pacific journal of mathematics Vol. 181, No., 1997 MOMENT MAPS ON SYMPLECTIC CONES Suzana Falc~ao B. de Moraes and Carlos Tomei We analyze some convexity properties of the image maps on symplectic cones, similar to the ones obtained by Guillemin- Sternberg and Atiyah for compact symplectic manifolds in the early 80's. We prove the image of the moment map associated to the symplectic action of an n-torus on a symplectic cone is a polytopic convex cone in R n : Then, we generalize these results to symplectic manifolds obtained by special perturbations of the symplectic structure of a cone: we obtain sucient (and essentially necessary) conditions for the image of the moment map associated to the perturbed form to remain unchanged. Hamiltonian actions of tori and the images of their moment maps have been intensely studied in the eighties. According to the fundamental result, obtained independently by Atiyah [] and Guillemin and Sternberg [4], the moment map of a Hamiltonian action of a torus on a compact symplectic manifold has for its image a convex polytope, spanned by the images of the xed points of the action. More recently, Prato [10] proved a convexity result concerning the image of moment maps of torus actions on non-compact symplectic manifolds. Theorem [10]. Let the torus T r act in a Hamiltonian fashion on the symplectic manifold (X;!) and denote by : X! (Lie T r ) = R r the corresponding moment map. Suppose that there exists a circle S 1 = fe t 0g T r for some 0 Lie T r such that 0 = h; 0 i is a proper function having a minimum as its unique critical value. Then (X) is the convex hull of a nite number of rays in (Lie T r ). In this paper, we prove a dierent kind of result, closer in spirit to perturbation theory: We start with a special non-compact symplectic manifold, described below, for which a similar convexity theorem holds, and consider the changes of the underlying symplectic structure which keep the image of the resulting moment maps unchanged. Let (X;!) be a symplectic cone with homothety group f t ; t R + g so that t! = t!, for positive t, and compact base X=R +. Suppose that the torus T r acts symplectically on (X;!) and that this action commutes with 357
358 SUZANA FALC ~ AO B. DE MORAES AND CARLOS TOMEI the R + -action. Since (X;!) is a symplectic cone, the torus action is Hamiltonian (Proposition 1.1), for some associated moment map : (X;!)! R r. Assume nally that there exists 0 R r such that 0 = h; 0 i > 0 (or, equivalently, that the image of lies in an open half-space of R r ). Then X is dieomorphic to M R + (again, from Proposition 1.1), where M = 0?1 (1) is a compact submanifold of X. We prove the following convexity theorem in Section 1. Theorem 1 (Convexity). polytopic convex cone. The image of the moment map, (X), is a We then consider in Section symplectic forms on X which do not necessarily satisfy the homogeneity hypothesis, t = t, but for which the same T r -action is still Hamiltonian, with moment map. We formulate hypothesis that are essentially necessary and sucient for the ranges of the moment maps to coincide, i.e., so that (X) = (X). Theorem (Rigidity). If lim t!0 (t; m) = 0 uniformly in m M and lim t!1 0(t; m) = 1 (where 0 = h 0 ; i) uniformly in m M, then the images of and coincide. Notice that Theorem only requires that the perturbed and unperturbed moment maps have similar behavior at the extremes of the cone. The essential necessity of these hypothesis will be shown by examples in the Appendix. The hypothesis at zero is equivalent to the integrability at zero of the oneform dened by the contraction of the perturbation?! with the vector elds induced by the torus action. The perturbation of the symplectic form at innity, on the other hand, is much more exible. An immediate consequence of the rigidity theorem is the result below, previously obtained by Guillemin (private communication) using G-invariant cohomology techniques. Corollary. Let the torus T r act symplectically on a symplectic cone (X;!), so that the torus action and the R + -action commute. Let : (X;!)! R r be the associated moment map. Assume that there exists a 0 R r such that h; 0 i > 0. Let be a symplectic form on X, such that the same T r - action is Hamiltonian with respect to it, with moment map : (X; )! R r. If?! is compactly supported, then the images of and coincide up to a translation. Theorem 1 can be obtained using techniques similar to those in the proof of Prato's theorem, but we prefer to provide the self-contained argument in Section 1, which follows more closely the original proof of Guillemin- Sternberg in [4]. More precisely, by homogeneity, the image of the moment
