Atiyah-like results for the gradient map on probability measures

0.21 MB·

by  Leonardo Biliotti

ATIYAH–LIKE RESULTS FOR THE GRADIENT MAP ON
PROBABILITY MEASURES
7
1
LEONARDOBILIOTTI AND ALBERTO RAFFERO
0
2

n
a
J Abstract. Given a K¨ahler manifold (Z,J,ω) and a compact real submanifold M ⊂ Z,

7
we study the properties of the gradient map associated with the action of a noncompact
1
real reductive Lie group G on the space of probability measures on M. In particular, we

proveconvexityresultsforsuchmapwhenGisAbelianandweinvestigatehowtoextend
]
G them to thenon-Abelian case.
D
. 1. Introduction
h
at Let (Z,J,ω) be a compact connected Ka¨hler manifold and let U be a compact connected
m Lie group with Lie algebra u. Assume that U acts on Z by holomorphic isometries and
[ in a Hamiltonian fashion with momentum mapping µ : Z → u∗. It is well-known that the
C
U-action extends to a holomorphic action of the complexification U of U. Moreover, the
1
C
v latter gives rise to a continuous action of U on the space of Borel probability measures on
9 Z endowed with the weak* topology. We denote such space by P(Z).
7 Recently, the first author and Ghigi studied the properties of the UC-action on P(Z)
7
using momentum mapping techniques [5]. Although it is still not clear whether any rea-
4
0 sonable symplectic structure on P(Z) may exist (but see [15] for something similar on
1. the Euclidean space), in this setting it is possible to define an analogue of the momentum
0 mapping, namely
7
1 F :P(Z) → u∗, F(ν):= µ(z)dν(z).
: ZZ
v
i F is called gradient map. Using it, the usual concepts of stability appearing in Ka¨hler
X
geometry [16, 19, 20, 21, 22, 29, 31, 33, 35, 36] can be defined for probability measures, too.
r C
In[5], theauthorswereinterested indeterminingtheconditions forwhichtheU -orbitof
a
a given probability measure ν ∈ P(Z) has non-empty intersection with F−1(0), whenever
0 belongs to the convex hull of µ(Z). This problem is motivated by an application to upper
boundsforthefirsteigenvalue oftheLaplacian actingonfunctions(seealso[1,3,4,10,28]).
Furthermore, they obtained various stability criteria for measures.
C
Stability theory for the action of a compatible subgroup G of U was analyzed by the
first author and Zedda in [9].
2010 Mathematics Subject Classification. 53D20.
Key words and phrases. gradient map, probability measures, convexity.
TheauthorswerepartiallysupportedbyFIRB“Geometriadifferenzialeeteoriageometricadellefunzioni”
of MIUR, and by GNSAGA of INdAM. The first author was also supported by PRIN “Variet`a reali e
complesse: geometria, topologia e analisi armonica” of MIUR.
1
2 LEONARDOBILIOTTIANDALBERTORAFFERO
C
Recall that a closed subgroup G of U with Lie algebra g is said to be compatible if
C
the Cartan decomposition U = Uexp(iu) induces a Cartan decomposition G = Kexp(p),
where K := G∩U and p:= g∩iu is a K-stable linear subspace of iu.
Identify u∗ with u by means of an Ad(U)-invariant scalar product on u. For each z ∈ Z,
letµ (z)denote−itimes thecomponentof µ(z)inthedirection ofip ⊂ u. ThisdefinesaK-
p
equivariant map µ : Z → p, which is called G-gradient map associated with µ [23, 25, 26].
p
Thename is dueto the fact that the fundamental vector field β ∈ X(Z) of any β ∈ p is the
Z
β
gradient of the function µ (·) := hµ (·),βi with respect to the Riemannian metric ω(·,J·),
p p
being h·,·i an Ad(K)-invariant scalar product on p.
