Tube formulas for valuations in complex space forms

Given an isometry invariant valuation on a complex space form we compute its value on the tubes of sufficiently small radii around a set of positive reach. This generalizes classical formulas of Weyl, Gray and others about the volume of tubes. We also develop a general framework on tube formulas for valuations in riemannian manifolds.


Introduction
For a compact convex set A ⊂ R m , the Steiner formula computes the volume of the set A t consisting of points at distance smaller than t from A as follows Here the functionals µ i are the so-called intrinsic volumes, and the normalizing constant ω k is the volume of the k-dimensional unit ball. By Hadwiger's characterization theorem, the intrinsic volumes span the space of valuations (finitely additive functionals on convex bodies) that are continous and invariant under rigid motions. The famous tube formula of H. Weyl is the assertion that (1) holds true for A ⊂ R m a smooth compact submanifold and t ≥ 0 small enough, with the additional insight that the coefficients µ i (A) depend only on the induced riemannian structure of A. Even more generally, Federer extended the validity of (1) to the class of compact sets of positive reach. Later on, the same formula has been proven to hold for bigger classes of sets (see e.g. [20,23]). As for the coefficients µ i , the current perspective is to view them as smooth valuations in the sense of Alesker's theory of valuations on manifolds (see [7]).
Already in Weyl's original work, the tube formula was extended to the sphere and to hyperbolic space. In that case, instead of a polynomial on the radius t one has a polynomial in certain functions sin λ (t), cos λ (t) whose definition we recall in (52). Later, Gray and Vanhecke computed the volume of tubes around submanifolds of rank one symmetric spaces (cf. [26]).
All these classical tube formulas are most naturally expressed in the language of valuations on manifolds. Furthermore, this theory has allowed for the determination of kinematic formulas (a far-reaching generalization of tube formulas) in isotropic spaces. These spaces are riemannian manifolds under the action of a group of isometries that is transitive on the sphere bundle. For instance, in [15] and [16] the kinematic formulas of complex complex space forms (i.e. complex euclidean, projective and hyperbolic spaces) were obtained, and Gray's tube formulas on such spaces were recovered.
Tube formulas, however, exist also for other valuations than the volume, and these do not follow from the kinematic formulas. For instance, differentiating the Steiner formula one easily obtains In real space forms (i.e. the sphere and hyperbolic space), Santaló obtained similar tube formulas for all isometry invariant valuations (see [40]). For rank one symmetric spaces, the tube formulas of a certain class of valuations (integrated mean curvatures) were found in [26], still with a differential-geometric viewpoint. There are however many invariant valuations on these spaces that were not considered.
In this paper we prove the existence of tube formulas for any smooth valuation in a riemannian manifold. Then we develop a method to determine these formulas for the invariant valuations of an isotropic space. Using this method we compute all tube formulas explicitly in the case of complex space forms. In fact, our approach also reveals some intersting aspects in the case of real space forms.
Let us briefly describe our results. First, given a riemannian manifold M we construct a family T t of tubular operators on the space V(M ) of smooth valuations of M such that for any µ ∈ V(M ) and every compact set of positive reach A ⊂ M one has for t ≥ 0 small enough (see Definition 4.1 and Corollary 4.7). Differentiating T t at t = 0 yields an operator ∂ : V(M ) → V(M ). If G is a group of isometries of M acting transitively on the sphere bundle SM , the subspace V(M ) G of G-invariant valuations is finite dimensional, and the determination of the tube operators T t reduces to the computation of the flow generated by ∂.
Once this general framework is established we concentrate on the complex space forms CP n λ . For λ = 0 this refers to complex euclidean space C n under the group of complex isometries, and for λ = 0 this is the n-dimensional complex projective or hyperbolic space of constant holomorphic curvature 4λ, under the full group of isometries G. We simply denote V n λ,C := V(CP n λ ) G . For λ = 0, we will readily obtain the tube formulas T t µ of all translationinvariant and U (n)-invariant continuous valuations µ thanks to the existence of an sl 2 -module structure on the space Val U(n) of such valuations. This structure, discovered by Bernig and Fu in [15], is induced by two natural operators Λ, L, the first of which is a normalization of ∂.
