cylindrical surfaces of delaunay

CYLINDRICAL SURFACES OF DELAUNAY 469

RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMOSerie II, Tomo LIII (2004), pp. 469-477

CYLINDRICAL SURFACES OF DELAUNAY

FRANK MORGAN

For the cylindrical norm on R3 , for which the isoperimetric shape is a cylinder ratherthan a round ball, there are analogs of the classical Delaunay surfaces of revolution ofconstant mean curvature.

1. Introduction.

Classical “Delaunay” surfaces of revolution of constant mean curvature1 are of the four types of Figure 1: nodoid, spheres, unduloid, and cylinder(see [3]). Instead of usual area, we consider the norm � which is the sumof areas of horizontal and vertical projections, for which the isoperimetricor “Wulff” shape is a cylinder instead of a round ball. For this norm � , ourTheorem 3.1 provides analogous “cylindrical surfaces of Delaunay,” picturedin Figure 2. For an expository account, see [9].

There are some complications in proving that these surfaces have unitmean curvature. Because the norm is not smooth, unit mean curvature isno longer a local condition, and the relevant first variation formula involvessingular terms.

The key to proving uniqueness is to show that the generating curves consistentirely of horizontal and vertical segments, which reduces the problem todiscrete geometry.

1.1. Surfaces of Kapouleas.

A sequel [6] provides nonrotationally symmetric unduloids and nodoidsbased on a hexagonal prism Wulff shape and uses them to construct compactimmersed “hexagonal” constant-mean-curvature surfaces. Together the papersprovide a simpler extension to other integrands of the more difficult and

470 FRANK MORGAN

technical results of Kapouleas [4, 5] which necessarily involved infinite-dimensional analysis.

1.2. Acknowledgements.

This work is partially supported by a National Science Foundation grant.

Figure 1 - The four classical types of surfaces of Delaunay (constant-mean-curvature surfacesof revolution): nodoid, spheres, unduloid, and cylinder. (B. White [11, Fig. 3].)

Figure 2 - Analogous cylindrical surfaces of Delaunay-nodoid, cylinders, unduloid, andcylinder-for our cylindrical norm � .


2. Cylindrical Norm and First Variation.

Section 2 defines our cylindrical norm � , and provides first variation for-mulas in terms of the associated variable-coefficient integrand� on generatingcurves for surfaces of revolution. For references see [8, Chapt. 10] and [10].

2.1. The cylindrical norm.

Let � be the “cylindrical” norm on R3, cheap on horizontal and verticalnormal vectors, defined by

�(α,β, γ ) =√α2 + β2 + |γ |.

A rectifiable surface S with unit normal n has an associated energy

�(S) =∫

S

�(n).

The dual 11 to the unit ball {� ≤ 1}, called the Wulff shape, providesthe least-area way to enclose given volume. For our cylindrical norm � , the(normalized) Wulff shape is the unit cylinder of height 2.

2.2. First variation.

Following [1] and [7], if S is oriented and V denotes volume insideS, we say that S has � mean curvature 1 if the first variation δ1(v) of�−2V is nonnegative for smooth variations v of compact support. (A smoothhypersurface in Rn+1 has classical mean curvature 1 if and only if the firstvariation of A − nV vanishes, where A denotes area). We say “nonnegative”rather than “zero” because since � is not smooth, δ1 is not linear, althoughit is well defined and homogeneous. For example a small smooth bump v onthe top of the Wulff shape (with exceptional vertical normal where � is notsmooth) will have δ1(v) and δ1(−v) both positive.

For a surface S of revolution, determined by curves C(s) in the halfplane{x ≥ 0},

(1)1

2π�(S) = �(C) =

∫

C

�(C(S),C ′(S))ds =∫

C

x(|dx | + |dy|),

where � is the variable-coefficient integrand

�((x, y), (α, β)) = x(|α| + |β|)on tangent vectors (α, β) = C ′(S) (the usual convention for curves, boundingarea on the left, with outward unit normal to the right), and dV = xd A.

