CHARACTERIZATIONS OF THE MEAN CURVATURE AND A PROBLEM OF G. CIMMINO^{∗}

Johannes C.C. Nitsche

University of Minnesota

1

Let T be a differential geometric surface immersed in Euclidean space R^{3}, p an interior point of T and Sp(r) the sphere of radius r centered at p. For sufficiently small values of r, Sp(r) intersects T in a simple closed curve Cp(r) dividing Sp(r) into two parts. Denote by σp(r) · r^{2} the area of that part which is penetrated by the normal vector of T at p. σp(r) is approximately equal to 2π. If T is a plane, then obviously σp(r) ≡ 2π. One may ask, as Gianfranco Cimmino did sixty-two years ago, to find all surfaces which share this property with the plane ([4], p. 288, [10], § 875 , [11], A7.3.):

Problem : To determine the class of all surfaces T which have the property that for every interior point p ∈ T there is a number rp > 0 such that σp(r) = 2π for 0 < r < rp.

The interest in this problem is heightened by the observation that the right helicoid is a member of this class, a fact which is easily confirmed. We shall prove below that there are no others:

Theorem: The plane and the right helicoid are the only differential geometric surfaces with the property formulated in the Problem above^{1}.

One is invariably reminded of another question, due to R. Osserman and not yet settled, in which, surely by sheer coincidence, plane and right helicoid are believed to be the sole possibilities^{2}; see e.g. [19].

∗1991 Mathematics Subject Classification. Primary 49Q05, 53A05, 53A10; Secondary 35J60, 53C42.

1The plane may be regarded as a degenerated helicoid of pitch 0.

2Of course, we have E. Catalan’s old discovery [3] of the right helicoid as the only skew ruled surface. For references to the literature see [11], § 79. Further characterizations of the right helicoid can be found in [10], §§ 712, 726, 730 and in section 8 below.

A promising approach to our problem will suggest itself if we take note of certain nontraditional characterizations^{3} of the mean curvature Hp of T at p that rely on the relation between the sphere Sp(r) and the curve Cp(r) and Hp. As we mention these characterizations, a few general comments seem to be apposite.

The name mean curvature (courbure moyenne) for the arithmetic mean of the principal curvatures k_{1} = 1/R_{1} and k_{2} = 1/R_{2} of T , as well as a discourse supporting its relevance, had been put forth by Sophie Germain ([8], p. 7ff). Earlier, more awkward descriptions were in use; for instance, a minimal surface, i.e. a surface of vanishing mean curvature, was said to be ‘a surface for which the principal curvature radii are equal and of opposite sign.’

Let us call l(r) the length of the curve Cp(r) and L(r) the length of its spherical image. R. Sturm, an ahherent of J. Steiner’s and proponent of synthetic geometry a century ago, stated in [16] that the quotient L(r)/l(r) converges for r → 0 to the suitably oriented mean curvature Hp of T at p^{4}. The similarity to O. Rodrigues’s characterization in 1815 of the Gauss curvature k_{1}k_{2} by comparing a surface element with its spherical image^{5} leads Sturm to the following comment^{6} ([16], p. 382):

“So it can be seen that Sophie Germain’s curvature measure \frac{1}{R_{1}} + \frac{1}{R_{2}} should not be totally disparaged vis-à-vis that of Gauss, and that its definition is not as arbitrary as it is for instance called by B. Cournot in his Theory of Functions. To be sure, the curvature measure of Gauss will remain absolutely the more valuable, and, in particular, the contemplations put forth by Miss Germain in support of her propostion are hardly acceptable.

In any event, the mean curvature has found its due significance also in the theory of minimal surfaces.”

Sturm might have mentioned as well the eminent physical significance of the quantity H which was recognized in 1805 and 1806 by T. Young [20] and P.S. Laplace [9], respectively, in their celebrated investigations concerning the rise of a liquid in a capillary tube: the pressure difference near an interface is proportional to the mean curvature of the interface at the point under consideration.

