New constructions of Hamiltonian-minimal Lagrangian submanifolds

Transcription

1 New constructions of Hamiltonian-minimal Lagrangian submanifolds based on joint works with Andrey E. Mironov Taras Panov Moscow State University Integrable Systems CSF Ascona, June 2016 Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

2 Basics Let M be a K ahler manifold with symplectic form ω, dim R M = 2n. An immersion i : N M of an n-manifold N is Lagrangian if i (ω) = 0. If i is an embedding, then i(n) is a Lagrangian submanifold of M. A vector eld ξ on M is Hamiltonian if the 1-form ω(, ξ) is exact. A Lagrangian immersion i : N M is Hamiltonian minimal (H-minimal) if the variations of the volume of i(n) along all Hamiltonian vector elds with compact support are zero, i.e. d dt vol(i t(n)) t=0 = 0, where i t (N) is a Hamiltonian deformation of i(n) = i 0 (N). Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

3 Overview Explicit examples of H-minimal Lagrangian submanifolds in C m and CP m were constructed in the work of Yong-Geun Oh, CastroUrbano, H eleinromon, AmarzayaOhnita, among others. In 2003 Mironov suggested a universal construction providing an H-minimal Lagrangian immersion in C m from an intersection of special real quadrics. The same intersections of real quadrics are known to toric geometers and topologists as (real) moment-angle manifolds. They appear, for instance, as the level sets of the moment map in the construction of Hamiltonian toric manifolds via symplectic reduction. Here we combine Mironov's construction with the methods of toric topology to produce new examples of H-minimal Lagrangian embeddings with interesting and complicated topology. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

4 Polytopes and moment-angle manifolds A convex polytope in R n is obtained by intersecting m halfspaces: P = { x R n : ai, x + b i 0 for i = 1,..., m }. Suppose each F i = P {x : ai, x + b i = 0} is a facet (m facets in total). Dene an ane map i P : R n R m, i P (x) = ( a 1, x + b 1,..., am, x + b m ). Then i P is monomorphic, and i P (P) R m is the intersection of an n-plane with R m = {y = (y 1,..., y m ): y i 0}. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

5 Dene the space Z P from the diagram Z P P i Z C m (z 1,..., z m ) µ i P R m ( z 1 2,..., z m 2 ) Z P has a T m -action, Z P /T m = P, and i Z is a T m -equivariant inclusion. A polytope P is simple if exactly n = dim P facets meet at each vertex. Proposition If P is simple, then Z P is a smooth manifold of dimension m + n. Proof. Write i P (R n ) by m n linear equations in (y 1,..., y m ) R m. Replace y k by z k 2 to obtain a presentation of Z P by quadrics. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

6 Z P is the moment-angle manifold corresponding to P. Similarly, by considering R P R m (u 1,..., u m ) µ P i P R m (u 2 1,..., u2 m) we obtain the real moment-angle manifold R P. Example P = {(x 1, x 2 ) R 2 : x 1 0, x 2 0, γ 1 x 1 γ 2 x }, γ 1, γ 2 > 0 (a 2-simplex). Then Z P = {(z 1, z 2, z 3 ) C 3 : γ 1 z γ 2 z z 3 2 = 1} (a 5-sphere), R P = {(u 1, u 2, u 3 ) R 3 : γ 1 u γ 2 u u 3 2 = 1} (a 2-sphere). Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

7 Torus actions Have intersections of quadrics Z P = {z = (z 1,..., z m ) C m : γ 1 z γ m z m 2 = c}, R P = {u = (u 1,..., u m ) R m : γ 1 u γ m u 2 m = c} where γ 1,..., γ m and c are vectors in R m n. Assume that the polytope P is rational. Then have two lattices: Λ = Z a 1,..., am R n and L = Z γ 1,..., γ m R m n. Consider the (m n)-torus T P = {( e 2πi γ 1,ϕ,..., e 2πi γm,ϕ ) T m}, i.e. T P = R m n /L, and set D P = 1 2 L /L = (Z 2 ) m n. Proposition The (m n)-torus T P acts on Z P almost freely. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

