Compact Hermitian symmetric spaces, coadjoint orbits, and the dynamical stability of the Ricci flow
aa r X i v : . [ m a t h . DG ] N ov COMPACT HERMITIAN SYMMETRIC SPACES,COADJOINT ORBITS, AND THE DYNAMICALSTABILITY OF THE RICCI FLOW
STUART JAMES HALL, THOMAS MURPHY, AND JAMES WALDRON
Abstract.
Using a stability criterion due to Kr¨oncke, we show, provid-ing n = 2 k , the K¨ahler–Einstein metric on the Grassmannian Gr k ( C n )of complex k -planes in an n -dimensional complex vector space is dy-namically unstable as a fixed point of the Ricci flow. This generalisesthe recent results of Kr¨oncke and Knopf–Sesum on the instability of theFubini–Study metric on CP n for n >
1. The key to the proof is usingthe description of Grassmannians as certain coadjoint orbits of SU ( n ).We are also able to prove that Kr¨oncke’s method will not work on anyof the other compact, irreducible, Hermitian symmetric spaces. Introduction
In 2013 Kr¨oncke proved the surprising result that the Fubini–Study K¨ahler–Einstein metric on CP n , n >
1, is unstable as a fixed point of the Ricci flow[24]. More precisely, he showed that there are certain conformal (and hencenon-K¨ahler) deformations of the Fubini–Study metric from which the Ricciflow never returns. This is in stark contrast to the behaviour of the K¨ahler–Ricci flow where Tian and Zhu [33] have shown that K¨ahler–Einstein metricsare essentially global attractors within their K¨ahler class. In [22] Knopf andSesum give an independent verification of Kr¨oncke’s result.The behaviour of Ricci flow on manifolds admitting K¨ahler metrics is a topicof current interest (see for example [15], [16], [20], and [26]). What Kr¨oncke’sresult suggests is that behaviour of the Ricci flow near the space of K¨ahlermetrics is more complicated than was initially believed. If a Fano manifold M with Hodge number h , ( M ) > h , ( M ) = 1. One such class of K¨ahler–Einstein manifolds are the compact,irreducible, Hermitian symmetric spaces. These manifolds were completelyclassified by E. Cartan into six types; there are four infinite families and twoexceptional spaces. Each of these spaces admits a K¨ahler–Einstein metricunique up to automorphisms of the complex structure; this metric is thesymmetric metric on each manifold. We will henceforth implicitly assumeall manifolds in this paper are equipped with their symmetric space (andK¨ahler–Einstein) metrics.The stability criterion employed by Kr¨oncke is very simple to state (c.f.Theorem 2.6); if an Einstein metric with Einstein constant τ > f say, with eigenvalue − τ and the ntegral over the manifold of f does not vanish, then the metric is dynam-ically unstable. It is a classical result of Matsushima [25] that, on a FanoK¨ahler–Einstein manifold, there is a bijection between Killing fields andthe eigenspace corresponding to − τ . Hence K¨ahler–Einstein manifolds withlarge symmetry groups are ideal candidates on which to attempt to useKr¨oncke’s result to investigate stability. The first theorem we prove saysthat, when the symmetric space is not a Grassmannian of complex k -planesin an n -dimensional complex vector space (which we denote Gr k ( C n )), theintegral of f will necessarily vanish. Theorem A.
Let ( M, g ) be a compact, irreducible, Hermitian symmetricspace other than a Grassmannian Gr k ( C n ) and let g be the canonical K¨ahler–Einstein metric normalised to have Einstein constant τ . Then any ( − τ )-eigenfunction of the Laplacian, f , satisfies Z M f dV g = 0 . The proof of this theorem uses the Chevalley–Shepherd–Todd theorem whichclassifies the degrees in which the generators of certain polynomial algebrascan exist when the polynomial is required to be invariant under the action ofa compact simple Lie group G . Applied to our stability problem, this clas-sification only permits a generator in the required degree when G = SU ( n )and so the only class of space where the integral can be non-zero is thecomplex Grassmannians.We are able to compute the stability integral for Gr k ( C n ) and show that,in some sense, ‘generic’ Grassmannians are unstable. This result generalisesthe CP n calculation of Kr¨oncke and Knopf–Sesum. Theorem B. If k, n ∈ N with ≤ k < n and n = 2 k , then Gr k ( C n ) isdynamically unstable as a fixed point of the Ricci flow. The integral of f vanishes for the spaces Gr k ( C k ); the proof of Theorem Bshows this directly but this can be seen without performing any calculation.The duality map Ψ : Gr k ( C n ) → Gr n − k ( C n ) (where a subspace is mappedto its complement) is an isometry and in the case when n = 2 k is also aninvolution. One can show (see, for example, [11]) that Ψ ∗ f = − f and thusthe integral of any odd power of f vanishes.We cannot yet say anything about the stability of the spaces Gr k ( C k ) apartfrom the case when k = 1 as then Gr ( C ) ∼ = CP ∼ = S . In this case, theK¨ahler–Einstein metric is the round metric and this is known to be dynam-ically stable by a result of Hamilton [17] (Chow later proved that the roundmetric on S is a global attractor for the normalised Ricci flow starting atany initial metric [7]).The methods used in [22] and [24] to show a destabilising eigenfunction existson CP n use the generalised Hopf fibration to lift the problem to finding cer-tain U (1)-invariant functions on the sphere S n +1 ⊂ C n +1 . This paper takesa totally different approach by viewing the Grassmannians as adjoint orbitsof SU ( n ) and using techniques coming from symplectic geometry (such as he Duistermaat–Heckman formula) to construct eigenfunctions and makecalculations of the relevant integrals.2. Background
Stability.
