Embedded constant mean curvature tori in the three-sphere
We prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric, and use this to give a complete classification of such surfaces for any given value of the mean curvature.
The study of constant mean curvature (CMC) surfaces in spaces of constant curvature (that is, , the sphere , and hyperbolic space ), is one of the oldest subjects in differential geometry. There are many beautiful results on this topic (see for example [Ho1], [Ho2], [Ch], [We], [Ab], [PS], [Bo] and many others).
The simplest examples of CMC surfaces in are the totally umbilic 2-spheres. Another basic example is the so-called Clifford torus . Identifying with the unit sphere in , the Clifford torus is defined for by
By constructing a holomorphic quadratic differential for CMC surfaces, H. Hopf showed that any CMC two-sphere in is totally umbilical (see [Ho1]). S. S. Chern extended Hopf’s result to CMC two-spheres in -dimensional space forms (see [Ch]). H. C. Wente was the first (see [We]) to show the existence of compact immersed CMC tori in . Wente’s examples solved the long standing problem of Hopf (see [Ho2]): Is a compact CMC surface in necessarily a round sphere? A. D. Alexandrov (see [Al]) showed that if a compact CMC surface is embedded in , or a hemisphere , then it must be totally umbilical. Wente’s paper was followed by a series of papers (see [Ab],[PS], [Bo] and many others), where Wente tori were investigated in detail and other examples of CMC tori were constructed. In particular A. I. Bobenko ([Bo]) constructed CMC tori in , and .
In principle, the construction in [PS, Bo] gives rise to all CMC tori in , but reading off properties such as embeddedness from this classification is difficult. It remains an interesting unsolved problem to classify all embedded CMC tori in . It is explicitly conjectured by Pinkall and Sterling [PS]*p.250 that such surfaces are surfaces of revolution, and our main result confirms this. We then give a complete classification of such embedded tori by completing a classification of rotationally symmetric CMC surfaces by Perdomo [Per]. Our result is as follows:
Every embedded CMC torus in is a surface of rotation: There exists a two-dimensional subspace of such that is invariant under the group of rotations fixing .
If is an embedded CMC torus which is not congruent to a Clifford torus, then there exists a maximal integer such that has -fold symmetry: Precisely, is invariant under the group generated by the rotation which fixes the orthogonal plane and rotates through angle .
For given , there exists at most one such CMC torus (up to congruence).
For given , there exists an embedded CMC torus with mean curvature and maximal symmetry if lies strictly between and .
If then every embedded torus with mean curvature is congruent to the Clifford torus.
Our contributions are items 1 and 3. Given these, the remaining results follow from [Per]. The case of item 5 is the Lawson conjecture proved recently by Brendle [Br]. The rigidity appearing for is unexpected, however. We note that the embeddedness assumption in Theorem 1 is crucial: Bobenko [Bo] has constructed an infinite family of non-rotationally symmetric immersed CMC tori in (see also Brito-Leite [BL], Wei-Cheng-Li [WCL], Perdomo [Per]).
Recently the first author proved in [An] a non-collapsing result for mean-convex hypersurfaces in Euclidean space evolving by mean curvature flow. The estimate was inspired by earlier work of Weimin Sheng and Xujia Wang [SW] (see also [Wh]), who in the course of a detailed analysis of singularity profiles in mean curvature flow proved that for any embedded compact mean-convex hypersurface moving under the mean curvature flow, there is a positive constant such that at every point of there is a sphere of radius enclosed by which touches at .
The main contribution of [An] was a very direct proof of this non-collapsing result using a maximum principle argument. In particular, the noncollapsing condition described in the last paragraph was expressed as the positivity of a certain function of pairs of points on the hypersurface, and this function was shown to admit a maximum principle argument to preserve initial positivity. We will discuss this expression for noncollapsing in section 2. The idea of working with functions of pairs of points was in turn inspired by earlier work of Huisken [Hu] and Hamilton [HamiltonCSFIsoperim, HamiltonRFIsoperim] for the curve shortening flow and for Ricci flow on surfaces.
In [ALM] this technique was adapted to a family of other flows, and it was shown that the mean curvature used to determine the scale of noncollapsing can be replaced by any function satisfying a certain differential inequality related to the linearisation of the evolution equation.
Recently, Brendle [Br] made a remarkable breakthrough: He reworked the technique of [An] in such a way that it can be applied to minimal surfaces, and succeeded in proving the Lawson conjecture, that the only embedded minimal torus in is the Clifford torus. Brendle used the same ‘non-collapsing’ quantity as in [An] (we describe the geometric meaning of this below in section 2), but exploited the additional special structure coming from the minimal surface condition to show that the maximum principle argument still works if the radii of the touching spheres are compared to the length of the second fundamental form rather than the mean curvature.
