Holomorphic Embedding of Complex Curves in Spaces of Constant Holomorphic Curvature (Wirtinger's Theorem/Kaehler Manifold/Riemann Surfaces) ISSAC CHAVEL and HARRY E

Proc. Nat. Acad. Sci. USA Vol. 69, No. 3, pp. 633-635, March 1972

Holomorphic Embedding of Complex Curves in Spaces of Constant Holomorphic Curvature (Wirtinger's theorem/Kaehler manifold/Riemann surfaces) ISSAC CHAVEL AND HARRY E. RAUCH* The City College of The City University of New York and * The Graduate Center of The City University of New York, 33 West 42nd St., New York, N.Y. 10036 Communicated by D. C. Spencer, December 21, 1971

ABSTRACT A special case of Wirtinger's theorem ever, the converse is not true in general: the complex curve asserts that a complex curve (two-dimensional) hob-o embedded as a real minimal surface is not necessarily holomorphically embedded in a Kaehler manifold is a minimal are obstructions. With the by morphically embedded-there surface. The converse is not necessarily true. Guided that we view the considerations from the theory of moduli of Riemann moduli problem in mind, this fact suggests surfaces, we discover (among other results) sufficient embedding of our complex curve as a real minimal surface topological aind differential-geometric conditions for a in a Kaehler manifold as the solution to the differential-geo- minimal (Riemannian) immersion of a 2-manifold in for our present mapping problem metric metric extremal problem complex projective space with the Fubini-Study as our infinitesimal moduli, differential- to be holomorphic. and that we then seek, geometric invariants on the curve whose vanishing forms necessary and sufficient conditions for the minimal embedding to be INTRODUCTION AND MOTIVATION holomorphic. Going further, we observe that minimal surface It is known [1-5] how Riemann's conception of moduli for con- evokes the notion of second fundamental forms; while the formal mapping of homeomorphic, multiply connected Rie- moduli problem, again, suggests quadratic differentials. mann surfaces on one another led to the conception of moduli, Hence, as our differential-geometric invariants, we seek qua- or, better, infinitesimal moduli, as obstacles or obstructions dratic differentials on the curve generated by the second funda- to the realization of such a map between two given surfaces. mental forms of the embedding. In another, simpler, context The question arises quite naturally as to whether analogous Heinz Hopf [7] has done this. phenomena occur in other, in a certain sense naturally re- Adopting an experimental approach, we take the simplest lated, mapping problems. examples for our Kaehler manifold, namely, those with con- Certainly a related problem is that of attempting to map a stant holomorphic curvature. Anticipating the necessity for Riemann surface (complex curve) complex analytically into some topological restrictions, since no local differential geo- a higher-dimensional complex manifold. That there are some metric conditions could force some pieces of minimal surface sorts of obstructions arising in general is clear. The question to be holomorphically embedded, we consider as a principal that presents itself is whether the obstructions appear as example a compact curve. Also, in view of the minimal area infinitesimal moduli related to, or resembling those of, the character of the extremal problem, we then assume the image moduli problem of the first paragraph. is not homologous to zero, and we choose as the most inter- We recall that the infinitesimal moduli there are precisely esting case of the ambient manifold the complex projective the everywhere-finite complex analytic quadratic differentials space with the Fubini-Study metric. (real on any boundaries) on the Riemann surfaces. More Armed with these heuristic considerations, the reader is precisely: in attempting to construct the desired map of sur- better prepared for our main theorem below. See also Remark face one on surface two, we transform the problem into a dif- 2 at the end of that section. ferential-geometric extremal problem by putting a Riemannian metric, consistent with the complex structure, on surface MAIN THEOREM AND PROOF two; we then construct a map of one on two that is extremal for THEOREM. Let M"(x) be an n-(complex) dimensional some differential-geometric functional and find that the pull- Kaehlerian manifold of constant holomorphic curvature x, back to one of the metric on two generates a quadratic differ- M a connected orientable 2-(real) dimensional Riemannian ential as described. The extremal map represents the attempt; manifold, and f:M M"(x) a minimal Riemannian immer- the success or failure is determined by whether the quadratic sion. If x = 0, then the second fundamental form of M in differential vanishes identically or not. M"(x) is formally holomorphic (compare definition below) In references [2-4], the extremal problem was suggested by relative to the complex structure on M induced by its Rie- Douglas's solution of the Plateau problem, so that the notion mannian metric. If x 0 then the second fundamental form of minimal surface is evoked. of M in Mi(x) is formally holomorphic if and only if: either The relevance of all this to our new mapping problem be- at every point p of M J [the complex structure of M"(x) I comes clearer when we recall Wirtinger's theorem [6], a special takes the tangent space to f(M) at f(p) into its orthogonal case of which asserts that a holomorphically embedded com- complement in the tangent space to M"(x) at f(p), or for plex curve in a Kaehler manifold is a minimal surface. How- every point p of M J leaves the tangent space to f(M) at f(p) 633 Downloaded by guest on September 28, 2021 634 Mathematics: Chavel and Rauch Proc. Nat. Acad. Sci. USA 69 (1972)