Einstein metrics g satisfying Ric( g ) = τ g for τ ∈ R , evolvevia homothetic scaling under the Ricci flow ∂g∂t = − g ) . It is therefore useful to view Einstein metrics as fixed points of the Ricciflow up to a normalisation of the volume of the metric by homothetic scal-ing. A natural question is whether a given Einstein metric is stable as afixed point in the sense that the Ricci flow starting at a small perturbationof the metric will return to the original Einstein metric. Perelman [29] in-troduced a functional ν ( g ) which is stationary at shrinking gradient Riccisolitons (every Einstein metric with τ > ν functional along potentially destabilising directions. If the entropyincreases along a particular direction then the corresponding perturbationof the Einstein metric will never return under the flow. This process wasfirst carried out for Einstein metrics by Cao, Hamilton, and Ilmanen [4] andlater generalised by Cao and Zhu [6]. Theorem 2.1 (Cao–Hamilton–Ilmanen [4]) . Let ( M, g ) be an Einstein met-ric with Einstein constant τ > . Let h ∈ s ( T M ∗ ) . Then d ds ν ( g + sh ) | s =0 = τ Vol(
M, g ) Z M h N ( h ) , h i dV g , where N ( h ) = 12 ∆ h + Rm( h, · ) + div ∗ div( h ) + 12 ∇ v h − g nτ Vol(
M, g ) Z M tr( h ) dV g , (2.1) and v h is the unique solution to ∆ v h + v h τ = div(div( h )) . The diffeomorphism and scale invariance of ν ( g ) means that to check linearstability one only needs to consider perturbations h ∈ s ( T M ∗ ) satisfyingdiv( h ) = 0 and h h, g i L = Z M tr( h ) dV g = 0 . In this case, the stability operator N in Equation (2.1) reduces to N ( h ) = 12 ∆ h + Rm( h, · ) = 12 (∆ L + 1 τ ) h, where ∆ L is the Lichnerowicz Laplacian∆ L h := ∆ h + 2Rm( h, · ) − Ric · h − h · Ric . We thus have the following definitions: efinition 2.2 (Linear stability of Einstein metrics) . Let (
M, g ) be a com-pact Einstein manifold satisfying Ric( g ) = 12 τ g and let − κ be the largesteigenvalue of the Lichnerowicz Laplacian restricted to the space of divergence-free, g -orthogonal tensors.(1) If κ > τ , g is called linearly stable .(2) If κ = 1 τ , g is called neutrally linearly stable .(3) If κ < τ , g is called linearly unstable . Definition 2.3 (Dynamical stability of Einstein metrics) . Let ( M n , g E ) bea compact Einstein manifold. The metric g E is said to be dynamically stable for the Ricci flow if for any m ≥ C m -neighbourhood U of g E inthe space of sections Γ( s ( T M ∗ )), there exists a C m +2 neighbourhood of g E , V ⊂ U , such that:(1) for any g ∈ V , the volume normalised Ricci flow ∂g∂t = − g ) + 2 n (cid:18)Z M scal( g ) dV g (cid:19) g, with g (0) = g exists for all time,(2) the metrics g ( t ) converge modulo diffeomorphism to an Einstein met-ric in U .If the metric g E is not dynamically stable then it is said to be dynamicallyunstable .If a metric is linearly unstable, then Sesum [31] showed that it is dynami-cally unstable. Similarly (though much more technically difficult to prove),under the assumption that all infinitesimal Einstein deformations are in-tegrable, Sesum proved that linear stability implies dynamical stability. In[23], Kr¨oncke built on the work of Haslhofer and M¨uller [18] and showed thatan Einstein metric with positive Einstein constant is dynamically stable ifand only if it is a local maximum of the ν functional. This characterisationdoes not require an infinitesimal perturbation to satisfy any integrabilityassumptions and forms the basis of Kr¨oncke’s stability criterion in Theorem2.6.In general, it is very difficult to analyse the spectrum of the Lichnerow-icz Laplacian for an arbitrary Einstein metric. If the metric has some extrastructure then more can be said. In the case the Einstein metric is K¨ahler–Einstein then there is the following topological condition (originally statedin [4] and proved for the more general class of K¨ahler–Ricci solitons in [14]) Theorem 2.4 (Cao–Hamilton–Ilmanen) . Let ( M, J, g ) be a K¨ahler–Einsteinmetric. If the Hodge number h , ( M ) > then g is linearly unstable. This proposition can be seen as generalising the fact that any product of Ein-stein metrics with fixed Einstein constant τ is unstable under the Ricci flow.The product of any two K¨ahler–Einstein metrics always has h , ( M ) > etric is K¨ahler–Einstein can be written in the form M = G/H where G isa connected compact simple Lie group and H is the isotropy subgroup. Wenote that the identity component of the isometry group Iso = G and so theLie algebra of Killing fields k = Lie (Iso ) is isomorphic to the Lie algebra g .All the manifolds in the following theorem have h , ( M ) = 1. Theorem 2.5 (Cao–He, c.f. Theorem 4.3 in [5]) . The linear stability of theirreducible compact Hermitian symmetric spaces M = G/H is as follows:(1) M is linearly unstable if: • M is the space of compatible complex structures on H n , M = Sp ( n ) /U ( n ) , for n > .