MOMENT MAPS ON SYMPLECTIC CONES 359 map on X is a cone over the image of its restriction to the compact manifold M. It is sucient then to show the convexity of the image of the restriction, and this is accomplished by making use of the by now standard ingredients: The local form of the moment map and Morse theory for clean functions. The fact that M is odd dimensional does not introduce any diculty in this approach. The proof of the rigidity theorem is by induction on the dimension of the cells associated to the usual stratication of the symplectic manifold by isotropy groups of the torus action. From the inductive hypothesis, the images of both unperturbed () and perturbed ( ) moment maps, restricted to the closure of each cell up to a given dimension, agree (as sets, not pointwise). The key point in the induction step is to combine the fact that the images agree on the boundaries of the closures of the cells of higher dimension, with the fact that the images of the interiors of these cells `leave no holes' in the convex span of their boundaries. The second fact follows from detailed study of the behavior of the moment map at interior faces (i.e., cells whose images are in the interior of the image of a larger cell). One of the nice features of the argument is that, without additional hypothesis, the perturbed moment map still omits the origin in its image. In the Appendix we describe some examples and counter-examples. First, for the natural action of T r on X = R n? f0g with the usual symplectic structure, we present all the symplectic structures for which the torus action remains Hamiltonian from the answer, we will see that the hypothesis of the rigidity theorem are also necessary in this case. In the second example, we show how the behavior of the moment map at the origin can alter the image at innity. The corollary above can be considered an additional example of both theorems applied to the more general symplectic cone. Acknowledgements. The authors would like to thank Victor Guillemin for suggesting the study of moment maps on symplectic cones and Regina Souza, Nicolau Saldanha and Eugene Lerman for fruitful conversations.this work was partially supported by CNPq, SCT and FAPERJ, Brazil. 1. A Convexity theorem for symplectic cones. Let (X;!) be a symplectic cone with compact base. Explicitly, X is a dierentiable manifold with symplectic form!, admitting a homothety group f t ; t R + g of dieomorphisms such that t! = t! and such that the base X=R + is a compact manifold. Suppose that the torus T r acts symplectically on (X;!) and that its action commutes with the R + -action. Given in the Lie algebra of T r = Lie T r ' R r, let ] be the symplectic vector eld on X associated to it. The conical structure of (X;!) gives
360 SUZANA FALC ~ AO B. DE MORAES AND CARLOS TOMEI rise to a convenient description of the moment map of the torus action, as we will see in Proposition 1.1: Dene : X! (Lie T r ) ' R r by = h; i = ( ] ); for Lie T r, where = ()!, the vector denotes the innitesimal generator of the R + -action and () is the contraction of the eld with the form. From now on, we assume that for some 0 Lie T r, 0 = h; 0 i > 0: In particular, 0 = (X): Under these assumptions, we have the main result of this section. Theorem 1 (Convexity). polytopic convex cone. The image of the moment map, (X), is a We make use of a number of simple properties of the moment map. Proposition 1.1. (1) The T r -action on (X;!) is Hamiltonian and : X! R r is indeed its moment map. () t = t, i.e., is homogeneous. (3) X is dieomorphic to the cartesian product M R +, where M =?1 0 (1) is a compact submanifold of X. In these coordinates, t (m; s) = (m; ts). In particular, the R + -action on X is free. (4) Let : G X! X be an action of a connected compact group G on X that commutes with the R + -action. For each x X, let G x be the stabilizer group of x and [x] denote the corresponding element in X=R +. Then G x = G [x] for x X. (5) Only a nite number of subgroups of T r occur as stabilizer groups of points in X. Proof. We rst show that d =?( ] )!. By the Weyl identity, D! = d(()!) + ()d!. But D! =! and! is closed, so! = d(()!) = d. Hence, if we dene = ( ] ), d = d(( ] )) = D ]? ( ] )d = 0? ( ] )! =?( ] )!. The homogeneity of the moment map is immediate. Now, consider the function 0 : X! R + and let M =?1 0 (1) X. Since 0 = ( 0), ] ( 0) ] x 6= 0, for all x X. Now, d 0 =?( 0)!, ] so 0 has no critical values and M is a manifold. Dene F : X! M R + to be F (x) = ( 1=0(x) (x); 0 (x)) with inverse F?1 (m; t) = t (m). Both are smooth functions and so X and M R + are dieomorphic and R + acts freely on X. Clearly, X=R + ' M and so M is a compact manifold. The proof of (4) splits in two parts. i) G x G [x]. Let g G x, i.e., g (x) = x. Then g ([x]) = [ g (x)] = [x] and g G [x].
MOMENT MAPS ON SYMPLECTIC CONES 361 ii) G [x] G x. Let g G [x], i.e., g ([x]) = [x]. In particular, g (x) = t (x), for some t R +. We want to show that t = 1. If we let g vary in G [x], we get a map A : G [x]! R + where A(g) = t dened by g (x) = t (x). The map A is a group homomorphism since if A(g 1 ) = t 1 and A(g ) = t then g1g (x) = g1 ( g (x)) = g1 ( t (x)) = t ( g1 (x)) = t ( t1 (x)) = t1 ( t (x)) = t1t (x). Since G [x] is compact, the image of the map A is a compact subgroup of R + and is therefore the identity. Hence A(g) = 1, for all g G [x] and t = 1. The proof of (5) is a direct consequence of (4) and the standard analogous result for compact group actions on compact manifolds (see, for example, [4] and [8]). From the proof of the last item, we conclude that the R + -orbits are transversal to the torus orbits. Notice that the description given for is not invariant under translations, as is frequent for moment maps. Let T 1 ; : : : ; T K be the subgroups of T r occurring as stabilizer groups of points of the symplectic cone X. Let X i = fx X j the isotropy group of x is T i g. By relabeling, we can assume the