only knew of minimal surface that were created by planer
curves. For a closed planer curve, the surface with the least area is the interior of the curve (the
area bounded by the curve) and thus the surface remains in the plane. These trivial examples
were of little insight and intrigue. However, in the later part of the century, Jean Baptiste
Meusnier discovered that by rotating the catenary curve (a curve made by a hanging chain)
around a circle, one creates a non-planer minimal surface (Polthier). This was the first of many
nontrivial examples of minimal surfaces.
In the 19th century, the Belgian physicist Joseph Antoine Ferdinand Plateau examined the
surfaces created by soap film on a variety of closed wire frames. For a given wire structure, the
surface that is produced by the film is in fact a minimal surface, due to the minimizing of the
surface tension of the film. The numerical data he collected became a strong support for theory
of minimal surfaces. He conjectured that “every closed boundary curve that neither touches itself
nor intersects itself can be spanned by a minimal surface” (Polthier). Another way of stating his
conjecture is that every simple closed curve bounds a minimal surface. Thus the term “Plateau’s
Problem” refers to finding these surfaces.
II.
Introduction and Preliminary Concepts
Before we begin the analysis of minimal surfaces, it will be beneficial to review some basic
concepts and definitions from differential geometry and complex variables that we will utilize in
our discussion. We will start with differential geometry. S, a subset of R3 is a surface if, for each
point in S, there exist an open set U in R2 and an open set V in R3 (containing that point), where
there is a continuous bijective map with a continuous inverse which maps the intersection of V
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with S onto the open set U. This map is called a homeomorphism (and is also called a surface
patch or parameterization of the surface S), and the sets are thus called homeomorphic. If the
map and its inverse are smooth functions (continuous and has continuous partial derivatives of
all orders), then we call the map a diffeomorphism and the sets are now called diffeomorphic. For
a given surface, we have a collection of surface patches that parameterizes that surface. We call
this collection an atlas for the surface S. If the surface patch is smooth and the partials are
linearly independent, we call it a regular surface patch. If the atlas, for a given surface S,
contains only regular surface patches, then S is a smooth surface. The standard unit normal of a
surface is the vector N, where N= σu x σv / ||σu x σv||.
There are two important results that help characterize surfaces, concerning the ideas of
arc length and curvature of the surface. These results are known as the two fundamental forms.
The first fundamental form derives from the concept of arc length (s) of a curve bound to the
surface. Given a surface patch, σ(u,v), of S, the first fundamental form is
ds2 = E du2 + 2 F du dv + G dv2 ,
where E= σu . σu , F= σu .σv , and G= σv . σv
The first fundamental form is the metric of the surface, which aids in the calculation of lengths,
angles, and areas on the surface. The second fundamental form derives from the idea of
curvature of a curve on a surface. Given a surface patch, σ(u,v), of S, the second fundamental
form is
L du2 + 2 M du dv + N dv2 ,
where L= σuu . N , M= σuv . N, and N= σvv . N .
The second fundamental form helps us look at how much the surface bends and curves.
Another way to write the fundamental forms is by taking advantage of 2x2 matrices. This
is done by writing FI (the First Fundamental Form) with (E,F) as its first row and (F, G) as the
second row and writing FII (the Second Fundamental Form) with (L,M) as its first row and (M,
N) as the second row. Recall in the one dimensional case, the curvature of a unit speed curve is
the norm of the second derivative. That is an adequate definition for curves, however, for the two
dimensional surfaces there are actually multiple concepts of curvature. The principal curvatures,
denoted by κ, are the roots of det(FII – κ FI )=0. The roots give us the maximum and minimum
curving of a surface. The principal curvatures are the essential building blocks from which we
may define many other useful concepts of curvature, such as geodesic curvature, normal
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curvature, et cetera. For the purpose of minimal surfaces, we will only use the principal
curvatures to define the Gaussian and Mean curvature. The Gaussian Curvature, K, is the
product of the principal curvatures and the Mean Curvature, H is the average of the principal
curvatures.
