The Geometry of Convex Affine Maximal Graphs

Math. Nachr. 232 (2001), 29 – 37 The Geometry of Convex Affine Maximal Graphs By Jose Antonio Gálvez, Antonio Martı́nez, and Francisco Milán of Granada (Received January 20, 2000; revised version January 16, 2001; accepted July 2, 2001) Abstract. Locally strongly convex surfaces which are extremal for the first variation of the equiaffine area integral have been investigated on several occasions. Here we are interested in the description of their behaviour at infinity. We consider an affine maximal annular end which is a graph of vertical flux and give a detailed representation of it when its affine conormal map works well at infinity. 1. Introduction At the beginning of the century Blaschke, see [B], studied the first variation of the equiaffine area integral. He found that the Euler–Lagrange equation is of fourth order and nonlinear, but it is equivalent to the vanishing of the affine mean curvature. When Calabi discovered in 1982 that the second variation is negative for extremal locally strongly convex surfaces he proposed to call them affine maximal surfaces. Although the elliptic paraboloid is still the only known example which is Euclidean complete, there are many properly embedded affine maximal annular ends. In fact, improper affine spheres are an important class of affine maximal surfaces that, up to equiaffine transformations, are, locally, graphs of solutions of the following Monge– Ampère equation
 (1.1) Det ∇2 f = 1 , on Ω . It was proven in [FMM1] and [FMM2] that the graph of a solution of (1.1) on Ω = {ζ ∈ C | |ζ| > 1} has always a properly embedded affine maximal end whose behaviour at infinity depends on five real numbers. Affine maximal surfaces of rotation, which were described in [K], have also Euclidean complete annular ends. Our aim in this paper is to study affine maximal graphs in a neighborhood of its end. We shall prove that if xf ≡ (x1 , x2 , f(x1 , x2)), with f proper, is an affine maximal annular end of vertical flux such that its affine conormal map is, locally, a vertical graph, then the end of xf resembles the end of an affine maximal surface of revolution. 1991 Mathematics Subject Classification. 53A15. Keywords and phrases. Affine maximal graphs, affine Bernstein problem. c
WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2001 0025-584X/01/23212-0029 $ 17.50+.50/0 30 Math. Nachr. 232 (2001) The paper is laid out as follows. In §2 we give a (brief) description about affine maximal surfaces, their fundamental equations and properties. The interested reader may consult [C3] and [LSZ] for further information. In §3 we introduce the concept of regular–balanced ends and see that a well–defined improper affine sphere at infinity can be associated with them. This fact enabled us to show some examples of affine maximal annular ends and prove that the elliptic paraboloid is the only Euclidean complete affine maximal surface with a regular– balanced end (Theorem 3.9). A general representation of regular–balanced ends is also obtained (Theorem 3.10). 2. Affine maximal surfaces Let S be a smooth surface and x : S → IR3 be an immersion with positive Gauss curvature K. Since S is orientable (the mean curvature vector field orientates it) we have a unit normal vector field n on S such that the Blaschke metric, g, gp (v, w) = −K −1/4 dnp (v), w , p ∈ S, v , w ∈ Tp S , is a Riemannian metric, where · , · denotes the usual inner product in IR3 . Since g is invariant under the equiaffine transformation group, we call it the equiaffine metric. From now on, S will be considered as a Riemann surface with the conformal structure induced by g. By considering the first variation of the equiaffine area integral of x one can find (see [B], [C2], [C3]) that the Euler–Lagrange equation for this variational problem is equivalent to the following system of differential equations