If M is a compact G-stable real submanifold of Z, we can restrict µ to M. Moreover,
p
the G-action on M extends in a natural way to a continuous action on P(M), and the map
F :P(M) → p, F (ν):= µ (x)dν(x),
p p p
Z
M
is the analogue of the G-gradient map in this setting. It is not hard to prove that its image
coincides with the convex hull of µ (M) in p (cf. Proposition 3.4).
p
Fix a probability measure ν ∈ P(M). Having in mind the classical convexity results for
the momentum mapping [2, 18, 30] and for the G-gradient map [23, 27], in this paper we
are interested in studying the behaviour of F on the orbit G·ν.
p
Let a ⊂ p be an Abelian subalgebra of g. Then, the Abelian Lie group A := exp(a) is
compatible and an A-gradient map is given by µ := π ◦µ , where π is the orthogonal
a a p a
projection onto a. In Section 4, we prove a result which can be regarded as the analogue of
an Atiyah’s theorem [2] in our setting (see also [27]):
Theorem. The image of the map F : A · ν → a is an open convex subset of an affine
a
subspace ofawithdirection a⊥. Moreover, F (A·ν)isthe convexhullofF (A·ν∩P(M)A),
ν a a
where P(M)A is the set of A-fixed measures.
As an immediate consequence of the above theorem, we get that F (A· ν) is an open
a
convex subset of a whenever the Lie algebra of the isotropy group A is trivial (Corollary
ν
4.4). The image F (A·ν) of the orbit A·ν is contained in the convex hull of µ (M). Hence,
a a
assuming that P := µ (M) is a polytope, it is natural to investigate under which conditions
a
F (A·ν) coincides with int(P). In Theorem 4.7, we show that this happens if A is trivial
a ν
and for any β ∈ p the unstable manifold corresponding to the (unique) maximum of the
Morse-Bott function µβ : M → R has full measure.
p
In Section 5, we focus our attention on the non-Abelian case. Let Ω(µ ) denote the
p
interior of the convex hull of µ (M) in p. In Theorem 5.2, we prove that F (G·ν) =Ω(µ )
p p p
under a mild regularity assumption on the probability measure ν, and that the map
F : G→ Ω(µ ), F (g) := F (g·ν),
ν p ν p
is a smooth fibration. We point out that the assumptions in Theorem 4.7 are weaker than
those of Theorem 5.2. Finally, if ν is a K-invariant smooth probability measure on M, we
show that the map F descends to a map on G/K which is a diffemorphism onto Ω(µ )
ν p
(Corollary 5.3).
The present paper is organized as follows. In Section 2, we review the main properties
of compatible subgroups and of the G-gradient map. In Section 3, we recall some useful
ATIYAH–LIKE RESULTS FOR THE GRADIENT MAP ON PROBABILITY MEASURES 3
results on measures and we introduce the gradient map. The convexity properties of the
gradient map in the Abelian and in the non-Abelian case are investigated in Section 4 and
in Section 5, respectively. Finally, in Appendix A, we prove a technical result which is of
interest in Section 5.
2. Preliminaries
2.1. Cartan decomposition and compatible subgroups. Let U be a compact con-
C
nected Lie group, denote by u its Lie algebra and by U its complexification. It is well-
C
known (see for instance [32]) that U is a complex reductive Lie group with Lie algebra
C
u = u⊕iu and that it is diffeomorphic to U×iu via the real analytic map
C
U×iu → U , (u,iξ) 7→ uexp(iξ).
C C
The resulting decomposition U = Uexp(iu) is called Cartan decomposition of U .
C
Aclosed connected subgroupG ⊆ U with Lie algebra g is said to becompatible with the
C
Cartan decomposition of U if G = Kexp(p), whereK := G∩Uandp := g∩iuis aK-stable
linear subspace of iu (cf. [25, 26]). In such a case, K is a maximal compact subgroup of G.
The Lie algebra of G splits as g = k⊕p, where k := Lie(K), and the following inclusions
hold
[k,k] ⊂ k, [k,p] ⊂ p, [p,p] ⊂ k.
C C
OntheLiealgebrau = u⊕iuthereexistsanondegenerate,Ad(U )-invariant, symmetric
R-bilinear form B :uC×uC → R which is positive definite on iu, negative definite on u and
such that the decomposition u⊕iu is B-orthogonal (see e.g. [6, p. 585]). In what follows,
we let h·,·i := B|iu×iu.
C
Whenever G = Kexp(p) is a compatible subgroup of U , the restriction of B to g is
Ad(K)-invariant, positive definite on p, negative definite on k, and fulfils B(k,p) = 0.