Remarkably, it turns out that also for λ = 0 the derivation operator ∂ is closely related to the operators Λ, L of the flat space. Indeed, in Theorem 4.11 we find an isomorphism Φ λ : Val U(n) → V n λ,C such that Using the decomposition of Val U(n) into irreducible components, the computation of the tubular operator boils down to the solution of a Cauchy problem in some abstract model spaces, yielding our main result.
Theorem. There exists a basis {σ λ k,r } of the space V n λ,C of invariant valuations of CP n λ such that where We describe the basis σ λ k,r explicitly in terms of the previously known valuations τ λ k,p of [16]. The tube formulas for the τ λ k,p can be easily obtained from the previous ones, as we also provide the expression of these valuations in terms of the σ λ k,r .
Curiously, the expressions (4) are extremely similar to those obtained by Santaló in the real space form S m λ of constant curvature λ. Indeed, for a certain basis The tube formula for σ m = vol is however quite different. As an explanation for these similarities, we show in Theorem 4.12 the existence of a phenomenon similar (but not completely analogous) to (3). The paper concludes with a detailed study of the spectrum and the eigenspaces of the derivative operator ∂ in V n λ,C and V m λ,R . In particular, we compute the kernel of ∂ in V n λ,C ; i.e. we determine the invariant valuations of CP n λ for which the tube formulas are constant. We also identify the images ∂(V n λ,C ) and ∂(V m λ,R ), and we compute the preimage by ∂ of any element belonging to these subspaces.

2.1.
Valuations. Let V be a finite-dimensional real vector space, and let K(V ) be the space of convex compact subsets of V , endowed with the Hausdorff metric. A valuation on V is a map ϕ : The notion of valuation was extended to smooth manifolds by Alesker (cf. [5,6,10,7]). For simplicity we will focus on the case of a riemannian manifold M n . It is also natural to consider here the class of compact sets of positive reach in M , which we denote R(M ). The definition and some basic properties of such sets are recalled in subsection 4.2.
Let SM be the sphere bundle of M consisiting of unit tangent vectors, and let π : SM → M be the canonical projection.
where ω ∈ Ω n−1 (SM ) and η ∈ Ω n (M ), are complex-valued differential forms, and N (A) is the normal cycle of A (cf. e.g. [20]). We will denote ϕ = ω, η in this case. For any subgroup G ≤ Diff(M ), we will denote by V G (M ) the space of G-invariant valuations; i.e. µ ∈ V(M ) such that µ(gA) = µ(A) for all A ∈ R(M ) and g ∈ G.
The kernel of the map (ω, η) → ω, η was determined by Bernig and Bröcker in [13] as follows. Given ω ∈ Ω n−1 (SM ), there exists ξ ∈ Ω n−2 (SM ) such that is a multiple of α, the canonical contact form on SM . The unique n-form Dω satisfying this condition is called the Rumin differential of ω. Then ω, η = 0 if and only if Dω + π * η = 0, and One of the most striking aspects of Alesker's theory of valuations on manifolds is the existence of a natural product on V(M ), which turns this space into an algebra with χ as the unit element. The realization by Fu that this product is closely tied to kinematic formulas opened the door to the recent development of integral geometry in several spaces, including the complex space forms [1,15,16].
Another important algebraic structure is the convolution of valuations found by Bernig and Fu in linear spaces (cf. [14], but also [9]). This is a product on the dense subspace Val ∞ (V ) := Val(V ) ∩ V(V ) characterized as follows. Given A ∈ K(V ), with smooth and positively curved boundary, we have µ A (·) := vol(· + A) ∈ Val ∞ (V ). The convolution is determined by where + refers to the Minkowski sum. In particular, vol is the unit element of this operation.

Real space forms.