 (2.1) △g K −1/4 n = 0 , where △g is the Laplace–Beltrami operator associated to g. The immersion U = K −1/4 n : S → IR3 is called the affine conormal map of x. From (2.1), x(S) is affine maximal if and only if U is a harmonic immersion. Associated to the affine conormal map U we also have the conjugate affine conormal map U ∗ which is well–defined only on some covering S of S. The relation dU ∗ = −dU ◦
Rotπ/2 shows that U and U ∗ are local immersions and Z = U + iU ∗ is holomorphic where Rotπ/2 denotes the operator on vector fields which rotates each tangent plane π/2 in a positive direction . The equiaffine invariant vector field, (2.2) ξ = 1 △g x , 2 is always transversal to the immersion and is called the affine normal vector field of x. The pair {U, ξ} is a relative normalization invariant under equiaffine transformations (see [LSZ]), and is called the equiaffine normalization of x. We choose a conformal parameter ζ = u + iv so that g = E |dζ|2 . Then, a straight computation gives: (2.3) E 2 = Det(xu , xv , xuu ) = Det(xu , xv , xvv ) , Det(xu , xv , xuv ) = 0 , Gálvez et al, Convex Affine Maximal Graphs (2.4) 31 E = Det(Uu , Uv , U ) = Det(xu , xv , ξ) , (2.5) U, xu = 0, Uu , xu = −E , Uv , xu = 0, U, xv = 0, Uu , xv = 0, Uv , xv = −E , U, ξ = 1, Uu , ξ = 0, Uv , ξ = 0, and dx = −U ∧ dU ∗ , (2.6) dU ∗ = ξ ∧ dx , where ∧ denotes the cross product, Det( · , · , · ) is the usual determinant form, ( · )u and ( · )v are, respectively, partial derivatives with respect to u and v. From (2.1), (2.5) and (2.6), we deduce that if x(S) is affine maximal, then ξ ∧ dx is a closed one–form on S and for closed curves γ on S,  ξ ∧ dx γ is a homology–invariant vector. Definition 2.1. The flux along γ is defined as the vector quantity  (2.7) F lux([γ]) = ξ ∧ dx . γ ∗ From (2.6), U is well–defined if and only if F lux([γ]) = 0 for any closed curve γ. When the affine normal ξ is a constant vector field the immersion is called an improper affine sphere. From (2.4), (2.5) and (2.6) x is an improper affine sphere if and only if it is affine maximal and U (S) lies on a plane. It is clear from (2.7) that improper affine spheres have vanishing flux. 3. Regular–balanced affine maximal graphs Let x : S → IR3 be a locally strongly convex affine maximal immersion and Σ ⊆ S such that x(Σ) is an annular end of x(S) with compact boundary x(γ). Definition 3.1. x(Σ) is called a regular–balanced end (in short, RB–end) if there exists a plane Π in IR3 with unit normal vector A such that the immersions x and U satisfy: (R1). x(Σ) is a graph on a domain in Π and x, A : Σ → IR is a proper map. (Regularity condition.) (R2). The orthogonal projection of U (Σ) on Π is a local diffeomorphism. (Regularity condition.) (B). A ∧ F = 0, where F is the flux of x(Σ) along γ. (Balanced condition.) Remark 3.2. If x : S → IR3 is a Euclidean complete affine maximal immersion, then, from Hadamard Theorem (see [W]), the conditions (R1) and (B) are always satisfied. 32 Math. Nachr. 