C
2.2. The G-gradient map. Let U and U be as in §2.1. Consider a compact connected
C
Ka¨hlermanifold(Z,J,ω),assumethatU actsholomorphicallyonitandthataHamiltonian
action of U on Z is defined. Then, the Ka¨hler form ω is U-invariant and there exists a
momentum mapping µ :Z → u∗. By definition, µ is U-equivariant and for each ξ ∈ u
dµξ = ι ω,
ξZ
where µξ ∈ C∞(Z) is defined by µξ(z) = µ(z)(ξ), for every point z ∈ Z, and ξ ∈ X(Z)
Z
is the fundamental vector field of ξ induced by the U-action, namely the vector field on Z
whose value at z ∈ Z is
d
ξ (z) = exp(tξ)·z.
Z
dt(cid:12)
(cid:12)t=0
Since U is compact, we can identify u∗ (cid:12)with u by means of an Ad(U)-invariant scalar
(cid:12)
product on u. Consequently, we can regard µ as a u-valued map.
C
Let G = Kexp(p) be a compatible subgroup of U . The composition of µ with the
orthogonal projection of u onto ip ⊂ u defines a K-equivariant map µ : Z → ip, which
ip
represents the analogue of µ for the G-action. Following [23, 25, 26], in place of µ we
ip
consider
µ : Z → p, µ (z) := −iµ (z).
p p ip
4 LEONARDOBILIOTTIANDALBERTORAFFERO
C
As the U -action on Z is holomorphic, for every β ∈ p the fundamental vector field β ∈
Z
X(Z) induced by the G-action is the gradient of the function
µβ : Z → R, µβ(z) := hµ (z),βi,
p p p
with respect to the Riemannian metric ω(·,J·). This motivates the following
Definition 2.1. µ is called G-gradient map associated with µ.
p
Let M be a G-stable submanifold of Z. We use the symbol µ to denote the G-gradient
p
map restricted to M, too. Then, for any β ∈ p the fundamental vector field β ∈ X(M)
M
is the gradient of µβ : M → R with respect to the induced Riemannian metric on M.
p
β
Moreover, if M is compact, µ is a Morse-Bott function (see e.g. [6, Cor. 2.3]). Thus,
p
β
denoted by c < ··· < c the critical values of µ , M decomposes as
1 r p
r
(2.1) M = W ,
j
jG=1
whereforeach j = 1,...,r,W istheunstablemanifoldofthecriticalcomponent(µβ)−1(c )
j p j
β
for the gradient flow of µ (cf. [24, 25]).
p
3. The gradient map on probability measures
In the first part of this section we recall some known results about measures. The reader
may refer for instance to [12, 14] for more details.
Let M be a compact manifold and let M(M) denote the vector space of finite signed
Borel measures on M. By [14, Thm. 7.8], such measures are Radon. Then, by the Riesz
Representation Theorem [14, Thm. 7.17], M(M) is the topological dual of the Banach
space (C(M),k·k ), namely the space of real valued continuous functions on M with the
∞
sup-norm. As a consequence, M(M) is endowed with the weak* topology [14, p. 169].
ThesetofBorelprobability measuresonM isthecompactconvexsubsetP(M) ⊂ M(M)
given by the intersection of the cone of positive measures on M and the affine hyperplane
{ν ∈ M(M) | ν(M) =1}. Observe that the weak* topology on P(M) is metrizable, since
C(M) is separable [12, p. 426].
Given a measurable map f : M → N between measurable spaces and a measure ν on M,
the image measure f∗ν of ν is the measure on N defined by f∗ν(A) := ν(f−1(A)) for every
measurable set A ⊆ N. f∗ν satisfies the following change of variables formula
(3.1) h(y)d(f∗ν)(y)= h(f(x))dν(x).
Z Z
N M
When a Lie group G acts continuously on a compact manifold M, it is possible to define
an action of G on P(M) as follows:
(3.2) G×P(M) → P(M), (g,ν) 7→ g∗ν := (Ag)∗ν,
where for each g ∈ G
A : M → M, A (x) := g·x,
g g
ATIYAH–LIKE RESULTS FOR THE GRADIENT MAP ON PROBABILITY MEASURES 5
is the homeomorphism induced by the G-action on M. By [5, Lemma 5.5], the action (3.2)
is continuous with respect to the weak* topology on P(M). In whatfollows, we denote this
action by a dot, i.e., g·ν := g∗ν whenever g ∈ G and ν ∈ P(M).
The next lemma is an immediate consequence of [5, Lemma 5.8].