In this paper, we again use the non-collapsing argument originating from [An], together with the modifications introduced by Brendle. A crucial point is to choose carefully the scale on which to compare the touching spheres: We show in section 3 that the difference of the curvatures of the touching spheres from the mean curvature of can be compared in ratio with the difference of the largest principal curvature from using a maximum principle argument. This implies (as in Brendle’s case) that every point of the surface is touched by a ball with boundary curvature equal to the maximum principal curvature at that point.
In Brendle’s argument in [Br] one can touch by such spheres on both sides of the surface, and deduce from this that the second fundamental form is parallel, from which the rigidity follows easily. In our case this is no longer true, and we can only deduce that some components of the derivative of second fundamental form vanish. However this is enough to conclude (as we do in Section 4) that the surface is rotationally symmetric.
Perdomo [Per] has given an analysis of rotationally symmetric surfaces with constant mean curvature in : These are constructed from the solutions of a certain differential equation, which are parametrized by a parameter for each . Embedded examples arise from those solutions for which an associated ‘period’ function equals for some positive integer . Perdomo showed that as varies the family of CMC surfaces deforms from the Clifford torus to a chain of ‘kissing spheres’, and found the limiting values of . Our second contribution in this paper is to complete the classification by proving that is monotone in , and so takes each value between the limits found by Perdomo exactly once.
It is a pleasure to thank Professor S. -T. Yau and Professor R. Schoen for their interest in this topic. We also express our thanks to graduate student Mr. Zhijie Huang for assistance with checking the proof of monotonicity of (see Proposition 13).
2. Touching interior balls
The key geometric idea in the non-collapsing argument from [An] is to compare the curvature of enclosed balls touching the surface to a suitable function (mean curvature in the setting of [An]) at the touching point. We recall here some of the expressions which arise from this picture (see also the discussion in [An] and [ALM]).
Let be an embedded hypersurface in given by an embedding , and bounding a region . We choose in such a way that the unit normal of points out of . For given we will derive an inequality which is equivalent to the geometric statement that there is a ball in of boundary curvature which touches at . A geodesic ball in is simply the intersection of a ball in with . In particular the ball in with boundary curvature which is tangent to at the point is , where , and is the unit normal to at in which points out of . The statement that this ball lies entirely in is equivalent to the statement that no other points of are inside . This is in turn equivalent to the statement that for any , , which can be written as follows:
Expanding the inner product (and multiplying through by ) this becomes
Since we have and , so that
In summary we have the following:
If is a smooth positive function, then the function is non-negative for every if and only if at every point there is a ball with boundary curvature with .
Following [ALM] we call the smallest for which this is true the interior ball curvature of the surface at , and denote it by . Since a ball of curvature less than the largest principal curvature cannot touch at , we always have , where is the largest principal curvature of the surface at . The main result of section 3 is that an embedded CMC torus in always has interior ball curvature equal to the maximum principal curvature at every point. The key result of [Br] is that an embedded minimal torus in always has interior ball curvature equal to the maximum principal curvature and exterior ball curvature equal to minus the minimum principal curvature.
3. Interior ball curvature equals maximum principal curvature
Let be an embedded torus in with constant mean curvature (note that we adopt the convention that is the average of the two principal curvatures, not their sum). By choosing the direction of the unit normal we can assume . Our aim in this section is to prove that the interior ball curvature equals the maximum principal curvature, by deriving a contradiction if this is not the case.
Let be a function on with . In the following we fix and denote for brevity (see equation (2)). Suppose that for all (so that by Proposition 2 there is a ball with boundary curvature touching at each point of the surface), and consider a pair of points such that . Then is a minimum point of and the differential of at the point vanishes.
We begin by elaborating the geometric picture: By Proposition 2, both and lie on the boundary of the ball in of boundary curvature , which is the intersection with of the ball of radius in centred at the point . Thus , so by construction lies in the exterior of and touches at the points and . It follows that the tangent spaces of the surface at these points agree with those of the two-dimensional sphere , and the outward unit normals and agree with those of the sphere also. Let . Then the symmetry of the sphere implies that the reflection maps the tangent space to at to that at , and that
where we used the equation in the last step.
