Compactness of Symplectic critical surfaces

Transcription

1 August 2016

2 Outline Symplectic surfaces 1 Symplectic surfaces

3 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 = cosα dµ Σ, where dµ Σ is the area element of Σ. Σ is a symplectic surface if cos α > 0.

4 Minimal symplectic surfaces 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 cosα = 2 α 2 cos α+ Rsin 2 α cosα. Therefore, if cos α > 0 and R 0, we have cosα 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.

5 Evolution equation of the Kähler angle The Kähler angle satisfies the following parabolic equation. Theorem Let Σ t be a mean curvature flow in the Kähler-Einstein surface M. Then the evolution equation of cos α is ( t )cosα = J Σ t 2 cosα+ Rsin 2 α cosα, where J Σt 2 = h 3 1k h4 2k 2 + h 3 2k + h4 1k 2,{e 1,e 2,e 3,e 4 } is any orthonormal basis for TM such that {e 1,e 2 } is the basis for TΣ t and {e 3,e 4 } is the basis for NΣ t. It is easy to check that J Σt 2 only depends on the orientation of the frame, and R is the scalar curvature of (M,g).

6 As a corollary, if the initial surface Σ 0 is symplectic, then along the flow, at each time t, Σ t is symplectic, and if the initial surface Σ 0 is Lagrangian, then along the flow, at each time t, Σ t is Lagrangian. The result was proved by Chen-Tian, Chen-Li, Wang.

7 Variational approach Since cos αdµ Σ = ω Σ, and dω = 0, one gets that l := cos αdµ Σ is homotopy invariant. Σ

8 Recall that the area functional is A= dµ Σ. Σ We (Han-Li) consider a new functional: 1 L β = Σ cos β α dµ Σ. It is obvious that holomorphic curves minimize the functional if β > 0.

9 The first variation formula Theorem Let M be a Kähler surface. The first variational formula of the functional L β is, for any smooth vector field X on Σ, δ X L β = (β + 1) +β(β + 1) Σ X H cos β dµ α (2.1) X (J(J cos α) )) cos β+3 dµ, α (2.2) Σ where H is the mean curvature vector of Σ in M, and () means tangential components of (), () means the normal components of (). The Euler-Lagrange equation of the functional L β is cos 3 αh β(j(j cos α) ) = 0. (2.3)

10 Remarks Symplectic surfaces We call it a β-symplectic critical surface. cos 3 αh β(j(j cos α) ) = 0 β = 0, we get minimal surface equation; If β, we get cosα = constant; A holomorphic curve satisfies the equation for all β.

11 Proposition A surface satisfies the equations for all β if and only if it is a minimal surface with Kähle angle constant. The surface is called a super minimal surface or a infinitesimal holomorphic curve.

12 Let {e 1,e 2,v 3,v 4 } be a orthonormal frame around p Σsuch that J takes the form J = Set V = 2 αv αv 4. 0 cos α sinα 0 cosα 0 0 sinα sinα 0 0 cos α 0 sin α cosα 0. V = (h h4 12 )v 3 (h h3 12 )v 4.

13 The Euler-Lagrange equation of the function L β is cos 2 αh β sin 2 αv = 0. (2.4) Proposition For a β-symplectic critical surface with β 0, the Euler-Lagrange equation is an elliptic system modulo tangential diffeomorphisms of Σ.

14 Examples in C 2 We consider the β-symplectic critical surfaces of the following form. F(r,θ)=(r cos θ,r sin θ,f(r),g(r)). (2.5) The equation is equivalent to cos 3 αh=β(j(j cos α) ) { r(1+(g ) 2 + β(f ) 2 )f + r(β 1)f g g +(1+(f ) 2 +(g ) 2 )f = 0 r(1+(f ) 2 + β(g ) 2 )g + r(β 1)f g f +(1+(f ) 2 +(g ) 2 )g = 0.

