Symplectic critical surfaces in Kähler surfaces

Transcription

1 Symplectic critical surfaces in Kähler surfaces Jiayu Li ( Joint work with X. Han) ICTP-UNESCO and AMSS-CAS November, 2008

2 Symplectic surfaces Let M be a compact Kähler surface, let ω be the Kähler form. For a compact oriented real surface Σ without boundary which is smoothly immersed in M, one defines, following Chern and Wolfson, the Kähler angle α of Σ in M as ω Σ = cos α dµ Σ, where dµ Σ is the area element of Σ. As a function on Σ, α is continuous everywhere and is smooth possibly except at the complex or anti-complex points of Σ, i.e. where α = 0 or π. We say that, Σ is a holomorphic curve if cos α 1, Σ is a Lagrangian surface if cos α 0, Σ is a symplectic surface if cos α > 0.

3 Symplectic critical surfaces We (Han-Li) consider a new functional: 1 L = cos α dµ Σ. Σ It is obvious that holomorphic curves minimize the functional. Recall that the area functional is A = Σ dµ Σ. If we consider the functional cos αdµ Σ A L. Σ

4 Since cos αdµ Σ = ω Σ, and dω = 0, one gets that l := cos αdµ Σ is homotopy invariant. Σ We have l A L.

5 The Euler-Lagrange equation of the functional is cos 3 α H (J(J cos α) ) = 0, (1) where H is the mean curvature vector of Σ in M, J is the complex structure on M, () denotes tangential components of (), and () denotes the normal components of (). We call it a symplectic critical surface. Proposition (Han-Li) If a symplectic critical surface is minimal, then cos α Constant.

6 Let {e 1, e 2, v 3, v 4 } be a orthonormal frame around p Σ such that J takes the form Then J = 0 cos α sin α 0 cos α 0 0 sin α sin α 0 0 cos α 0 sin α cos α 0 (J(J cos α) ) Set V = 2 αv αv 4.. = cos α sin 2 α 1 αv 4 + cos α sin 2 α 2 αv 3.

7 And consequently the Euler-Lagrange equation of the function L is cos 2 αh sin 2 αv = 0. (2) or equivalently, H sin 2 α(h + V ) = 0. It is not difficult to see, roughly, the symbol of the equation is ( (1 sin 2 α)ξ σ := 2 + η 2 (sin 2 ) α)ξη (sin 2 α)ξη ξ 2 + (1 sin 2 α)η 2, which enables one believes that the equation (1) is an elliptic equation. Proposition (Han-Li) The equation (1) is elliptic.

8 Assume that Σ is immersed in M by F. Without loss of generality, we assume that Σ is a graph of two functions f, g in a small neighborhood U, i.e., F = (x, y, f (x, y), g(x, y)) in U. Suppose that the complex structure in the neighborhood F (U) is standard, i.e., J = We choose and and e 1 = F x = (1, 0, f x, g x ), e 2 = F y = (0, 1, f y, g y ) v 3 = ( f x, f y, 1, 0) v 4 = ( g x, g y, 0, 1) Then {e 1, e 2, v 3, v 4 } is a basis of M.

9 The metric of Σ in the basis is ( 1 + f 2 (g ij ) 1 i,j 2 = x + gx 2 ) f x f y + g x g y f x f y + g x g y 1 + fy 2 + gy 2 and the inverse matrix is (g ij ) 1 i,j 2 = 1 det(g ij ) ( 1 + f 2 y + g 2 y f x f y g x g y f x f y g x g y 1 + fx 2 + gx 2 ). A direct computation yields the determinant of the symbol of the equation is det(σ) = cos2 α det g (g 22ξ 2 2g 12 ξη + g 11 η 2 ) 2. Because (g ij ) is positive definite and cos α > 0, we see that det(σ) > 0 if (ξ, η) (0, 0), which implies that the equation (1) is elliptic.

10 Theorem (Han-Li) If M is a Kähler-Einstein surface and Σ is a symplectic critical surface, then we have cos α = 3 sin2 α 2 α 2 R cos 3 α sin 2 α, cos α where R is the scalar curvature of M. Corollary If the scalar curvature R of M is nonnegative, then a symplectic critical surface is holomorphic.