232 (2001) Proposition 3.3. Let x(Σ) be an RB-end of x. Then Σ is conformally a punctured disk and Y : Σ → IR3 , Y = U ∗ ∧ A + x, A A , (3.1) is a well–defined improper affine sphere with the end on the boundary of a convex body in IR3 (that is, Y is regular at infinity in the sense of [FMM1]). Pro o f . From (2.6), (2.7) and (B), it is clear that Y is well–defined. Let ζ = u + iv be a conformal parameter of x such that g = E |dζ|2 . From (3.1), (3.2) Yu = −Uv ∧ A + xu , A A , Yv = Uu ∧ A + xv , A A . Because of (R2) and (3.2), Yu ∧ Yv , A = Det(Uu , Uv , A) does not vanish on Σ and Y is an immersion with transversal vector A. From (2.4), (2.5), (2.6) and (3.2), we obtain, Det(Yu , Yv , Yuu ) = Det(Yu , Yv , Yvv ) = Det(Uu , Uv , A)2 , Det(Yu , Yv , Yuv ) = 0 . It is not a restriction to assume Det(Uu , Uv , A) > 0. Then, from (2.3), (2.4), (2.5) and the above expressions, Y is a locally strongly convex immersion with Blaschke metric, (3.3) h = Det(Uu , Uv , A) |dζ|2 = ξ, A E |dζ|2 . Since Yuu + Yvv = xuu + xvv , A A, from (2.2), (3.2) and (3.3), we conclude that Y is an improper affine sphere with affine normal A. The fact that x is a graph and x, A a proper map, the level curves x(γc ) = x(Σ) ∩ { x, A = c} must be strictly convex Jordan curves for a large enough c. Moreover from (2.6), (3.1) and (3.2), if T is a tangent vector along x(γc ) then ξ, A T is a tangent vector along Y (γc ). Thus, the end of Y (Σ) is also fibred by strictly convex Jordan curves. Since from (3.3) the affine metric h of Y and g are conformal metrics and Y is regular at infinity we conclude, see [FMM1], that Σ must be conformally a punctured disk. ✷ Definition 3.4. The improper affine sphere Y given by (3.1) is called the tangent improper affine sphere at infinity of x(Σ). 3.1. Some examples Let us consider f as a solution of
  = {ζ ∈ C | |ζ| > 1} . det ∇2 f = 1 on Ω  R = R Ω.  Then the graph xf of f is an improper affine We also use the notation Ω  Moreover, see [FMM2], it can be sphere and its end is an RB–end conformal to Ω. represented as    1 1 1 G+F 1 2 2 R , (3.4) xf ≡ , |G| − |F | + ℜ(GF ) − ℜ F dG , on Ω 2 8 8 4 2 Gálvez et al, Convex Affine Maximal Graphs 33 for some R > 1, with (3.5) G(ζ) = ζ , ∞  an F (ζ) = µζ + ν + , ζn n=1 R , ζ ∈Ω where by ℜ we denote the real part and µ, ν, an ∈ C , for n ≥ 2, a1 ∈ IR and |µ| < 1. The affine metric and the affine conormal map of xf are given, respectively, by  1
|dG|2 − |dF |2 , 4   F −G = ,1 . 2 (3.6) hf = (3.7) Uf  R → R be a  R → R be a bounded harmonic function and a∗ : Ω Now, let a : Ω conjugated harmonic of a (which always exists because a is bounded). Assume that Residue [a + ia∗ , ∞] = 0. If we consider the following harmonic immersion,  F (ζ) − ζ , a(ζ) + b log |ζ| , (3.8) N (ζ) = 2 with b ∈ IR, b ≥ 0 and lim|ζ |→∞ a(ζ) > 0 if b = 0, then from (3.8) one has


 4 Det(N, Nu , Nv ) = a(u, v) + b log u2 + v2 1 − |F ′ |2   2bu + au + 2 ((1 + F2v )(F1 − u) − (v + F2 )F1v ) u + v2   2bv + av + 2 ((v + F2 )(F1u − 1) − F2u (F1 − u)) , u + v2  R . Hence, it is clear, from the above assumptions, where F = F1 +iF2 and ζ = u+iv ∈ Ω that there exists R1 > R, such that Det (N, Nu , Nv ) > 0  R1 . Consequently, using (2.6), then for all ζ = u + iv ∈ Ω (3.9) X(ζ) = − is an affine maximal surface.  ζ ζ0 N ∧ dN ∗ , R , ζ ∈Ω 1 Proposition 3.5. X is a well–defined affine maximal surface with an RB–end. Moreover,   ζ  F (ζ) + ζ 2 X(ζ) = (a(ζ) + b log |ζ| − bζ + 2 α(w)dw + ρ̄(w) d w , 2 ζ0  (3.10) ζ 1 2 1 1 1 2 |ζ| − |F (ζ)| + ℜ(ζF (ζ)) − ℜ F (w) dw , 8 8 4 2 ζ0 34 Math. Nachr. 232 (2001)  R2 , for some R2 > R1 , and ζ = u + iv ∈ Ω α(ζ) = − with ∂ ∂ζ =