15 This is equivalent to a nice equation { (rf (1+(f ) 2 +(g ) 2 ) β 1 2 ) = 0 (rg (1+(f ) 2 +(g ) 2 ) β 1 2 ) = 0. i.e. { for any constant C 1,C 2. rf (1+(f ) 2 +(g ) 2 ) β 1 2 = C 1 rg (1+(f ) 2 +(g ) 2 ) β 1 2 = C 2, (2.6)

16 Theorem Let Σ be a 1-symplectic critical surface (i.e., β = 1) in C 2 of the form (2.5), then there exists constants C 1, C 2, C 3 and C 4, such that f = C 1 lnr+c 3, g=c 2 lnr+c 4. (2.7)

17 When β = 0, i.e., Σ is a minimal surface inc 2, we have Theorem Let Σ be a minimal surface (i.e., β = 0) inc 2 of the form (2.5). If g constant, we suppose C 1 = 1 in (2.6). Then f(r)=±cosh 1 r+c 3, and Σ is the catenoid. If both f and g are not constants, we suppose C 1 = C 2 = 1 in (2.6). Then there exist constants C 3 and C 4, such that f =±cosh 1 r + C 3, g=±cosh 1 r + C 4. (2.8) 2 2

18 Now we assume β 1,0 and C 1 0,C 2 0 in the following. The constants C 1,C 2 are determined by the initial data of f and g. Without loss of generality, we assume C 1 = C 2 = 1. Theorem For any β > 0 and ε > 0, the equations rf (1+(f ) 2 +(g ) 2 ) β 1 2 = 1, rg (1+(f ) 2 +(g ) 2 ) β 1 2 = 1, f(ε) = f 0, g(ε) = g 0, have a unique C -solution on [ε,+ ).

19 Moreover, f = g. As r, we have the asymptotic expansion, f = 1 r β 1 r 3 + o(r 3 ). As r 0, we have the asymptotic expansion, f = 2 1 β 2β r 1 β β 1 3β+1 β 2 2β r 1 β + o(r β). 1

20 Proposition f 0 as β in [ε,+ ) for any ε > 0. Corollary Let Σ β be a family of smooth complete β-symplectic critical surface which are rotationally symmetric. Then there exists a subsequence Σ βi, such that Σ βi converges to a plane locally on R 4 \{0}.

21 Figure: Comparison of different values of β

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

23 Theorem If Σ is a closed symplectic surface which is smoothly immersed in M with the Kähler angle α, then α satisfies the following equation, cosα = cosα( h 3 1k h4 2k 2 h 4 1k+ h 3 2k 2 ) +sinα(h 4,1 + H3,2 ) sin2 α cosα (K K 1234 ). (3.1) where K is the curvature operator of M and H α,i = N e i H,v α.

24 Theorem Suppose that M is Kähler surface and Σ is a β-symplectic critical surface in M with Kähler angle α, then cosα satisfies, cosα = 2β sin 2 α cosα(cos 2 α+ β sin 2 α) α 2 2cosα α 2 cos2 α sin 2 α cos 2 α+ β sin 2 α Ric(Je 1,e 2 ). (3.2)

25 Corollary Assume M is Kahler-Einstein surface with scalar curvature K, then cos α satisfies, cosα = 2β sin 2 α cosα(cos 2 α+ β sin 2 α) α 2 2cosα α 2 K cos3 α sin 2 α cos 2 α+ β sin 2 α. (3.3) Corollary Any β-symplectic critical surface in a Kähler-Einstein surface with nonnegative scalar curvature is a holomorphic curve for β 0.

26 By the equations obtained by Micallef-Wolfson, we see that, on a β-symplectic critical surface we have sinα ζ =(sinα)h, where h is a smooth complex function, ζ is a local complex coordinate on Σ, and consequently, we have Proposition A non holomorphic β-symplectic critical surface in a Kähler surface has at most finite complex points.

27 Theorem Suppose that Σ is a non holomorphic β-symplectic critical surface in a Kähler surface M. Then and χ(σ)+ χ(ν)= P, c 1 (M)([Σ])= P, 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.

28 Theorem gives a proof of Webster s formula for β-symplectic surfaces: Corollary Suppose that Σ is a β-symplectic critical surface in a Kähler surface M. Then χ(σ)+ χ(ν)=c 1 (M)([Σ]).