Since the principal curvatures are dependent on the two fundamental forms, we can “redefine” K and H as:
K= (L N - M2) / (E G - F2)
H= (L G – 2 M F + N E) / 2(E G - F2).
As mentioned before the area of the surface is defined by the first fundamental form, namely by
the double integral of the det FI over a region. If the surface patch preserves angles, we call it a
conformal map, which translates to the first fundamental form of the surface being E (du2 + dv2)
(where E=G, F=0).
Now we move on to a review of some complex analysis that will be important to the
discussion of minimal surfaces. The derivative of a complex valued function, φ(ς)=f(u,v)+i
g(u,v) , where ς = u + i v ε C and f and g are real functions, exists if the partial are continuous
and satisfy both fu = gv & fv = -gu. The pair of partial differential equation are known as the
Cauchy-Riemann equations. Thus, if the function satisfies the Cauchy-Riemann equations and
the partials are all continuous, then the function is (complex) differentiable. Also, if the function
φ is differentiable throughout a neighborhood of a point then φ is analytic at that point. We say
that φ is analytic in a region if it is analytic at all points within the region. A function is
holomorphic if it is complex differentiable at all points in an open subset of the complex plane
and it is meromorphic if it is complex differentiable at all but a finite number of points in an
open subset of the complex plane. A useful property of holomorphic and meromorphic functions
is that their zeros are isolated, meaning that there are no other zero in a neighborhood of a zero.
For a given function, we define the Laplacian of a function, σ, as Δσ = σuu + σvv . If the
Laplacian of σ is zero then σ is called harmonic. It is clear that a complex valued function that
satisfies the Cauchy-Riemann equations has harmonic real and imaginary parts.
There are two very important results in complex analysis that will be used in our
discussion. The first is Cauchy’s Theorem. Cauchy’s Theorem states that for a simple closed
curve, if a function is analytic in the interior of the curve then the complex line integral of that
function around the closed curve is always zero. This also means that the line integral of an
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analytic function is path-independent, which is very useful result. The second important result is
the Laurent Series or Laurent Expansion for a complex function. For a given function f(z) that is
analytic in an open disc centered around zo in the complex plane, the Laurent Series
f(z)=Σ aj (z - zo ) j , where j goes from -∞ to ∞,
converges uniformally to the function, f(z). If the series does not have poles in the region, the its
Laurent Expansion will start at a nonnegative j, if it does have poles, then it will start at a
negative j.
III.
Analysis of Minimal Surface
Given a surface, Σ, and its surface patch or parameterization, σ(u,v), we shall look at a
family of parameterizations σt :U-> R3 , where U is an open subset of R2 independent of t, and t
lies in (-d,d) for some d>0. We define the surface variation of this family as φ :U-> R3, φ= Dt σt
(where Dt denotes the derivative with respect to t). Let π be a simple closed curve so that π and
its interior, int(π), is in U. Then mapping π to the surface, we get γt = σt (π). Define the area of Σ
that is bound by the curve γt as:
A(t)=
∫∫ int(π) dAσ
t
Since we are looking for surfaces that give us the least area for a given fixed boundary, we have
that γt = γ for all t. This means that there is no variation for points on the boundary, therefore, the
surface variation φt is zero everywhere on the boundary.
Theorem 1: For a fixed boundary, where φt = 0 on the boundary. Then
Dt A(0)= -2∫∫ int(π) H (EG-F2)1/2 α du dv
where H is the mean curvature of the surface patch, and E, G, and F are the coefficients of the
first fundamental form. Also, α= φ.N where N is the standard unit normal of the surface patch.
The proof is a bit tedious and is not particularly enlightening for this discussion so we refer you
to Elementary Differential Geometry by Pressley or Differential Geometry of Curves and
Surfaces by Carmo for further reading. Note that the term (EG-F2)1/2 (aka det FI), was used in the
define of area, so it would be expected to also arise in this theorem.
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We know that a minimal surface is the surface that minimizes area for a fixed boundary.