X i 's to be connected. Then X can be written as X = t N i=1x i (disjoint union): We now prove the analogous of Theorem 3.7 and 3.8 in [4]. Proposition 1.. Each X i in the decomposition X = t N i=1x i is a T r - invariant conic symplectic submanifold of X of strictly positive dimension and maps each X i submersively onto an open subset of (Lie T i )?, the annihilator space of Lie T i in (Lie T i ). Moreover, (X i ) is a union of a nite number of polytopic convex cones. Proof. Since T r is abelian, T r X i X i. Since the actions of R + and T r commute and R + acts freely on X, R + also acts freely on X i and so X i is a conic manifold of non-zero dimension. By Theorem 3.5 in [4], X i is symplectic (and so even dimensional). The map i = j Xi is the moment map associated to the action of T r on X i. Also as in [4], (d i ) x maps T x X i onto (Lie T i )? for all x X i and hence maps X i submersively onto an open subset of p i +(Lie T i )? where p i = i (x). From the homogeneity of, (X i ) is an open cone of the vector space (Lie T i )?. Since 0 = (X), is homogeneous and X i is conic, we have that 0 < dim((x i )) = dim((lie T i )? ) induction starts with k = 1. If the codimension of Lie T i is 1, (X i ) is an open and conic subset in the one-dimensional space (Lie T i )?. Since 0 = (X); 0 = (X i ) and (X i )
36 SUZANA FALC ~ AO B. DE MORAES AND CARLOS TOMEI is an open half-line starting at 0. Assume that the result is true when codim (Lie T i ) < k. If codim (Lie T i ) = k, (X i ) is a k-dimensional open subset of the vector space (Lie T i )? and its boundary components, if they exist, are the connected components of [ j (X j ), T j T i. By induction, each (X j ) is a nite union of polytopic convex cones. Therefore the boundary of (X i ) is a nite union of polytopic convex cones and so the same holds for (X i ) itself. We continue using the same techniques as in [4], but instead of inspecting the properties of the maps : X! R, Lie T r, we consider the restrictions j M : M! R; Lie T r, where M =?1 0 (1), the compact submanifold of X dened in Proposition 1.1. Lemma 1.3. The function j M : M! R has a unique local maximum. Before proving this lemma, we use it to prove Theorem 1. Proof of Theorem 1. We want to show that (X) is a polytopic convex cone. By Proposition 1., it is enough to show that (X) is convex. Since X = t (M) and the moment map satises (X) = ( t (M)) = t (M), t R +, we only need to show that (M) is convex. Let p be a point in the boundary of (M) (X) and m M a preimage of p. From the local form of the moment map (again, as in [4]), there is an open neighborhood U of m in X taken to the intersection U 0 of a convex polytopic cone with vertex p with an open neighborhood of p in (Lie T r ). Consider now the ane hyperplane H 1 = fv R r j hv; 0 i = 1g: Clearly, the restricted neighborhoods V = U \ M M and V = V 0 \ H 1 satisfy (V ) = V 0, where V 0 is also the intersection of a convex polytopic cone with vertex p with a neighborhood of p in H 1. Let S i be a boundary component of V 0 containing p. We can choose Lie T r such that h; fi = 0, forf S i and h; fi < 0 in the interior of V 0. Then (y) = h; (y)i 0; for all y V. So, 0 is a local maximum of j M and by Lemma 1.3, 0 is a global maximum of j M, i.e. (M) 0. Mimicking again the argument in [4], (M) behaves as a convex set relative to its boundary points and must then be convex. Lemma 1.3 follows from Morse theory as expanded for clean functions by Bott ([3], [4]). Let K be a compact connected manifold of dimension n. Recall that, for a clean function f : K! R, the index (i.e., the number of strictly negative eigenvalues of the Hessian of f at a point) is constant along a connected component C i of the critical set of f. Also, a clean function f : K! R with indices on critical components dierent from 1 or n? 1 has a unique local maximum. Lemma 1.3 is an immediate consequence of the theorem below.
MOMENT MAPS ON SYMPLECTIC CONES 363 Proposition 1.4. The function j M : M! R is clean. The indices of the critical manifolds of j M and the number of positive eigenvalues of the Hessian at a critical point are even. The proof of this result requires some preparation. Lemma 1.5. (X) [ f0g is closed in R r. Proof. Suppose that there is a sequence f(x n )g converging to v R r. We need to show that v (X) [ f0g. It is enough to consider v6=0. Since (x n )! v, there exists a sequence f(m n ; t n )g M R + such that (m n ; t n )! v. But (m n ; t n ) = t n (m n ; 1), so j t n j jj (m n ; 1) jj! jjvjj. Since 0 = (X) and (m n ; 1) is in a compact set, jj(m n ; 1)jj is bounded away from zero and so t n is in a in a compact subset of R +. Then the sequence fx n g = f(m n ; t n )g has a convergent subsequence in X and 06=v = lim n!1 (x n ) (X). Because of Lemma 1.5, fx j hx; 0 i > 0gn(X) has a non-empty interior, so we can assume (after a small perturbation if necessary) that 0 is in the set 1 m Zr for some integer m (here, Z r is the kernel of exp : Lie T r! T r ). Thus 0 generates an action of the circle group S 1 on X with moment map 0 = h; 0 i : X! R +. Since 0 < h; 0 i = 0 = ( ] 0), the S 1 - action is locally free. The proof of Proposition 1.4 makes use of a normal form for the moment map of a torus action in a neighborhood of a point. This problem has been solved by Guillemin and