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Finiteness theorems for equifocal hypersurfaces

In this paper, we give a finiteness result on the diffeomorphism types of curvature-adapted equifocal hypersurfaces in a simply connected compact symmetric space. Furthermore, the condition curvature-adapted can be dropped if the symmetric space is of rank one.


Introduction
A hypersurface M n in a real space form N n+1 (c) with constant sectional curvature c is said to be isoparametric if it has constant principal curvatures. Since the work of Cartan and Münzner, the subject of isoparametric hypersurfaces especially in the spherical case is rather fascinating to geometers. Hitherto, the classification problem has been almost completed except for one case (see [17] and [1] for excellent surveys and [2], [10], [3], [11], [8] for recent progresses and applications).
In a general Riemannian manifold, a hypersurface is called isoparametric if its nearby parallel hypersurfaces have constant mean curvature. Note that this definition coincides with that in the case of real space forms above by a theorem of Cartan (cf. [7]). In particular, isoparametric hypersurfaces in a simply connected compact symmetric space have been found identical with equifocal hypersurfaces that introduced by Terng and Thorbergsson [14]. In fact, they also introduced equifocal submanifolds of high codimensions and established similar structural results as the classical case of isoparametric hypersurfaces and submanifolds. It is worth mentioning that [13] obtained the possible values of the multiplicities (m 1 , m 2 ) for equifocal hypersurfaces in rank two symmetric spaces , and [4] generalized Thorbergsson's result of the homogeneity of isoparametric submanifolds in Euclidean spaces of codimension at least two (see [16]) by showing that equifocal submanifolds in simply connected compact symmetric spaces of high codimensions must be homogeneous, and thus can be classified. However, the classification of equifocal hypersurfaces is still far from being reached.
In this paper, we endeavor to make some progress towards this classification problem by proving the finiteness of the diffeomorphism types of equifocal hypersurfaces. Our finiteness result relies on an additional condition that the equifocal hypersurfaces should be curvatureadapted. Nevertheless, our Theorem 1.1 generalizes the finiteness theorem for isoparametric hypersurfaces in spheres with four distinct principal curvatures given by Wu [19], since every hypersurface in a sphere is curvature-adapted. Notice that our result also covers the finiteness conclusion for isoparametric hypersurfaces in spheres with six distinct principal curvatures which can not be derived by the method of Wu [19]. Now we state the finiteness theorem as the following.
Theorem 1.1. Given a simply connected compact symmetric space N , there are only finitely many diffeomorphism classes of curvature-adapted equifocal hypersurfaces in N .
Furthermore, the condition curvature-adapted above can be dropped if the symmetric space is of rank one, i.e., Theorem 1.2. Given a simply connected compact rank one symmetric space N , there are only finitely many diffeomorphism classes of equifocal hypersurfaces in N .
In section 4, we will give some examples of equifocal hypersurfaces in CP n which are the images of one isoparametric hypersurface in S 2n+1 under the Hopf fibrations with different S 1actions. It turns out that these equifocal hypersurfaces are of different diffeomorphism types, which illustrates the non-triviality of Theorem 1.2 since now one can not expect to prove this finiteness result directly from that in spheres by using the Hopf fibrations.
2. Focal structure of equifocal hypersurfaces 2.1. Preliminaries. In this subsection, we firstly recall some fundamental definitions and results of [14].   Throughout this paper, we assume that N = G/K is a simply connected compact symmetric space, g = k + p is its Cartan decomposition, and N is equipped with the G-invariant metric g given by the restriction of the negative of the Killing form , of g to p. Then we restate a part of Theorem 1.6 of [14] in the following which will play a crucial role in the proof of our theorems later.
Theorem 2.1. ( [14]) Let M be an immersed, compact, equifocal hypersurface in the simply connected compact symmetric space N , and v a unit normal vector field. Then the following hold: (a) Normal geodesics are circles of constant length, which will be denoted by l. (b) There exist integers m 1 , m 2 , an even number 2g and 0 < θ < l 2g such that the focal points on the normal circle T and their multiplicities are m 1 if j is odd and m 2 if j is even.

2.2.
Relation between focal points and shape operators. In this subsection, we discuss the relation between focal points and shape operators of submanifolds with abelian normal bundle in the simply connected compact symmetric space N = G/K. In fact, this has been done in Section 3 of [14]. For completeness, we repeat it as follows.
Let G = Iso(N ) and M be a submanifold with abelian normal bundle in N . Let x 0 ∈ M , K = G x 0 and g = k + p be the Cartan decomposition. Let π : G → G/K = N be the canonical projection. For simplicity, we will denote by π * the restriction π * | p . Then π * : p → T N x 0 is an isomorphism, and T N x 0 is identified with p by π * . Moreover, the curvature tensor of N can be expressed as the following: Proposition 2.1. With notations as above, we have where the last equality follows from the assumption that ν(M ) is abelian.