Using Theorem 1, it seems natural to define a minimal surface as one with mean curvature zero
everywhere. Setting H=0 in the equation above, we see that the area is a minimum.
Example 1: The trivial examples, as stated in the part I, are the surfaces that are the interior of
curves in the plane. They are minimal surfaces since all surface restricted to the plane have H=0.
For the unit circle, the minimal surface is the unit disk. For any boundary in the plane (as long as
it is a simple closed curve), the interior will create a regular smooth surface with zero mean
curvature and thus be a minimal surface.
Recall that a conformal map is one whose first fundamental form is E (du2 + dv2), where
E is a smooth positive function of (u,v) in U. From the definition of the first fundamental forms it
is clear that if a regular parameterization, σ, is a conformal map, then (i) σu . σu = σv . σv and (ii)
σu . σv = 0. If a regular surface patch satisfies both of these condition, then σ is conformal or
sometimes referred to as isothermal.
Proposition 1: Every surface has an atlas of surface patches that are conformal.
We shall take this proposition without a proof, for it is only a tool that we will use throughout
this discussion.
Proposition 2: Let σ be a regular conformal parameterization. Then
Δσ = σuu + σvv = 2 E H
where E= σu . σu = σv . σv (since it is conformal) and H=H N is the mean curvature vector,
where H is the mean curvature and N is the standard unit normal of the surface.
Proof: Since σ is conformal, σ satifies the two conditions stated in the last paragraph. By taking
the derivative of σu . σu = σv . σv with respect to u we get
2 σuu . σu = 2 σvu . σv
σuu . σu = σvu . σv
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By taking the derivative of σu . σv = 0 with respect to v we get
σvu . σv = - σu . σvv
σuu . σu = - σu . σvv
So,
(σuu + σvv ) . σu = 0.
By the same manner we also obtain,
(σuu + σvv ) . σv = 0. Therefore Δσ = σuu + σvv is
perpendicular to the partials of σ. Recall that { σu , σv , N } forms a basis for R3, so Δσ must be
parallel to N. Also recall that for conformal maps the mean curvature is equal to N + L / 2 E. So
(σuu + σvv ) . N = N + L=2 E H, which implies σuu + σvv = 2 E H. QED
As was said earlier, if the Laplacian of σ, Δσ = σuu + σvv , is zero then σ is a harmonic
function. The last proposition gives us a generalized formula for the Laplacian of any surface
patch, but when we apply this result to minimal surfaces we see a very interesting consequence.
Corollary 1: Let σ be a regular conformal parameterization, then σ is a minimal surface if and
only if σ is harmonic.
The proof is simple. Since σ is a minimal surface, H=0, therefore H=0. Thus the equation from
proposition 2 implies that Δσ = 0. So clearly σ is harmonic. QED
With the introduction of harmonic functions, the use of complex variables in this analysis
now seems very natural. We define a new complex-valued smooth function φ, φ(ς)= σu – i σv ,
where the complex coordinate ς = u + i v ε C for all (u,v) ε U. The choice of defining a function
in this way may not initial seem clear, but if one recalls from part II that the complex derivative
of an analytic function f(u,v)= a + i b, where a and b are real functions of u and v, satisfy the
Cauchy-Riemann Equations, namely, au= bv and av= - bu , one sees that if σ is harmonic than φ
satisfy the Cauchy-Riemann Equations. Therefore φ is a holomophic function. So we may
“restate” corollary 1 as:
Corollary 2: Let σ be a regular conformal parameterization then σ is a minimal surface if and
only if φ is a holomorphic function on U.
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With this result, we have converted the examination of minimal surfaces into a problem
that can be dealt with complex analysis. We know redirect our attention to φ.
Theorem 2: If σ is a regular conformal parameterization (isothermal) from an open set U in R2 to
R3 then the complex vector-valued function φ = ( φ1 , φ2 , φ3 ) as define before must satisfy the
two condition below:
(i)
φ.φ=0
(ii)
φ ≠ 0 for all points in U.