invariant, i.e., f(M) is a complex submanifold of M"(x). In structure, we have particular, if M'(x) is complex projective space with its Fubini-Study metric, and M is compact, connected, ori- k-wlhklaea = (1/2){ReW2(hi a- h22a - 2i~h2a) + entable, f:M -- CP, minimal such that f(M) is not homol- owC(hiia + h22a)Jea. ogous to zero, then the formal holomorphicity of the second fundamental form implies f(M) is a complex submanifold We henceforth assume that the immersion is minimal, i.e., of CP,. hi,,a + h22, = 0; therefore, Our proof is suggested by S. S. proof of a Chern's [8, 9] Wk1lh41a = {Re w2(hila'- ih,2i) lea, theorem of F. J. Almgren [10] and E. Calabi [11] that states that a minimal immersion of the 2-sphere into the standard i.e., the second fundamental form is the real part of a vector- 3-sphere of constant curvature 1 is totally geodesic. In par- valued quadratic differential. We define the second funda- ticular, we follow Chern's notation and leave it to the reader mental form to be formally holomorphic if to fill in the necessary details by reference to [8]. Our indices A,B,C,D will range from 1 to 2n; j,k,l from 1 to 2; a,#,'y from 0 co A (Dhiia - Dh,2,a) a 3, ... ., 2n 3 to 2n; and we always sum repeated indices unless otherwise on M. One easily checks that the formula is invariant under indicated. We now turn to the formulas needed for subsequent change of adapted frame. We remark that if we know a calculation. priori that M is holomorphically embedded, then formally Let {eA } denote a local orthonormal frame field on M"(x) holomorphic is holomorphic. with dual basis of local 1-forms {coI}W. The complex structure on M"(x) will be given locally by JeA = JABeB, where J2 = LEMMA. Forf:M -- M'(x) minimal we have -1, and J is orthogonal (in particular, J is skew-symmetric). (3x/4) (Jia - J12W1 A CW2 = W A (Dhiia - The Levi-Civita connection of M"(x) is given locally by a iJ2a) iDhl2a). unique skew-symmetric matrix of local 1-forms (COAB) which Proof: Set a. = Cla + ?W2a, Ha = hila + ih,21. Then by satisfy the equations (4,2,5) we have