472 FRANK MORGAN

Assume that initially C is parametrized by arclength, so that C ′ is the unittangent and C ′′ is the curvature vector κ . Differentiating (1) (see [8, 10.4]) andsubtracting the volume term yields

(2) δ1(v) =∫

D1�(v)+ D2�(v′)− 2

∫xv · n.

Whenever � is smooth (i.e., whenever C ′ lies in the interior of one of thefour quadrants), it follows by integration by parts that

(3)

δ1(v) =∫

D1� · v+ D2� · v′ − 2

∫xv · n

=∫(D1�− D12�(C

′)− D22�(κ)− 2xn) · v.

The vector D22�(κ) is called the � curvature vector. Since our � is locallylinear in (α, β), D22� = 0 and (3) becomes

(4) δ1(v) =∫(±1− 2x)n · v.

according to whether C ′ is in the upper or lower halfplane, i.e., whether theenclosed area is to the left or right. (It suffices to do the computation in thefirst quadrant; the others follow by symmetry. The first and third quadrants areidentical, except that n has changed sign and the volume has moved from theleft to the right, flipping back the sign of the −2x term. The first and secondquadrants are completely symmetric.)

For a vertical segment {x = R} with boundary, for a smooth variationv = (u, v), (2) yields

(5) δ1(u, v) =∫

u + R

∫|du| + R(v1 − v0)− 2R

∫u.

The first term reflects the increase in � as you move right. The next two termsreflect the fact that D2�((x, y), (0, 1))(α,β) = |α|+ |β|. The final term is thevolume term.

For a horizontal segment {r ≤ x ≤ R} with boundary, (2) yields

(6) δ1(u, v) =∫

x |dv| + R

∫|du| + Ru1 − Ru0 − 2

∫xv.


3. Cylindrical Surfaces of Delaunay.

Theorem 3.1 characterizes the surfaces of revolution of generalized con-stant mean curvature for our cylindrical norm.

THEOREM 3.1. (Cylindrical surfaces of Delaunay). For the cylindricalWulff shape, there are precisely two one-parameter families of “Delaunay”surfaces of revolution of generalized constant mean curvature 1, as in Figure 2.

The “unduloids” are embedded chains of similar cylinders of radiusalternating between R and r < R and height alternating between H andh < H , with R + r = 1 and

H

R= h

r= 2

R − r

As r approaches1

2, they approach an infinite cylinder; as r approaches 0,

they approach a chain of unit cylinders of height 2 (the Wulff shape).The “nodoids” are immersed chains of similar cylinders of alternating

orientation, radius R or r < R, height H or h < H , with R − r = 1 and

H

R= h

r= 2

R + r

As r approaches 0, they approach a chain of unit cylinders.There are no other connected complete piecewise smooth surfaces of

revolution of generalized mean curvature 1.

3.2. Remark on curvature 0. For the cylindrical Wulff shape, much easiersimilar analysis shows that there are precisely two types of connected surfacesof revolution of generalized constant mean curvature 0: (1) a horizontal planeand (2) the complement of a finite cylinder of radius R and height 2R (a Wulffshape) in a horizontal slab of thickness 2R, a “cylindrical catenoid.”

3.3. Remark on stable surfaces. Just as round spheres and planes arethe only stable surfaces of revolution of classical constant mean curvature(for fixed volume), Wulff cylinders and planes are the only stable surfacesof revolution of cylindrical constant mean curvature, with one remarkableexception. In contrast to the classical Rayleigh instability of a long cylinder,the infinite cylinder is stable for cylindrical energy, as I learned from J. Taylorand J. Cahn (see [2]). Indeed, as in the following proof, the first variation ofany nonzero normal variation is strictly positive.