Cournot’s remark to which Sturm alludes is as follows ([5], p. 474):

“Miss Sophie Germain has in her investigations concerning elastic surfaces and, above all, in the memoir ([8]), the final work of this remarkable woman, taxed her energies to establish the fact that the curvature of a surface has as measure the function \frac{1}{R_{1}} + \frac{1}{R_{2}} ; but her arguments do not convince at all. Very recently, the same idea has been taken up again by a young professor, Mr. Renard, in a thesis defended before the Faculty of Sciences of Paris, and that is known to us only through an abstract. It is only by analogy that one can pass from the definition for the curvature of a curve, which is generally unchallenged, to the definition for the curvature of a surface; and in geometry as well, analogy suggests itself rather than proves itself. The best definition for the curvature of a surface will be that which allows us to link the largest possible number of surface properties by means of the simplest possible expression.”

3i.e. characterizations not mentioned in the standard texts on differential geometry.

4Intriguing as Sturms statement is – he proves it with the tools of synthetic geometry in the spirit of J. Steiner – it appears to be incorrect.

5Rodrigues missed, however, the theorema egregium that Gauss obtained in its full generality in the course of 1826.

6Translation of the quotes are by the author.

The above quotes are examples of the strong (and strange) sentiments, to the point of proselytizing, that earlier geometers harbored concerning the value of various curvature measures for surfaces, in particular the relative merits of the total (now called Gaussian) curvature K over the mean curvature H. But other curvature measures, to complement Gaussian and mean curvature, have been proposed as well. For example, in 1889 F. Casorati made the case for the measure C = (k_{1}^{2} + k_{2}^{2})/2 (the letter C presumably standing for Casorati). We have 2H^{2} = K + C. In fact, Casorati puts forth a fervent plea ([2], p. 109) recommending to the aspiring young mathematicians that they pursue research dedicated to the new curvature measure and urging the authors of textbooks to accord it a proper place in their work^{7}. One of his arguments was the fact that only the vanishing of C, but not that of K or H, characterizes the plane. Bianchi ([1], p. 197) rejects this argument as unsound pointing out that it could be continued ad infinitum: to the curvature measure k_{1}^{4} + k_{2}^{4} etc.^{8}. Also of interest in this connection is Darboux’s balanced assessment ([6], sec. 497):

“Memoirs have been written in search of the question which of the two quantities [k_{1}k_{2} or (k_{1} + k_{2})/2] should serve as measure for the curvature of a surface at a given point ... .”

Of course, the expression H^{2} and the variational problem

δ ∫ ∫_{T} H^{2} dA = 0  (1)

had been introduced and investigated by S.D. Poisson [14] and S. Germain [7] in their investigations of problems in the theory of elasticity^{9}.

Another characterization of the mean curvature in a similar vein as Sturm’s is due to G.Cimmino [4]:

Hp = \frac{1}{π} lim_{r→0} \frac{2π − σp(r)}{r} .  (2)

It should be mentioned that the existence of the limit in (2) does not require much of T by way of regularity. Thus, relation (2) is suitable for the definition of a generalized mean curvature. If the surface T has a representation of class C^{2}, however, it follows from (2) that the only candidates for the specified class of surfaces are minimal surfaces. Whether a surface with vanishing generalized mean curvature everywhere must be a minimal surface in the classical sense is the other question raised by Cimmino in [4].

7If one surveys the recent literature concerning variational problems for curvature functions (see e.g. [18]), then one realizes that today’s mathematicians have heeded his plea; to be sure, they have forgotten Casorati.

8In recent years, this expression has been discussed in connection with engineering applications; for references see [13].

9For appraisals of these contributions by experts in the theory of elasticity, see e.g. [12], p. 83.