The converse is true also; if U is simply-connected (every simple closed curve in U can be
shrunk to a point in U) and if φ1 , φ2 , and φ3 are holomorphic in the open set U and φ meet the
two conditions above, then there exists a regular conformal parameterized minimal surface σ
from U to R3 such that the function φ is equal to φ(ς)= σu – i σv and σ is uniquely determined by
φ1 , φ2 , and φ3 up to translation.
Proof: For the first part of the theorem it is clear that φ . φ = σu . σu - σv . σv – 2i σu . σv , and
since σu . σu = σv . σv and σu . σv = 0, then φ . φ = 0. Since σ is regular, its partials are never both
zero, therefore φ is never zero. This completes the first part of the proof.
The converse is a bit more involved. We have that φ satisfies the two conditions above.
Lets take a fixed point in U and call it (x,y). We define σ by a complex line integral:
σ(u, v)= Re[ ∫π φ(ς) dς ]
where π is a curve in U from (x,y) to (u,v). By Cauchy’s Theorem, the integral is independent of
the curve π and thus the surface patch is path-independent as well. We define the total complex
line integral as Φ(ς)=
∫π φ(ς) dς , which is a holomorphic function in U since φ is, by
assumption, holomorphic, so Φ’(ς)= φ(ς). From our complex line integral definition of σ we
have
σu = Re[Φu ]=Re[Φ’]=Re[φ]
σv = Re[Φv ]=Re[ iΦ’]= -Im[φ].
Therefore,
σu – i σv = Re[φ] +i Im[φ]= φ.
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All that remains is showing that the surface patch is conformal. Condition (ii) shows that
since φ is never zero, then the partials of σ are never both zero. By condition (i), we see that
σu . σu - σv . σv – 2i σu . σv = 0
Real part: σu . σu - σv . σv = 0 which implies σu . σu = σv . σv
Imagery part: -2σu . σv = 0 which implies σu . σv = 0
We have shown that the partials are never both zero, so the imagery part implies that the partials
are linearly independent, and therefore σ is regular. And since σu . σu = σv . σv and σu . σv = 0, σ is
a regular conformal surface patch.
Let’s say we find another surface patch, ρ, that satisfies these conditions and also defines
the same φ. Then they will have equal partial derivatives: ρu = σu and ρv = σv, thus ρu - σu =0 and
ρv – σv =0. Therefore integration with respect to u and v we get that the difference, ρ – σ is a
constant, thus the surface patches are defined up to translations. QED
Example 2: Scherk’s Minimal Surface: The equation for Scherk’s minimal surface is
z= log (cos y/ cos x).
A parameterization for this surface is given by the following equations:
σ(u,v) = ( arg ( ς+i/ ς-i ), arg ( ς+1/ ς-1 ), log | ς2+1/ ς2-1 | )
where ς = u + i v ε C and ς≠ ±1, ±i. Also recall that arg ς is defined as the angle that is made by ς
with the real axis.
Therfore,
arg ( ς+i/ ς-i ) = tan-1 (2 u / u2 + v2 -1)
arg ( ς+1/ ς-1 ) = tan-1 (- 2 v / u2 + v2 -1)
log | ς2+1/ ς2-1 | = ½ log [ (u2 - v2 +1) 2 + 4 u2 v2 / (u2 - v2 -1) 2 + 4 u2 v2].
Now to apply the theorem, we have φ(ς)= σu – i σv, so we have
φ(ς)=( -2 / 1+ς2, -2i / 1-ς2, 4 ς / 1-ς4 ).
We see that all the components of φ are holomorphic in the defined domain, φ ≠ 0 everywhere,
and lastly that:
φ . φ = 4 / ( 1+ς2 ) 2 - 4 / ( 1-ς2 ) 2+16 ς2 / (1-ς4 ) 2
= 4/ (1-ς4 ) 2 *[ ( 1-ς2 ) 2 - ( 1+ς2 ) 2 + 4 ς2 ]
= 4/ (1-ς4 ) 2 *[ 1- 2 ς2 + ς4 - 1- 2 ς2 - ς4 + 4 ς2 ]
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= 0.