Sternberg in [5] and by Marle in [8]. We now outline the construction in Lerman [7], from which the results in [5] and [8] obtain the requested information. Let (X;!) be a n-dimensional symplectic manifold on which the torus T r acts in Hamiltonian fashion and let X : X! R r be the associated moment mapping. Let Z be the action orbit through x and consider the (symplectic) vector space V = T Z x? =T Z x, where the orthogonality sign is to be understood with respect to! at the point x. For x X, let T x T r be its isotropy group, and denote its connected component containing the identity by Tx o. Finally, let K be a subtorus such that T r = K Tx. o (Since we are interested strictly in a local normal form at x, the possible existence of other connected components of the isotropy group is irrelevant, but both cases are treated in [7].) Consider now the model symplectic manifold Y = T K V on which T r = K Tx o acts as a product of a free (right) action of K in the cotangent bundle T K and a linear symplectic action of Tx o in V. Denote by Y the moment map of this action. From [5] or [8], in an appropriate T r -invariant neighborhood of x in X, there is a symplectic change of coordinates to a neighborhood
364 SUZANA FALC ~ AO B. DE MORAES AND CARLOS TOMEI of the origin of T K V taking one torus action to another. In particular, the moment maps X and Y are related by X = Y. Also, it is simple to obtain variables so that (x 1 ; y 1 ; : : : ; x k ; y k ; x k+1 ; y k+1 ; : : : ; x n ; y n ) = e 1 y 1 : : : + e k (y k + 1) + k+1 (x k+1 + y k+1)= + : : : + n (x n + y n)=; where the e i 's are canonical vectors and the j 's are weights of the linearization of the action of the isotropy T x on V, following the usual procedures for normal forms as in [6] (also [5]). The subspaces spanned by f 1 ; : : : ; k g and fe k+1 ; : : : ; e n g have trivial intersection. Notice that the coordinate system can be taken so that the moment map 0 corresponding to the locally free S 1 -action is 0 = y k + 1. Proof of Proposition 1.4. Let C M be the critical set of j M. It suces to show that the Hessian of j M at x C M has an even number of positive (negative) eigenvalues and that its kernel at x is the tangent space of C M at x. Clearly, it suces to check both statements for a convenient coordinate system we consider in the local normal form described above on a neighborhood fx 1 ; y 1 ; : : : ; x n ; y n g of the origin. For points in M near the origin, we must have y k = 0. A simple computation shows that, locally, the critical component containing zero is a subspace. Also, for the nondegenerate part of the Hessian, the double partial derivatives in variables x j and y j, for j = k; : : : ; n; give rise to eigenvalues of the same sign.. A Rigidity theorem. Let (X;!) be a symplectic cone with compact base, admitting a symplectic T r action which commutes with the R + -action. We can assume, without loss, that T r acts locally freely on a conic open subset X 0 of X and that the complement of X 0 is of codimension in X. Let : (X;!)! R r be the moment map associated to the T r -action. Assume that h; 0 i > 0 for some 0 1 m Zr, for some integer m, as remarked after Lemma 1.5. By Theorem 1, (X) is a polytopic convex cone. Consider now a symplectic form on X, not necessarily satisfying t = t, but such that the same T r -action on X admits a moment map with respect to. We make the following assumptions about. 1. lim t!0 (t; m) = 0 uniformly in m M. (This is equivalent to saying that the contraction of?! with any vector eld induced by the torus action is integrable close to the vertex of X, t = 0.). lim t!1 0(t; m) = 1, where 0 = h ; 0 i, uniformly in m M.
MOMENT MAPS ON SYMPLECTIC CONES 365 We will show that under these assumptions, the following rigidity theorem holds. Theorem. The images of and coincide. Corollary. Let : (X;!)! R r and : (X; )! R r be moment maps as above. Suppose also that 0 = (X) and that there exists a 0 R r such that h; 0 i > 0 on X. If?! is compactly supported, then the images of and coincide up to a translation. In the notation of Section 1, X can be written as a disjoint union X = t N i=1x i, where X i = fx X j the stabilizer of x is T i g is connected and (X) = [ N i=1(x i ), (X) = [ N i=1 (X i ). The boundaries of both moment maps are formed by faces of the form (X k ) or (X k ), with T k dierent from the identity. On the other hand, some (interior) faces do not lie on the boundaries of the images, but in the interiors of the images they will receive special attention in the proof of Theorem. We will need some preparatory lemmas. Lemma.1. 0 = (X). Proof. Consider 0(x) = h 0 ; (x)i; x X. We rst show that 0 has an open image. This follows from the simple fact below, together with the openness of j Xi in (Lie T i )?. Fact: h 0 ; ij (Lie Ti )? is not identically zero for any (Lie T i )?. Proof of fact. If h 0 ; ij (Lie Ti )? were identically zero, (Lie T i )? would be contained in the hyperplane H 0 = fv (Lie T r ) j h 0 ; vi = 0g (since the fact that (X) has no vertices implies that dim (Lie T i )? 1). But (X i ) (Lie T i )? and, by hypothesis, (X) and so (X i ) are contained in the interior of H + = fv (Lie T r ) j h 0 ; vi > 0g. Hence h 0 ; ij (Lie Ti )? cannot be identically zero. Now, notice that 0 is bounded from below. Indeed, suppose it is not. Then there would be a sequence fx n g in X such that lim n!1 0(x n ) =?1. The sequence would not converge in X, for if it did, f 0(x n )g would converge too. Since we can write x n = (m n ; t n ), with m n M, we have two possibilities: For a subsequence, either t n! 1 and 0(x n )! 1 (by hypothesis) or t n! 0 and 0(x n )! 0 (by hypothesis). Hence we never have 0(x n )!?1 and 0 is bounded from below. If 0 (X), 0 attains a negative value? by the openness of its image and by using the hypothesis at the extremes of the cone again, the set