3

In what follows we shall work with differential geometric surfaces in the sense of [11], § 47; the objects of our investigation can then in fact be assumed to be surfaces of the regularity class C^{ω}. It will be necessary to find an expansion for the expression on the right hand side of (2) in the form

\frac{2π − σp(r)}{2πr} = ∑_{n=0}^{∞} σ^{(n)}r^{n} ,  (3)

so that σ^{(0)} = Hp/2. It had already been shown in [10], [11], § 64 that

σ^{(1)} = 0 ,  (4)
σ^{(2)} = \frac{1}{32}[ΔH + 2H(H^{2} − K)]p .  (5)

Here Δ denotes the Beltrami operator of the surface T . For details, see section 4 below. We are then led to the following extension of the limit relation (2):

[ΔH + 2H(H^{2} − K)]p = \frac{16}{π} lim_{r→0} \frac{2π − σp(r) − rπHp}{r^{3}} .  (6)

The appearance of the differential operator in (5), (6) is challenging: the differential equation ΔH +2H(H^{2} −K) = 0 , a fourth order equation for the position vector of T , is the Euler-Lagrange equation for the variational problem (1). Its nonparametric form had been derived by S.D. Poisson; see [14], pp. 224, 215. Its invariant form is due to W. Schadow; see [17], p. 56. The differential equation vanishes, as does the limit (6), for all extremal surfaces of the variational problem (1), in particular for all Willmore surfaces, i.e. compact differential geometric extremal surfaces without boundary immersed in R^{3}; see [18], p. 131.

Of, course, the term σ^{(2)} vanishes automatically for minimal surfaces. Thus we will have to turn our attention to the higher terms, at least to σ^{(4)}, as it can be seen without difficulty that σ^{(3)} = 0.

4

Let T be a nonplanar surface of class C^{ω} and let us consider the neigborhood of a nonumbilic point on T (the umbilic points on a nonplanar minimal surface are isolated; see [10] , [11], § 125). It shall be assumed that this point is the origin of our coordinate system and that the surface is locally represented in nonparametric form z = f(x, y) satisfying the conditions f(0, 0) = f_{x}(0, 0) = f_{y}(0, 0) = f_{xy}(0, 0) = 0. Then

f(x, y) = ∑_{n=2}^{∞} P_{n}(x, y) , P_{n}(x, y) = ∑_{m=0}^{n} a_{n−m,m}x^{n−m}y^{m} ,  (7)

with P_{2}(x, y) = a_{2,0}x^{2} + a_{0,2}y^{2}. (For a minimal surface, we have obviously a_{2,0} + a_{0,2} = 0, but a_{2,0} ≠ 0.)

Using spherical coordinates x = r sin ϑ cos φ , y = r sin ϑ sin φ , z = r cos ϑ, we set

P_{n}(x, y) = r^{n} sin^{n} ϑ ∑_{m=0}^{n} a_{m−n,m} cos^{n−m} φ sin^{m} φ
= r^{n} sin^{n} ϑ p_{n}(φ) ,

The curve Co(r) is given by the equation r cos ϑ = f(r sin ϑ cos φ, r sin ϑ sin φ) . Thus, for small values of r, the angle ϑ is nearly equal to π/2 and the equation for Co(r) can be solved in the form ϑ = ϑ(r, φ); we also set α(r, φ) = π/2 − ϑ(r, φ). Then

sin α(r, φ) = ∑_{n=2}^{∞} r^{n−1}p_{n}(φ) cos^{n} α ,

so that

sin α(r, φ) = rq_{2}(φ) + r^{2}q_{3}(φ) + ….

The first coefficient functions q_{n} = q_{n}(φ) are

q_{2} = p_{2} ,
q_{3} = p_{3} ,
q_{4} = p_{4} − p_{2}^{3} ,
q_{5} = p_{5} − \frac{7}{2}p_{2}^{2}p_{3} ,
q_{6} = p_{6} − 4p_{2}p_{3}^{2} − 4p_{2}^{2}p_{4} + 2p_{2}^{5} .

Now

σo(r) · r^{2} = r^{2} ∫_{0}^{2π} dφ ∫_{0}^{ϑ(r,φ)} dϑ sin ϑ = r^{2} ∫_{0}^{2π}(1 − cos ϑ(r, φ)) dφ

and

σo(r) = 2π − ∫_{0}^{2π} sin α(r, φ) dφ .