11 Theorem (Chern-Wolfson) Assume that M is a Kähler-Einstein surface with scalar curvature R. Let α be the Kähler angle of a minimal surface Σ in M. Then cos α = 2 α 2 cos α + R sin 2 α cos α. Therefore, if cos α > 0 and R 0, we have cos α Constant. So, every symplectic minimal surface in a K-E surface with nonnegative scalar curvature, is a holomorphic curve. Note that, if R < 0 the result is not true! Claudio Arrezo gave the examples.

12 Symplectic critical surfaces in C 2 We first consider the surface Σ which is defined by F (u, v) = (v cos u, v sin u, ln v, 0), u [0, 2π], v > 0. It is in fact a rotational symmetric surface z = 1 2 log(x 2 + y 2 ) in R 3, we consider it as a surface in C 2. Using the standard complex structure J of C 2, J = (3)

13 The Kähler angle of Σ is cos α = v 1 + v 2. It is easy to compute that H 1 = 1, v(1 + v 2 ) 3 2 H 2 = 0, and (J cos α) = cos α (1 + v 2 ) 2 F u, (J(J cos α) ) v = cos α v (1 + v 2 )

14 Therefore cos 3 α H = (J(J cos α) ). So Σ is a symplectic critical surface which is not minimal. The Gauss curvature of Σ is K = 1 (1 + v 2 ) 2. It is not difficult to see that Σ is a simply connected complete surface.

15 Figure: symplectic critical surface

16 We then consider the surface Σ F (u, v) = (v sin u, v cos u, f (v), u), u (, ), v > 0. Using the standard complex structure of C 2, we have, cos α = f v (1 + v 2 )(1 + (f ) 2 ). Straight computation yields that, and H 1 = 0, H 2 = f vf, (1 + (f ) 2 ) 3 2 (1 + v 2 )(1 + (f ) 2 ) 1 2 (J(J cos α) ) cos α(1 + vf ) = (1 + (f ) 2 )(1 + v 2 ) 1 2 cos α. v

17 One checks that F is a symplectic critical surface in C 2 if and only if f (v) satisfies the equation (1 + v 2 ) 2 f = 1 + (2v v 3 )f + 3v 2 (f ) 2 v(f ) 3. It is clear that f 1 (v) = 1 2 v 2, f 2 (v) = ln v and f 3 (v) = v 2 + ln v (v > 0) are solutions. The surface Σ with f = f 1 is Lagrangian which is not minimal. The surface Σ with f = f 2 is a holomorphic curve. If we choose f = f 3, then Σ is a symplectic critical surface which is not holomorphic.

18 We prove a Liouville theorem for symplectic critical surfaces in C 2. Theorem If Σ is a complete symplectic critical surface in C 2 with area quadratic growth, and cos α δ > 0, then it is a holomorphic curve.

19 Topological property By the equations obtained by Wolfson and (1), we see that, on a symplectic critical surface we have sin α ζ = (sin α)h, where h is a smooth complex function, ζ is a local complex coordinate on Σ, and consequently, we have Proposition (Han-Li) A non holomorphic symplectic critical surface in a Kähler surface has at most finite complex points.

20 Theorem (Han-Li) Suppose that Σ is a non holomorphic symplectic critical surface in a Kähler surface M. Then and χ(σ) + χ(ν) = P 1 2π Σ α 2 cos 2 α dµ, c 1 (M)([Σ]) = P 1 α 2 2π Σ cos 3 α dµ, where χ(σ) is the Euler characteristic of Σ, χ(ν) is the Euler characteristic of the normal bundle of Σ in M, c 1 (M) is the first Chern class of M, [Σ] H 2 (M, Z) is the homology class of Σ in M, and P is the number of complex tangent points.

21 Corollary (Han-Li) Suppose that Σ is a symplectic critical surface in a Kähler surface M. Then χ(σ) + χ(ν) c 1 (M)([Σ]), the equality holds if and only if Σ is a holomorphic curve. Corollary (Han-Li) Suppose that Σ is a non holomorphic symplectic critical surface in a Kähler-Einstein surface M with scalar curvature R > 0. Then 1 2 R deg Σ = P + 1 α 2 2π Σ cos 3 α dµ.