Therefore, by the previous theorem, σ is the surface patch for a minimal surface. From a
geometric perspective, the minimal surface is a saddle resting inside a bent rectangular boundary
(see attached graphic). It repeats itself in a lattice type pattern where only every other boundary
contains a surface and the remaining space is empty, giving a checkerboard like image.
We see, from example 2, that theorem removes much of the differential geometry behind
minimal surfaces, and instead deals with the analysis of the holomorphic function φ. By taking a
deeper look at the conditions in the previous theorem, we can actually find a general formulation
for φ, called Weierstrass’s representation.
Proposition 3: Let f (ς) be holomorphic on an open set U in C, where f (ς) ≠ 0 everywhere and
let g(ς) be a meromorphic function on the same set U where, if ςo ε U is a pole of order m≥1 of g,
then ςo is also a zero of order n≥ 2m of f. Then,
φ = ( ½ f (1 - g2 ), i/2 f (1 + g2 ), fg)
satisfies the two conditions in theorem 2.
Conversely every holomorphic function satisfying those two conditions is of this form.
Proof: Let f and g satisfy the assumption in this proposition. By Laurent expansion of the
functions around ςo we get:
f(ς) = an (ς - ςo ) n + O(n+1)
and
g(ς)= b-m (ς - ςo ) -m + O(-m+1)
where f starts its series expansion at n (similar to a Taylor series, since f is holomorphic) and g
starts its series expansion at –m, since it has poles in U, and an and b-m are nonzero complex
numbers. Then,
f (1 ± g2 ) = ± an b2-m (ς - ςo ) n-2m + O(n-2m+1)
and
fg= an b-m (ς - ςo ) n-m + O(n-m+1)
and all exponents of (ς - ςo ) are nonzero by assumption. So since φ’s components can be written
as a Laurent expansion of positive powers, φ is holomorphic around ςo . We can see that φ is
holomorphic when g is holomorphic and by the expansion we see φ is holomorphic around all of
g’s poles, so φ is holomorphic in U. We can also easily check that φ satisfies theorem 2.
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For the converse, assume φ = ( φ1 , φ2 , φ3 ) is holomorphic in U satisfying theorem 2. If
φ1 - iφ2 is not zero everywhere, we define
f= φ1 - iφ2 and g= φ3 / φ1 - iφ2 .
We know that f is holomorphic and g is meromorphic since φ is holomorphic. Since φ . φ = 0 we
have (φ1 + iφ2 )( φ1 - iφ2 ) = φ12 + φ22 = -φ32 and thus -fg2 =(φ1 + iφ2 ). The previous few
equations imply that the general formula in the theorem is correct and also imply that -fg2 is
holomorphic. Taking a look back at the Laurent series we see that the constraints on the orders of
the zeros and poles were necessary.
We also know that φ1 + iφ2 and φ1 - iφ2 cannot both be zero because that would mean φ1 and φ2
are both zero and then φ3 is also zero, contradicting the fact that we have assumed φ is nonzero
everywhere. So if φ1 - iφ2 is zero we can just switch it with φ1 + iφ2 and vice versa in this proof,
using the same logical arguments. QED
This formulation also gives a clean way to define the Gaussian curvature of these
surfaces from the complex functions f and g.
Proposition 4: The Gaussian curvature of a minimal surface with the functions f and g in
Weierstrass’s representation is
K= - 16 | dg/dς |2
|f|2 (1 + |g|2 )4
The proof is computational and reveals nothing of significance.
However this proposition gives rise to the last result.
Corollary 3: Let Σ be a minimal surface that is not part of a plane. Then, the Gaussian curvature
of Σ has isolated zeros.