8 Main construction Consider the map f : R P T P C m, (u, ϕ) u ϕ = (u 1 e 2πi γ 1,ϕ,..., u m e 2πi γm,ϕ ). Note f (R P T P ) Z P is the set of T P -orbits through R P C m. Have an m-dimensional manifold N P = R P DP T P. Lemma f : R P T P C m induces an immersion j : N P C m. Theorem (Mironov) The immersion j : N P C m is H-minimal Lagrangian. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

9 Question When j : N P C m is an embedding? A simple rational polytope P is Delzant if for any vertex v P the set of vectors ai 1,..., ai n normal to the facets meeting at v forms a basis of the lattice Λ = Z a 1,..., am : Z a 1,..., am = Z ai 1,..., ai n for any v = F i1 F in. Theorem The following conditions are equivalent: 1 j : N P C m is an embedding of an H-minimal Lagrangian submanifold; 2 the (m n)-torus T P acts on Z P freely. 3 P is a Delzant polytope. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

10 Get an H-minimal Lagrangian submanifold N P in C m for any Delzant n-polytope P with m facets! Explicit constructions of families of Delzant polytopes are known in toric geometry and topology: simplices and cubes in all dimensions; products and face truncations; associahedra (Stashe polytopes), permutahedra, and generalisations. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

11 Examples Example (one quadric) Let P = m 1 (a simplex), i.e. m n = 1. R m 1 is given by a single quadric γ 1 u γ m u 2 m = c with γ i > 0, i.e. R m 1 = S m 1. Then N = S m 1 Z2 S 1 = { S m 1 S 1 if τ preserves the orient. of S m 1, K m if τ reverses the orient. of S m 1, where τ is the involution and K m is an m-dimensional Klein bottle. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

12 Proposition (one quadric) We obtain an H-minimal Lagrangian embedding of N m 1 = S n 1 Z2 S 1 in C m whenever γ 1 = = γ m in γ 1 u γ mu 2 m = c. The topology of N m 1 = N(m) depends on the parity of m: N(m) = S m 1 S 1 N(m) = K m if m is even, if m is odd. The Klein bottle K m with even m does not admit Lagrangian embeddings in C m [Nemirovsky, Shevchishin]. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

13 Theorem (two quadrics) Let m n = 2, i.e. P p 1 q 1. R P is dieomorphic to R(p, q) = S p 1 S q 1 given by u u 2 k + u2 k u2 p = 1, u u 2 k +u 2 p u 2 m = 2, where p + q = m, 0 < p < m and 0 k p. If N P C m is an embedding, then N P is dieomorphic to N k (p, q) = R(p, q) Z2 Z 2 (S 1 S 1 ), where the two involutions act on R(p, q) by ψ 1 : (u 1,..., u m ) ( u 1,..., u k, u k+1,..., u p, u p+1,..., u m ), ψ 2 : (u 1,..., u m ) ( u 1,..., u k, u k+1,..., u p, u p+1,..., u m ). There is a bration N k (p, q) S q 1 Z2 S 1 = N(q) with bre N(p). Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

14 Example (three quadrics) In the case m n = 3 the topology of compact manifolds R P and Z P was fully described by [Lopez de Medrano]. Each manifold is dieomorphic to a product of three spheres, or to a connected sum of products of spheres, with two spheres in each product. The simplest case is n = 2 and m = 5: a Delzant pentagon. In this case R P is an oriented surface of genus 5, and Z P is dieomorphic to a connected sum of 5 copies of S 3 S 4. Get an H-minimal Lagrangian submanifold N P C 5 which is the total space of a bundle over T 3 with bre a surface of genus 5. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

15 Proposition Let P be an m-gon. Then R P is an orientable surface S g of genus g = m 3 (m 4). Get an H-minimal Lagrangian submanifold N P C m which is the total space of a bundle over T m 2 with bre S g. It is an aspherical manifold (for m 4) whose fundamental group enters into the short exact sequence 1 π 1 (S g ) π 1 (N) Z m 2 1. For n > 2 and m n > 3 the topology of R P and Z P is even more complicated. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