Next we recall an elementary fact concerning the tangential map dη and Jacobi fields.
where J is the Jacobi field along γ(t) = exp x(0) (tv(0)) satisfying the initial condition is the shape operator with respect to v(0) and ∇ ⊥ is the normal connection of the submanifold M .
Then the Jacobi equation for J gives rise to the following equation for Y : Let a be a maximal abelian subalgebra in p containing a and p = a ⊕ α∈∆ p α its root space decomposition, where ad(a) 2 (z α ) = −α(a) 2 z α for z α ∈ p α . By direct computation, we know that the solution of the equation where p 0 , q 0 ∈ a, p α , q α ∈ p α and λ −1 sin λ is defined to be 1 if λ = 0. For convenience, let D 1 (a) and D 2 (a) be the operators defined as follows: for z = p 0 + α p α ∈ p with p 0 ∈ a and p α ∈ p α , Put R a (z) := π −1 * R(π * (a), π * (z))π * (a) = −ad(a) 2 (z). Then we have R a ≥ 0, D 1 (a) = cos( √ R a ) and D 2 (a) = ( √ R a ) −1 sin( √ R a ), which imply that D 1 and D 2 depend only on a, but not on the choice of the maximal abelian subalgebra a.
by the definitions of D 1 (a) and D 2 (a).
Now we can prove the following theorem of [14] where the case of N = G is explicitly presented and the general case has been abbreviated.
Proof. Since part (1) follows straightforward from Proposition 2.1, it suffices to prove part (2). For the tangent space T ν(M ) (x 0 ,v) , we can choose a natural basis which consists of vectors of the form v ′ (0) as in Lemma 2.1 and σ ′ (0) with σ(t) = exp x 0 (π * (a + tb)) = η(π * (a + tb)), where e a * denotes the tangential map of the G-action e a : N → N . Now the theorem follows from Lemma 2.1.

Hypersurfaces in spheres.
In this subsection, to warm up we apply Theorem 2.2 to investigate the focal structure of hypersurfaces in spheres. For simplicity, henceforth we identify p with T x 0 N without referring to π * .
Let M n be a hypersurface in the sphere S n+1 , G = Iso(S n+1 ) = O(n + 2). Note that the normal bundle ν(M ) is 1-dimensional and thus abelian. Given x 0 ∈ M, let K = G x 0 ∼ = O(n + 1), g = k + p be the Cartan decomposition, and a ∈ p be a unit vector normal to M at x 0 . Then Applying Theorem 2.2, we get the following proposition immediately.
Proposition 2.3. With notations as above and for t ∈ R, where A a is the shape operator of M with respect to a ∈ ν(M ) x 0 ⊂ p.
Remark 2.1. Denote by L the number of focal points along the normal geodesic exp x 0 (ta), t ∈ [0, π), of M with respect to x 0 . The proposition above implies which is the Theorem 1 of [12]. Combining this with Theorem 2.1 shows that, for any given equifocal (isoparametric) hypersurface M in S n+1 , the distance between the two focal submanifolds satisfies d(M + , M − ) ≥ π n+1 , which says that d(M + , M − ) has a lower bound that depends only on S n+1 . Such type fact is crucial for our proof of the finiteness theorem later.
2.4. Hypersurfaces in simply connected compact symmetric spaces. In this subsection, we will firstly apply Theorem 2.2 to investigate the focal structure of hypersurfaces in the simply connected compact symmetric space N . From this focal structure we derive some corollaries when the hypersurface is curvature-adapted, or in addition it is equifocal.
Let N n+1 be a simply connected compact symmetric space of dimension n + 1 and rank r, G = Iso(N ), and M n be a hypersurface in N . Observe that the normal bundle of a hypersurface is always abelian. Let x 0 ∈ M , K = G x 0 , g = k + p be the Cartan decomposition, and a ∈ p be a unit vector normal to M at x 0 . Let a be a maximal abelian subalgebra in p containing a, and p = a ⊕ α∈∆ p α its root space decomposition. One can choose a basis for each p α and a so as to constitute a basis of p = T N x 0 including a. Then since R a = −ad(a) 2 , under this basis we can diagonalize the operator where A a is the shape operator of M with respect to a and d i cot Proof. It follows from the discussions above that under some basis of T M x 0 , we have Note that A ta = tA a and thus which immediately implies the conclusion by Theorem 2.2.
Recall that a hypersurface M is called curvature-adapted if its shape operator A a commutes with the normal Jacobi operator R a | T M for a ∈ ν(M ), or equivalently, they are simultaneously diagonalizable. Then it follows from the proposition above the following corollary which can be regarded as a generalization of the theorem of Pinkall [12] in the spherical case to more general ambient spaces (see Remark 2.1).