366 SUZANA FALC ~ AO B. DE MORAES AND CARLOS TOMEI?1 0 (?1;?=] is contained in a compact set of the form M [a; b]; 0 < a < b < 1. Then 0 attains its inmum, contradicting the openness of its image. In the next lemma we will show that the map is open at points that are taken into the interior faces. In Lemma.3 we will show an analogous result for. Lemma.. Assume that there exists x k in X k such that (x k ) Interior((X)). Then is an open map at the point x k i.e., there is an open neighborhood of x k in X taken to an open neighborhood of (x k ) in (Lie T r ). Similarly, is also open at x k. Proof. By the usual local convexity of the moment map as in [4], the image of a suciently small neighborhood U of x k is taken to an intersection of halfspaces whose bounding hyperplanes contain (x k ), which without loss in this argument we assume to be the origin of (Lie T r ). Let one such bounding hyperplane be the annihilator of some Lie T r and orient so that the half-space containing (U) is of the form h; fi 0, for f (Lie T r ). By Lemma 1.3, the local maximum x k of the function j M is a point in which j M attains its maximum, contradicting the fact that (x k ) is an interior point of (X). Thus, there can be no such bounding hyperplanes and is open at x k. Moreover, from the local form of at x k, we have that the weights 1 ; : : : ; n of the linear isotropy action of T k on (T x X;! xk ) span (Lie T r ). Now, T k induces the same (linear) action on the vector space T x X, which can be interpreted in two dierent ways as a symplectic action, depending if the bilinear form considered in T x X is! xk or xk. By Darboux theorem, both actions are conjugate by a symplectic transformation and since the weights of the rst action span (Lie T r ), the same must hold for the weights of the second action, implying the openness of at x k. We now start the proof of Theorem. We rst prove that (X i ) (X i ) (where X i is the closure of X i in X and so (X i ) does not contain 0) and conclude that (X) (X). Then we prove that (X) covers an open dense set of (X) and by taking closures we conclude that both images are equal. Step 1. (X i ) (X i ). Proof. The proof will be by induction on k = dim (Lie T i )?, which is the dimension of the interior of either (X i ) or (X i ). As in the proof of Proposition 1., k 1. From Theorem 1, for (X), there exists X i such that the interior of (X i ) has dimension equal to 1. If k = 1, then (X i ) is an open half-line starting at 0, contained in the straight line (Lie T i )?, and (X i ) is an open subset in (Lie T i )? +p, for some
MOMENT MAPS ON SYMPLECTIC CONES 367 p R r. Now, X i is a conic manifold, so ( t (x)) (X i ), for positive t and x X i. By hypothesis, lim t!0 ( t (x)) = 0, so (X i ) is also a subset of (Lie T i )?. Moreover, 0 = (X) by Lemma.1, (hence 0 = (X i )), and the limits along the ber t (x) are equal to 0 and 1 by hypothesis. So we must have that the connected set (X i ), which is open in (Lie T i )?, is also an open half-line beginning at 0 contained in (Lie T i )?. The case k = 1 is proved. Suppose now that (X i ) (X i ) for all X i such that dim (Lie T i )? < k. We will show that the same is true for all X i such that dim (Lie T i )? = k. As shown in Section 7 of [6], X i is a symplectic manifold, hence a symplectic cone with compact base and (X i ) is a polytopic convex cone in (Lie T i )? (Theorem 1). Set A = (X i ) [ f0g and B = A c, the complement of A in (Lie T i )? which is open by Lemma 1.5. We rst show that (X i ) \ B = B or the empty set, using a connectedness argument. i) (X i ) \ B is open in B. (X i ) = (X i ) [ ([ j (X j )), where T j T i, and by induction (X j ) (X j ) A. So (X i ) \ B = (X i ) \ B and (X i ) \ B is open since both (X i ) and B are. ii) (X i ) \ B is closed in B. Consider a sequence fy n = (x n )g in (X i ) \ B converging to y B. Since X ' M R +, we can write x n = (m n ; t n ). Since fy n g converges to y B and 0 = B, t n is far from zero and from 1. Hence the sequence fx n g is in a compact set in X and admits a subsequence converging to a point x X i. Therefore fy n g converges to y = (x) (X i ) and (X i ) \ B is closed in B. Since B is connected, (X i ) \ B = B or (X i ) \ B = ;. But 0(X) is bounded from below (by the proof of Lemma.1) so we cannot have (X i ) \ B = B. Therefore (X i ) A = (X i ) [ f0g. Since 0 = (X) ; 0 = (X i ) and then (X i ) (X i ), completing the proof of Step 1. In particular, (X) (X). We now show that (X) covers an open dense set of (X). Let X m = fx X j the isotropy group of x is the identityg. By Section 3 of [4], X m is a connected open set in X. Consider the set C = f (X i ) for X i 6=X m j (X i ) Int((X))g: Recall that for all X i in the stratication X = t N i=1x i, (X i ) (X i ) (X).