Lemma 3.1. Let M be a compact manifold endowed with a smooth action of a Lie group
G. Consider ν ∈M(M), ξ ∈ g, and suppose that ξ vanishes ν-almost everywhere. Then,
M
exp(Rξ) is contained in the isotropy group G of ν.
ν
Proof. Since ξ vanishes ν-almost everywhere, its flow
M
ϕ :M → M, ϕ (x) = exp(tξ)·x,
t t
satisfies ϕ ν = ν for any t ∈ R by [5, Lemma 5.8]. (cid:3)
t∗
Let us focus on the setting (M,G,K,µ ) introduced at the end of §2.2. From now on,
p
we assume that the G-stable submanifold M ⊂ Z is compact. By the above results, the
group G = Kexp(p) acts continuously on P(M). Moreover, albeit a reasonable symplectic
structure on P(M) does not seem to exist, it is possible to define a map which can be
regarded as the analogue of the G-gradient map µ for the action of G on P(M).
p
Definition 3.2. The gradient map associated with the action of G on P(M) is
F :P(M) → p, F (ν):= µ (x)dν(x).
p p p
Z
M
Remark 3.3. By [9, Prop. 45], F is precisely the gradient map of a Kempf-Ness function
p
for (P(M),G,K). Thus, it is continuous and K-equivariant (cf. [9, Sect. 3]).
Using F , the usual concepts of stability [16, 19, 20, 21, 22, 29, 31, 33, 35, 36] can be
p
defined for probability measures, too (see also [5, 9]). For instance, a measure ν ∈ P(M)
is said to be stable if
G·ν ∩F−1(0) 6= ∅,
p
and g := Lie(G ) is conjugate to a subalgebra of k. In such a case, G is compact [5,
ν ν ν
Cor. 3.5].
In the light of previous considerations, it is natural to ask whether established results for
the G-gradient map [2, 11, 18, 23, 27] can be proved also for the gradient map. Here, we
focus our attention on convexity properties of F . We begin with the following observation.
p
Proposition 3.4. The image of the gradient map F :P(M) → p coincides with the convex
p
hull E(µ ) of µ (M) in p.
p p
Proof. Consider ν ∈ P(M). Observe that F (ν) is the barycenter of the measure µ ν ∈
p p∗
P(µ (M)), since by the change of variables formula (3.1) we have
p
F (ν)= µ (x)dν(x) = βd(µ ν)(β).
p p p∗
Z Z
M p
Thus, F (ν) lies in E(µ ). Conversely, for any γ ∈ E(µ ), we can write
p p p
m
γ = λ γ ,
j j
Xj=1
6 LEONARDOBILIOTTIANDALBERTORAFFERO
m
for a suitable m, where λ = 1, λ ≥ 0 and γ ∈ µ (M). For each j = 1,...,m, let
j=1 j j j p
xj ∈ M be a point in thPe preimage of γj and let δxj denote the Dirac measure supported
at x . Then, γ = F (ν), where
j p
m
e ν := λ δ .
j xj
Xj=1
e
(cid:3)
Due to the previous result, in the next sections we shall study the behaviour of F on the
p
orbits of the G-action.
4. Convexity properties of F : Abelian case
p
Let a ⊂ p be a Lie subalgebra of g. Since [p,p] ⊂ k and g = k ⊕ p, a is Abelian.
The corresponding Abelian Lie group A := exp(a) ⊂ G is compatible with the Cartan
C
decomposition of U and an A-gradient map µ : M → a is given by µ := π ◦µ , where
a a a p
π is the orthogonal projection onto a. Therefore, the gradient map associated with the
a
A-action on P(M) is
F :P(M) → a, F (ν) = µ (x)dν(x).
a a a
Z
M
Fix a probability measureν ∈ P(M). We want to study thebehaviour of F on the orbit
a
A·ν. First of all, we show that A is always compatible.
ν
Lemma 4.1. The isotropy group A of ν is compatible, namely A = exp(a ).
ν ν ν
Proof. Let α := F (ν) ∈ a. Since a is Abelian, µ := µ −α is still an A-gradient map and
a a a
the corresponding gradient map F : P(M) → a satisfies
a
e
e
F (ν)= µ (x)dν(x) = F (ν)−αν(M) = 0.
a a a
Z
M
Then, A is compatiblee by [9, Prope. 20]. (cid:3)
ν
Consider the decomposition
⊥
a = a ⊕a ,
ν ν
where a⊥ν is the orthogonal complement of aν in a with respect to B|a×a. We denote by
π : a → a⊥ the orthogonal projection onto a⊥ and we let Aˆ := exp(a⊥). Since exp : a → A
ν ν ν
is an isomorphism of Abelian Lie groups, we have A = AˆA and A·ν = Aˆ ·ν.