The fundamental examples of valuations in Euclidean space R m are the intrinsic volumes µ k . These are implicitly defined by the Steiner formula where B m in the unit ball and ω i is the volume of the i-dimensional unit ball. In particular µ 0 = χ, µ m−1 = 2 perimeter, and µ n = vol m are intrinsic volumes. We will denote by S m λ the m-dimensional complete and simply connected riemannian manifold of constant curvature λ. That is, the sphere S m ( √ λ) for λ > 0, Euclidean space R n for λ = 0, and hyperbolic space H m ( √ −λ) for λ < 0. Let G λ,R be the group of orientation preserving isometries of S m λ ; i.e. G λ,R ∼ = SO(m + 1) for λ > 0, and G λ,R ∼ = SO(m) ⋊ R m for λ = 0, while G λ,R ∼ = P SO(m, 1) for λ < 0. We will denote by V m λ,R the space of G λ,R -invariant valuations of S m λ . Let κ 0 , . . . , κ m−1 ∈ Ω m−1 (SS m λ ) G λ,R be the differential forms defined in [19, §0.4.4]. In the same paper it was shown that the R-algebra of G λ,R -invariant differential forms is generated by κ 0 , . . . , κ m−1 , α, dα. It follows by [16,Prop. 2.6] that the following valuations constitute a basis of V m λ,R In euclidean space R m these valuations are proportional to the intrinsic volumes: For general λ, the σ λ i are proportional to the valuations τ λ i appearing in [11,24] As we will see, the normalization taken for the σ λ i makes the tube formulas in V m λ,R specially simple. A stronger reason in favor of this normalization is Theorem 4.12.
2.3. Complex space forms. We denote by CP n λ the complete, simply connected n-dimensional Kähler manifold of constant holomorphic curvature 4λ; i.e. the complex projective space (with the suitably normalized Fubini-Study metric) for λ > 0, the complex euclidean space C n for λ = 0, and the complex hyperbolic space for λ < 0. For λ = 0 we let G λ,C be the full isometry group of CP n λ . For λ = 0 we put G λ,C = U (n) ⋊ C n . We denote by V n λ,C the space of G λ,C -invariant valuations on CP n λ . Let {β k,q , γ k,q } ⊂ Ω 2n−1 (SCP n λ ) G λ,C be the differential forms introduced in [15] for λ = 0, and extended to the curved case λ = 0 in [16]. Let also where dvol is the riemannian volume element. It was shown in [15,16] that these valuations µ λ k,q with max{0, k − n} ≤ q ≤ k 2 ≤ n consitute a basis of V n λ,C . It is convenient to emphasize that the µ λ k,q do not coincide with the hermitian intrinsic volumes µ M k,q for M = CP n λ introduced in [17]. For λ = 0 we simply write µ k,q instead of µ 0 k,q . We will also use the so-called Tasaki valuations It will be useful to consider the following linear isomorphisms: More generally, whenever we have a valuation ν in Val U(n) we will denote ν λ := F λ,C (ν). For instance τ λ k,q = F λ,C (τ k,q ).

Tube formulas in linear spaces
Let V be an m-dimensional euclidean vector space. Given t ≥ 0, let T t : Val(V ) → Val(V ) be given by where B m is the unit ball. We will call T t the tubular operator. Let also ∂ : Val(V ) → Val(V ) be the operator given by This operator has sometimes been denoted by Λ in the literature, but following [15] we reserve the symbol Λ for a certain normalization of ∂ (see (18)). The properties of the Minkowski sum ensure that T t+s = T t • T s = T s • T t . Differentating with respect to s at zero yields It follows that For each µ ∈ Val(V ), the map t → T t µ is a polynomial in t of degree m by (12) and the Steiner formula (7) (or by [35]). Hence Note also that, by (15) and (16), the derivative operator ∂ is (m + 1)-nilpotent; i.e. ∂ m+1 = 0. Let us compute the tube formula for the intrinsic volume µ i for each 0 ≤ i ≤ m using (17). For that purpose we first compute ∂. Since T t+s = T s • T t we have On the other hand Differentiating at s = 0 and comparing coefficients yields Finally, using (17), we get which is (2).