(2) M is neutrally linearly stable if: • M is a complex Grassmannian Gr k ( C n ) = SU ( n ) /S ( U ( k ) × U ( n − k )) , where n > and < k < n , • M is a complex hyperquadric Q n = SO ( n + 2) / ( SO ( n ) × SO (2)) , where n ≥ , • M is a space of orthogonal almost complex structures on R n , M = SO (2 n ) /U ( n ) , for n > , • M is one of the exceptional spaces, M = E / ( SO (10) × SO (2)) , or M = E / ( E × SO (2)) . (3) If M is the sphere S ∼ = Gr ( C ) ∼ = SO (4) U (2) then M is dynamicallystable and so linearly stable. Missing from this list (as it is not irreducible) is the hyperquadric Q = SO (4) SO (2) × SO (2) ∼ = CP × CP . It is unstable as it is a product. The hyperquadric Q ∼ = Sp (2) U (2) has h , = 1but is nevertheless linearly unstable by a result of Gasqui and Goldschmidt[11].What Theorem 2.5 shows is that most of the Hermitian symmetric spacesare neutrally linearly stable. In particular, the complex projective spaces CP n = Gr ( C n +1 ) with n > M, g ) with Einstein constant τ , if there is aneigenfunction f satisfying ∆ f = − τ f then we define the tensor h f := (∆ f ) g − Hess( f ) + f τ g. It can be shown ([4], [5]) that h f is divergence free, L -orthogonal to g andsatisfies ∆ L h f = − τ h f . In 2013 Kr¨oncke proved the following stability criterion by computing thethird variation of the ν functional. heorem 2.6 (Kr¨oncke, Theorem 1.7 in [24]) . Let ( M, g ) be an Einsteinmetric with Einstein constant τ and let f be an eigenfunction of the Lapla-cian with eigenvalue − τ . If the integral Z M f dV g = 0 , (2.2) then g is dynamically unstable as a fixed point of the Ricci flow and is desta-bilised by the tensor h f . Kr¨oncke then constructed a eigenfunction satisfying the condition (2.2) forthe spaces CP n with n > Corollary 2.7.
The Fubini–Study metrics on CP n , n > are dynamicallyunstable as fixed points of the Ricci flow. This result was somewhat unexpected as a long-standing conjecture in thefield had included CP on the list of stable, four-dimensional geometries forthe Ricci flow. Theorem B can be seen as a generalisation of the CP n resultsof Kr¨oncke and Knopf–Sesum; however, as mentioned in the introduction,our construction of eigenfunctions and method of evaluating the integral istotally different from the methods used in [22] and [24].Given Theorem B, it seems likely that the other compact irreducible Her-mitian symmetric spaces are also unstable but that they represent metricswith high degrees of degeneracy as critical points of the ν functional. Fur-ther evidence for their instability might come from the behaviour of othernon-Hermitian compact symmetric spaces. The first author [13] has in-vestigated the stability of the canonical metric on the compact simple Liegroup G which is also neutrally linearly stable with the neutral directionscoming from conformal perturbations corresponding to eigenfunctions of theLaplacian. In this case, Kr¨oncke’s stability integral is non-zero for a certaineigenfunction and so G is unstable.2.2. Geometry of coadjoint orbits.
In this section G is a compact,semisimple Lie group. Hence the Killing form h· , ·i is non-degenerate and sothe adjoint representation of G is orthogonal. We can also use the Killingform (or any Ad G -invariant inner product) to identify g and g ∗ . The coad-joint action of G on g ∗ is defined byAd ∗ g ( ξ )( X ) := ξ (Ad g − ( X ))for g ∈ G , ξ ∈ g ∗ and X ∈ g . If ξ ( · ) = h· , X i then Ad ∗ g ( ξ )( · ) = h· , Ad g ( X ) i and we have a straightforward identification of coadjoint and adjoint orbitsvia the Killing form.For ξ ∈ g we consider the orbit O ξ of ξ under the adjoint action of G .Denote by H stabiliser of ξ and let h ⊂ g be its Lie algebra. Then g = h ⊕ m where m can be identified with the tangent space to O ξ at ξ . The subalgebra h is the kernel of the map ad( ξ ) : g → g and thus m is the image.We let T be a maximal torus of G and take t = Lie ( T ) to be its Lie al-gebra. The Weyl group W = N G ( T ) /T where N G ( T ) is the normaliser of T in G . A classical theorem (see for example Bott [3]) yields:(1) O ξ ∩ t = ∅ , O ξ ∩ t is a W -orbit.This means, without loss of generality, we can take the element representingthe orbit ξ ∈ t .The orbits have the structure of a complex manifold. Decomposing thecomplexified Lie algebra g ⊗ C we get g ⊗ C = t C ⊕ M α : h α,ξ i =0 R α ⊕ A ⊕ ¯ A where α ∈ t C are the roots of G , R α is the root space of α , and A is thespan of the root spaces satisfying[ ξ, r α ] = i h α, ξ i r α , with h α, ξ i > r α ∈ R α . One can identify m C ∼ = A ⊕ ¯ A and show that A and t C ⊕ A are Lie subalgebrasof g ⊗ C . By defining m (1 , = A we get a G -invariant complex structure on O ξ .The Kirillov–Kostant–Souriau symplectic form is defined as ω ξ ( x, y ) = −h ξ, [ x, y ] i , for x, y ∈ m . This is extended over O ξ using the adjoint action. This form iscompatible with the complex structure and gives the orbit the structure of aK¨ahler manifold. In the case when the center of H has dimension 1 (whichwill be the cases that we wish to consider in this article), the induced metricis K¨ahler–Einstein (c.f.[2] Proposition 8.85).2.3. Properties of the eigenfunctions.
