ISOPERIMETRIC INEQUALITY FOR FLAT SURFACES
JAIGYOUNG CHOE
Korea Institute for Advanced Study, Seoul, 130-722, Korea e-mail : [email protected]
RICHARD SCHOEN
Department of Mathematics, Stanford University, Stanford, CA 94305, USA e-mail : [email protected]
1 Introduction
Given a domainD in R2, it is well known that the areaAofD and the length Lof ∂Dsatisfy
(1.1) 4πA≤L2
and that equality holds if and only if D is a disk. This isoperimetric inequality is perhaps the most beautiful inequality in geometry. In the hope of generalizing (1) one can ask the following question. Are there any surfaces that satisfy (1)? There are two natural candidates for this question: flat surfaces and minimal surfaces.
A flat surface is, by definition, a two-dimensional surface with flat metric.
Therefore a flat surface can be obtained from a generalized domain in R2 with some identifications along the boundary. Then, does every flat surface satisfy the isoperimetric inequality (1.1)? To this very natural question one can easily find a counterexample: a long cylinder. LetABCDbe a rectangle inR2. Then identifying the parallel sidesADandBCgives rise to a cylinder. Identifying the sides decreases the circumference of the rectangle, thereby causing the isoperimetric inequality (1.1) to fail if ADis sufficiently longer thanAB.
Being locally area minimizing and having zero mean curvature, minimal surfaces in Rn are thought of as generalized planes. Therefore it has long been conjectured that minimal surfaces should satisfy (1.1) as well.
In this note we will prove the isoperimetric inequality for some flat surfaces.
And we will see how the isoperimetric inequality for some minimal surfaces in R3 can be derived from that of the associated flat surfaces.
2 flat surfaces
There are dozens of proofs for the isoperimetric inequality inR2. To cite a few, see Steiner [Sp, p.439], [Cv, p.283], Bonnesen [Os, p.1199], Hurwitz [Os, p.1184], Brunn-Minkowski [Os, p.1190], Hadwiger [Ha, p.153], Knothe [T, p.22], Schmidt
103
[dC, p.32], Gromov [Cv, p.276], and H´elein [He]. Among these Knothe’s proof will be used in this section.
Definition. Aflat surfaceis an open surface with or without boundary which has a flat metric. Every flat surface can be isometrically immersed as a generalized domain in R2. This generalized domain may have multiplicity and branch points, and some parts of its boundary may be identified. Therefore a flat surface can be obtained as a layered surface by applying cutting and pasting to separate pieces of paper(=domain) and by identifying some parts of the boundary.
Theorem 1. Let F be a flat surface with nonempty boundary and suppose that the rays emanating from each boundary point ofF never intersect each other. ThenF satisfies the classical isoperimetric inequality 4πA ≤L2, and equality holds if and only ifF is a disk inR2.
Proof. There are two ways to find the area of a domain by scanning the interior from its boundary points. First, let D be a convex domain in R2, pa boundary point, and ρ a ray tangent to ∂D at p in the counterclockwise direction along ∂D. Let (r, θ) be the polar coordinates withrthe distance frompandθthe angle measured from ρ. Define r(θ) = r if there exists a boundary point with polar coordinates (r, θ). Then
(2.2) A=1
2 Z π
0
r(θ)2dθ.
IfD is not convex, definer(θ) = min{r: (r, θ) are polar coordinates of a boundary point}. Then we just have
(2.3) A≥1
2 Z π
0
r(θ)2dθ.
The second way is by Crofton’s formula [Sa]. Parametrize Γ = ∂D by arclength s, 0 ≤ s ≤ L = Length(Γ). Let (r(s), θ(s)) be the polar coordinates measured with respect to the point Γ(s) and the ray tangent to Γ at Γ(s). Definer(s, θ) = min{r(s) : (r(s), θ(s)) are polar coordinates of a point in Γ}. Then Crofton’s formula states, whetherD is convex or not,
(2.4) 2πA=
Z π
0
Z L
0
r(s, θ) sinθ ds dθ.
In general, (2.2) and (2.3) do not hold on a flat surface. This is because the scanning map from a fixed boundary point may not be one-to-one on a flat surface. But if no two rays emanating from any boundary point intersect each other, which is guaranteed by the hypothesis of this theorem, then (2.3) holds. However, (2.4) holds even without this hypothesis for flat surfaces. Therefore, using (2.3) and (2.4) and integrating on∂D×∂D, we have
0 ≤
Z
Γ
Z
Γ
Z π
0
Z π
0
(r1sinθ2−r2sinθ1)2dθ1dθ2ds1ds2
= 2
Z
Γ
Z π
0
Z π
0
Z
Γ
r12sin2θ2ds1dθ1dθ2ds2
−2 Z
Γ
Z
Γ
Z π
0
Z π
0
r1sinθ1r2sinθ2dθ1dθ2ds1ds2
≤ 4A Z
Γ
Z π
0
Z
Γ
sin2θ2ds1dθ2ds2−2 Z
Γ
Z π
0
r1sinθ1dθ1ds1
2
= 2πA(L2−4πA).