As a result we obtain the expansion (3), where

σ^{(n)} = \frac{1}{2π} ∫_{0}^{2π} q_{n+2}(φ)dφ .

It is desirable to express the expansion coefficients σ^{(n)} in terms of geometric invariants of the surface T . For this purpose, note that the mean and Gaussian curvatures of T are

H = \frac{(1 + f_{y}^{2})f_{xx} − 2f_{x}f_{y}f_{x,y} + (1 + f_{x}^{2})}{2\sqrt{1 + f_{x}^{2} + f_{y}^{2}}^{3}} ,

K = \frac{f_{xx}f_{yy} − f_{xy}^{2}}{(1 + f_{x}^{2} + f_{y}^{2})^{2}} ,

while the Beltrami operator applied to a function ζ(x, y) defined on T is given by

Δζ = \frac{(1 + f_{y}^{2})ζ_{xx} − 2f_{x}f_{y}ζ_{xy} + (1 + f_{x}^{2})ζ_{yy}}{1 + f_{x}^{2} + f_{y}^{2}} − \frac{2H(f_{x}ζ_{x} + f_{y}ζ_{y})}{\sqrt{1 + f_{x}^{2} + f_{y}^{2}}} ;

see [11], § 61. It follows that

Ho = a_{2,0} + a_{0,2} ,
Ko = 4a_{2,0}a_{0,2}
(H^{2} − K)o = (a_{2,0} − a_{0,2})^{2} ,
(ΔH)o = 4 [3a_{4,0} + a_{2,2} + 3a_{0,4}]
+4(a_{2,0} + a_{0,2}) [2a_{2,0}a_{0,2} − 3(a_{2,0}^{2} + a_{0,2}^{2}] .

A computation then shows that that the first expansion terms σ^{(0)} , σ^{(1)} , σ^{(2)} have the values already laid down in (2), (4), (5) above.

5

For the determination of σ^{(4)}, we shall restrict ourselves here to the consideration of minimal surfaces, with the normalization a_{2,0} = −a_{0,2} = 1. The explicit expressions for the first homogeneous polynomials in the expansion (7) are

P_{2}(x, y) = x^{2} − y^{2} ,  (8)
P_{3}(x, y) = Re{(A_{3} − iB_{3})(x + iy)^{3})} ,  (9)
P_{4}(x, y) = Re\{(A_{4} − iB_{4})(x + iy)^{4}\} + \frac{2}{3}(x^{4} − y^{4}) ,  (10)
P_{5}(x, y) = Re{(A_{5} − iB_{5})(x + iy)^{5}}
+\frac{4}{5}A_{3}(3x^{5} − 5xy^{4}) + \frac{4}{5}B_{3}(5x^{4}y − 3y^{5}) ,  (11)
P_{6}(x, y) = Re\{(A_{6} − iA_{6})(x + iy)^{6}\} + \frac{4}{15}A_{4}(11x^{6} − 15x^{4}y^{2} − 15x^{2}y^{4} + 11y^{6})

+\frac{32}{5}B_{4}(x^{5}y + xy^{5}) + \frac{1}{10}(33A_{3}^{2} + 15B_{3}^{2})x^{6} + \frac{36}{5}A_{3}B_{3}(x^{5}y − xy^{5})

+\frac{9}{2}(B_{3}^{2} − A_{3}^{2})(x^{4}y^{2} + x^{2}y^{4}) − \frac{1}{10}(15A_{3}^{2} + 33B_{3}^{2})y^{6} + \frac{32}{45}(x^{6} − y^{6}) . (12)

The coefficients A_{n} , B_{n} denote arbitarary real parameters. For instance, in the case of Enneper’s minimal surface we have A_{3} = B_{3} = 0 , A_{4} = B_{4} = 0 , A_{5} = B_{5} = 0 , A_{6} = 1/15 , B_{6} = 0; see [10] , [11], § 89.