We will now show how to con-struct eigenfunctions for the Laplacian of the K¨ahler–Einstein metric on O ξ .We begin by defining functions f η ∈ C ∞ ( O ξ ) by f η ( Z ) := h Z, η i , (2.3)where Z ∈ O ξ and h· , ·i is the inner product on g coming from the Killingform. These functions satisfy some important properties. Lemma 2.8.
Suppose that η ∈ g and ˜ η ∈ g are in the same G -orbit. Thenthe functions f η , f ˜ η ∈ C ∞ ( O ξ ) defined by Equation (2.3) satisfy Z O ξ f kη ω n = Z O ξ f k ˜ η ω n , where k ∈ N , n is the compex dimension of O ξ , and ω is the Kirillov–Kostant–Souriau symplectic form. Furthermore, in the case that k = 1 wehave Z O ξ f η ω n = 0 , for all η ∈ g .Proof. The conditions of the lemma mean there is a g ∈ G such that ˜ η =Ad g ( η ). Hence by the Ad-invariance of the inner product we have f η ( Z ) = h η, Z i = h Ad g − (˜ η ) , Z i = h ˜ η, Ad g ( Z ) i = f ˜ η (Ad g ( Z )) . n other words, f η = Ad ∗ g ( f ˜ η ). As Ad g : O ξ → O ξ is an orientation preserv-ing isometry, we have Z O ξ f kη ω n = Z O ξ Ad ∗ g ( f ˜ η ) k ω n = Z O ξ Ad ∗ g ( f ˜ η ) k (Ad ∗ g ω n ) = Z O ξ f k ˜ η ω n . To prove the second part of the lemma, we note that the function F : G → R given by F ( g ) = Z O ξ f Ad g ( η ) ω n , is constant. Taking the derivative at the identity yields Z O ξ f [ η,ζ ] ω n = 0 , for all η, ζ ∈ g . This means that the map η → Z O ξ f η ω n is a Lie algebra homomorphism. The fact that G is simple means that thismap must be the trivial homomorphism and so the result follows. (cid:3) Next we recall a theorem of Matsushima [25] which says that for any FanoK¨ahler–Einstein manifold (
M, g, J ) there is an isomorphism between the( − /τ )-eigenspace, E ( − /τ ) , and the Lie algebra of Killing vector fields k given by φ → − J ∇ φ, where J is the complex structure.All the connected, compact, irreducible Riemannian symmetric spaces canbe constructed in the form M = Iso / Iso p where Iso is the connected com-ponent of the identity of the isometry group of ( M, g ) and Iso p is the isotropygroup of isometries fixing a point. For the spaces G/H in Theorem 2.5 wehave g ∼ = Lie(Iso) ∼ = k . This map can be realised by the assignment η → ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Ad exp( tη ) ( Z ) = [ η, Z ] , for η ∈ g and Z ∈ O ξ . Lemma 2.9.
If the K¨ahler–Einstein metric on O ξ has Einstein constant τ , then the functions f η defined in Equation (2.3), satisfy ∆ f η = − τ f η . Proof.
Let X ∈ m and consider F ( t ) = f η (Ad exp( tX ) ξ ) = h η, Ad exp( tX ) ξ i = h Ad exp( − tX ) η, ξ i . Taking derivatives we see dFdt (cid:12)(cid:12)(cid:12) t =0 = −h ξ, [ X, η ] i = − ω ( X, η ) = g ( X, J η ) . ence ∇ f η = J η (here we identify η with the Killing field it generates on O ξ ). As η is a holomorphic Killing field we invoke Matsushima’s theoremwhich says the map φ → ∇ φ, is an isomorphism between the eigenspace E − /τ and J k where k is the spaceof Killing fields. Hence, as f η has mean value zero by Lemma 2.8, we see f η is an eigenfunction of the Laplacian with eigenfunction − τ . (cid:3) An element X ∈ g is said to be regular if its centraliser in g is of smallestpossible dimension. The regular elements of t we will denote by t reg . Thefollowing lemma will be vital in our analysis. Lemma 2.10 (Bott [3]) . Let D ∈ t reg be regular. Then the function f D defined by Equation (2.3) has the following properties.(1) The critical points of f D are non-degenerate and of even index.(2) The critical points are the orbit of ξ ∈ t under the action of Weylgroup W . We remark that f D is a Hamiltonian function for the action on O ξ generatedby D . If Λ ⊂ t is the weight lattice, then choosing D ∈ Λ ⊗ Z Q will generatean S -action. We denote the set of elements in t that generate closed orbitsby t c . The set t reg ∩ t c is dense in t (with respect to the Euclidean topology).It turns out that the eigenfunctions f D generated by D ∈ t reg ∩ t c are theonly ones that one needs to check the stability condition (2.2) on. Proposition 2.11.
Let
G/H = O ξ be one of the symmetric spaces in The-orem 2.5 and let T be a maximal torus in G with Lie algebra t . Suppose thatthere exists f ∈ E − (1 /τ ) such that Z O ξ f ω n = 0 , then there exists D ∈ t reg ∩ t c such that Z O ξ f D ω n = 0 . Proof.
