2010 Mathematics Subject Classification:Primary 53C43; Secondary 53C45
We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of rotation (but not the radius). Such surfaces generalize the Enneper surface. We show that they are necessarily of constant mean curvature, and that the rotational speed of the principal curvature directions is constant. We classify the minimal case. The (non-zero) constant mean curvature case has been classified by Smyth.
The Enneper surface (see Figure 1) is given in conformal polar coordinates as
It was discovered in 1871 by Alfred Enneper . Its first fundamental form is given by
This means that the Enneper surface is intrinsically a surface of revolution (but obviously not extrinsically).
An intrinsic surface of revolution is a surface with first fundamental form of the shape
where is a positive function.
Of course any surface of revolution is also intrinsically a surface of revolution.
The shape operator of the Enneper surface is also rather special:
is the counterclockwise rotation by . This is in contrast to the shape operator of a surface of revolution which always takes diagonal form in polar coordinates. It is, however, rather special, because the principal curvature directions rotate with constant speed independent of and the principal curvatures are independent of .
We are generalizing this property of the Enneper surface by introducing the following concept:
Let be a -function. We say a surface has twist if its shape operator is of the form
Note that this precisely means that the principal curvature directions are independent of , and the principal curvatures are independent of .
In summary, the Enneper surface is an example of an intrinsic surface of revolution with twist . A standard surface of revolution, on the other hand, has twist .
Now we can formulate our main theorem, which is a consequence of the Codazzi equations. We note that we generally assume our surfaces to be three times continuously differentiable.
Let be an intrinsic surface of revolution with twist function . Assume that is not identically equal to 0 or any other integral multiple of on any open interval. Assume furthermore that the surface has no open set of umbilic points. Then has constant mean curvature, and the twist function is linear .
Constant mean curvature surfaces that are intrinsic surfaces of revolution have been studied by Smyth, see . Thus our result complements Smyth’s result by replacing his assumption about constant mean curvature with a geometric assumption.
In order to begin a complete classification, we will invoke Bonnet’s theorem  to prove:
Let be an intrinsic surface of revolution of constant mean curvature , first fundamental form for , and linear twist . Then satisfies the differential equation
for a constant .
In the special case of minimal surfaces, we can achieve a complete classification.
Let be an intrinsic surface of revolution that is also minimal with constant twist with . Then belongs to an explicit 2-parameter family of minimal surfaces with Weierstrass data given by
with parameters and .
In the case that the twist function is , we prove:
Given a conformal factor on an interval and a constant such that for all . Then there is an intrinsic surface of revolution defined on the domain with first fundamental form and twist . Moreover, this surface can be realized as an actual surface of revolution in of the form
with suitable functions .
The paper is organized as follows:
In section 4, we briefly discuss the constant mean curvature case by connecting our approach to Smyth’s. While we are not able to find explicit solutions for the Smyth surfaces, we can find numerical solutions and make images.
2. Gauss- and Codazzi equations for intrinsic surfaces of revolution with twist
We will apply Bonnet’s theorem to determine when the first fundamental form and shape operator of an intrinsic surface of revolution with twist function are induced by an actual surface in .
In order to derive the Gauss- and Codazzi equations we first determine the relevant covariant derivatives. Much of this preparation is standard.
as the normalized coordinate vector fields. Then we have
The Levi-Civita connection of the first fundamental form is given by
By the -invariance of the first fundamental form, the curves are geodesics, and has length with respect to the first fundamental form. This implies . Next and intrinsic rotations are parallel, so that as well.
Using that is torsion-free and metric, we compute
The Gauss equation is equivalent to
The Gauss equation gives us:
Let us calculate .
The claim follows. ∎
In order to use the Codazzi equations we continue to compute the relevant covariant derivatives.
The covariant derivatives of the twist rotation are given by
The first equations follows because intrinsic rotations by a constant angle are parallel and is independent of . For the second, we use the chain rule and observe that . ∎
The covariant derivatives of the eigenvalue endomorphism are given by
The first equation is immediate because the frame is parallel in the -direction. For the second, we compute
The third equation is proven the same way. ∎
The covariant derivatives of the shape operator are given by
In the statement, we have indicated the dependence of each functions by their respective variables. To improve legibility, we will drop the variables in the computations below. Observe, however, that derivatives like and are always taken with respect to the proper variables.
The first equation is again immediate. For the second, we need to work harder. We begin by differentiating the definition of according to the product rule, and then use the lemmas above:
The Codazzi equations are equivalent to
The Codazzi equations state that for any pair of tangent vectors and . As we are in dimension 2 and the equation is symmetric, it suffices to verify this for and . By the previous theorem, this is equivalent to
Pairing both sides with and respectively gives
The first equation simplifies to
and the second to
as claimed. ∎
We are now ready to prove Theorem 1.3
By assumption, the twist function is not identically equal to an integral multiple of on any open interval. By the first Codazzi equation, the mean curvature is constant except possibly at isolated points. As we assume that is at least , this implies that is constant.
This simplifies the second Codazzi equation to
As the right hand side is independent of , so is the left hand side. This can only be the case if is a constant as claimed, or that on an open interval. In the latter case we have on the same interval that for a constant . But this means that this portion of the surface is umbilic, which we have excluded. ∎
Observe that we have not used the Gauss equations in the proof above. We will now use the second Codazzi equation to eliminate and from the Gauss equations.
For and a constant, the second Codazzi equation has the general solution
where is any real number.
The second Codazzi equation is then equivalent to
Integrating and substituting back gives the claim. ∎
The following corollary proves Theorem 1.4.