A lengthy computation shows that

σ^{(4)} = \frac{1}{3} A_{4} .  (13)

As a consequence, the requirement σ^{(4)} = 0 implies the condition

A_{4} = 0  (14)

or the equivalent relations

f_{xxxx}(0, 0) = −f_{yyyy}(0, 0) = 16 , f_{xxyy}(0, 0) = 0 .  (15)

Note that condition (13) is formulated here in terms of a local coordinate system; it must be satisfied in all nonumbilic points of T . (In the case of Enneper’s surface, as well as many other minimal surfaces, it turns out that condition (13) holds only at the point of highest symmetry.)

6

For the geometric interpretation of condition (13), the surface T is assumed to be before us in parametric representation T = {x = x(u, v) ; (u, v) ∈ P } , where the position vector x(u, v) = (x(u, v), y(u, v), z(u, v)) is defined in a neigborhood P of the nonumbilic reference point in the (u, v)-parameter plane. The unit normal vector of T will be denoted by X(u, v) = (X(u, v), Y (u, v), Z(u, v)). We shall take u and v as curvature line parameters which for a minimal surface are known to be isothermal; see [10], [11], § 172. The first fundamental form of T has the form ds^{2} = E(u, v)(du^{2} + dv^{2}), i.e. E(u, v) ≡ x_{u}^{2} = G(u, v) ≡ x_{v}^{2} and F (u, v) ≡ x_{u} ·x_{v} = 0. The coefficients L(u, v), M(u, v), N(u, v) of the second fundamental form of T satisfy the conditions L = −N = const , M = 0. Without loss of generality, we may assume that L(u, v) = 1.

It is convenient to employ the complex variable w = u + iv and the complex-valued differentiations ∂/∂w = (∂/∂u−i ∂/∂v)/2 and ∂/∂\overline{w} = (∂/∂u+i ∂/∂v)/2. The fundamental equations of differential geometry become

x_{ww} = \frac{E_{w}}{E} x_{w} + \frac{1}{2} X ,  (16)

x_{w\overline{w}} = 0 ,  (17)
X_{w} = −\frac{1}{E} x\overline{w} ,  (18)

while the theorema egregium is expressed by the equation

E_{w\overline{w}} = \frac{1}{E} |E_{w}|^{2} + \frac{1}{2} or Δ log E = \frac{2}{E} .  (19)

The position vector of T is the real part of an analytic vector Φ(w) = (ϕ_{1}(w), ϕ_{2}(w), ϕ_{3}(w)) which satisfies in P the relations

Φ′ · Φ′ = 0 , |Φ′|^{2} = 2E .  (20)

It follows from (16) – (20) that

Φ′ · Φ′′ = 0 , Φ′ · Φ′′′ = −Φ′′ · Φ′′ = −1  (21)

and

\overline{Φ′} · Φ′′ = 2E_{w} , \overline{Φ′} · Φ′′′ = 2E_{ww} .  (22)

We also have the relations

X · Φ′′ = 1 , X · Φ′′′ = \frac{E_{w}}{E} , X · Φ^{iv} = 2\frac{E_{ww}}{E} − \frac{E_{w}^{2}}{E^{2}} .  (23)

Let now be w^{0} = u^{0} + iv^{0} be a nonumbilic point in P; we also write x(u^{0}, v^{0}) = x^{0} , E(u^{0}, v^{0}) = E^{0} etc. In a neighborhood of this point, the surface T will be referred to a local coordinate system with axes in the directions of the orthonormal unit vectors

e_{1} = \frac{1}{\sqrt{E^{0}}} x_{u}^{0} , e_{2} = \frac{1}{\sqrt{E^{0}}} x_{v}^{0} , e_{3} = X^{0} ,

and new coordinates (ξ, η, ζ) will be introduced according to

ξ = \frac{1}{2E^{0}} e_{1} · (x(u, v) − x^{0}) ,
η = \frac{1}{2E_{0}} e_{2} · (x(u, v) − x^{0}) ,
ζ = \frac{1}{2E^{0}} e_{3} · (x(u, v) − x^{0}) .