Corollary 2.1. With notations as above, suppose that M n is a curvature-adapted hypersurface and denote by L the number of focal points along the normal geodesic exp x 0 (ta), t ∈ [0, π B ), of M with respect to x 0 , then where s ≤ n + 1 − r.
Proof. The first equality follows immediately from Proposition 2.4. Since M is curvatureadapted, A a and √ R a | a ⊥ can be diagonalized simultaneously. Therefore, the equality (2.2) holds if and only if d i cot(td i ) or 1 t equals some eigenvalue of A a , which can occur at most (s + 1)n times for t ∈ [0, π B ). This proves the second inequality of the corollary.
As a direct application we obtain the following estimate for a universal lower bound of the distance between the two focal submanifolds of any curvature-adapted equifocal hypersurface.
Corollary 2.2. With notations as above, suppose that M n is a curvature-adapted equifocal hypersurface in N and M ± are the focal submanifolds defined in Theorem 2.1, then the distance between the two focal submanifolds satisfies i.e., d(M + , M − ) has a lower bound which only depends on N .
Proof. One can conclude from Theorem 2.1 that along a normal geodesic exp x 0 (ta), the focal points of M in N with respect to x 0 occur alternately and equidistantly in the two focal submanifolds M + and M − . Therefore, the distance between any two succeeding focal points occurring in exp x 0 (ta), t ∈ [0, π B ), equals d(M + , M − ) and the distance between the last focal point and exp x 0 ( π B a) is no more than d(M + , M − ). Hence we have where L is the number of focal points along the normal geodesic exp x 0 (ta), t ∈ [0, π B ), of M with respect to x 0 . Then applying the inequality of Corollary 2.1 will complete the proof.
As another corollary of proposition 2.4, we observe a direct proof of the following result which is a part of Theorem 1.4 of [7] proved by some knowledge of algebraic geometry. Proof. Given a unit normal field v for a curvature-adapted equifocal hypersurface M n in the simply connected compact symmetric space N n+1 , we have n continuous functions, the principal curvatures λ 1 ≥ λ 2 ≥ ... ≥ λ n on M . For x ∈ M and t ∈ R, we know from Proposition 2.4 that, exp x (tv) is a focal point of M with respect to x if and only if det( where A v is the shape operator of M at x with respect to v. Since now M is curvature-adapted, A v can be diagonalized simultaneously with √ R v | v ⊥ . Hence the equation above holds if and only if for some t = t(x) ∈ R, 1 ≤ i ≤ s and 1 ≤ k ≤ n, λ k (x) = d i cot(td i ), or λ k (x) = 1 t . On the other hand, since M is equifocal, such functions t = t(x) should be constant on M by Theorem 2.1. In fact, by Theorem 2.1, each normal geodesic exp x (tv) is a circle of constant length l and there exists an even number 2g and 0 < θ < l 2g , such that the focal points on each normal circle T x = exp(ν(M ) x ) are x(j) = exp((θ + (j−1)l 2g )v(x)), 1 ≤ j ≤ 2g, which means that the functions t(x) = θ + (j−1)l 2g are constant on M . In conclusion, we have Now by the assumption that N is a rank one symmetric space, we know that the numbers d i , i = 1, ..., s, are constant on M independent of the choice of x ∈ M . Finally, since the principal curvature functions are continuous and the right set above is totally discontinuous, it follows that λ 1 , λ 2 , ..., λ n are constant on M .
The proof is now completed.

Proof of Theorem 1.1
In this section, based on the results in previous sections we are now able to prove the finiteness Theorem 1.1. Firstly we recall a general finiteness theorem for submanifolds proved in [5].  Then I contains only finitely many equivalence classes of immersions.
Next we recall the following lemma in [9] on estimating the principal curvatures.
Let M be a submanifold in a Riemannian manifold N . For any point x ∈ M and unit normal vector v ∈ ν(M ) x , define where A v denotes the shape operator of M with respect to v at x. Also, we recall that the cut-focal radius of M at x in the direction v is defined by This shows V ol(M ξ ) ≤ V ol(N ) D . In addition, by the choice of ξ, we have On the other hand, the structural results in Theorem 2.1 show that at any point x in an equifocal hypersurface with respect to a unit normal vector v, the focal points coincide with the cut-focal points and both are the points in the intersection of the normal geodesic circle exp x (ν(M ) x ) with the focal submanifolds M ± . Hence the cut-focal radius e c (x, v) is nothing but the distance from x to M + or M − according to exp x (e c (x, v)v) ∈ M + or exp x (e c (x, v)v) ∈ M − respectively. In particular, by the discussion above, on the equifocal hypersurface M ξ we have for any point x ∈ M ξ and unit normal vector v ∈ ν(M ξ ) x , As it is well known, a compact simply connected symmetric space has nonnegative sectional curvatures. It follows from Lemma 3.1 that the maximal eigenvalue κ(v) of the shape operator Since this inequality holds for any unit normal vector v ∈ ν(M ξ ) x , it follows that each eigenvalue This shows that the operator norm |Π| of the second fundamental form at x ∈ M ξ satisfies Hence M ξ satisfies all the conditions of Theorem 3.1 so that there are only finitely many equivalence classes of such immersions of curvature-adapted equifocal hypersurfaces.