ν
We are now ready to state the main result of this section.
Theorem 4.2. The image F (A·ν) of the orbit A·ν is an open convex subset of an affine
a
subspace of a with direction a⊥.
ν
Before proving Theorem 4.2, we show a preliminary lemma.
Lemma 4.3. The projection of F (Aˆ ·ν) onto a⊥ is convex.
a ν
ATIYAH–LIKE RESULTS FOR THE GRADIENT MAP ON PROBABILITY MEASURES 7
Proof. By [9, Thm.39], there exists a Kempf-Ness function Ψ :M×A→ R for (M,A,{e}),
where e ∈ A is the identity element. Recall that for each point x ∈ M the function Ψ(x,·)
is smooth on A, and that for every γ ∈ a
d2
(4.1) Ψ(x,exp(tγ)) ≥ 0,
dt2
and it vanishes identically if and only if exp(Rγ) ⊂ A . Moreover, for every a,b ∈ A, the
x
following condition is satisfied
(4.2) Ψ(x,ab) = Ψ(x,b)+Ψ(b·x,a).
Ψ is related to the A-gradient map µ by
a
d
(4.3) Ψ(x,exp(tγ)) = hµ (x),γi.
a
dt(cid:12)
(cid:12)t=0
We define a function f :a⊥ → R(cid:12)(cid:12) as follows
ν
f(α) := Ψ(x,exp(α))dν(x).
Z
M
We claim that f is strictly convex. By (4.1) and (4.2), for every α,β ∈a⊥ we have
ν
d2 d2
f(tβ+α) = Ψ(exp(α)·x,exp(tβ))dν(x) ≥ 0.
dt2 Z dt2
M
If it was identically zero, then d2 Ψ(exp(α)·x,exp(tβ)) would vanish ν-almost everywhere.
dt2
Asaconsequence,foreverypointxoutsideasetofν-measurezerowewouldhaveexp(Rβ) ⊂
A = A , which implies that β (x) = 0. Therefore, exp(Rβ) ⊂ A by Lemma 3.1,
exp(α)·x x M ν
which is a contradiction. By a standard result in convex analysis (see for instance [17,
p. 122]), the pushforward df : a⊥ → (a⊥)∗ is a diffeomorphism onto an open convex subset
ν ν
of (a⊥)∗. Now, using (3.1), (4.2), (4.3), for each α,β ∈ a⊥ we have
ν ν
d
df(α)(β) = f(tβ+α)
dt(cid:12)
(cid:12)t=0
(cid:12)
d
(cid:12)
= Ψ(exp(α)·x,exp(tβ))dν(x)
Z dt(cid:12)
M (cid:12)t=0
(cid:12)
(cid:12)
= h µ (exp(α)·x)dν(x),βi
a
Z
M
= h µ (y)d(exp(α)·ν)(y),βi
a
Z
M
= hF (exp(α)·ν),βi
a
= hπ(F (exp(α)·ν)),βi,
a
from which the assertion follows. (cid:3)
Corollary 4.4. If a = {0}, then F (A·ν) is convex in a and the map
ν a
FA :A → a, FA(a) := F (a·ν),
ν ν a
is a diffeomorphism onto F (A·ν).
a
8 LEONARDOBILIOTTIANDALBERTORAFFERO
Proof of Theorem 4.2. SinceA is compatible, itfollows from theproofof [9, Prop.52] that
ν
ν is supported on
Maν := {x ∈ M | ξ (x) = 0 ∀ ξ ∈ a }.