29 Consider F t,ε : Σ ( δ,δ) ( a,a) M with F 0,0 = F, where F : Σ M is a β-symplectic critical surface. Let Denote so that F t,0 t t=0 = X, ν β,t,ε = F 0,ε ε=0 = Y, and 2 F t,ε ε t ε t=0,ε=0= Z. det (β+1)/2 (g t,ε ) ω β ( F t,ε / x 1, F t,ε / x 2 ) L β (φ t,ε )= ν β,t,ε dx 1 dx 2. Σ

30 It is easy to see that t t=0,ε=0 g ij = ei X,e j + e i, ej X, (4.1) and ε t=0,ε=0 g ij = ei Y,e j + e i, ej Y, (4.2) 2 t ε t=0,ε=0 g ij = ei Z+ R(Y,e i )X,e j + e i, ej Z+ R(Y,e j )X + ei X, ej Y + ei Y, ej X. (4.3) Here, R is the curvature tensor on M.

31 We have the general second variation formula: 2 t ε t=0,ε=0 L β (φ t,ε ) e = (β + 1) i X, e i Y R(X,e i,y,e i ) Σ cos β dµ (β + 1) α Σ cos β dµ α +(β + 1) 2 div Σ Xdiv Σ Y Σ cos β dµ α β + 1 e i, ej X e j, ei Y + e j, ei X e i, ej Y 2 Σ cos β dµ α div Σ X(ω( e1 Y,e 2 )+ω(e 1, e2 Y)) β(β + 1) Σ cos β+1 dµ α div Σ Y(ω( e1 X,e 2 )+ω(e 1, e2 X)) β(β + 1) cos β+1 dµ (4.4) α Σ

32 β ω(k(y,e 1 )X,e 2 )ω(e 1, R(Y,e 2 )X) Σ cos β+1 dµ α ω( e1 X, e2 Y)+ω( e1 Y, e2 X) β Σ cos β+1 dµ α +β(β + 1) Σ (ω( e1 X,e 2 )+ω(e 1, e2 X))(ω( e1 Y,e 2 )+ω(e 1, e2 Y)) cos β+2 ( α where {e 1,e 2 } is a local orthonormal basis of Σ, div Σ X= ei X,e i and R is the curvature tensor on M.

33 If we choose X=Yto be normal vector fields, = β + 1 cos β α J β(β + 1) 0(X) cos β+1 α cos αx β 2 (β + 1) 2 cos β+6 α X,(J(J cos α) ) (J(J cos α) ) 2 β 2 (β + 1) cos β+4 α [ω( e 1 X,e 2 )+ω(e 1, e2 X)](J(J cos α) ) β(β + 1) cos β+2 α [ e 1 cos α(j e2 X) e2 cosα(j e1 X) ] ω( e1 X,e 2 )+ ω(e 1, e2 X) β(β + 1) e1 cos β+2 (Je 2 ) α ω( e1 X,e 2 )+ ω(e 1, e2 X) +β(β + 1) e2 cos β+2 (Je 1 ) α := J β X,X dµ, Σ

34 Theorem If we choose X=x 3 e 3 + x 4 e 4 and Y= J ν X=x 4 e 3 x 3 e 4, then the second variation formula is II β (X)+II β (Y) R X 2 sin 2 α = 2(β + 1) Σ cos β α dµ +(β + 1) X 2 (2cos 2 α+ β sin 2 α) Σ cos β+2 dµ α (2cos 2 α+ β sin 2 α)(cos 2 α+ β sin 2 α) (β + 1) cos β+4 X 2 α 2 dµ α Σ where R is the scalar curvature of M.

35 As applications of the stability inequality above, we can obtain some rigidity results for stable β-symplectic critical surfaces. Corollary Let M be a Kähler surface with positive scalar curvature R. If Σ is a stable β-symplectic critical surface in M with β 0, whose normal bundle admits a nontrivial section X with X 2 X 2 cos2 α+ β sin 2 α cos 2 α 2, α then Σ is a holomorphic curve.

36 Corollary Let M be a Kähler surface with positive scalar curvature R. If Σ is a stable β-symplectic critical surface in M with β 0 and 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.