A first fundamental form with and shape operator as in Equation (1) such that is constant and satisfy the Gauss and Codazzi equations if and only if
In particular, by Bonnet’s theorem, these data determine an intrinsic surface of revolution, and every such surface arises this way.
To classify all intrinsic surfaces of revolution, we would need to find all solutions to the differential equation 4, and then to integrate the surface equation to obtain a parametrization. We will discuss the solutions of 4 for in Section 3.
We end this section by carrying out the first integration step of the surface equation, which is quite explicit and shows that special coordinate curves are planar.
Assume that is a solution of 4. To determine the surface parametrization, we will first determine a differential equation for the curve with .
are a parallel frame field along with respect to the first fundamental form.
Following the proof of Bonnet’s theorem, we derive a Frenet-type differential equation for the orthonormal frame , , and .
In our case, using the explicit formula for the shape operator and the principal curvatures in terms of and , this simplifies to give the following lemma:
The space curve is planar.
This is immediate because is constant. Note that this only works because . ∎
This is as far as we can get in the general case. For the minimal case, we will solve the Equation (4) explicitly and be able to integrate the surface equations further.
3. The minimal case
In the minimal case the differential equation for simplifies to
Without much loss of generality, we can assume by scaling by a positive constant. There is one exception, namely when . In this case, , so that the surface is a plane, which we disregard.
All positive solutions of
defined in any open interval are given by
for arbitrary .
It is easy to check that satisfies Equation (6). To show that every local solution is of this form, it suffices to show that for any fixed real , the initial values and are equal to the initial data and for a suitable choice of and . Then the local uniqueness theorem for ordinary differential equations implies that near and hence everywhere.
To this end, we have to solve
for and . Surprisingly, this is explicitly possible.
The strategy is to solve the first equation for , choosing the larger solution of the two. This gives
Inserting this into the second equation and simplifying gives
which can be solved for . Again choosing the positive solution gives
Note that as we are only interested in positive conformal factors. This in turn makes the radicand in the preliminary expression for , and hence itself, positive. Explicity:
The Enneper solution corresponds to , .
Integrating gives the following lemma:
Up to a motion in space, the solution to this equation is given by
We have normalized the frame to that for , and in agreement with our parametrization of the Enneper surface.
The space curve is given by
if . If (say, the other case being similar), we have
This follows by integrating
using the previous lemma, and simplifying. ∎
Instead of now integrating the surface equations likewise along the curves for fixed , we will use the Björling formula  to obtain the parametrization more easily.
Recall that given a real analytic curve and a real analytic unit normal field satisfying , the unique minimal surface containing and having surface normal along can be given in a neighborhood of by
where we write and have extended and to holomorphic maps into .
In our case, we obtain for
Note that in the last case scaling by a constant and by the reciprocal only scales the surface, so we can as well assume that in this case.
The Weierstrass data  of these surfaces are particularly simple. Using , let (also for )
be the Gauss map and height differential of the Weierstrass representation formula
This gives the surfaces above. This can be verified either by evaluating the integral or by solving the Björling integrand for and .
Of particular interest are the cases when and are integers. Then the substitution changes the Weierstrass data into
defined on the punctured plane and being minimal surfaces of finite total curvature.
A substitution in the domain of the form will scale and by powers of , so we can assume without loss of generality that .
Some of the minimal surfaces we have obtained are described in . We will now discuss examples.
In case that , we obtain the Enneper surfaces of cyclic symmetry of order , see Figure 3. For , we obtain the original Enneper surface.
The planar Enneper surfaces of order are given by choosing and . See Figure 4 for the cases and . These surfaces feature an Enneper type end and a planar end. Remarkably, in the non-zero CMC case, there are only one-ended solutions .
Other choices of and lead to more wildly immersed examples. In Figure 5 we show images of thin annuli .
There is one case that deserves attention: If , the Weierstrass 1-forms have residues, and hence the surface can become periodic. The prototype case here is and (see Figure 6) which leads to a translation invariant surface that hasn’t made it into the literature to our knowledge. It deserves attention because it is in the potentially classifiable list of minimal surfaces in the space form (where acts through a cyclic group of translations) of finite total curvature . Other surfaces in this list include the helicoid and the singly periodic Scherk surfaces.
4. Constant Mean Curvature
intrinscial? In , Smyth considers intrinsical surfaces of revolution under a different viewpoint: He assumes from the beginning that his surfaces have constant mean curvature, but does not make further assumptions about the shape operator. Nevertheless, we both end up with the same class of surfaces. Therefore we would like to connect our approach with Smyth’s in the CMC case.
is indeed holomorphic and agrees with Smyth’s computation. Secondly, to show that our equation for is equivalent to Smyth’s equation, we substitute
in the case that (which is Smyth’s case ). This again agrees with Smyth’s equation, up to a normalization of constants.
In general, there are apparently no explicit solutions to Equation (4) for in the literature. There is, however, one explicit solution given by
By Lemma 2.7, the principal curvatures become simply and . This implies that the surface under consideration is in fact a cylinder. This is somewhat surprising, as the standard parametrization of a cylinder over a circle of radius as an extrinsic surface of revolution has twist 0. In our case, however, the cylinder is parametrized using geodesic polar coordinates (see the left image in Figure 7) as
For other initial data of the Equation (4), only numerical solutions are available. These can be obtained easily by integrating the surface equations. The right image in Figure 7 was obtained using , , , and .
5. The untwisted case
Thus we are given a first fundamental form and shape operator
depending on the congruence class of modulo . Without loss of generality, we will assume and therefore
The Gauss- and Codazzi equations become