d&A = WB A WBA. (1) dw= iW12 A co The curvature matrix R&B = (-1/2)RBCDc A WD (where &a= -iC12 A Q, + Qu A cvw - RABCD + RABDC = 0) is skew-symmetric and satisfies the (3x/4) (Jia + iJ2a)Ji2,w A C02 (7) equations and the minimality of the immersion implies 12a = CoH, d4'AB = WAC A WCB + UAB, (2) which implies the of and metric, being constant holomorphic curvature x, is dua = iW12 A a - c-0 A dHa. (8) characterized by By substracting (8) from (7), we obtain RABCD = (X/4) (6AC5BD - 6AD5BC + 0 = o A dHa - 2iW2 A Qa + A wpa- JACJBD - JADJBC + 2JABJCD) (3) Up (3x/4)(Jha + iU2a)Ji2Wi A W2 Now let f:M - Mn(x) be an immersion. We then use adapted frames el, ..., e2. along f(M), i.e., along f(M) we - X A (dHa + 2iWi2Ha + Hpwpa) - assume eie2 are tangent to f(M), and e3, ..., e2,, are normal to (3x/4) (Jia + iU2a)Jl2Wl A f(M). Then on M (we do not indicate f* since there is no con- W02. fusion) we have 0 = w3 = ... = W2U = 0, which implies by Hence, taking conjugates, we get (1) 0 = W A (dfla - 2icviiHa + Harm) - dk= Wj A Wik (4) (3x/4)(Jia - iJ2c,)Jiic2 A W2 0 = Wj A Wia - W A (Dhila - Dhl2a) - (3x/4)(Ja -iJU2)Ji21W A W2 Therefore, by E. Cartan's lemma, wja can be written as by (6), and the lemma is proven. Wja = WkhCja Proof of the Theorem: If x = 0 then the theorem follows hkja = hika directly from the lemma. If x z 0 then the second funda- if and and is the second fundamental form of M in Mn(x). mental form of M in Mn(x) is formally holomporhic csojhkjaea = .. 2n on all of M. Now On M we also have by (3) onlyif (Jo-iJ2)J12=, .a 3, ., J12 = 0 is equivalent to saying that J takes the tangent space Gja = (-x/8) (JjkJal - J§tJazk + 2JjaJkl)Wk A WI (5) to f(M) into its orthogonal complement, and Ja -iJ2a = 0, a = 3,... ,2n is equivalent to saying that J leaves the tangent and the covariant differential of the second fundamental form space to f(M) invariant. By the nonsingularity, and continu- is given by ity of J, we have the second statement of the theorem. For we remark is that = 0 on all Dhjkoa = dhika - W0jlhlka - Wkihjla + (6) the final statement, all need J12 hjfwfta. of f(M) implies that the fundamental Kaehler form vanishes The metric on M induces a complex structure on M such that identically on f(M) (indeed, the fundamental Kaehler form ) = WI, + iW2 is a (1,0)-form, and relative to this complex is given by 1/2WA A JWA). Downloaded by guest on September 28, 2021 Proc. Nat. Acad. Sci. USA 69 (1972) Holomorphic Embedding of Complex Curves 635

Remark 1: We leave it to the reader to check that for quasiconformal mappings. I," Proc. Nat. Acad. Sci. USA arbitrary Riemannian manifolds MI the 40, 808-812. holomorphicity of 3. Rauch, H. E. (1955) "On the transcendental moduli of the second fundamental form is equivalent to the invariance algebraic Riemann surfaces," Proc. Nat. Acad. Sci. USA 41, of the tangent space to M under the action of the curvature 42-49. tensor as a map of 2-vectors to 2-vectors. 4. Rauch, H. E. (1955) "On moduli in conformal mapping," Proc. Nat. Acad. Sci. USA 41, 176-180. Remark 2: Standard Riemannian function theory shows 5. Rauch, H. E. (1965) "A transcendental view of the space of algebraic Riemann surfaces," Bull. Amer. Math. Soc. 71, (pluricanonical embedding) that any compact Riemann sur- 1-39. face can be embedded one-one and nonsingularly in complex 6. Wirtinger, W. (1936) "Eine Determinantenidentitat und projective space of sufficiently high dimension. The point of ihre Anwendung auf analytische Gebilde in Euklidischer the note, however, is not the existence of an embedding but und Hermitischer Maasbestimmung," Monatsh. Math. Physik 44, 343-365. the attempt to "reverse" Wirtinger's theorem, and the point 7. Hopf, H. (1950-1951) "Uber FlAchen mit einer Relation of the first section is to show the heuristic arguments that led zwischen der Hauptkriimmungen," Math. Nach. 4, 232-249. to the main theorem. 8. Chern, S. S. (1968) "Minimal submanifolds in a Riemannian manifold," Technical Report 19 (New Series) (University of This research was partially sponsored by the Air Force Office Kansas, Department of Mathematics). of Scientific Research, Air Force Systems Command, U.S.A.F., 9. Chern, S. S. (1969) "Simple proofs of two theorems on under grant no. AFOSR 71-2063. minimal surfaces," L'Ens. Math. XV, 53-61. 10. Almgren, F., Jr. (1966) "Some interior regularity theorems 1. Teichmueller, 0. (1939) "Extremale quasikonforme Abbil- for minimal surfaces and an extension of Bernstein's dungen und quadratische Differentiale," Abh. Preuss. Theorem," Ann. Math. 84, 277-292. Akad. Wiss. Math.-Naturwiss. Kl. 22, 197 p. 11. Calabi, E. (1967) "Minimal immersions of surfaces in 2. Gerstenhaber, M. & Rauch, H. E. (1954) "On extremal Euclidean spheres," J. Diff. Geom. 1, 111-125. Downloaded by guest on September 28, 2021