1 ∂ 2 ∂u ζ ∂a , 2 ∂ζ ρ(ζ) = − F (ζ) ∂a b F (ζ) − , 2 ∂ζ 2 ζ  ∂ − i ∂v . Pro o f . Since F is a holomorphic function, from (3.8) and (3.9) then the coordinate functions (X1 , X2 , X3 ) of X satisfy,    ∂ bF F1 + u
F +ζ 2 a + b log |ζ| + bu = − aζ − = α + ρ, X1 − ∂ζ 2 2 2ζ    ∂ F2 − v
F −ζ b F a + b log |ζ|2 + bv = −i aζ − i = i(−α + ρ) . X2 + ∂ζ 2 2 2 ζ Consequently, X1 X2 and    F1 + u
2 = a + b log |ζ| − bu + (α + ρ)dw + (α + ρ) d w , 2    v − F2
a + b log |ζ|2 − bv + i (−α + ρ) dw + i (α − ρ) d w , = 2 X1 + iX2  F +ζ
= a + b log |ζ|2 − bζ + 2 2  αdw + 2  ρdw. gives that X1u X2v − X1v X2u is of the same order as
A straight computation 
a + b log |ζ|2 − 2b + a + b log |ζ|2 . Thus, when R3 is large enough, X1 + iX2 is  R3 and also a covering map. a local diffeomorphism on Ω Now, as    F
a + b log |ζ|2 + 2 α dw + 2 ρ̄ d w 2  R3 by a constant C and the fact that is bounded on Ω   a + b log |ζ|2 g(ζ) = −b ζ 2 winds around the origin once for |ζ| = R4 with R4 large, and |g(ζ)| > C, we conclude  R4 ) must be a graph on Π ≡ x3 = 0.  R4 , and X(Ω that X1 + iX2 is one–to–one on Ω Analogously, from (3.8) and (3.9),   ∂ |F |2 u2 1 X3 + = − (uF1u − vF2u − F1 ) , ∂u 8 8 4   |F |2 v2 ∂ 1 X3 + = − (uF1v − vF2v + F2 ) , ∂v 8 8 4 and Gálvez et al, Convex Affine Maximal Graphs X3 = 35 1 2 1 1 1 |ζ| − |F (ζ)|2 + ℜ(ζF (ζ)) − ℜ 8 8 4 2   F (w) dw . From (2.5) and bearing in mind that (3.11) lim |(u,v)|→∞ Nu (u, v) = (−1, 0, 0) , lim |(u,v)|→∞ Nv (u, v) = (0, −1, 0) , we can choose R2 > R4 large enough in order to get ξ, (0, 0, 1) > 0, that is (R2) is also satisfied. 
 R is (0, 0, 2πb) and the end of X Finally, from (2.7) and (3.8), the flux along X ∂ Ω is an RB–end. ✷ Remark 3.6. The tangent improper affine sphere at infinity of (3.10) is the graph xf given by (3.4). Remark 3.7. Every end of an elliptic revolution affine maximal surface can be represented as in (3.10) by taking a ≡ constant and F (ζ) = c/ζ for some c ∈ IR.  R → IR3 given by Remark 3.8. There exists R > 1 such that the immersion x : Ω
 (3.12) x(ζ) = 4ζ(1 − log(|ζ|), 2 |ζ|2 + 8ℜ(ζ)(1 − log(|ζ|) is a well–defined affine maximal vertical graph which has not got an RB–end because condition (B) fails. 3.2. On the affine Bernstein problem Theorem 3.9. Let x : S → IR3 be an Euclidean complete affine maximal surface with an RB–end. Then x(S) is an elliptic paraboloid. Pro o f . From the Euclidean completeness x(S) is the boundary of an unbounded convex set in IR3 and by using (R1) has to be a global graph on a convex domain Ω in Π. Since x has an RB–end x(Σ), then, from Proposition 3.3, Σ is conformally a punctured disk and consequently S must be conformally C . Now, as x is a graph, then one can assume that U, A is a positive harmonic function on C . Thus, U, A must be constant and x is an improper affine sphere. The result follows from Jörgens Theorem, see [J]. ✷ 3.3. The general case Theorem 3.10. Let x : S → IR3 be an affine maximal surface with an RB–end. Then, there exists a representation of its end as in (3.10). Pro o f . It is not a restriction to assume that Π ≡ x3 = 0 and A = (0, 0, 1). 36 Math. Nachr. 232 (2001) Let Y : Σ → IR3 be the tangent improper affine sphere at infinity of x. From  R = {ζ ∈ C | |ζ| > R}. Moreover, Proposition 3.3, Σ is conformally equivalent to Ω see [FMM1], there exists a conformal representation of Y as   ζ + F (ζ) 1 2 1 1 1 2 (3.13) Y (ζ) = , |ζ| − |F (ζ| + ℜ(ζF (ζ)) − ℜ F (ζ) dz , 2 8 8 4 2 with F (ζ) = µ ζ + ν + ∞  an , ζn n=1 R . ζ ∈Ω By using (3.1) and (3.13), we can obtain that the affine conormal map of x : Σ → IR3 is given by  F (ζ) − ζ (3.14) U (ζ) = , U3 , 2  R because x is a graph. where U3 = U, A is a positive harmonic function on Ω