In order to compute the tube formulas for invariant valuations in C n (i.e. to determine T t on Val U(n) ), it will be useful to recall the sl 2 -module structure of Val U(n) found in [15]. Consider the linear maps Λ, L, H : Val(V ) → Val(V ), defined as follows where · refers to the Alesker product.
while on Val U(n) one has which implies Proof.  The decomposition into irreducible components is as follows where V (m) is the (m+1)−dimensional irreducible sl 2 -representation. In particular, for 0 ≤ 2r ≤ n, there exists a unique, up to a multiplicative constant, primitive element (i.e. anihilated by Λ) in each irreducible component of Val U(n) . By the socalled Lefschetz decomposition, the L-orbits of these primitive elements consitute a basis of Val U(n) . This basis was explictly computed in [15] as follows.
In particular the irreducible components of Val U(n) are the following subspaces We are now able to compute the tube formulas in the complex case using (17).
Proof. By [15,Lemma 5.6], and then Using (17), we obtain the tube formula These tube formulas can also be given in terms of the valuations τ k,q . To this end, we next compute their Lefschetz decomposition.
Proposition 3.5. The Lefschetz decomposition of τ k,r is given by Proof. Consider the linear map ψ : Val U(n) → Val U(n) mapping τ k,r to the left hand side of (33). We need to show that ψ = id. Let us check that this endomorphism commutes with both Λ and L. To check commutation with Λ, we only need to verify the following Comparing term by term, the previous identities boil down to which is trivial. Commutation with L is straightforward using Lπ k,i = π k+1,i . Given that ψ commutes with the operators Λ and L and Val U(n) is multiplicityfree, Schur's lemma implies that for each 0 ≤ 2r ≤ n, there exists a constant c r such that ψ| I n,r 0 = c r id.
By plugging (28) and (33) in (30) one gets the tube formulas T t τ k,p in terms of the τ i,j .
→ SM is the projection on the second factor, and φ t = φ(·, t). We define the derivative operator ∂ = ∂ M by To show that these definitions are consistent, suppose µ = ω, η = 0, and let us check that T t µ = 0 for all t ≥ 0, i.e.
as α vanishes on N (A). Since ω, η = 0, we have N (A) ω = − A η. Therefore T t µ = 0. Let us next establish some basic properties of these operators.
Proof. Given a compact smooth submanifold N ⊂ SM , Since i T and φ * t commute, the result follows.
Together with Lemma 4.1, this yields Evaluating at t = 0, this gives i).
In order to prove ii), it is enough to check that both sides have the same derivative with respect to s, as they clearly agree for s = 0. By (35), we have Since φ * t and i T commute, it follows from (35) If µ ∈ V(M ) G for a group G acting on M by isometries, then also T t µ ∈ V(M ) G . Hence, in case V G (M ) is finite-dimensional, computing T t µ boilws down to solving the Cauchy problem (36) with initial condition T 0 µ = µ; i.e.
This is the approach we will follow to obtain the tube formulas for invariant valuations in complex space forms. Note that (37) coincides with (16) except that ∂ does not need to be nilpotent for general M . For such sets A we will prove that T t µ(A) = µ(A t ) for any µ ∈ V(M ) and sufficiently small t. By the previous definition, there is a well-defined map where γ is the unique minimizing geodesic such that γ(0) = f A (p) and γ(d A (p)) = p.  i) for 0 < t < r the restriction F A | ∂At gives a bilipschitz homeomorphism between ∂A t and N (A), preserving the natural orientations, ii) the distance function d A is of class C 1 in A r \ A and φ dA(p) (F A (p)) = (p, ∇d A (p)), ∂A t = d −1 A ({t}) for 0 < t < r. In particular, each level set ∂A t with 0 < t < r is a C 1 -regular hypersurface with unit normal vector field ∇d A .
The following propositions are certainly well-known.