37 we define the set S : = {β [0, ) strictly stable β symplecitc critical surface Σ}. Theorem The set S is open. Conjecture Let M be a Kähler surface. There is a holomorphic curve in the homotopy class of a symplectic stable minimal surface in M. There does exist symplectic stable minimal surfaces in a Kähler-Einstein surface which are not holomorphic (Claudio).

38 for minimal surfaces Theorem (Choi-Schoen) Let M be a 3-dimensional Riemannian manifold and Σ i M a sequence of closed minimal surfaces with Area(Σ i ) C. Then there exists a finite set of points S M and a subsequence Σ i that converges in the C l topology (for any l< ) on compact subsets of M\S to a smooth minimal surface Σ M. The subsequence also converges to Σ in (extrinsic) Hausdorff distance.

39 for minimal surfaces Theorem Let M be a 3-dimensional Riemannian manifold and Σ i M a sequence of closed stable minimal surfaces with Area(Σ i ) C. Then there exists a finite set of points S M and a subsequence Σ i that converges in the C l topology (for any l< ) on compact subsets of M\S to a smooth stable minimal surface Σ M. The subsequence also converges to Σ in (extrinsic) Hausdorff distance. Furthermore, the bubble at each p γ S is a plane.

40 for β -SCS Theorem Let M be a closed Kähler surface and Σ i M a sequence of closed β i -symplectic critical surfaces with β i β 0 (0, ), suppose that Area(Σ i ) C 1, Σ i A i 2 C 2, and min Σi cosα i δ > 0. Then there exists a finite set of points S M and a subsequence Σ i that converges in the C l topology (for any l< ) on compact subsets of M\S to a β 0 -symplectic critical surface Σ M. The subsequence also converges to Σ in (extrinsic) Hausdorff distance. Furthermore, the bubble at each p γ S is a holomorphic curve in C 2.

41 Theorem Suppose that Σ is a limiting surface in the above theorem. The tangent cone of Σ S at p γ is a flat cone consists of union of planes in C 2 which intersects at a single point. Each connected components of Σ\{p γ } can be extended smoothly to p γ.

42 Theorem Let {Σ βi } be a sequence of connected β i -symplectic critical surface in B 4 (r 0 ) M with β i β 0 and cosα i δ > 0, such that Σ βi B 4 (r 0 ). Suppose that there is a uniform constant C 0 > 0, such that for all i. sup Σ βi A βi (x) C 0, (5.1) Then there is a subsequence of Σ βi, still denote by Σ βi, which converges in the C loc topology to a smooth β 0-critical surface Σ β0 in B 4 (r 0 ) with sup Σβ0 A β0 (x) C 0.

43 Theorem Let 0 A<B<. There exist constants ε > 0 and ρ > 0 (depending on M), such that if r 0 ρ, Σ is a β-critical surface with Σ β B 4 (r 0 ), β [A,B], cos α δ > 0, and then we have B r0 Σ max σ 2 σ [0,r 0 ] A 2 dµ ε, sup B r0 σ Σ A 2 4.

44 Proposition Let (M n,ḡ) be a closed Riemannian manifold with sectional curvature bounded by K 0 (i.e., K M K 0 ) and injectivity radius bounded below by i 0 > 0. Suppose that Σ 2 M n is a smooth surface and x 0 M, then for any 0<s 1 < s 2 < i 0 e 4 K 0 s 1 Area(B s1 (x 0 ) Σ) s 2 1 ( 2e 4 K 0 s Area(Bs2 2 (x 0 ) Σ) s B s2 (x 0 ) Σ ) H 2 dµ. (5.2)

45 we define the set S : = {β [0, ) stable β symplecitc critical surface Σ}. Theorem The set S is closed.

46 Lower bound for cosα on Σ β Theorem We have on Σ β. cos α C(L β (Σ β )) Applying Sobolev inequality ( ) n 1 h n 1 2n n dµ Σ using Moser iteration. c(n) [ h 2 + h 2 + h 2 H ]dµ, (5.3) Σ

47 Upper bound for A on Σ β Theorem We have A 2 dµ β C(β). Σ β where C(β) depends on β. Applying Gauss equation, the equation for β-symplectic critical surfaces, lower bound for cosα, and the second variation formula.

48 Thanks for your attention