4πA=L2 implies that
r1sinθ2−r2sinθ1= 0
for all values ofθ1,θ2,s1 ands2. Hence fors1=s2= 0 and θ2=π/2 we have r1(0, θ1) =r2(0, π/2) sinθ1,
therefore∂D is a circle with diameterr2(0, π/2).
We now turn to the second condition which guarantees the isoperimetric in- equality for the flat surfaces.
LetCbe a smooth immersion of a circle in a flat surfaceF. Therotation number ofC in F is defined as follows. Suppose Cis parametrized by arclength 0≤s≤`.
Since F is locally inR2,C0(s) is a well-defined map from [0, `] into the unit circle S1.
Definition. Therotation number ofC inF is defined to be the degree of the map C0(s) : [0, `]→S1. The rotation number is not necessarily an integer.
Theorem 2. If a flat surface with integer rotation number F contains no straight loop (i.e., a loop with vanishing curvature), then 4πA ≤L2 holds for F, equality holding if and only if F is a disk.
Proof. See [CS].
3 Minimal Surfaces
A minimal surface is not flat; its Gaussian curvature is negative except possibly at isolated flat points. However, a minimal surface is locally area minimizing away from isolated singular points. It is in this sense that a minimal surface is called a generalized plane and it is for this reason that one conjectures that a minimal surface should satisfy the same isoperimetric inequality as inR2: 4πA≤L2.
In this note we will show how to obtain from a minimal surface two flat surfaces whose area and boundary length appropriately control those of the minimal surface.
Given a minimal surface Σ in R3, one can construct three flat surfaces from Σ by way of the Weierstrass representation formula as we shall see below.
Let Σ be a surface in R3 and let x1, x2, x3 be the rectangular coordinates of R3. If we denote the vector-valued function Ψ on Σ by Ψ = (x1, x2, x3), then
∆Ψ = (∆x1,∆x2,∆x3) =H,~
where the Laplacian is taken with respect to the metricds2of Σ andH~ is the mean curvature vector of Σ. Hence if Σ is a minimal surface, thenx1,x2,x3are harmonic on Σ. Letz=x+iy be a complex coordinate on Σ such that
ds2= 2λ|dz|2. Then
(3.5) ∆Ψ = 2
λΨzz= 0.
Define
(ϕ1, ϕ2, ϕ3) = 2 dx1
dz ,dx2 dz ,dx3
dz
.
From (3.5) it follows thatϕ1,ϕ2,ϕ3are holomorphic in z. Furthermore, since Ψ : (x, y)7→(x1, x2, x3) is a conformal map, we have
ϕ21+ϕ22+ϕ23=|Ψx|2− |Ψy|2−2i <Ψx,Ψy>= 0.
Now, defining a holomorphic functionf and a meromorphic functiongby f =ϕ1−iϕ2, g= ϕ3
ϕ1−iϕ2
,
we obtain theWeierstrass representation formulafor Σ : (x1, x2, x3) = Re
1 2
Z z
f(1−g2)dz, i 2
Z z
f(1 +g2)dz, Z z
f g dz
.
Note here that
(3.6) 4λ=|ϕ1|2+|ϕ2|2+|ϕ3|2=|f|2(1 +|g|2)2 2
and that the Gaussian curvature K of Σ is given by K =−1
2∆ logλ≤0.
However, since log|f|2, log|f g|2 and log|f g2|2 are harmonic we can define three flat metricsg1,g2,g3 on Σ:
g1=1
4|f|2|dz|2, g2= 1
2|f g|2|dz|2, g3=1
4|f g2|2|dz|2.
LetFibe the flat surface Σ with metricgi,i= 1,2,3. Fi can be constructed as a generalized domain in R2 by complex integration on Σ as follows.
(3.7) F1:
Z z1
2f dz, F2: Z z1
2f g dz, F3: Z z1
2f g2dz.
In fact, F1 and F3 are the limits of the sequences of minimal surfaces {Σ1n} and {Σ3n}, respectively, defined as follows.