368 SUZANA FALC ~ AO B. DE MORAES AND CARLOS TOMEI Let ~ C = Int((X))? C, which is clearly an open set, and A1 ; : : : ; A s be the open connected components of ~ C. Step. (X) \ A j 6=; j = 1; : : : ; s: Proof. If s = 1, C ~ has only one connected component A1. Since (X) (X), the result follows in this case. Consider now s > 1. For each A j, there exists at least one (X i ) C such that (X i ) is in the boundary of A j. By Lemma., given x X i such that (x) Int((X)), there exists an open set Vw 0 of (X) around w = (x) in (Lie T r ). So Vw 0 intersects all the A j 's that have (X i ) as a boundary component and then Vw 0 \A j6=; and (X) \ A j 6=;. We conclude that for any A j, there is an open set in (X) that intersects it. Hence (X) \ A j 6=; for j = 1; : : : ; s: Step 3. (X) \ A j = A j ; j = 1; : : : ; s. Proof. This is connectedness again. i) (X) \ A j is open in A j. Since (X) = (X m ) [ ([ i (X i )), (X) \ A j = (X m ) \ A j. Therefore, (X) \ A j is open, since both (X m ) and A j are. ii) (X) \ A j is closed in A j. Let fy n = (x n )g be a sequence in (X) \ A j converging to y A j. We want to show that y (X) \ A j, i.e., that y (X). We now make use of compactness as above. Again, since y6=0, fx n g has a subsequence that converges in X. Then fy n g converges to y (X). Therefore (X) \ A j is not empty (from Step ), open and closed in the connected set A j, and so (X) \ A j = A j and A j (X). We conclude then that (X) covers an open dense subset of (X) since Int((X))? C = [ j A j (X). The closure of Int((X))? C is (X) [ f0g and from a by now standard compactness argument the closure of (X) is also its union with f0g. So, taking closures, (X) [ f0g (X) [ f0g: Since 0 = (X) and 0 = (X) we conclude that (X) (X). By Step 1, (X) = (X).
MOMENT MAPS ON SYMPLECTIC CONES 369 3. Appendix. Example 1. and Let X = R n? f0g with symplectic form! = P n i=1 dx i ^ dy i t (x 1 ; y 1 ; : : : ; x n ; y n ) = ( p tx 1 ; p ty 1 ; : : : ; p tx n ; p ty n ) t R + an action of R + on X. It is easy to see that X=R + = S n?1 and that ( t )! = t!. Hence (X;!) with this R + -action is a symplectic cone. Consider the following T n -action on X: (1;::: ; n)(x 1 ; y 1 ; : : : ; x n ; y n )= 0 B@ cos 1? sen 1 sen 1 cos 1... cos n? sen n sen n cos n 10 CA B@ x 1 y 1. x n y n It is clear that =. Moreover, this T n -action is Hamiltonian with moment map : X! R n, where x (x 1 ; y 1 ; : : : ; x n ; y n ) = 1 + y 1 ; : : : ; x n + y n and Im = f(r 1 ; : : : ; r n ) R n? f0g j r i 0 i = 1; : : : ; ng: Let be a T n -invariant symplectic form in X obtained by a perturbation of!, i.e., =! + where is a closed -form and is non-degenerate. It is possible to show that the most general satisfying these properties is given by 1 CA : X =! + X i + X i<j + X i<j + X i<j + i<j c ii dx i ^ dy i x i y j x j y i a ij x i x j? c ij + c x j + yj ji x i + yi x i x j a ij x i y j + c ij x j + yj?a ij x j y i + c ij y i y j x j + y j x j y i a ij y i y j + c ij x j + yj + c ji y i y j x i + y i x i x j + c ji x i + yi? c ji x i y j x i + y i! dx i ^ dx j! dx i ^ dy j! dx j ^ dy i! dy i ^ dy j where, if u i = x i +y i ;
370 SUZANA FALC ~ AO B. DE MORAES AND CARLOS TOMEI 1) c ij = u j ij (u 1 ; : : : ; u n ) with ij C 1 (R n? f0g) and (c ij ) uk = (c kj ) ui 8i6=k; i6=j ) a ij is a function of u 1 ; : : : ; u n and (a ij ) uk? (a kj ) ui + (a ik ) uj 6=0 8i6=j6=k with being non-degenerate. The formulae in (1) and () must be interpreted in the variables x i ; y i. It is easy to see that does not necessarily satisfy ( t ) = t : Let us check that satises the properties above, i.e., it is T n -invariant and closed. First, we will write in a simpler form in Y, an open and dense subset of X, to be dened below. Let u i = x i +y i ; i = arctg yi x i and Y = f(u i ; i ) i = 1; : : : ; n; u i > 0g. Notice that, by the inclusion of Y in X, every in X gives rise to a form in Y but a form in Y is not naturally associated to an extension in X. For this, it is necessary (and sucient) that the coecients c ij and a ij when written in the coordinates (x i ; y i ) have a smooth extension from Y to