Proc. NatL Acad. Sci. USA Vol. 79, pp. 3931-3932, June 1982 Mathematics

Examples of constant mean curvature immersions of the 3-sphere into euclidean 4-space (global differential geometry of submanifolds) Wu-yi HSIANG*, ZHEN-HUAN TENG*t, AND WEN-CI YU*t *Department of Mathematics, University of California, Berkeley, California 94720; tPeking University, Peking, China; and tFudan University, China Communicated by S. S. Chern, December 14, 1981

ABSTRACT Mean curvature is one of the simplest and most is probably the only possible shape of closed constant-mean- basic of local differential geometric invariants. Therefore, closed curvature hypersurface in euclidean spaces ofarbitrary dimen- hypersurfaces of constant mean curvature in euclidean spaces of sion or, at least, that Hopf's theorem should be generalizable high dimension are basic objects of fundamental importance in to higher dimensions. global differential geometry. Before the examples of this paper, We announce here the construction of the following family the only known example was the obvious one ofthe round sphere. of examples of constant mean curvature immersions of the 3- Indeed, the theorems ofH. Hopf(for immersion ofS2 into E3) and sphere into euclidean 4-space. A. D. Alexandrov (for imbedded hypersurfaces of E") have gone MAIN THEOREM: There exist infinitely many distinct differ- a long way toward characterizing the round sphere as the only entiable immersions ofthe 3-sphere into euclidean 4-space hav- example ofa closed hypersurface ofconstant mean curvature with ing a given positive constant mean curvature. The total surface some added assumptions. Examples ofthis paper seem surprising area ofsuch examples can be as large as one wants. and are constructed in the framework of equivariant differential Such examples, on the one hand, demonstrate that Hopf's geometry. theorem is, indeed, a special phenomenon valid only in euclid- ean 3-space and, on the other hand, the assumption of no es- Historical background and statement of main results sential self-intersection in the theorem ofAlexandrov is, in fact, The study of closed hypersurfaces of constant mean curvature a necessity that cannot be replaced by immersion. in euclidean space has a long and interesting history. Among the local differential geometric invariants of a given hypersur- Method of construction face, the mean curvature is undoubtedly the simplest basic in- In the isometry group of euclidean 4-space, ISO(E4), there are variant having important geometric meaning-i.e., it is the first two types ofcompact subgroups with codimension two principal variation of surface "area." Therefore, closed hypersurfaces of orbits-i.e., 0(3) and 0(2) x 0(2) acting with S2 and T' as their constant mean curvature are exactly the simplest global geo- principal orbit types, respectively. Therefore, it is rather natural metric objects that can be physically described as possible to investigate the constant-mean-curvature hypersurfaces of shapes of free soap bubbles in equilibrium. Obviously, the E4, which are invariant under the action of either 0(3) or 0(2) round sphere is an example ofa closed hypersurface ofconstant x 0(2). The 0(3) case was thoroughly studied in ref. 5 and the mean curvature. Moreover, it is extremely difficult to imagine only closed one it provides are those round spheres. In our con- any other closed constant-mean-curvature hypersurface. One tinuing effort of studying generalized rotational hypersurfaces is thus led to the natural question "Is a closed hypersurface of of constant mean curvature in euclidean spaces (see ref. 5), it constant mean curvature in euclidean space necessarily a round is a pleasant surprise to discover that one can construct infinitely sphere?" many smooth immersions of the 3-sphere into E4 with a given In 1900, Liebmann (1) proved that a strictly convex closed constant mean curvature that are 0(2) X 0(2) invariant. surface ofconstant mean curvature in E3 is necessarily a round It is easy to see that the orbit space E4/0(2) X 0(2) can be sphere. In 1951, Hopf(2) proved a much stronger theorem-i.e., parametrized by {(x,y), x 2 0, y 2 0} with orbital distance met- the only possible differentiable immersions of s2 into E3 with ric ds2 = dx- + dy2. Let M3 be an arbitrary 0(2) X 0(2) invariant constant mean curvature are exactly the round sphere. Due to hypersurface in E4. Then, M3/0(2) X 0(2) is a curve in the the fact that Hopf's proof uses the isothermal coordinate and above orbit space. The mean curvature ofM' is equal to a given complex analysis on S2, his method cannot be generalized to constant value h if and only if the above image curve satisfies higher dimensions. In 1958, Alexandrov (3) proved another far- the following differential equation: reaching generalization of Liebmann's result. By an ingenious geometric argument using the reflectional symmetries of eu- a= xy - yx = 3h + i- , [1] clidean spaces, he showed that any imbedded closed hyper- y x surface of constant mean curvature (with no assumption on its where of is the angle between the x axis and the tangent vector topological type) must be a round sphere. In a later paper, Al- and cr, i, y and i, y are, respectively, their first and second de- exandrov (4) generalized the above remarkable theorem in var- rivatives with respect to the arc length s. The functions ious ways, including the admission of a certain type of self-in- tersection in the assumption. In view of the above results of 1 1 IT x = cos(hs), Y= sin(hs) (0 s_s 2) Liebmann, Hopf, and Alexandrov, the general feeling among h h 2h differential geometers on this matter is that the round sphere are clearly solutions of E7. 1 whose inverse image is a round The publication costs ofthis article were defrayed in part by page charge sphere of radius 1/h in E . Differentiably, it is not difficult to payment. This article must therefore be hereby marked "advertise- see that the inverse image of a smooth curve with its two ends ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. perpendicular to the x and y axes, respectively, is a smooth im- 3931 Downloaded by guest on September 29, 2021 3932 Mathematics: Hsiang et aL Proc. NatL Acad. Sci. USA 79 (1982) mersion of S3 into E4. Analytically, it is easy to show that any One of the useful properties of I and 1 is the following fact: solution curve of Eq. 1 ending up at the x or y axis must be perpendicular to the x or yaxis. Therefore, the proofofour main dl xi2 = 2:-0, d = x o theorem can be reduced to showing the following analytical ds x theorem. ds y ANALYTICAL THEOREM: For each positive integer n, there The proofand further discussion of its significance in geometry exists a solution curve ofEq. 1 that starts at the x axis and ends elsewhere. at the y axis and its total change ofdirection (ao) is 2nfr + (XI/ and variation theory will be published 2). To prove this analytical theorem, one needs a careful study This work was partially supported by National Science Foundation ofthe geometric behavior ofsolution curves ofEq. 1. We found Grant MCS-77-23579. that the following two quantities play important roles in the 1. Liebmann, H. (1900) Math. Ann. 53, 81-112. study oftheir geometry; i.e., 2. Hopf, H. (1950-1951) Math. Nachr. 4, 232-249. 3. Alexandrov, A. D. Vest. Leningr. Univ. 13, 000-000. 3h2 4. A. D. Ann. Mat. Pura 303-315. = Alexandrov, (1962) AppL 58, I J = yX + 2 xy2 5. Hsiang, W. Y. & Yu, W. (1982)J. Differ. Geom., in press. Downloaded by guest on September 29, 2021