Proof:
For proposition 4, we see K is zero only when dg/dς is zero. Since g is meromorphic, dg/dς is
also meromorphic. This means that unless dg/dς is identically zero, dg/dς has a finite number of
poles. dg/dς is not identically zero for that would mean Σ is a part of the plane, which is, by
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assumption, not the case. As mentioned in part II, meromorphic function have the property that
their zeros are isolated, so dg/dς has isolated zeros which implies that K does also. QED
Example 3: Catenoid/Hanging Chain: The catenary curve discussed in part I is the curve that is
made when a chain with fixed endpoints is left to hang freely and the surface of rotation made by
this curve is easily parameterized by the function:
σ(u,v) = ( cosh u cos v, cosh u sin v, u).
It is not hard to show that this is a conformal parameterization of the curve. The corresponding
holomorphic function, φ(ς)= σu – i σv , can also be easily found:
σu = ( sinh u cos v, sinh u sin v, 1)
σv= ( -cosh u sin v, cosh u cos v, 0)
Thus,
φ(ς)= σu – i σv = ( sinh u cos v, sinh u sin v, 1) – i ( -cosh u sin v, cosh u cos v, 0)
= ( sinh u cos v + i cosh u sin v, sinh u sin v- i cosh u cos v, 1).
With a bit of trigonometric simplification we get:
φ(ς)= ( sinh (u+iv), - icosh (u+iv), 1) = ( sinh ς , - icosh ς, 1).
We can clearly see that φ(ς) is never zero since its third component is always 1. Also,
φ . φ = sinh2 ς - cosh2 ς + 1= 0.
So by theorem 2, it is a minimal surface. From the proof of proposition 3 we can find the f and g
that are in Weierstrass’s representation since:
f = φ1 - iφ2 and g = φ3 / φ1 - iφ2
Therefore,
f = sinh ς –i (- icosh ς) = sinh ς –cosh ς = - e-ς (nonzero)
g = 1/- e-ς = - eς
Thus
φ(ς)= ( ½ f (1 - g2 ), i/2 f (1 + g2 ), fg) = (- ½ e-ς (1 - e2ς ), - i/2 e-ς (1 - e2ς ), 1)
Which we can show gives us our original description of φ(ς).
Also since,
dg/dς = - eς
| dg/dς |2 = e2u
and
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|f|2 (1 + |g|2 )4 = e-2u (1 + e2u)4
the Gaussian curvature as defined in proposition 4 is:
K= -16/ e-4u (1 + e2u)4 = -16/ (e-u + eu)4 = -[ 2 / ( e-u + eu ) ]4 = - sech4 u.
and of course the mean curvature is zero. The boundary of this surface is the two equal circles as
seen in the graphic below:
From: http://en.wikipedia.org/wiki/Image:Catenoid.png
Now the true beauty of minimal surfaces is exposed. Originating as a curious physics
problem involving hanging chains and soap films, strictly defined and examined using calculus
of variations and differential geometry, elegantly reduced to a simple inquiry by the analysis of
complex variables, and producing very interesting geometric objects, one sees that the study of
minimal surfaces is an example of many distinct branches of mathematics weaving together one
concept.
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Works Cited:
"Catenoid." Wikipedia, The Free Encyclopedia. 16 December 2005. Wikimedia Foundation, Inc.
23 April 2007 < http://en.wikipedia.org/wiki/Image:Catenoid.png >.
Do Carmo, Manferdo P. Differential Geometry of Curves and Surfaces. New Jersey: Prentice
Hall, 1976.
Levinson, Norman, and Raymond M. Redheffer. Complex Variables. San Francisco: HoldenDay, 1970.
Polthier, Konrad. “Touching Soap Film.” 10 January 2007. Freie Universität Berlin. 23 April
2007 < http://page.mi.fu-berlin.de/polthier/booklet/intro.html>.
Pressley, Andrew. Elementary Differential Geometry. Great Britain: Springer, 2006.
Weisstein, Eric W. "Meromorphic Function."12 March 2000. MathWorld--A Wolfram Web
Resource. 23 April 2007 <http://mathworld.wolfram.com/MeromorphicFunction.html>