M ν
By [24, 25], there exists a decomposition
Maν = M ⊔···⊔M ,
1 n
where each M is an A-stable connected submanifold of M. Consequently,
j
n
ν = λ ν ,
j j
Xj=1
n
wherefor j = 1,...,n, ν is a probability measureon M , λ ≥ 0 and λ = 1. By [26],
j j j j=1 j
for every x ∈ M the image µ (A·x) of A·x is contained in an affine sPubspace α +a⊥ of a.
j a j ν
Then, since M is A-stable, there is a map µ :M → a⊥ such that µ (a·x) = α +µ (a·x),
j j j ν a j j
for every a ∈A. Now, we have
e e
F (a·ν) = µ (y)d(a·ν)(y)
a a
Z
M
= µ (a·x)dν(x)
a
Z
M
n
= λ µ (a·x)dν (x)
j a j
Z
Xj=1 Mj
n n
= λ α + λ µ˜ (a·x)dν (x).
j j j j j
Z
Xj=1 Xj=1 Mj
Hence, F (A·ν) ⊆ α+a⊥, where α := n λ α . Using Lemma 4.3, we can conclude that
a ν j=1 j j
F (A·ν) is an open convex subset of thPe affine subspace α+a⊥ of a. (cid:3)
a ν
From the previous result and the compactness of P(M), it follows that F (A·ν) =
a
F (A·ν) is a compact convex subset of a. Moreover, if we denote by
a
P(M)A := {ν ∈P(M) | A·ν = ν}
the set of A-fixed measures, then we have the following
Proposition 4.5. F A·ν is the convex hull of F (A·ν ∩P(M)A).
a a
(cid:0) (cid:1)
Proof. By [34, Cor. 1.4.5], it is sufficient to show that every extremal point β ∈F A·ν is
a
the image of an A-fixed measure. Consider ν ∈ A·ν such that F (ν)= β. By Theo(cid:0)rem 4(cid:1).2,
a
F (A·ν) is an open convex subset of an affine subspace α+a⊥ ⊂ a. Since β is an extremal
a νe
point, we have necessarily a⊥ = {0}. Thus,eν ∈P(M)A. e (cid:3)
νe
e
Let P := µ (M). It was shown in [23, Sect. 5] that P is a finite union of polytopes.
a e
When P is a single polytope (cf. [2, 7, 18, 23]), it is closely related to E(µ ) [8]. Moreover,
p
it follows from the previous results that the image F (A·ν) of A·ν is an open convex subset
a
of P whenever the isotropy group of ν is trivial. Therefore, it is interesting to investigate
when F (A·ν) coincides with the interior int(P) of P in a. For this reason, from now on we
a
ATIYAH–LIKE RESULTS FOR THE GRADIENT MAP ON PROBABILITY MEASURES 9
assume that P is a polytope. Under such hypothesis, for each β ∈ p every local maximum
β
of the Morse-Bott function µ is a global maximum. As a consequence, the Morse-Bott
p
β
decomposition (2.1) of M with respect to µ has a unique unstable manifold which is open
p
and dense, namely W , while the remaining unstable manifolds are submanifolds of positive
r
codimension.
Definition 4.6. Let W (M,p) denote the set of probability measures on M for which the
open unstable manifold W has full measure for every β ∈ p.
r
Atypical example of probability measures belongingto W(M,p) isgiven by smooth ones,
namely those having a smooth positive density in any chart of the manifold with respect to
the Lebesgue measure of the chart (cf. [14, Sect. 11.4]).
In a similar way as in [5, Prop. 6.8], we can prove the following
Theorem 4.7. Let ν ∈ W(M,p) and assume that A = {e}. Then, F (A·ν) coincides with
ν a
int(P).
Proof. For simplicity of notation, let O := F (A·ν) ⊂ a. We already know that O ⊆ int(P).
a
SupposebycontradictionthatOisstrictlycontainedinint(P). Then,O ⊂ P,sinceOandP
arebothconvex. Considerα ∈ P−O,α ∈ O andthelinesegment σ(t) := (1−t)α +tα .
0 1 0 1
Let t := inf{t ∈ [0,1] | σ(t) ∈ O} and α := σ(t). As O is closed, α ∈ O and t ∈ (0,1). We
claim that α ∈ ∂O∩int(P). Indeed, it is clear that α ∈ ∂O, while α ∈ int(P) follows from
α ∈ O ⊂ int(P) and t > 0. By [34], every boundary point of a compact convex set lies on
1
an exposed face, that is, it admits a supporthyperplane. Therefore, there exists β ∈ a such
that
hα,βi = maxhα,βi = suphα,βi = suphF (exp(γ)·ν),βi.
a
α∈O α∈O γ∈a
Since ν ∈ W(M,p) and µβ = µβ for every β ∈ a, it follows from [9, Cor. 54] and from the
p a
proof of [9, Thm. 53] that
maxµβ = lim µβ(exp(tβ)·x)dν(x)= lim hF (exp(tβ)·ν),βi.