Let be geodesic normal coordinates around , and let be geodesic normal coordinates around . We specify further that the tangent vectors diagonalize the second fundamental form, so that , , and , where . We use the following notations:
We also assume that the coordinate tangent vectors at are defined by reflecting those at :
The key computation is the following:
If for all , then at the point we have
We first compute the derivatives of :
In particular this vanishes at the point , so we have
Now we begin computing : Differentiating the expression (1) in the direction we obtain the following:
Differentiating again and taking a sum over , we obtain the following:
The result follows directly. ∎
At the point we have
We have . ∎
Now we can prove the main result of this section:
Suppose that is an embedded torus with constant mean curvature in . Then the interior ball curvature of is equal to the maximum principal curvature at every point.
The case was proved in [Br], so we assume that . We will apply the maximum principle using the formula in Proposition 4, with chosen as follows: We denote by the largest principal curvature at , and by the difference . Then we choose
where is a positive constant. We require the following variant of Simons’ identity:
Suppose that is an embedded CMC torus in . Then the function is strictly positive and satisfies the partial differential equation
Note that , so vanishes only at umbilical points. It follows from work of Hopf that a CMC torus in has no umbilical points (see [Ho2] or [Ch]), so is strictly positive and smooth everywhere. Using the Simons identity (cf. [Si], [NS], [Ya]), we have the known result (for example, see (2.7) and (2.8) in [Li])
which is equivalent to the proposition 8. ∎
Since is compact and is positive, has a positive lower bound. Since is embedded, the interior ball curvature is bounded above. Therefore for sufficiently large we have and the function is non-negative.
Along any geodesic in through we have , and hence .
In particular, if then we can choose to be in the direction of the largest principal curvature, so that . Then , so that takes negative values.
On the other hand if then in every direction from we have , and it follows that is positive in a neighbourhood of the diagonal in .
We choose . The considerations above then show that . Note that if then the theorem is proved, since then we have and hence as claimed. We complete the proof of the theorem by assuming that and deriving a contradiction:
If , then since is positive in a neighbourhood of the diagonal in , there must exist a point in with such that , while for every . Also, we have for all , so Proposition 3 applies. Since this is a minimum point of , the second derivatives are non-negative. However, using Proposition 6, the identity of Corollary 4 becomes the following:
and we have a contradiction since the left hand side of (12) is non-negative while the right-hand side is strictly negative. ∎
4. Rotational symmetry
Now we prove that embedded CMC tori in have rotational symmetry:
Let be a CMC embedding for which for every (equivalently, everywhere). Then is rotationally symmetric.
In this case, we have
for all points . For simplicity, we identify the surface with its image under the embedding , so that . Since is a CMC torus and therefore has no umbilical points, we have global smooth eigenvector fields and such that and , and . Exactly as in [Br] we can deduce that , and consequently also everywhere on (we repeat the argument here for convenience): For any , let be the geodesic which passes through in direction , and define . Then as in [Br] we have
In particular . Differentiating we have
so that also. A further differentiation gives
which again vanishes when . Since for all we must have :
Note that , so that and similarly and , where .
Now we choose a local orthonormal basis with along , and let be the dual coframe of . We recall that the Levi-Civita connection is defined by
We have , . The derivatives of the components of the second fundamental form are defined by
The Codazzi equations give
Choosing and in (14), we have
Thus we have
It follows from (19) that the flow lines of are geodesic in .
we have from (20)
Multiplying by , we obtain
where is a constant.
Let us denote by the position vector, and by the Euclidean connection on . Using the fact that , and , we get that
Let us fix a point , and denote by the geodesic in such that and . We write . Equation (23) implies that
where is a constant greater than and . The polynomial
is positive on an interval with and . The roots can be explicitly calculated:
We have that is a periodic function with period
We can solve from (25)
From the expression of , we get that its period .
The vector fields and commute.
The plane generated by the vectors and is constant on .
We show that the derivatives of the given basis are in : Differentiating in the direction we clearly have
Noting that , we compute
Next we check the derivatives in the direction: We have
by (18). Also we have
where we used lemma 8 in the second line. It follows that is locally constant, hence constant on . ∎
We now parametrize by two parameters and so that corresponds to the point , , and (this is possible since and commute by lemma 8).
where is the constant in (24). Note that only depends on , since . We have (here ) and
It follows from lemma 9 that the two-dimensional plane perpendicular to is also constant. This is generated by the orthonormal basis where and . We then have a convenient orthonormal basis for given as follows:
forming an orthonormal basis for ; and
forming an orthonormal basis for . We compute the rates of change of these bases along the vector fields and : We have
and (since the plane