Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/people/taras/talks/2011osaka-talk.pdf Äàòà èçìåíåíèÿ: Fri Dec 2 21:35:44 2011 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:03:47 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 13

Intersections of quadrics and H-minimal Lagrangian submanifolds
Taras Panov Moscow State University based on joint work with with Andrey Mironov
The 10th Pacific Rim Geometry Conference Osaka­Fukuoka, 1­9 Decemb er 2011.


(M , ) a symplectic Riemannian 2n-manifold. An immersion i : N M of an n-manifold N is Lagrangian if i( ) = 0. If i is an emb edding, then i(N ) is a Lagrangian submanifold of M . A vector field on M is Hamiltonian if the 1-form ( · , ) 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 fields with compact supp ort are zero, i.e. d vol(it(N )) = 0, t=0 dt where it(N ) is a Hamiltonian deformation of i(N ) = i0(N ), and vol(it(N )) is the volume of the deformed part of it(N ).
2


Explicit examples of H-minimal Lagrangian submanifolds in Cm and CP m were constructed in the work of Yong-Geun Oh, Castro­Urbano, H´ elein­Romon, Amarzaya­Ohnita, among others. In 2003 A. Mironov suggested a universal construction providing an H-minimal Lagrangian immersion in Cm from an intersection of sp ecial real quadrics. The same intersections of real quadrics are known to toric geometers and top ologists as (real) moment-angle manifolds. They app ear, for instance, as level sets of the moment map in the symplectic reduction construction of Hamiltonian toric manifolds. Here we combine Mironov's construction with the metho ds of toric top ology to pro duce new examples of H-minimal Lagrangian emb eddings with interesting and complicated top ology.
3


A convex p olyhedron in Rn obtained by intersecting m halfspaces: P=
{

x Rn : a i, x + bi
(

0

for i = 1, . . . , m .
)

}

Define an affine map iP : Rn Rm, iP (x ) = a 1, x + b1, . . . , a m, x + bm .

If P has a vertex, then iP is monomorphic, and iP (P ) is the intersection of an n-plane with Rm = {y = (y1, . . . , ym) : yi 0}. Define the space ZP from the diagram ZP - Cm
µ
i -P Rm iZ

(z1, . . . , zm)


P

(|z1|2, . . . , |zm|2)

ZP has a Tm-action, ZP /Tm = P , and iZ is a Tm-equivariant inclusion.
4


Prop osition 1. If P is a simple p olytop e (more generally, if the presentation of P by inequalities is generic), then ZP is a smo oth manifold of dimension m + n. Pro of. Write iP (Rn) by m - n linear equations in (y1, . . . , ym) Rm. Replace yk by |zk |2 to obtain a presentation of ZP by m - n quadrics. ZP : p olytopal moment-angle manifold corresp onding to P . Similarly, by considering the projection µ : Rm Rm instead of µ : Cm Rm we obtain the real moment-angle manifold RP Rm. Example 1. P = {(x1, x2) R2 : x1 1, 2 > 0 (a 2-simplex). Then 0, x 2 0, -1x1 - 2x2 + 1 0},

ZP = {(z1, z2, z3) C3 : 1|z1|2 + 2|z2|2 + |z3|2 = 1} (a 5-sphere), RP = {(u1, u2, u3) R3 : 1|u1|2 + 2|u2|2 + |u3|2 = 1} (a 2-sphere).
5


m m: ZP = z = (z1, . . . , zm) C j k |zk |2 = cj , for 1 j m - k=1 { m m: RP = u = (u1, . . . , um) R j k u2 = cj , for 1 j m - n k k=1 Set k = (1k , . . . , m-n,k ) Rm-n for 1 k m.

{

}

n
}

.

Assume that the p olytop e P is rational. Then have two lattices: = Za 1, . . . , a m Rn Consider the (m - n)-torus TP = i.e. TP = Rm-n/L, and set
{(

and

L = Z1, . . . , m Rm-n.
) } m, T

e2 i1,, . . . , e2im,

1 DP = L /L = (Z2)m-n. 2 Prop osition 2. The (m - n)-torus TP acts on ZP almost freely.
6


Consider the map f : RP â TP - Cm, (u , ) u · = (u1e2i1,, . . . , ume2im,). Note f (RP â TP ) ZP is the set of TP -orbits through RP Cm. Have an m-dimensional manifold NP = RP âDP TP . Lemma 1. f : RP â TP Cm induces an immersion j : NP Cm .

Theorem 1 (Mironov). The immersion i : N Lagrangian. When it is an emb edding?

Cm is H-minimal

7


A simple rational p olytop e P is set of vectors a i1 , . . . , a in normal basis of the lattice = Za 1, . . . ,

Delzant if for any vertex v P the to the facets meeting at v forms a a m : for any v = Fi1 · · · Fin .