It is necessary to express the equation of T in the form

ζ = f(ξ, η) = ∑_{n=2}^{∞} P_{n}(ξ, η) ,

as in (7). For this purpose, observe that x_{u} + ix_{v} = \overline{Φ′}, so that

λ ≡ 2\sqrt{E^{0}}(ξ + iη) = \frac{1}{E^{0}} \overline{Φ′} · (x(u, v) − x^{0}) ,

while

x(u, v) − x^{0} = Re\{(w − w^{0})Φ′(w^{0}) + \frac{1}{2}(w − w^{0})^{2}Φ′′(w^{0}) + …\} .

A computation utilizing the relations (16) – (23) leads to

w − w^{0} = λ − \frac{1}{2} \frac{E_{w}^{0}}{E^{0}} λ^{2} + \{\frac{1}{2}(\frac{E_{w}^{0}}{E^{0}} )^{2} − \frac{1}{6} \frac{E_{ww}^{0}}{E^{0}} \}λ^{3} + \frac{1}{12} \frac{1}{E^{0}} \overline{λ}^{3} + …

and ultimately to the expansion (6), where, as in (8), (9), (10),

P_{2}(ξ, η) = ξ^{2} − η^{2} ,  (24)
P_{3}(ξ, η) = Re{(A_{3} − iB_{3})(ξ + iη)^{3})} ,  (25)
P_{4}(ξ, η) = Re\{(A_{4} − iB_{4})(ξ + iη)^{4}\} + \frac{2}{3}(ξ^{4} − η^{4}) .  (26)

Here

A_{3} − iB_{3} = −\frac{4}{3}(\frac{E_{w}}{E^{3/2}} )^{0} ,  (27)
A_{4} − iB_{4} = −\frac{2}{3}(E_{ww} − 5\frac{E_{w}^{2}}{E})^{0}  (28)
= \frac{1}{6}\{E^{5}(E^{−4})_{ww})\}^{0} .  (29)

In view of condition (14) and the remarks at the end of section 5, we must have

Re{(E^{−4})_{ww}} = 0 or (E^{−4})_{uu} − (E^{−4})_{vv} = 0  (30)

at all nonumbilic points w = u + iv in the domain P. This differential equation joins the theorema egregium (19) as condition for the first fundamental form of our surface T . Their implications will be discussed in the next section. For this purpose, it is convenient to switch from the parametrization of T by lines of curvature to the parametrization by asymptotic lines, which is also an isothermal parametrization; see [10], [11], § 172. It may be assumed that the parameter domain P contains the origin u = v = 0, but no umbilic points of T . Setting p(u, v) = E^{−4}(u, v), our conditions then become

p(u, v) > 0 , p_{uv} = 0 , Δ log p + 8p^{1/4} = 0  (31)

for all points (u, v) ∈ P.

We shall prove the following

Lemma: A function p(u, v) satisfying conditions (31) depends only on one of its variables, i.e. p = p(u) or p = p(v) .

It is a consequence of this lemma that the coefficient E of the first fundamental form of T is constant on each member of one of the two families of asymptotic lines of T .

Set q(u, v) = p(u, v)/p(0, 0). Then q(0, 0) = 1 and Δ log q = −kq^{1/4} , k = 8p^{1/4}(0, 0) > 0. It is clear that q(u, v) has the form q(u, v) = 1 + f(u) + g(v), where

f(u) = a_{1}u + a_{2}u^{2} + …,
g(v) = b_{1}v + b_{2}v^{2} + ….