 So, H = U3 − F, A log |ζ|2 is the real part of a holomorphic function well–defined  R . Since H is harmonic, we have the following Laurent expansion at infinity, on Ω
 0 < U3 (ζ) = F, A log |ζ|2 + a(ζ) + Γ(ζ) ,  R , and Γ(ζ) is harmonic in the finite plane C . where a(ζ) is harmonic and bounded on Ω If n is any positive entire number greater than F, A , it follows that for |ζ| > R1 > R, 1 ≤ M ζ n expG(ζ ) , where M is a suitable constant and G(ζ) is an entire function with real part Γ(ζ). Thus, the entire function ζ n expG(ζ ) has not got an essential singularity at infinity and is polinomial. Hence, G(ζ) must be constant and from (3.14) the affine conormal map of the end of x is given as in (3.8), which concludes the proof. ✷ Remark 3.11. We deduce from (3.11) and Theorem 3.10 that if an annular affine maximal end x(Σ) is an RB–end, then ξ I) It is conformally a punctured disk and the affine Gauss map |ξ| extends to the puncture as the unit vector A. II) The affine conormal map U (Σ) lies in a half–space of IR3 . III) F ∧ A = 0, where F is the flux of x. Conversely, one can prove that if x(Σ) satisfies I), II) and III), then x(Σ) is an RB–end. Acknowledgements The research of the authors was partially supported by DGICYT Grant No. PB97–0785. Gálvez et al, Convex Affine Maximal Graphs 37 References [A] Ahlfors, A.: Complex Analysis, McGraw–Hill, New York, 1979 [B] Blaschke, W.: Vorlesungen über Differentialgeometrie, II., Berlin, Springer–Verlag, 1923 [C1] Calabi, E.: Improper Affine Hyperspheres of Convex Type and a Generalization of a Theorem by K. Jörgens, Mich. Math. J. 5 (1958), 108 – 126 [C2] Calabi, E.: Hypersurfaces with Maximal Affinely Invariant Area, Amer. J. Math. 104 (1984), 91 – 126 [C3] Calabi, E.: Convex Affine Maximal Surfaces, Result in Math. 13 (1988), 209 – 223 [CY] Cheng, S. Y., and Yau, S. T.: Complete Affine Hypersurfaces, Part I. The Completeness of Affine Metrics, Comm. Pure Appl. Math. 39 (1986), 839 – 866 [FMM1] Ferrer, L., Martı́nez, A., and Milán, F.: Symmetry and Uniqueness of Parabolic Affine Spheres, Math. Ann. 305 (1996), 311 – 327 [FMM2] Ferrer, L., Martı́nez, A., and Milán, F.: An Extension of a Theorem by K. Jörgens and a Maximum Principle at Infinity for Parabolic Affine Spheres, Math. Z. 230 (1999), 471 – 486 [J] Jörgens, K.: Über die Lösungen der Differentialgleichung rt − s2 = 1, Math. Ann. 127 (1954), 130 – 134 [K] Krauter, P.: Affine Minimal Hypersurfaces of Rotation, Geometriae Dedicata 51 (1994), 287 – 303 [LSZ] Li, A. M., Simon, U., and Zhao, G.: Global Affine Differential Geometry of Hypersurfaces, Walter de Gruyter, Berlin – New York, 1993 [NS] Nomizu, K., and Sasaki, T.: Affine Differential Geometry, Cambridge Univ. Press, 1994 [W] Wu, H. H.: The Spherical Images of Convex Hypersurfaces, J. Differential Geometry 9 (1974), 279 – 290 Departamento de Geometrı́a y Topologı́a Facultad de Ciencias Universidad de Granada 18071 Granada Spain E–mails: [email protected] [email protected] [email protected]