Σ1n : Re 1
2 Z z
f(1−g2 n2)dz, i
2 Z z
f(1 + g2 n2)dz,
Z zf g n dz
, Σ3n : Re
1 2
Z z f
n2(1−n2g2)dz, i 2
Z z f
n2(1 +n2g2)dz, Z zf g
n dz
, which are obtained from Σ by changing its Weierstrass data from f and g to f andg/n,f /n2andng, respectively. Alternatively,F1 andF3 can be introduced as follows. Let ¯xj be the harmonic conjugate ofxj, j= 1,2, and definex∗j =xj+i¯xj. Then we have the two maps x∗1−ix∗2 and−x∗1−ix∗2 from Σ into C, and we can easily see that the images of these maps are the flat surfaces F1 andF3.
The following theorem states thatF1andF3 play a pivotal role for the isoperi- metric inequality of the minimal surface Σ.
Theorem 3. If the flat surfaces F1 and F3 of a minimal surface Σ in R3 both satisfy the isoperimetric inequality4πA≤L2, then so doesΣ.
Proof. Let A = Area(Σ), Ai = Area(Fi), L = Length(∂Σ), Li = Length(∂Fi).
Then
A =
Z
Σ
1
4|f|2(1 +|g|2)2dx dy
= Z
Σ
1
4|f|2dx dy+ Z
Σ
1
2|f g|2dx dy+ Z
Σ
1
4|f g2|2dx dy
= A1+A2+A3
and L=
Z
∂Σ
1
2|f|(1 +|g|2)ds= Z
∂Σ
1
2|f|ds+ Z
∂Σ
1
2|f g2|ds=L1+L3. Moreover, by the H¨older inequality,
(A2)2= Z
Σ
1
2|f g|2dx dy 2
≤ Z
Σ
1
2|f|2dx dy Z
Σ
1
2|f g2|2dx dy= 4A1A3. Hence
4πA ≤ 4πA1+ 8πp
A1A3+ 4πA3≤(L1)2+ 2L1L3+ (L3)2= (L1+L3)2
= L2.
Using this theorem we will prove the isoperimetric inequality for some minimal surfaces inR3 in the following theorems. See [CS] for the proofs.
Theorem 4. If Σ⊂R3 is a minimal surface with fluxes parallel to a line, then Σ satisfies4πA≤L2.
Theorem 5. LetΣbe a triply connected minimal surface inR3, that is,Σhas three boundary components and no genus. Then Σ satisfies the isiperimetric inequality 4πA≤L2.
Definition. LetS⊂R3 be a compact orientable surface whose boundary∂S is the union of Jordan curves γ1, ..., γn. We say that ∂S is non-twisted if an ε-tubular neighborhood ofγk inΣcan be deformed to a trivial strip for all k= 1, ..., n.
Theorem 6. If Σis a compact minimal surface inR3 with non-twisted boundary, thenΣsatisfies 4πA≤L2.
References
[Ca] T. Carleman,Zur theorie der Minimalfl¨achen, Math. Z.9(1921), 154–160.
[dC] M. do Carmo,Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, 1976.
[Cv] I. Chavel,Riemannian Geometry: A Modern Introduction, Cambridge Uni- versity Press, 1993.
[Ch] J. Choe, The isoperimetric inequality for a minimal surface with radially connected boundary, Ann. Scuola Norm. Sup. Pisa Cl. Sci.17(1990), 583–
593.
[CS] J. Choe and R. Schoen, The sharp isoperimetric inequality for a minimal surface in Euclidean space.
[Fe] J. Feinberg,The isoperimetric inequality for doubly connected minimal sur- faces inRn, J. d’Anal. Math.32(1977), 249–278.
[Ha] H. Hadwiger, Vorlesungen ¨uber Inhalt, Oberfl¨ache und Isoperimetrie, Springer, Berlin, 1957.
[He] Fr´ed´eric H´elein,In´egalit´e isop´erim´etrique et calibration, Ann. Inst. Fourier 44(1994), 1211–1218.
[Hs] C. C. Hsiung, Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary, Ann. Math.73(1961), 213–220.
[LSY] P. Li, R. Schoen and S.-T. Yau,On the isoperimetric inequality for minimal surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci.11(1984), 237–244.
[Os] R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc.
84(1978), 1182–1238.
[Re] W. T. Reid,The isoperimetric inequality and associated boundary problems, J. Math. Mech.8(1959), 897–906.
[Sa] L. A. Santal´o,Integral Geometry and Geometric Probability, Encyclopedia of Mathematics and its Applications1, Addison-Wesley, Reading, 1976.
[Sp] M. Spivak,A comprehensive Introduction to Differential Geometry,4, Pub- lish or Perish, Berkeley, 1979.
[T] A. Treibergs, Inequalities that Imply the Isoperimetric Inequality, Lecture Note, University of Utah,
http://www.math.utah.edu/∼treiberg/isoperim/isop.pdf