Proof of Theorem 3.1. We give the proof that the unduloids have gener-alized constant mean curvature 1. A similar proof holds for the nodoids, and

474 FRANK MORGAN

more easily for the unit and infinite cylinders (which alternatively are equilib-ria as limits of equilibria). Finally we’ll show that there are no other surfacesof revolution of cylindrical constant mean curvature 1.

By standard symmetrization arguments, it suffices to consider only sur-faces of revolution, determined by curves C(s) in the halfspace {x ≥ 0} withrenormalized energy

�(C) =∫

C

x(|dx | + |dy|)

and volume dV = xd A.Consider one of our generating curves C as in Figure 3. By the defini-

tion of unit mean curvature, we must show that for any smooth, compactlysupported variation vectorfield v(s) = (u, v)(s), the first variation δ 1(v) of�− 2A is nonnegative. The first variation formulas 2.2(5,6) show that the ef-fects of u and v are decoupled, so we may consider them separately, startingwith u. It suffices to consider u on one vertical segment (and boundary effectson the adjacent horizontal segments). For an outer vertical segment,

δ1(u) = R

[u0 + u1 +

∫|du|

]+ (1− 2R)

∫u

≥ 2R max(u)+ (1− 2R)H max(u) = 0

because H = 2R

(R − r)= 2R

(2R − 1). For an inner vertical segment,

δ1(u) = r

[−u0 − u1 +

∫|du|

]+ (1− 2r)

∫u

≥ −2r min(u)+ (1− 2r)h min(u) = 0

because h = 2r

(R − r)= 2r

(1− 2r). For v on a horizontal segment (with say

outward normal upward),

δ1(v) = Rv1 − rv0 +∫

x |dv| − 2

∫xv.

Since replacing v(x) by sup {v(ξ) : ξ ≥ x} does not increase δ1(v), we may


assume that v is nonincreasing. Then

δ1(v) = Rv1 − rv0 −∫

xdv − 2

∫xv

= Rv1 − rv0 − [xv]Rr +

∫vdx − 2

∫xv

= 2

∫ (1

2− x

)v.

Since r + R = 1,1

2is the midpoint of [r, R]. Since v is nonincreasing, by

symmetry about1

2, δ1(v) ≥ 0.

Figure 3 - A generating curve for a cylindrical unduloid.

To prove uniqueness, consider any piecewise smooth surface of revolutionof mean curvature 1, determined by a curve C(s) in the halfplane {x ≥ 0}. IfC has a smooth nonlinear pieces, it has a smooth nonlinear piece with C ′(s)strictly within one quadrant, where � is smooth. By the first variation formula

2.2(4), equilibrium implies that x = ± 1

2, a contradiction. We conclude that

C consists of horizontal and vertical segments.

There cannot be an infinite horizontal segment. Otherwise a unit outwardvariation over a long interval, with nice transition to zero over unit intervals atthe ends, would have negative first variation. A similar argument shows that

(1) an infinite vertical segment must have x = 1

2with the enclosed region

on the left.

Hence if there is no horizontal segment, we have a cylinder of radius1

2.

476 FRANK MORGAN

If a horizontal segment H1 goes to the axis {x = 0},H1 = {0 ≤ x ≤ R, y constant},

with say the enclosed region below, for equilibrium the adjacent verticalsegment V1 must go downward. Equilibrium under vertical translation of H1

implies that R = 1. By (1), V1 is finite. If the next horizontal segment H2

goes to the right, equilibrium under horizontal translation of V1 implies that

R = 1

2, a contradiction. Hence H2 goes to the left. It is in equilibrium under

vertical translation only if it goes to the axis. Equilibrium under horizontaltranslation of V1 implies that its height is 2. The surface is the Wulff shape.

Henceforth we may assume that horizontal segments do not go to the axis{x = 0}.

Now we may assume that there is a horizontal segment

H1 = {r1 ≤ x ≤ R1, y constant},with the adjacent segment on the left downward.