As the map ̺ : g → E − /τ given by ̺ ( X ) = f X is an isomorphism,there must exist η ∈ g such that f = f η . As remarked in Section 2.2, given afixed maximal torus T with Lie algebra t , any element of g is in the adjointorbit of some element in t . If D = Ad g ( η ) ∈ t for some g ∈ G then, byLemma 2.8 Z O ξ f D ω n = 0 . Finally we see we can take D ∈ t reg ∩ t c as this set is dense in t . (cid:3) Computing integrals.
We will need to be able to compute integralsof powers of f over the orbit. This is achieved by the famous Duistermaat–Heckman formula [9] (see [27] for the form we are using). On a symplec-tic manifold ( M n , ω ) with a Hamiltonian circle action that has an as-sociated Hamiltonian function ϕ with non-degenerate critical points, theDuistermaat–Heckman formula is Z M e − tϕ ( p ) ω n ( p ) = n ! t n X q critical e − tϕ ( q ) ̟ ( q ) , here the ̟ ( q ) is the product of the weights of the circle action that isinduced on the tangent space at each fixed point q .The holomorphic tangent space at a critical point q is identified with thespan of the root spaces R α satisfying h q, α i >
0, we will denote this set P ( q ).The derivative of r α → ad exp( tD ) r α at t = 0 is h α, D i r α . Hence the weightat each fixed point q ∈ t is given by ̟ ( q ) = Y α ∈ P ( q ) h α, D i . Proposition 2.12.
Let D ∈ t reg ∩ t c and let f D ( Z ) = h Z, D i . Then Z O ξ e − tf ( p ) ω n ( p ) = n ! t n X w ∈ W/ stab ( ξ ) e − tf ( w · ξ ) Q α ∈ P ( w · ξ ) h α, D i , (2.4) where n is the complex dimension of the orbit O ξ .Proof. We simply apply the Duistermaat–Heckman formula to the function f D which is the Hamiltonian for the circle action generated by D ∈ t reg ∩ t c .As the element D is regular, Lemma 2.10, says the critical points are non-degenerate precisely the orbit of ξ under the action of the Weyl group W .The value of the weights follows from the previous discussion. (cid:3) Formulae similar to (2.4) occur in the theory of the orbit method developedby Kirillov [21]. In this theory integrals over certain coadjoint orbits cor-respond to the characters of the representation corresponding to the orbit.Similar expressions also appear elsewhere in the literature, for example inthe papers of Berline and Vergne [1], Paradan [28], and Rossman [30].3.
The proof of Theorem A
The proof of Theorem A is based on the classification of certain invari-ant polynomial algebras. If we let G be one of the compact simple Liegroups appearing as the larger group in Theorem 2.5, then we denote by R [ g ] G the graded algebra of G -invariant polynomials functions on g . If T is a maximal torus of G with Lie algebra t and Weyl group W , then theChevalley restriction theorem (see for example Section 23 in [19]) yields anisomorphism r : R [ g ] G → R [ t ] W where r ( P ) is simply restriction of a poly-nomial P ∈ R [ g ] G to t . By the Chevalley–Shephard–Todd theorem and theShephard–Todd classification of complex reflection groups, see [8] and [32],the algebra R [ t ] W is a polynomial algebra, with generators of well defineddegrees. We list them in the following table with the groups in bracketshaving Lie algebra with the corresponding root system:Root system Degree of generators in R [ t ] W A n ( SU ( n + 1)) 2 , , . . . , n + 1 B n ( SO (2 n + 1)) 2 , , . . . , nC n ( Sp ( n )) 2 , , . . . , nD n ( SO (2 n )) 2 , , . . . , n − nE , , , , , E , , , , , , R [ g ] G . emma 3.1. Let I k : g → R be defined by I k ( η ) = Z O ξ ( f η ) k ω n . Then I k is a (possibly trivial) Ad G -invariant, homogenous, degree − k poly-nomial.Proof. The Ad G -invariance was demonstrated in Lemma 2.8. The fact thatthe function is a homogenous, degree- k polynomial is straightforward if onepicks a basis of g and then calculates in coordinates. (cid:3) We now give the proof of Theorem A.
Proof.