The proof is now completed.

Equifocal hypersurfaces in compact rank one symmetric spaces
In this section, we give a more detailed study for equifocal hypersurface in compact rank one symmetric spaces.
First, we investigate some examples of equifocal hypersurfaces in complex projective spaces. We will construct them through Hopf fibrations by projecting the OT-FKM-type isoparametric hypersurfaces in spheres which almost cover all isoparametric hypersurfaces with four distinct principal curvatures in spheres (cf. [2], [3], [10]). Now we recall some fundamental definitions. For a symmetric Clifford system A 0 , ..., A m on R 2l , i.e., A i 's are symmetric matrices satisfying A i A j + A j A i = 2δ ij I 2l , the OT-FKM-type isoparametric polynomial F on R 2l is then defined as (cf. [6]): where we take the coordinate system z = (x t , y t ) t = (x 1 , ..., x l , y 1 , ..., y l ) t ∈ R 2l . By orthogonal transformations, without loss of generality we can write where {E 2 , ..., E m } is a skew-symmetric Clifford system on R l , i.e., E i 's are skew-symmetric matrices satisfying E i E j + E j E i = −2δ ij I l . It can be verified that the level hypersurfaces of this polynomial restricted to the unit sphere have 4 distinct constant principal curvatures with multiplicities m 1 = m and m 2 = l − m − 1, provided l − m − 1 > 0. By using the well-known Hopf fibration π : S 2n+1 → CP n , [18] proved that a hypersurface M in CP n is isoparametric (equifocal) if and only if its inverse image π −1 (M ) under the Hopf fibration π is an isoparametric hypersurface in S 2n+1 . However, given one isoparametric hypersurface in S 2n+1 , it can be neither projectable nor projected uniquely up to isometry through the Hopf fibrations with different S 1 -actions. What is more, the induced equifocal hypersurfaces in CP n may have different diffeomorphism types as the following examples will show.
Remark 4.1. Even if the isoparametric hypersurfaces in spheres are classified completely, we can not get the classification for isoparametric hypersurfaces in CP n or in HP n directly by using the Hopf fibrations. We should be careful that different complex structures may induce different S 1 -actions, and different S 1 -actions may give non-diffeomorphic quotient submanifolds in CP n or in HP n . Thereby Theorem 1.2 of us really makes sense in this viewpoint.
To prepare for the proof of Theorem 1.2 we need the following remarkable equality established by Thorbergsson involving g, m 1 , m 2 for an equifocal hypersurfce M in a simply connected symmetric space N . Proposition 4.1. ( [15]) Let i denote the index of γ| [0,2π] as a critical point of the energy functional E in the path space Ω pp , where γ| [0,2π] is a closed geodesic normal to M and γ is parameterized such that its minimal period is 2π, and let v denote its nullity. Then we have g(m 1 + m 2 ) = i + v.
Remark 4.2. For N = S n , CP n , HP n and the Cayley projective plane CaP 2 , the equality of Thorbergsson will give the well-known formulas: g(m 1 +m 2 ) = 2(n−1) for S n , g(m 1 +m 2 ) = 2n for CP n , g(m 1 + m 2 ) = 4n + 2 for HP n and g(m 1 + m 2 ) = 22 for CaP 2 . Now we are ready to prove Theorem 1.2. Proof of Theorem 1.2. Let M be an equifocal hypersurface in a simply connected compact rank one symmetric space N , M + and M − be the focal submanifolds. One can conclude from Theorem 2.1 that along a normal geodesic exp x (tv(x)), the focal points of M in N with respect to x ∈ M occur alternately and equidistantly in the two focal submanifolds M + and M − and normal geodesics are closed. Note that a simply connected compact symmetric space has rank one if and only if all its geodesics are closed, and we will assume that the Riemannian metrics on these spaces are normalized such that their closed geodesics are of length 2π. By Remark 4.2 and 2gd(M + , M − ) = 2π, we can get a lower bound on d(M + , M − ) depends only on N . In fact, we have d(M + , M − ) = π/g ≥ π/n for CP n , d(M + , M − ) = π/g ≥ π/(2n + 1) for HP n and d(M + , M − ) = π/g ≥ π/11 for CaP 2 . Then mimicking the proof of Theorem 1.1, we finally complete the proof of Theorem 1.2. ✷