Proposition 4.4. For 0 < s < r = r A the set A s has positive reach and on A r \ A s we have In particular (A s ) t = A t+s for t + s < r.
To check surjectivity, given p ∈ A t \ A take ξ = F A (p), s = d A (p) and note that π • φ(ξ, s) = p.
where it is understood that σ λ −1 = 0. Let us emphasize that (40) would make formal sense but does not hold for i = m − 1.
Note that by (18) Remarkably, a similar identity holds for all λ, which will be crucial for our determination of tube formulas in CP n λ . Theorem 4.11. The linear isomorphism By combining Proposition 4.10, Proposition 3.1 and the fact ωn ωn−2 = 2π n , this is straightforward to check: . A similar phenomenon holds in real space forms, but restricted to a hyperplane of V m λ,R . Theorem 4.12. The linear monomorphism Proof. By Proposition 4.8 and Theorem 3.4 Note the difference of dimensions between the source and the target of Ψ λ . We will show that there is no isomorphism between Val O(m) and V m λ,R intertwining ∂ and Λ − λL. This is essentially due to the fact that (41) and (42) differ from (40).

A model space for tube formulas
We next perform some abstract computations that will easily lead to the tube formulas in both complex and real space forms via (62) and (64). The same approach will allow us to determine the kernel, the image, and the spectrum of the derivative operator ∂ on these spaces.

A system of differential equations. It is well-known that the operators
is the subspace of m-homogeneous polynomials: Motivated by Theorem 4.11, we consider Y λ = Y − λX, which is a derivation on C[x, y]. It will be sometimes convenient to consider the monomials m k x k y m−k . In these terms Our goal here is to solve the following Cauchy problem: find p k : R → V (m) such that i.e. to compute We will use the standard notation which is an analytic function in both λ and t, and cos λ (t) := d dt sin λ (t). Proposition 5.1. For any λ, t ∈ R, we have exp(tY λ )x = x cos λ (t) + y sin λ (t)=: u, exp(tY λ )y = y cos λ (t) − λx sin λ (t)=: v.
Proof. Since clearly In the same way we can compute exp(tY λ )y.
The following standard and elementary fact will be useful.
Lemma 5.2. Let A be an algebra. A vector field on A is a derivation iff its flow φ t satisfies φ t (pq) = φ t (p)φ t (q), ∀p, q ∈ A, ∀t ∈ R.
In other words, each φ t is an A-morphism.

5.2.
Eigenvalues and eigenvectors of Y λ . Given f : V → V an endomorphism of C-vector spaces, we denote by spec(f ) the set of eigenvalues of f and by E α (f ) the eigenspace associated to each α ∈ spec(f ).
Proof. The result is trivial to check for m = 1 as Since Y λ is a derivation as stated.
Remark. It is interesting to notice that the spectra of Y λ and √ −λH, when restricted to each V (m) , are identical. These two operators are thus intertwined e.g. by the linear isomorphism x k y m−k → e k 1 e m−k 2 . 5.3. Image of Y λ . Using Lemma 5.4, we can conclude that Y λ | V (m) is bijective if and only if m is odd. If m is even, then the kernel is one-dimensional. An explicit description is the following.
Proposition 5.5. If m is even, then where Z m,λ := Proof. By the binomial formula if k is even, and Z m,λ (x k y m−k ) = 0 if k is odd. Therefore This shows that im(Y λ ) is a subspace of ker Z m,λ . Given that Z m,λ is not zero, we have dim ker Z m,λ = m, and by Lemma 5.4, we know that the image of Y λ | V (m) has the same dimension. This yields (58) Next we compute, for even m and given ϕ in the image of Y λ | V (m) , the preimage A simple computation using (49) shows where c m,k = 0 if m − k is even, and otherwise With these ingredients at hand, for even m, we can now compute a preimage by Y λ of any element in im Y λ as follows.
6. Tube formulas in S m λ and CP n λ Here we will obtain our main result: the tube formulas for invariant valuations of CP n λ (i.e. the tubular operator T t on V n λ,C ). We will also recover Santaló's tube formulas for V m λ,R (cf. [39]) in a way that explains the similarities between the real and the complex space forms.