Case 1: the segment V1 on the right is upward. Equilibrium under ver-tical translation of H1 implies that the enclosed volume lies above and thatR1 + r1 = 1. By (1), V1 is finite.

If the next horizontal segment H2 goes to the right, equilibrium under

horizontal translation of V1 implies that R1 = 1

2, a contradiction. Hence

H2 goes to the left from R1 to say r2 < R1. If the next vertical segmentV2 goes downward, equilibrium under vertical translation of H2 implies thatR1 − r2 = 1, a contradiction. Hence V2 goes upward and R1 + r2 = 1, sothat r2 = r1. By (1), V2 is finite. If the next horizontal segment H3 goes

to the left, equilibrium of V2 implies r1 = 1

2, a contradiction. Hence H3

goes to the right, from r1 to say R3, which now equals R1 as before andthe pattern continues. Similarly the pattern continues downward. Equilibriumunder horizontal translation of the vertical segments implies that the outer ones

have height H = 2R1

R1 − r1and that the inner ones have height h = 2r1

(R1 − r1).

The surface is an unduloid.

Case 2: the segment V1 on the right is downward. Equilibrium undervertical translation of H1 implies that the enclosed volume lies below and thatR1 − r1 = 1. By (1), V1 is finite.

If the next horizontal segment H2 goes to the right, equilibrium under

horizontal translation of V1 implies that R1 = 1

2, a contradiction. Hence


H2 goes to the left from R1 to say r2 < R1. If V2 goes down, equilibriumunder vertical translation of H2 implies that R1 + r2 = 1, a contradiction.Hence V2 goes up, and equilibrium under vertical translation of H2 impliesthat R1 − r2 = 1, so that r2 = r1. By (1), V2 is finite. If H3 goes to the left,equilibrium under horizontal translation of V2 yields a contradiction. HenceH3 goes to the right. If V3 goes up, equilibrium under vertical translation ofH3 yields a contradiction. Hence V3 goes down. Equilibrium under verticaltranslation of H3 implies that R3 − r1 = 1, so that R3 = R1, and thepattern continues. Similarly the pattern continues in the opposite direction.Equilibrium under horizontal translation of the vertical segments implies that

the outer ones have height H = 2R1

R1 + r1and that the inner ones have height

h = 2r1

R1 + r1. The surface is an nodoid.

REFERENCES

[1] Barbosa Joao Lucas, do Carmo Manfredo, Stability of hypersurfaces with constant meancurvature, Math. Z., 185 (1984), 339-353.

[2] Cahn J. W., Stability of rods with anisotropic surface free energy, Scripta Met., 13 (1979),1069-1071.

[3] Eells James, The surfaces of Delaunay, Math. Intel., 9 (1987), 53-57.[4] Kapouleas Nicolaos, Compact constant mean curvature surfaces in Euclidean three-

space, J. Diff. Geom., 33 (1991), 683-715.[5] Kapouleas Nicolaos, Constant mean curvature surfaces in Euclidean three-space, Bull.

AMS, 17 (1987), 318-326.[6] Morgan Frank, Hexagonal surfaces of Kapouleas, Pacific J. Math., to appear, preprint at

arxiv.org.[7] Morgan Frank, Planar Wulff shape is unique equilibrium, Proc. Amer. Math. Soc., 133

(2005), 809-813.[8] Morgan Frank, Riemannian Geometry: a Beginner’s Guide, A. K. Peters, Ltd, 1998.[9] Morgan Frank, Streams of cylindrical water, Math. Intel., 26 (2004), 70-72.

[10] Taylor Jean, Crystalline variational problems, Bull. AMS, 84 (1978), 568-588.[11] White Brian, Some recent developments in differential geometry, Math. Intel., 11 (1989),

41-47.

Pervenuto il 3 aprile 2003.

Department of Mathematics and StatisticsWilliams College

Williamstown, Massachusetts [email protected]

cylindrical surfaces of delaunay

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