We note by Proposition 2.11, we might as well assume that a destabil-ising eigenfunction is of the form f D for D ∈ t reg ∩ t c . If the symmetric spaceis not a Grassmanian then it is of the form G/H with G being one of thegroups B n , D n , E or E . Lemma 3.1 shows that I is a degree 3, homoge-nous G -invariant polynomial and so by the Chevalley restriction theoremyields a degree 3 element of R [ t ] W . However, the above table shows thisfunction must vanish unless G = D = SO (6) = A . One can show that SO (6) /U (3) ∼ = CP and Q ∼ = Gr (2 ,
4) and so the stability of these spacesfollows from the type A n consideration. (cid:3) The fact that the integral 2.2 vanishes on the spaces Gr k ( C k ) is a conse-quence of the calculation in section 5, specifically Equation (5.2).We also note that this method of proof also shows that the functions f η defined by Equation (2.3) have mean value 0 (which was demonstrated ex-plicitly in Lemma 2.8). This follows as there are no non-zero homogeneous,degree 1 polynomials that are invariant under the Weyl groups of the com-pact connected Lie groups we are considering in this article.4. Combinatorial properties of certain determinants
In order to prove Theorem B we first need to collect some results on thepower series of certain matrices that will appear after the manipulation of therighthand side of Equation (2.4). For m , m , . . . , m n ∈ R and 0 < k ≤ [ n/ M : R → Mat n × n R given by M ( t ) = e − m t e − m t . . . e − m n t e − m t m e − m t m . . . e − m n t m n ... ... ... ... e − m t m k − e − m t m k − . . . e − m n t m k − n . . . m m . . . m n ... ... ... ... m n − k − m n − k − . . . m n − k − n . (4.1)We will be interested in computing the derivatives of det( M ( t )) when t = 0;it is therefore useful to think of det( M ( t )) as the sum of various productsof k functions. To compute the p th derivative we can use the formula d p dt p det( M ( t )) = X d + d + ··· + d k = p (cid:18) p ! d ! d ! . . . d k ! (cid:19) det( M ( R ( d )1 , R ( d )2 , . . . , R ( d k ) k )) , here d i ∈ N ∪ { } and M ( R ( d )1 , R ( d )2 , . . . , R ( d k ) k ) is the matrix formed bytaking d i derivatives of the i th row.It is clear that, evaluating at t = 0, this formula is going to require thecalculation of determinants of matrices of the form A = m e m e . . . m e n m e m e . . . m e n ... ... ... ... m e k m e k . . . m e k n m n − k − m n − k − . . . m n − k − n m n − k − m n − k − . . . m n − k − n ... ... ... ... m m . . . m n . . . , for exponents e , e , . . . , e k ∈ N . In fact, a matrix of the form A that gives anon-zero determinant can be written (after possibly reordering rows) in theform A ij = m λ i + n − ij , (4.2)for some vector λ = ( λ , λ , . . . , λ n ) ∈ Z n ≥ with λ i ≥ λ i +1 . Such determi-nants are all multiples of the Vandermonde determinant V given by V = | m n − ij | = Y ≤ i 0) case of Equation (4.2). Thisleads to the definition of the Schur polynomials. Definition 4.1 (Schur polynomial) . Given λ = ( λ , λ , . . . , λ n ) ∈ Z n ≥ with λ i ≥ λ i +1 the Schur polynomial S λ is given by S λ ( m , m , . . . , m n ) = | m λ i + n − ij | V . The Schur polynomial S λ is a homogeneous, Sym n -invariant multinomial ofdegree P i = ni =1 λ i . It is straightforward to write a short list of these in degrees to 3: S (0 , ,..., = 1 ,S (1 , ,..., = m + m + · · · + m n ,S (2 , ,..., = i = n X i =1 m i + X ≤ i Let k, n, and M ( t ) be as in Equation (4.1) and let V bethe Vandemonde determinant. Then the power series expansion about of det( M ( t )) begins det( M ( t )) = ε k,n V ( k ( n − k ))! c t k ( n − k ) + . . . , where ε k,n = ( − k ( n − k )+ σ ( k )+ σ ( n − k ) , and c = ( k ( n − k ))! i = k Y i =1 ( i − n − k + i − . (4.3) Proof. In order to get a non-zero determinant when t = 0, we need the first k rows to yield the powers m n − kj , m n − k +1 j , . . . , m n − j . Given that there areexisting powers m i , m i , . . . m k − i we require j = n − X j = n − k j − j = k − X j =0 = k (2 n − k − − k ( k − k ( n − k ) , derivatives. The ( k ( n − k )) th derivative of det( M ( t )) evaluated at t = 0 isthen some multiple of the Vandemonde determinant V .For 1 ≤ i ≤ k let l i = ( n − k ) + k − i = n − i . For each σ ∈ Sym k , letthe number of derivatives of the σ ( i ) th line be given by( l i − σ ( i ) + 1) . (Hence the power of m contributed by the σ ( i ) th row is l i ). Using themultinomial version of the Leibniz rule and weighting the resulting matrixby the sign of the permutation σ (needed to reorder the rows so the powers f m i run from 1 to n − 1) we obtain the ( k ( n − k )) th derivative of det( M ( t ))at t = 0 is given by ε k,n V X σ ∈ Sym k sgn( σ ) [ k ( n − k )]!( l − σ (1) + 1)!