6.1. Tube formulas in complex space forms. Recalling (25) and Proposition 3.3, we get an isomorphism I : W n → Val U(n) of sl 2 -modules from to Val U(n) by putting I(y 2n−4r ) = π 2r,r (i.e. mapping Y -primitive elements to Λ-primitive elements) and By Theorem 4.11, the map J λ,C := Φ λ • I : We define and arrive at our main theorem.
Remark. An interesting feature of the previous tube formulas is the following selfsimilarity property, which is explained by (62). Let G n,j λ : V n λ,C −→ V n+2j λ,C , G n,j (σ λ k,r ) = σ λ k+2j,r+j . Then one has T t • G n,j = G n,j • T t .
Theorem 6.2. The tubular operator on V m+1 λ,R is given as follows. For i = 0, . . . , m, In particular and thus These formulas where first obtained by Santaló [39].
Remark. It is worth pointing out the similarity between tube formulas in real and complex space forms. More precisely, note that the isomorphism F n,r : H 2n−4r+1 between the subspaces H 2n−4r+1 λ ⊂ V 2n−4r+1 λ,R and I n,r λ ⊂ V n λ,C commutes with the tubular operator T t . This is explained by (62) and (64). 6.3. Spectral analysis of the derivative map. Here we compute the eigenvalues and eigenvectors of ∂ λ,R and ∂ λ,C . Note that the tube formulas for such valuations are extremely simple: if ∂µ = aµ with a ∈ C, then T t µ = e at µ. Proposition 6.3. For 0 ≤ 2r ≤ n, the restriction of ∂ λ,C to I n,r λ has the following (simple) eigenvalues and eigenspaces: Hence ∂ λ,C diagonalizes on V n λ,C with the following eigenspaces: for −n ≤ j ≤ n.
Remark. We conclude from Prosposition 6.4 and Lemma 5.4 that there is no isomorphism between Val O(m) and V m λ,R intertwining Λ − λL and ∂ λ,R . Indeed, these two operators have different spectra no matter the parity of m.
6.4. Stable valuations in complex space forms. We say that a valuation ϕ ∈ V(M ) on a riemannian manifold M is stable if ∂µ = 0, or equivalently, if T t µ = µ for all t. By Proposition 6.4 and 6.3, up to multiplicative constants, the Euler characteristic is the unique isometry-invariant stable valuation in S m λ . The complex case is more interesting. ) , for each 0 ≤ 2r ≤ n.
Next we express the Euler characteristic as a combination of the stable valuations ψ 2r . Note in particular that χ is not confined to any ∂-invariant subspace I n,r λ .
Proof. Since χ is stable, it can be expressed as χ = j a j ψ 2j . By [16,Theorem 3.11] The coefficient of τ λ 2r,r in this expansion is Hence a r = λ π r 2r r r! 4 r ω 2n−2r and the result follows. 6.5. Image of ∂ λ,C and ∂ λ,R . Next we describe the image of the operators ∂ λ,C and ∂ λ,R , and we compute the preimage of any element belonging to them. Proposition 6.7. Given any ϕ = k,r a k,r σ λ k,r ∈ V n λ,C , we have ϕ ∈ im ∂ λ,C if and only if n−2r l=r a 2l,r n − 2r l − r λ n−l−r = 0, for 0 ≤ 2r ≤ n.
Proof. Note that ϕ = r ϕ r with ϕ r = k a k,r σ λ k,r is the decomposition of ϕ corresponding to V n λ,C = ⌊n/2⌋ r=0 I n,r λ . By (62) and Proposition 5.5 we have ϕ ∈ where we used (59).
Proof. This follows at once from Proposition 5.6 after decomposing ϕ = r ϕ r as in the previous proof.
Proof. By (8) and (9) where the term between brackets appears only if m − k is even. Using Proposition 4.8, this yields (80). The rest of the statement follows.