X. Rewriting in polar coordinates (in Y ) we get: = X i du i ^ d i + X i c ii du i ^ d i + X i6=j X c ij du i ^ d j + a ij du i ^ du j : i<j The action of T n on Y is given by: (1;::: ; n)(u 1 ; : : : ; u n ; 1 ; : : : ; n ) = (u 1 ; : : : ; u n ; 1 + 1 ; : : : ; n + n ): Since c ij and a ij are functions of the u i 's only, is T n -invariant in Y. Moreover, by the relations (1) and (), we get that is closed in Y. The fact that is closed and invariant by in Y implies in closure and -invariance of in X if the coecients of have a smooth continuation in X, and this is assured by (1) and (). Now, writing in matrix form we get: A = 1 0 B@ 0 a 1 : : : a 1n 1 + c 11 c 1 : : : c 1n?a 1 0 : : : a n c 1 1 + c : : : c n......?a 1n?a n : : : 0 c n1 c n : : : 1 + c nn?1? c 11?c 1 : : :?c n1 0 0 : : : 0?c 1?1? c : : :?c n 0 0 : : : 0......?c 1n?c n : : :?1? c nn 0 0 : : : 0 We have that is non-degenerate () det(a )6=0. Therefore, in order for to be non-degenerate, the coecients c ij 's must satisfy some relations 1 CA :
MOMENT MAPS ON SYMPLECTIC CONES 371 that must also hold in the limit when the u i 's go to 0 (not all of them simultaneously). Let : X! R n be the moment map associated to the action of of T n on (X; ). It is easy to see that if X = ( 1; : : : ; n) then d k = (1 + c kk )du k + c jk du j k = 1; : : : ; n: j6=k Let us look at some examples of and for X = R 4? f0g and compare the images of and. In this case, the non-degeneracy of is equivalent to det 0 B@ 0 a 1 1 + c 11 c 1?a 1 0 c 1 1 + c?1? c 11?c 1 0 0 CA 6=0 in R4? f0g?c 1?1? c 0 0 1 that is, 1 + c 11 + c + c 11 c? c 1 c 1 6=0; 8 u 1 ; u 6=0 and lim u 1!0 (1 + c 11 + c + c 11 c? c 1 c 1 )6=0; lim u!0 (1 + c 11 + c + c 11 c? c 1 c 1 )6=0: Example 1.1. Let X = R 4?f0g and =! + (x 1 +y 1 ) dx 1 ^dy 1 + (x +y ) dx ^ dy. The -form is non-degenerate in X since 1+c 11 +c +c 11 c?c 1 c 1 = 1 + u 1 + u + u 1 u 6=0: Let = u 1 + u 1 ; u + u. It is easy to see that Image( ) = Image() = f(x; y) R? f0g j x; y 0g: Note that the perturbation =?! satises ( t ) = t and lim t!0 t (u 1 ; u ) = (0; 0): Example 1.. Let X = R 4? f0g and (x + y =! + ) (x 1 + y1 + x + y) dx (x 1 + y 1 ^ dy 1 + 1) (x 1 + y1 + x + y) dx ^ dy x 1 y? x y 1 + (x 1 + y1 + x + y) dx x 1 x + y 1 y 1 ^ dx? (x 1 + y1 + x + y) dx 1 ^ dy x 1 x + y 1 y? (x 1 + y1 + x + y) dx x 1 y? x y 1 ^ dy 1 + (x 1 + y1 + x + y) dy 1 ^ dy i.e., c 11 = u (u 1+u) ; c = u1 (u 1+u) ; c 1 =?u (u 1+u) ; c 1 =?u1 (u 1+u) and 1 + c 11 + c + c 11 c? c 1 c 1 = 1 + u 1 + u (u 1 + u ) 6=0 in R4? f0g:
37 SUZANA FALC ~ AO B. DE MORAES AND CARLOS TOMEI Consider = ( 1; ) given by 1 = u 1 + u1 u 1+u ; = u + u u 1+u : The perturbation =?! is homogeneous of degree zero, that is, ( t ) =. Moreover, t (u 1 ; u ) = tu 1 + u1 u 1+u ; tu + u u 1+u and lim t!0 t (u 1 ; u ) = u1 u u 1+u ; u 1+u = (f 1 ; f ) where (f 1 ; f ) are points on the line f 1 + f = 1 with f 1 ; f 0. Hence, the image of is the convex region in R shown in Figure 1 and the images of and are dierent. Example 1.3. Let X = R 4? f0g and (x + y =! + ) (x 1 + y dx (x 1 + y1 + x + y) 3 1 ^ dy 1 + 1) dx (x 1 + y1 + x + y) 3 ^ dy! dx 1 ^ dx x 1 y (x 1 + y + 1) x y 1 (x + y? ) (x 1 + y 1 + x + y) 3 (x 1 + y 1 + x + y) 3 x 1 x (x 1 + y? 1) y 1 y (x + y + ) (x 1 + y 1 + x + y) 3 (x 1 + y 1 + x + y) 3 y 1 y (x 1 + y? 1) x 1 x (x + y + ) (x 1 + y1 + x + y) 3 (x 1 + y1 + x + y) 3 x y 1 (x 1 + y +? 1) x 1 y (x + y + ) (x 1 + y1 + x + y) 3 (x 1 + y1 + x + y) 3 i.e. c 11 = u (u 1 +u ) 3 ; c = u 1 (u 1 +u ) 3! dx 1 ^ dy! dx ^ dy 1! dy 1 ^ dy ; c 1 = c 1 =? u1u : The form is (u 1 +u ) 3 non-degenerate since 1+c 11 +c +c 11 c?c 1 c 1 = 1+ u 1 +u (u 1 +u ) 3 Let = ( 1; ) where 1 = u 1 + p u1 and u 1 +u = u + 6=0 in R 4?f0g p u : u 1 +u The perturbation =?! is such that ( t ) = and the moment map satises t (u 1 ; u ) = tu 1 + p u1 u 1 +u ; tu + p u u 1 +u : Therefore, u lim t!0 t (u 1 ; u ) = p 1 ; p u = (g u 1 +u u 1 +u 1 ; g ) where (g 1 ; g ) are points on the circle g 1 + g = 1 with g 1 ; g 0. It follows then that the image of is the non-convex region in R indicated in Figure and the images of and are dierent.