22 Let g be the genus of Σ, I Σ the self-intersection number of Σ, D Σ the number of double points of Σ. Then χ(σ) = 2 2g, χ(ν) = I Σ 2D Σ. Set we have C 1 (Σ) = C 1 (M)([Σ]), Theorem (Han-Li) Suppose that Σ is a symplectic critical surface in a Kähler surface M. Then 2 2g c 1 (Σ) + I Σ 2D Σ = 1 2π Σ (1 cos α) α 2 cos 3 dµ. α

23 Main steps in the computation Set g(α) = ln(sin 2 α). We obtain g(α) = 2 α 2 2 cos α sin 2 α cos α α 4cos2 sin 2 α α 2 = 4 α cos 2 α(k K 1234 ). This equation is valid away from the complex tangent points of M. By the Gauss equation and Ricci equation, we have, R 1212 = K h α 11h α 22 (h α 12) 2 R 1234 = K h 3 1k h4 2k h4 1k h3 2k, where R 1212 is the curvature of T Σ and R 1234 is the curvature of NΣ.

24 Adding these two equations together, we get that, K K 1234 = R R H ((h3 1k h4 2k )2 + (h 4 1k + h3 2k )2 ) = R R V 2 + H V = R R cos 2 α α 2. Thus, R R 1234 = 1 g(α) 2 cos 2 α + α 2 cos 2 α.

25 Integrating the above equality over Σ we have, α 2 2π(χ(T Σ) + χ(nσ)) = 2πP Σ cos 2 α dµ Σ, where χ(t Σ) is the Euler characteristic of Σ and χ(nσ) is the Euler characteristic of the normal bundle of Σ in M, P is the sum of the orders of complex tangent points.

26 We also get that, g(α) = 4 α cos 3 αric(je 1, e 2 ). Note that Ric(Je 1, e 2 )dµ Σ is the pulled back Ricci 2-form of M by the immersion F to Σ, i.e, F (Ric M ) = Ric(Je 1, e 2 )dµ Σ. Thus, F (Ric M ) = ( 1 g(α) 2 cos 3 α + 2 α 2 cos 3 α )dµ Σ. Integrating it over Σ, we obtain that, α 2 2πc 1 (M)[Σ] = 2πP Σ cos 3 α dµ Σ.

27 The geometry of the normal bundle We choose a local orthonormal frames {e 1, e 2, e 3, e 4 } in M such that {e 1, e 2 } is a local orthonormal frames of Σ and {e 3, e 4 } is a local orthonormal frames of the normal bundle ν of Σ in M.

28 Denote the complexification of ν by ν C = ν C = {ν + iν}. Extend the covariant differential to ν C complex linearly, i.e, (x + iy) = x + i y, x, y are sections of ν. The surface Σ admits a complex structure defined by the (1, 0) form φ = θ 1 + iθ 2 and the (0, 1) form φ = θ 1 iθ 2. We can decompose to (1, 0) and (0, 1) parts and write it as = (1,0) + (0,1). Using the Newlander-Nirenberg theorem ν C can be made into a holomorphic bundle whose -operator is (0,1).

29 There exists a natural almost complex structure on ν, defined by J ν (e 3 ) = e 4, J ν (e 4 ) = e 3. Extend J ν to ν C complex linearly. For a normal vector x = x 3 e 3 + x 4 e 4 we have x = (x 31 x 42 ) φ e 3 + (x 41 + x 32 ) φ e 4, then x is a holomorphic section of ν if and only if x = 0.

30 Second variation formula Suppose that φ t,ε is a symplectic variation of φ. For simplicity, we write φ 0,0 (Σ) = φ(σ) = Σ. Let φ t,0 φ 0,ε t = X, t=0 ε = Y, ε=0 and 2 φ t,ε t ε = Z. t=0,ε=0

31 Lemma We have the general second variation formula: 2 div Σ Xdiv Σ Y t ε L(φ t,ε ) = 4 t=0,ε=0 Σ cos α R(X, e i, Y, e i ) e 2 dµ Σ + 2 i X, e i Y dµ Σ Σ cos α Σ cos α e i, ej X e j, ei Y + e j, ei X e i, ej Y dµ Σ Σ cos α div Σ X (ω( e1 Y, e 2 ) + ω(e 1, e2 Y )) 2 Σ cos 2 dµ Σ α div Σ Y (ω( e1 X, e 2 ) + ω(e 1, e2 X )) 2 Σ cos 2 dµ Σ α ω(r(y, e 1 )X, e 2 ) + ω(e 1, R(Y, e 2 )X ) cos 2 dµ Σ α Σ