( l − σ (2) + 1)! . . . ( l k − σ ( k ) + 1)! . The result now follows from the discussion in [10] where the quantity insidethe brackets is shown to compute the number of standard Young tableau ofrow structure ( n − k, n − k, . . . , n − k ) ∈ N k . The formula for c can be computed by the famous Hook Length formula(see Section 4.1 in [10]). This gives the result. (cid:3) The formula for c (up to factors of π ) recovers the volume of Gr k ( C n ) asfirst computed by Schubert [12]. We will see that c is essentially the firstterm in the expansion of the Duistermaat–Heckman integral (2.4) which in-deed should be the volume of the manifold computed with respect to thesymplectic form.In order to compute the stability integral (2.2), we will require the coef-ficient of t k ( n − k )+3 in the power series expansion about 0 of det( M ( t )).After factoring out V , this coefficient will be a combination of the Schurpolynomials S (3 , ,..., , S (2 , , ..., , and S (1 , , , ..., . The coefficient of eachpolynomial can be computed in terms constant c given by Equation (4.3). Lemma 4.3. Let c ( k ( n − k ) + 3)! be the coefficient of t k ( n − k )+3 in the powerseries expansion about of det( M ( t )) and let ε k,n be as in Lemma 4.2. Then c = ε k,n V ( c (3 , , S (3 , ,..., + c (2 , , S (2 , , ..., + c (1 , , S (1 , , , ..., ) , where c (3 , , = [ k ( n − k ) + 1][ k ( n − k ) + 2][ k ( n − k ) + 3]6 k ( k + 1)( k + 2) n ( n + 1)( n + 2) c , (4.4) c (2 , , = [ k ( n − k ) + 1][ k ( n − k ) + 2][ k ( n − k ) + 3]3 ( k − k ( k + 1)( n − n ( n + 1) c , (4.5) c (1 , , = [ k ( n − k ) + 1][ k ( n − k ) + 2][ k ( n − k ) + 3]6 ( k − k − k ( n − n − n c . (4.6) Proof. Let χ = (3 , , . . . , , χ = (2 , , , . . . , , and χ = (1 , , , , . . . , ∈ Z k .For i = 1 , υ = ( n − k, n − k, . . . , n − k ) + χ i . Furthermore, for j = 1 , , . . . , k let l j = ( υ j + k − j ) . Let row σ ( j ) have ( l j − σ ( j ) + 1) derivatives so, as in the proof of Lemma4.3, the power of the m s in row σ ( j ) is l j . For a fixed element σ ∈ Sym k ,computing the derivative of det( M ( t )) at 0 and after dividing through by V , the distribution of the powers shows we get (abusing notation slightly)the Schur polynomial S χ i .Using the multinomial version of the Leibniz rule where we sum only overthe distribution of derivatives that yield powers of m running 1 , , . . . , n + 2and weighting the resulting matrix by the sign of the permutation σ we btain that this part of the ( k ( n − k ) + 3) rd derivative of det( M ( t )) /V at t = 0 (and hence the coefficient of S χ i ) is given by ε k,n V X σ ∈ Sym k sgn( σ ) [ k ( n − k ) + 3]!( l − σ (1) + 1)!( l − σ (2) + 1)! . . . ( l k − σ ( k ) + 1)! . Again, the discussion in [10] shows that the quantity inside the bracketscomputes the number of standard Young tableau of row structure ( n − k, n − k, . . . , n − k ) + χ i . The result follows from the Hook Length formula. (cid:3) Proof of Theorem B The Lie algebra of SU ( n ), su ( n ), is identified with trace-free n × n skew-Hermitian matrices. The rank of SU ( n ) is n − T being given by diagonal matrices. Hence the Lie algebra of T , t , can beidentified withDiag( √− µ , √− µ , . . . , √− µ n ) with X i µ i = 0 . The roots can identified with e j − e l for j = l where e j is the diagonal ma-trix with the entry √− j th coefficient and we are using the innerproduct h X, Y i = tr( X ∗ Y ) to identify su ( n ) and su ∗ ( n ). The Weyl group W ∼ = Sym n acts on T by permuting the elements of the diagonal and W acts on t permuting the √− µ i .The adjoint orbits we consider can be represented by an element ξ ∈ t .We let ξ = Diag( √− µ , . . . , √− µ | {z } k entries , √− µ , √− µ , . . . , √− µ | {z } n − k entries )where µ > kµ + ( n − k ) µ = 0. In fact, it will be useful to fix µ = n − kn and µ = − kn , so that µ − µ = 1 . We get an identification of O ξ with Gr k ( C n ) by consid-ering the k -plane generated by the span of the √− µ eigenspace at eachpoint in the orbit.The vector D ∈ t given by D = Diag(2 π √− m , π √− m , . . . , π √− m n ),where m j ∈ Z , m j = m l for j = l , and P j m j = 0, generates a circle ac-tion on O ξ . Recall the function f D : O ξ → R defined by Equation (2.3). ByLemma 2.9, f D is an eigenfunction of the Laplacian and by Lemma 2.10, thefixed points of the circle action (or equivalently the critical points of f D ) arethe orbit of ξ under the Weyl group Sym n which are the vectors consistingof to the n C k possible placements of the µ s. If we index a fixed point ofthe circle action by the k -element set J ⊂ { , , . . . , n } corresponding to thisplacement and denoting such a fixed point q J , then the value of the function f D at this point is f D ( q J ) = 2 π µ X j ∈ J m j + µ X j ∈ J c m j . he set of roots α ∈ Λ R such that h α, q J i > e j − e j where j ∈ J and j ∈ J c . Hence the weight of the induced action on the holomorphictangent space at q J is ̟ ( q J ) = Y j ∈ J, l ∈ J c ( m j − m l ) . The Duistermaat–Heckman formula yields Z O ξ e − tf D ( p ) ω n ( n − k ) ( p ) = [ k ( n − k )]! t n ( n − k ) X J ⊂{ , ,...,n } : | J | = k e − ( µ P j ∈ J m j + µ P j ∈ Jc m j ) πt Q j ∈ J, l ∈ J c ( m j − m l ) . We manipulate this expression by pulling out the Vandemonde factor V = Y ≤ j Berline, N., and Vergne, M. Fourier transforms of orbits of the coadjoint rep-resentation. In Representation theory of reductive groups (Park City, Utah, 1982) ,vol. 40 of Progr. Math. Birkh¨auser Boston, Boston, MA, 1983, pp. 53–67.