MOMENT MAPS ON SYMPLECTIC CONES 373 y y x Figure 1. Figure. x In the next example, the images of the moment maps close to innity are dierent polytopic cones. This is a consequence of the behaviour of the perturbation close to the origin. Example. Consider C 4? f0g with symplectic form p?1! = (dz 1 ^ dz 1 + dz ^ dz? dz 3 ^ dz 3? dz 4 ^ dz 4 ): The Hamiltonian action of S 1 on (C 4? f0g;!) given by e i (z 1 ; z ; z 3 ; z 3 ; z 4 ) = (e i z 1 ; e i z ; e i z 3 ; e i z 4 ) admits H(z 1 ; z ; z 3 ; z 4 ) = 1(jz 1j + jz j? jz 3 j? jz 4 j ) as a moment map and H?1 (0) = f(z 1 ; z ; z 3 ; z 4 ) C 4? f0g j jz 1 j + jz j = jz 3 j + jz 4 j g is a manifold. Let t (z 1 ; z ; z 3 ; z 4 ) = ( p t z 1 ; p t z ; p t z 3 ; p t z 4 ) for t R + be an action of R + on C 4? f0g. It is easy to see that R + and S 1 act on H?1 (0), that these actions commute and that ( t )! = t!. Therefore, if X is the reduced space H?1 (0)=S 1, R + also acts on X. If! 0 is the symplectic form on X induced by!, then ( t )! 0 = t! 0 and X=R + ' H?1 (0)=C is a compact manifold. Hence, (X;! 0 ) is a symplectic cone with base H?1 (0)=C. Consider the Hamiltonian action of the torus T 3 on (X;! 0 ) given by (e i1 ; e i ; e i3 ) [z 1 ; z ; z 3 ; z 4 ] = [e i1 z 1 ; e i z ; e i3 z 3 ; z 4 ] where (z 1 ; z ; z 3 ; z 4 ) is a point in H?1 (0) and [z 1 ; z ; z 3 ; z 4 ] is the corresponding point in X.
374 SUZANA FALC ~ AO B. DE MORAES AND CARLOS TOMEI Let : X! R 3 be the associated moment map (normalized in zero). Then jz1 j ([z 1 ; z ; z 3 ; z 4 ]) = ; jz j ;?jz 3j with image Im = f (x; y;?z) R 3? f0g j x + y z; x; y; z 0 g a polytopic convex cone with vertex at the origin, bounded by the planes x = 0; y = 0; x + y =?z: Notice that the intersection of Im() with the family of planes x + y = c (c real and greater than zero) is a family of rectangles with sides c and c p. Dene now in C 4? f(z 1 ; z ; z 3 ; z 4 ) such that z 1 = z = 0 or z 3 = z 4 = 0g the symplectic form p?1 jz j =! + (jz 1 j + jz j ) dz jz 1 j 1 ^ dz 1 + (jz 1 j + jz j ) dz ^ dz jz 4 j? (jz 3 j + jz 4 j ) dz jz 3 j 3 ^ dz 3? (jz 3 j + jz 4 j ) dz 4 ^ dz 4 z 1 z? (jz 1 j + jz j ) dz z 1 z 1 ^ dz? (jz 1 j + jz j ) dz ^ dz 1 + z 3 z 4 (jz 3 j + jz 4 j ) dz z 3 z 4 3 ^ dz 4 + (jz 3 j + jz 4 j ) dz 4 ^ dz 3 : Let 0 be the symplectic form induced by in X. It is easy to see that the action of T 3 on (X; 0 ) is still Hamiltonian with associated moment map : X! R 3 where ([z 1 ; z ; z 3 ; z 4 ]) jz 1 j + = 1 with image jz 1 j jz 1 j + jz j ; jz j + jz j jz 1 j + jz j ;?jz 3j? jz 3 j jz 3 j + jz 4 j Im = f(x; y;?z) R 3 j x; y; z 0; x + y > 1; x + y + 1 zg a truncated cone (still polytopic and convex) in R 3 bounded by the planes x = 0; y = 0; x + y = z + 1; x + y = 1. The intersection of Im( ) with the family of planes x+y = c (c real and greater than 1) is a family of rectangles with sides c p and c + 1. In this example, the two images are dierent and it is not possible to include by translation one into the other. The images are shown in Figure 3.
MOMENT MAPS ON SYMPLECTIC CONES 375 -z -z the original image at innity the perturbed image at innity Figure 3. References [1] R. Abraham and J. Marsden, Foundations of mechanics, Benjamin, 1978, Reading, MA. [] M.F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (198), 1-15. [3] R. Bott, Nondegenerate critical manifolds, Ann. Math., 60() (1959), 48-61. [4] V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (198), 491-513. [5], A Normal form for the moment map, Di. Geom. Methods, (ed. S.Sternberg), Mathematical Physical Studies, Vol. 6, (1984), 161-175, D. Reidel, Boston. [6], Symplectic techniques in physics, (1984), Cambridge University Press, Cambridge. [7] E. Lerman, On the centralizer of invariant functions on a Hamiltonian G-space, J. of Di. Geom., 30 (1989), 805-815. [8] C.M. Marle, Classication des actions hamiltoniennes au voisinage d'une orbite, C. R. Acad. Sci. Paris, Ser. I, Math., 99 (1984) 49-5. [9] G.D. Mostow, On a conjecture of Montgomery, Ann. of Math., 65() (1957), 513-516. [10] E. Prato, Convexity properties of the moment map for certain non - compact manifolds, preprint. Received May 0, 1993 and revised November 17, 1996. PUC-Rio R. Marqu^es de S. Vicente, 5 Rio de Janeiro, 453-900 Brazil E-mail address: tomei@mat.puc-rio.br