M a t→+∞ZM a t→+∞ a
Consequently,
hα,βi = suphF (exp(γ)·ν),βi ≥ maxµβ = maxhρ,βi.
a a
γ∈a M ρ∈P
That being so, the linear function α 7→ hα,βi attains it maximum on P at α ∈ int(P).
Since P is convex, β must be zero, which is a contradiction. (cid:3)
5. Convexity properties of F : general case
p
Thegoal of this section is to prove a result similar to Theorem 4.7 when the group acting
on P(M) is non-Abelian.
Let G = Kexp(p) be a compatible subgroup of UC and fix ν ∈ P(M). To our purpose,
it is useful to consider the map [4, 5, 10, 28]
F :G → p, F (g) := F (g·ν),
ν ν p
10 LEONARDOBILIOTTIANDALBERTORAFFERO
where F : P(M) → p is the gradient map associated with the action of G on P(M). In
p
C
[5, Thm. 6.4], the authors showed that F is a smooth submersion when G = U and G is
ν ν
C
compact. This is true for a compatible subgroup of U , too.
Proposition 5.1. If G is compact, then F is a smooth submersion.
ν ν
Proof. We have to prove that the pushforward dF (g) : T G → p of F is surjective for
ν g ν
every g ∈ G. Let us consider the curve σ(t) := exp(tβ)·g in G, where β ∈ p. Using the
change of variables formula (3.1), we can write
F (σ(t)) = µ (exp(tβ)·x)dν(x),
ν p
Z
M
where ν := g · ν ∈ P(M). Suppose that dF (g)(σ˙(0)) =e 0. Then, denoted by k·k the
ν
Riemannian norm on M, we have
e
d
0= hdF (σ˙(0)),βi = µβ(exp(tβ)·x)dν(x) = kβ k2(x)dν(x),
ν p M
Z dt(cid:12) Z
M (cid:12)t=0 M
sincegrad(µβ) = β .Therefore,β (cid:12)(cid:12)vanishesν-almosteveerywhere. ByLemma3e.1,exp(Rβ)
p M M
is contained in Gνe = gGνg−1, which is compact. Thus, β = 0. We can conclude that
e
dF (g) is injective on the subspace dR (e)(p) of T G, being R the right translation on G.
ν g g g
By dimension reasons, dF (g) is surjective. (cid:3)
ν
Asintheprevioussection, whenevera ⊂ pisamaximalAbeliansubalgebraofgwithcor-
respondingAbelianLiegroupA := exp(a), weassumethattheimageoftheA-gradientmap
µ is a polytope. In the non-Abelian case, we can exploit the so-called KAK decomposition
a
of G (cf. [32, Thm. 7.39]) to show the following
Theorem 5.2. Let ν ∈ P(M) be a probability measure which is absolutely continuous with
respect to a K-invariant smooth probability measure ν ∈ P(M) and assume that 0 belongs
0
to the interior Ω(µ ) of E(µ ) in p. Then, F (G ·ν) = Ω(µ ) and F : G → Ω(µ ) is a
p p p p ν p
smooth fibration with compact connected fibres diffeomorphic to K.
Before proving the theorem, we make some remarks on its content. First, we observe
that the hypothesis on ν is satisfied by smooth probability measures, which constitute a
dense subset of P(M) (see for instance [12]). Moreover, it guarantees that whenever {k }
n
is a sequence in K converging to some k ∈ K, then the sequence {k ·ν} ⊂ P(M) converges
n
to k·ν in the norm
kνk := sup hdν | h ∈ C(M), sup|h| ≤ 1 ,
(cid:26)Z (cid:27)
M M
by [5, Lemma 6.11]. Finally, we underline that the assumption 0∈ Ω(µ ) is not restrictive,
p
assuchconditionisalwayssatisfieduptoreplaceGwithacompatiblegroupG′ =K′exp(p′)
such that µp′(M) = µp(M) and up to shift µp′. We will show this assertion in Proposition
A.1 of Appendix A, since most of its proof is rather technical.
Proof of Theorem 5.2. We can reason as in the proof of [5, Prop. 6.12]. First of all, notice
that ν ∈ W(M,p), since it is absolutely continuous with respect to the smooth probability
measure ν . As 0∈ Ω(µ ), for every β ∈ p the function µβ has a strictly positive maximum.
0 p p