Substitution of the expression for q into (19) yields

(1 + f + g)(f′′ + g′′) − f^{′2} − g^{′2} + k(1 + f + g)^{9/4} = ∑_{m,n=0}^{∞} c_{m,n}u^{m}v^{n} = 0 .  (32)

Some of the first coefficients c_{m,n} are as follows:

c_{0,0} = 2(a_{2} + b_{2}) − a_{1}^{2} − b_{1}^{2} + k ,  (33)
c_{1,0} = 6a_{3} − 2a_{1}(a_{2} − b_{2}) + \frac{9}{4}ka_{1} ,  (34)
c_{0,1} = 6b_{3} + 2b_{1}(a_{2} − b_{2}) + \frac{9}{4}kb_{1} ,  (35)
c_{2,0} = 12a_{4} − 2a_{2}(a_{a} − b_{2}) + \frac{9}{32}k(8a_{2} + 5a_{1}^{2}) ,  (36)
c_{1,1} = 6a_{1}b_{3} + 6a_{3}b_{1} + \frac{45}{16}ka_{1}b_{1} ,  (37)
c_{0,2} = 12b_{4} + 2b_{2}(a_{2} − b_{2}) + \frac{9}{32}k(8b_{2} + 5b_{1}^{2}) ,  (38)
c_{2,1} = 12a_{4}b_{1} + 6a_{2}b_{3} + \frac{45}{128}k(8a_{2}b_{1} + a_{1}^{2}b_{1}) ,  (39)
c_{1,2} = 12a_{1}b_{4} + 6a_{3}b_{2} + \frac{45}{128}k(8a_{1}b_{2} + a_{1}b_{1}^{2}) .  (40)

Combination of the equations (34), (35), (37), the equations (35), (36), (39) and the equations (34), (38), (40), respectively, leads to the three conditions

a_{1}b_{1} = 0 , a_{1}(8b_{2} + 5b_{1}^{2}) = 0 , b_{1}(8a_{2} + 5a_{1}^{2}) = 0 .

Assume first that b_{1} = 0. Then we must have a_{1}b_{2} = 0. Inspection of (35) also shows that b_{3} = 0. Two cases have to be considered:

Case 1: b_{2} = 0. The differential equation (32) for the value u = 0 becomes

(1 + g)(2a_{2} + g′′) − a_{1}^{2} − g^{′2} + k(1 + g)^{9/4} = 0 .  (41)

Since b_{1} = b_{2} = 0 and, in view of (33), 2a_{2} − a_{1}^{2} + k = 0, we see that g(v) ≡ 0 is the only solution of (41). It follows that q(u, v) = 1 + f(u).

Case 2: a_{1} = 0 , b_{2} ≠ 0. By a previous argument, also a_{3} = 0. The coefficient c_{2,2} becomes

c_{2,2} = 12(a_{2}b_{4} + a_{4}b_{2}) + \frac{45}{16}ka_{2}b_{2} = −\frac{819}{16}ka_{2}b_{2} .

The condition c_{2,2} = 0 implies a_{2} = 0, and an argument similar to the one used before shows now that q(u, v) = 1 + g(v).

The initial assumption a_{1} = 0 leads to the same conclusion. The lemma is proved.

8

Consider a portion of the minimal surface T in an isothermal parametrization by asymptotic lines and free of umbilic points. We know now that the coefficient E(u, v) of the first fundamental form depends only on one of its two variables, say E = E(u). The function h(u) = \sqrt{E(u)} > 0 satisfies the differential equation

hh′′ = 1 + h^{′2} .  (42)

The general solution of (42) is

h(u) = a cosh (κu) + b sinh (κu) ,  (43)

where κ and b denote arbitrary constants and the coefficient a > 0 satisfies the relation a^{2} = b^{2} + 1/κ^{2}. Integration of the fundamental equations of differential geometry whose details are omitted here leads to the following conclusion: In a suitably chosen (y^{1}, y^{2}, y^{3})- coordinate system, the equation of T takes the form {y = y(u, v) ; (u, v) ∈ P } where

y = (h′(u) cos(κv), h′(u) sin(κv), λv) .  (44)

(44) shows that T is a right helicoid.With this, Cimmino’s question has been answered

References

[1] L. Bianchi: Lezioni di geometria differenziale, 3rd ed., vol. 1, part 1. Nicola Zanicelli, Bologna, 1914 .

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