32 Σ +2 Σ ω( e1 X, e2 Y ) + ω( e1 Y, e2 X ) cos 2 dµ Σ α (ω( e1 X, e 2 ) + ω(e 1, e2 X )) cos 3 α (ω( e1 Y, e 2 ) + ω(e 1, e2 Y ))dµ Σ, where {e 1, e 2 } is a bases of Σ, is covariant differential of M, and div Σ X = ei X, e i.

33 Corollary If X = Y is a normal variation, we have 2 t 2 L(φ t ) =: II (X ) t=0 (X H) 2 = 4 Σ cos α dµ R(X, e i, X, e i ) Σ 2 dµ Σ Σ cos α e +2 i X 2 Σ cos α dµ (X A(e i, e j )) 2 Σ 2 dµ Σ Σ cos α X H(ω( e1 X, e 2 ) + ω(e 1, e2 X )) +4 Σ cos 2 dµ Σ α [ω( e1 X, e 2 ) + ω(e 1, e2 X )] 2 +2 Σ cos 3 dµ Σ α ω(x, e2 X ) e1 cos α 2 Σ cos 3 dµ Σ. α ω( e1 X, X ) e2 cos α 2 cos 3 dµ Σ. α Σ

34 Theorem If we choose X = x 3 e 3 + x 4 e 4, Y = J ν X = x 4 e 3 x 3 e 4, then the second variation of the functional L of a symplectic critical surface is II (X ) + II (Y ) = Σ Σ Σ R X 2 sin 2 α dµ Σ cos α X 2 (1 + cos 2 α) cos 3 dµ Σ α (1 + cos 2 α) cos 5 X 2 α 2 dµ Σ. α In the computation, we used techniques which J. Micallef and J. Wolfson used in the computation of the second variation of area of minimal surfaces.

35 Proposition Let M be a Kähler surface with positive scalar curvature R. If Σ is a stable symplectic critical surface in M whose normal bundle admits a section X with then Σ is a holomorphic curve. X 2 X 2 α 2 cos 2 α,

36 Corollary Let M be a Kähler surface with positive scalar curvature R. If Σ is a stable symplectic critical surface in M with χ(ν) g where χ(ν) is the Euler characteristic of the normal bundle ν of Σ in M and g is the genus of Σ, then Σ is a holomorphic curve. Let K be the canonical line bundle over Σ, let H 0 (Σ, O(ν)) denote the space of holomorphic sections of ν and let H 0 (Σ, O(ν K)) denote the space of holomorphic sections of ν K, then by Riemann-Roch theorem, we have dim H 0 (Σ, O(ν)) = χ(ν) g dim H 0 (Σ, O(ν K)) 1. We therefore have a holomorphic section on ν, so the corollary follows from Proposition 4.

37 Conjecture: In each homotopy class of a symplectic surface in a Kähler surface, there is a symplectic critical surface.

38 The gradient flow We consider the gradient flow of the function L, i. e, df dt = cos2 αh 1 cos α (J(J cos α) ). (4) We set f = cos 2 αh 1 cos α (J(J cos α) ). It is clear that, if f = 0, Σ is a symplectic critical surface. By the first variational formula of the functional L, we see that, along the flow, dl 1 = 2 dt Σ cos 3 α f 2 dµ. One checks that the equation (4) is a parabolic equation, and the short time existence can be shown by a standard argument. We set Σ t = F (Σ, t) with Σ 0 = Σ.

39 We derive the evolution equation of cos α along the flow (4), which can be seen as a start point of the study of the flow. Theorem Let M be a Kähler-Einstein surface with scalar curvature R. Assume that α is the Kähler angle of Σ t which evolves by the flow (4). Then cos α satisfies the equation ( d dt ) cos α = cos3 α( h 3 1k h4 2k 2 + h 3 2k + h4 1k 2 ) +R cos 3 α sin 2 α + cos α sin 2 α H 2 cos α sin 2 α V + H 2, and consequently, if Σ is symplectic, along the flow (4), at each time t, Σ t is symplectic.