16 Generalisation to toric manifolds Consider 2 sets of quadrics: Z Γ = {z C m : m γ k z k 2 = c}, γ k, c R m n ; k=1 Z = {z C m : m } δ k z k 2 = d, δ k, d R m l ; k=1 s. t. the polytopes corresponding to Z Γ, Z and Z Γ Z are Delzant. Dene R Γ, T Γ = T m n, D Γ = Z m n 2, R, T = T m l, D = Z m l 2 as before. The idea is to use the rst set of quadrics to produce a toric manifold M via symplectic reduction, and then use the second set of quadrics to dene an H-minimal Lagrangian submanifold in M. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

17 M := C m / T Γ = Z Γ /T Γ a toric manifold, dim M = 2n. Real points R Γ /D Γ Z Γ /T Γ = M. R := (R Γ R )/D Γ subset of real points of M, dim R = n + l m. Dene N := R D T M, dim N = n. Theorem N is an H-minimal Lagrangian submanifold in M. Idea of proof. Consider M := M /T = (Z Γ Z )/(T Γ T ). Then Ñ := N/T = (R Γ R )/(D Γ D ) (Z Γ Z )/(T Γ T ) = M is a minimal (totally geodesic) submanifold. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

18 Example 1 1. If m n = 0, then the set of quadrics dening Z Γ is void, so Z Γ = M = C m and we get the original construction of H-minimal Lagrangian submanifolds N in C m. 2 If m l = 0, then the set of quadrics dening Z is void, so N = R = R Γ /D Γ is set of real points of M = Z Γ /T Γ. The submanifold N is minimal (totally geodesic) in M. 3 If m n = 1 and Z Γ is compact, then Z Γ = S 2m 1, and we obtain H-minimal Lagrangian submanifolds in M = Z Γ /T Γ = CP m 1. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

19 Integrable systems and dynamics General fact: the above constructed H-minimal Lagrangian submanifolds N C m are `special' degenerations of Liouville tori for certain integrable systems. Have R N Z, where Z is the level set for m n Hermitian quadrics and R is its real part H 1 = = H m n = 0, H R 1 = = H R m n = 0, Then N can be written by H 1 = = H m n = 0, F 1 =... = F n = 0, where {H i, H j } = {H i, F k } = {F k, F l } = 0 and F 1,..., F n are polynomial integrals. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

20 Example Consider the quadric x1 2 + x 2 2 = 1 giving rise to an embedded H-minimal Lagrangian 2-torus N = T 2 C 2. It is given by two equations H = x y x y 2 2 = 1, F = x 1 y 2 x 2 y 1 = 0, with {H, F } = 0. This torus is included to the family of Liouville tori given by H = x y x y 2 2 = C, F = x 1 y 2 x 2 y 1 = D, which degenerates to a circle at D = C/2. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

21 Example Now consider the quadric x x 2 2 H-minimal Klein bottle K C 2. It is given by two equations = 1 giving rise to an immersed H = x y x y 2 2 = 1, F = x 2 1 y 2 y 2 1 y 2 2x 1 x 2 y 1 = 0, with {H, F } = 0. Have a family of Liouville tori H = x y x y 2 2 = C, F = x 2 1 y 2 y 2 1 y 2 2x 1 x 2 y 1 = D, which degenerates to a Klein bottle at D = 0. Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22

22 References Victor Buchstaber and Taras Panov. Toric Topology. Math. Surv. and Monogr., 204, Amer. Math. Soc., Providence, RI, Andrey E. Mironov. New examples of Hamilton-minimal and minimal Lagrangian submanifolds in C n and CP n. Sbornik Math. 195 (2004), no. 12, Andrey E. Mironov and Taras Panov. Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings. Funct. Anal. and Appl. 47 (2013), no. 1, Andrey E. Mironov and Taras Panov. Hamiltonian minimal Lagrangian submanifolds in toric varieties. Russian Math. Surveys 68 (2013), no. 2, Taras Panov. Geometric structures on moment-angle manifolds. Russian Math. Surveys 68 (2013), no. 3, Taras Panov (MSU) H-minimal Lagrangian submanifolds 22 June / 22