Za 1, . . . , a m = Za i1 , . . . , a in

Theorem 2. The following conditions are equivalent: 1) j : NP Cm is an emb edding of an H-minimal Lagrangian submanifold; 2) the (m - n)-torus TP acts on ZP freely. 3) P is a Delzant p olytop e. Explicit constructions of families of Delzant p olytop es are known in toric geometry and top ology: - simplices and cub es in all dimensions; - pro ducts and face cuts; - asso ciahedra (Stasheff ptop es), p ermutahedra, and generalisations.
8


Example 2 (one quadric). Let P = m-1 (a simplex), i.e. m - n = 1 and Rm-1 is given by a single quadric with i > 0, i.e. Rm
-1

1u2 + · · · + mu2 = c m 1 = S m-1. Then

(1)

m -1 â S 1 S m-1 â S 1 = S N= Z2 Km

if preserves the orient. of S m-1, if reverses the orient. of S m-1,

where is the involution and Km is an m-dimensional Klein b ottle. Prop osition 3. We obtain an H-minimal Lagrangian emb edding of Nm-1 = S n-1 âZ2 S 1 in Cm if and only if 1 = · · · = m in (1). The top ological typ e of Nm-1 = N (m) dep ends only on the parity of m: N (m) = S m-1 â S 1 if m is even, N (m) = Km if m is o dd. The Klein b ottle Km with even m do es not admit Lagrangian emb eddings in Cm [Nemirovsky, Shevchishin].
9


Example 3 (two quadrics). Theorem 3. Let m - n = 2, i.e. P p-1 â q-1. (a) RP is diffeomorphic to R(p, q ) = S p-1 â S q-1 given by u2 + . . . + u2 + u2+1 + · · · + u2 = 1, p 1 k k +u2+1 + · · · + u2 = 2, u2 + . . . + u2 m 1 p k where p + q = m, 0 < p < m and 0 k p. (b) If NP Cm is an emb edding, then NP is diffeomorphic to Nk (p, q ) = R(p, q ) âZ2âZ2 (S 1 â S 1), where the two involutions act on R(p, q ) by 1 : (u1, . . . , um) (- 2 : (u1, . . . , um) (- u1, . . . , -uk , -uk+1, . . . , -up, up+1, . . . , um), (2) u1, . . . , -uk , uk+1, . . . , up, -up+1, . . . , -um).

There is a fibration Nk (p, q ) S q -1 âZ2 S 1 = N (q ) with fibre N (p) (the manifold from the previous example), which is trivial for k = 0.
10


Example 4 (three quadrics). In the case m - n = 3 the top ology of compact manifolds RP and ZP was fully describ ed by [Lop ez de Medrano]. Each manifold is diffeomorphic to a pro duct of three spheres, or to a connected sum of pro ducts of spheres, with two spheres in each pro duct. The simplest P with m - n = 3 is a (Delzant) p entagon, e.g. P=
{

(x1, x2) R2 : x1

0, x2

0, -x1+2

0, -x2+2

0, -x1-x2+3

0.

}

In this case RP is an oriented surface of genus 5, and ZP is diffeomorphic to a connected sum of 5 copies of S 3 â S 4. Get an H-minimal Lagrangian submanifold NP C5 which is the total space of a bundle over T 3 with fibre a surface of genus 5.
11


Prop osition 4. Let P b e an m-gon. Then RP is an orientable surface Sg of genus g = 1 + 2m-3(m - 4). Get an H-minimal Lagrangian submanifold NP Cm which is the total space of a bundle over T m-2 with fibre Sg . It is an aspherical manifold (for m 4) whose fundamental group enters into the short exact sequence 1 - 1(Sg ) - 1(N ) - Zm-2 - 1. For n > 2 and m - n > 3 the top ology of RP and ZP is even more complicated.

12


Generalisation to toric varieties: Consider 2 sets of quadrics:
m m: Z = z C k=1 { m m: Z = z C k=1 {

k |zk |2 = c , k |zk |2 = d ,
}

}

k , c Rm-n; k , d Rm-;

s. t. n +

m, and Z, Z and Z Z satisfy the conditions ab ove.

m Define R, T = Tm-n, D = Zm-n, R, T = Tm-, D = Z2 - as 2 b efore. The idea is to use the first set of quadrics to pro duce a toric variety M via symplectic reduction, and then use the second set of quadrics to define an H-minimal Lagrangian submanifold in M .
13


M := Cm/ T = Z/T /

toric variety, dim M = 2n.

It contains (R R)/D =: R as a subset of real p oints, dim R = n + - m. Define N := R âD


T M ,

dim N = n.

Theorem 4. N is an H-minimal Lagrangian submanifold in M . Idea of pro of. Consider M := M / = (Z Z)/(T â T). /T Then N := N/T = (R R)/(D â D) (Z Z)/(T â T) = M is a minimal (totally geo desic) submanifold. Therefore, N M is H-minimal by a result of Y. Dong.
14


Example 5. 1. If m - = 0, i.e. Z = , then M = Cm and we get the original construction of H-minimal Lagrangian submanifolds N in Cm. 2. If m - n = 0, i.e. Z = , then N is set of real p oints of M . It is minimal (totally geo desic). 3. m - = 1, i.e. Z = S 2m-1, then we get H-minimal Lagrangian submanifolds in M = CP m-1.

15


Reference: Andrey Mironov and Taras Panov. Intersections of quadrics, momentangle manifolds, and Hamiltonian-minimal Lagrangian emb eddings. Preprint (2011); arXiv:1103.4970.

16