[2] Besse, A. L. Einstein manifolds . Classics in Mathematics. Springer-Verlag, Berlin,2008. Reprint of the 1987 edition.[3] Bott, R. The geometry and representation theory of compact Lie groups. In Proceed-ings of the SRC/LMS Research Symposium held in Oxford, June 28–July 15, 1977 (1979), G. L. Luke, Ed., vol. 34 of London Mathematical Society Lecture Note Series ,Cambridge University Press, Cambridge-New York, pp. v+341.[4] Cao, H.-D., Hamilton, R., and Ilmanen, T. Gaussian densities and stability forsome Ricci solitons. - (2004). preprint, arXiv:math/0404165 [math.DG].[5] Cao, H.-D., and He, C. Linear stability of Perelman’s ν -entropy on symmetricspaces of compact type. J. Reine Angew. Math. 709 (2015), 229–246.[6] Cao, H.-D., and Zhu, M. On second variation of Perelman’s Ricci shrinker entropy. Math. Ann. 353 , 3 (2012), 747–763.[7] Chow, B. The Ricci flow on the 2-sphere. J. Differential Geom. 33 , 2 (1991), 325–334. Coxeter, H. S. M. The product of the generators of a finite group generated byreflections. Duke Math. J. 18 (1951), 765–782.[9] Duistermaat, J., and Heckman, G. On the variation in the cohomology of thesymplectic form of the reduced phase space. Inventiones Mathematicae 69 (1982),259–269.[10] Fulton, W., and Harris, J. Representation theory , vol. 129 of Graduate Texts inMathematics . Springer-Verlag, New York, 1991. A first course, Readings in Mathe-matics.[11] Gasqui, J., and Goldschmidt, H. Radon transforms and the rigidity of the Grass-mannians , vol. 156 of Annals of Mathematics Studies . Princeton University Press,Princeton, NJ, 2004.[12] Griffiths, P., and Harris, J. Principles of algebraic geometry . Wiley-Interscience[John Wiley & Sons], New York, 1978. Pure and Applied Mathematics.[13] Hall, S. J. The canonical Einstein metric on G is dynamically unstable under theRicci flow. Bull. Lond. Math. Soc. 51 , 3 (2019), 399–405.[14] Hall, S. J., and Murphy, T. On the linear stability of K¨ahler-Ricci solitons. Proc.Amer. Math. Soc. 139 , 9 (2011), 3327–3337.[15] Hall, S. J., and Murphy, T. Variation of complex structures and the stability ofK¨ahler-Ricci solitons. Pacific J. Math. 265 , 2 (2013), 441–454.[16] Hall, S. J., and Murphy, T. On the spectrum of the Page and the Chen-LeBrun-Weber metrics. Ann. Global Anal. Geom. 46 , 1 (2014), 87–101.[17] Hamilton, R. S. The Ricci flow on surfaces. In Mathematics and general relativity(Santa Cruz, CA, 1986) , vol. 71 of Contemp. Math. Amer. Math. Soc., Providence,RI, 1988, pp. 237–262.[18] Haslhofer, R., and M¨uller, R. Dynamical stability and instability of Ricci-flatmetrics. Math. Ann. 360 , 1-2 (2014), 547–553.[19] Humphreys, J. E. Introduction to Lie algebras and representation theory , vol. 9of Graduate Texts in Mathematics . Springer-Verlag, New York-Berlin, 1978. Secondprinting, revised.[20] Isenberg, J., Knopf, D., and Sesum, N. Non-K¨ahler Ricci flow singularities thatconverge to K¨ahler-Ricci solitons. preprint, arXiv:1703.02918 [math.DG].[21] Kirillov, A. A. Merits and demerits of the orbit method. Bull. Amer. Math. Soc.(N.S.) 36 , 4 (1999), 433–488.[22] Knopf, D., and ˇSeˇsum, N. Dynamic instability of CP N under Ricci flow. J. Geom.Anal. 29 , 1 (2019), 902–916.[23] Kr¨oncke, K. Stability and instability of Ricci solitons. Calc. Var. Partial DifferentialEquations 53 , 1-2 (2015), 265–287.[24] Kr¨oncke, K. Stability of Einstein metrics under Ricci flow. Comm. Anal. Geom. (toappear). arXiv:1312.2224 [math.DG].[25] Matsushima, Y. Remarks on K¨ahler–Einstein manifolds. Nagoya Math. J. 46 (1972),161–173.[26] M´aximo, D. On the blow-up of four-dimensional Ricci flow singularities. J. ReineAngew. Math. 692 (2014), 153–171.[27] McDuff, D., and Salamon, D. Introduction to symplectic topology , second ed.Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press,New York, 1998.[28] Paradan, P.-E. The Fourier transform of semi-simple coadjoint orbits. J. Funct.Anal. 163 , 1 (1999), 152–179.[29] Perelman, G. The entropy formula for the Ricci flow and its geometric applications. - (2002). preprint, arXiv:math/0211159 [math.DG].[30] Rossmann, W. Kirillov’s character formula for reductive Lie groups. Invent. Math.48 , 3 (1978), 207–220.[31] Sesum, N. Linear and dynamical stability of Ricci-flat metrics. Duke MathematicalJournal 133 , 1 (2006), 1–26.[32] Shephard, G. C., and Todd, J. A. Finite unitary reflection groups. Canadian J.Math. 6 (1954), 274–304.[33] Tian, G., and Zhu, X. Convergence of the K¨ahler-Ricci flow on Fano manifolds. J.Reine Angew. Math. 678 (2013), 223–245. chool of Mathematics and Statistics, Herschel Building, Newcastle Univer-sity, Newcastle-upon-Tyne, NE1 7RU E-mail address : [email protected] Department of Mathematics, California State University Fullerton, 800 N.State College Bld., Fullerton, CA 92831, USA. E-mail address : [email protected] School of Mathematics and Statistics, Herschel Building, Newcastle Univer-sity, Newcastle-upon-Tyne, NE1 7RU E-mail address : [email protected]@ncl.ac.uk