Polyharmonic hypersurfaces into complex space forms

We characterize polyharmonic Hopf hypersurfaces with constant principal curvatures as solutions of a fourth-order algebraic equation. We construct six different families of proper polyharmonic hypersurfaces in $ \mathbb{ C P }^n $, and prove that such solutions cannot exist in $ \mathbb{ C H }^n $. Moreover, we classify all biharmonic Hopf hypersurfaces with constant principal curvatures in complex space forms and study their stability.


Introduction
Harmonic maps are critical points of the energy functional where ψ : M → N is a smooth map between two closed Riemannian manifolds (M, g) and (N, h).Equivalently, ψ is harmonic if and only if it is a solution to the Euler-Lagrange equation associated with this functional, namely: τ (ψ) := Tr ∇dψ = 0.
The section τ (ψ) ∈ ψ * T N is called the tension field of ψ.In particular, if ψ is an isometric immersion, it is harmonic if and only if ψ(M ) is a minimal submanifold of N .We refer the reader to the work of Eells and Lemaire [9,10] for background on the theory of harmonic maps.
The study of higher-order functionals has already been proposed in [9] as a generalization to the classic energy.In the last decades, this topic has raised the interest of many mathematicians, leading to intriguing results from both the analytic and the geometric perspective, see for instance [3,4,5,15,18].If r = 2s and s ≥ 1, we define the r-energy functional as If, on the other hand, r = 2s + 1, then where {e i } m i=1 is a local orthonormal frame on M and ∆ = d * d denotes the rough Laplacian acting on sections of ψ * T N .A polyharmonic map of order r (in short, r-harmonic) is a critical point for the r-energy functional.In particular, if ψ is an isometric immersion, we say that M is an r-harmonic submanifold of N .The biharmonic case (r = 2) has shown itself to be of special interest, and we refer to [8,11,20,22] for an introduction to this field.Note that any harmonic map is automatically polyharmonic of any order.If, on the contrary, a map is a critical point for the r-energy functional but not for the classic energy, we refer to it as a proper r-harmonic map.Specifically, an r-harmonic submanifold is said to be proper if it is not minimal.Recently, Montaldo, Oniciuc, and Ratto [18] addressed the problem of existence of polyharmonic immersions into real space forms, obtaining different families of proper r-harmonic isoparametric hypersurfaces in the sphere.In this manuscript, we consider Hopf hypersurfaces with constant principal curvatures in a complex space form (N, h, J).A hypersurface M of N is said to be Hopf if −Jξ is an eigenvector for the shape operator of M , that is to say, S(−Jξ) = −αJξ where ξ is a local choice of unit normal vector and α ∈ R. Hopf hypersurfaces with constant principal curvatures in CH n or CP n have been classified in [2,13].They are all tubes of radius t ∈ I over a complex submanifold of N , where I is an open interval that only depends on the choice of the complex submanifold.We reduce the condition for a Hopf hypersurface with constant principal curvatures to be rharmonic, which in general would lead to a semilinear system of partial differential equations of order 2r, to a fourth-order algebraic equation.More specifically, the main results we present here are the following.
Theorem A. Let M be a Hopf hypersurface with constant principal curvatures in CP n or CH n , n ≥ 2. Then M is r-harmonic for r ≥ 2 if and only if The study of this equation leads us to the following result.
Theorem B. Let {M t } t∈I be a family of Hopf hypersurfaces with constant principal curvatures in CP n .There exist two natural numbers r ′ and r ′′ such that (1) If r > r ′ , the family {M t } t∈I contains at least two proper r-harmonic hypersurfaces.
(2) If r > r ′′ , the family {M t } t∈I contains exactly four proper r-harmonic hypersurfaces.
Explicit bounds for r ′ and r ′′ are given.Moreover, we classify biharmonic Hopf hypersurfaces with constant principal curvatures, as the following theorem shows.
Theorem C. A Hopf hypersurface with constant principal curvatures in CP n is proper biharmonic if and only if it is a tube over a totally geodesic CP k , 1 ≤ k ≤ n − 1, with radius t given by Furthermore, CH n does not admit any proper biharmonic Hopf hypersurface with constant principal curvatures.
Regarding stability, we obtain the following sufficient condition for a biharmonic hypersurface to be normally stable.Recall that a biharmonic hypersurface is normally stable if the second variation of the bienergy is non-negative for any normal variation with compact support.With this result, we are able to ensure the normal stability of some of the biharmonic hypersurfaces presented in Theorem C.
The organization of the document is as follows.In Section 2 we introduce some basic notions on polyharmonic maps, Hopf hypersurfaces, and quartic polynomials.In Section 3 we prove Theorem A. Theorem B and explicit upper bounds for r ′ and r ′′ are given in Section 4. Theorems C and D are proved in Section 5.

Preliminaries
2.1.Polyharmonic maps.We introduce here some basic concepts on polyharmonic maps and the Laplace operator used throughout the manuscript.The r-tension field, τ r , is a higher-order analog of the tension field, in the sense that the system of partial differential equations τ r (ψ) = 0 characterizes polyharmonic maps of order r.The following equations depict an explicit formula for the r-tension field, see [15] for a reference.
A proper biharmonic hypersurface M in N is said to be normally stable if the second variation of the bienergy functional is non-negative for any normal variation with compact support.Ou [21] showed that a complete orientable biharmonic hypersurface M of a Riemannian manifold N is normally stable if and only if for any compactly supported function f on M we have Q(f ) ≥ 0, where The Laplace operator ∆ has proved to be useful in the study of the second variation of biharmonic submanifolds, as shown in [21,17].For convenience we recall some properties of this elliptic operator and its spectrum, we use [7] as a reference.
Let M be a compact and connected manifold.The set of eigenvalues for ∆f + µf = 0, and each associated eigenspace is finite-dimensional.Eigenspaces belonging to distinct eigenvalues are orthogonal in L 2 (M ), and L 2 (M ) is the direct sum of all the eigenspaces.Furthermore, each eigenfunction is in C ∞ (M ).
In addition, the following result turned out to be useful for our purposes 2.2.Hopf hypersurfaces.We give in this subsection a brief introduction to the theory of Hopf hypersurfaces.We use [6] as a reference.Let (N, •, • , J) be a Kähler manifold and σ be a plane in the tangent space T p N , p ∈ N .We write for the sectional curvature, where {X, Y } is an orthonormal basis of σ.If σ is invariant by the almost complex structure J, then K p (σ) is called the holomorphic sectional curvature of σ.In particular, if K(σ) is constant for all planes σ in T p N invariant by J and for all points p ∈ N , then N is called a space of constant holomorphic sectional curvature.
The Riemannian curvature tensor for a space of constant holomorphic sectional curvature c = 0 can be expressed as The sectional curvature for any plane σ in T p N spanned by two orthonormal vectors X, Y reads The complex projective space endowed with the Fubini-Study metric or the complex hyperbolic space with the Bergman metric are examples of spaces with constant holomorphic sectional curvature.Moreover, any simply connected complete 2n-dimensional Kähler manifold of constant holomorphic sectional curvature c is holomorphically isometric to CP n (c), C n or CH n (c) depending if c > 0, c = 0 or c < 0, respectively.We refer the reader to [14] for an overview of the general aspects of complex geometry.
In what follows we use N n to denote either CH n or CP n .Let M be an (2n − k)-dimensional Riemannian manifold, k < 2n, and let f : M → N n , be a Riemannian immersion.Write BM for the bundle of unit normal vectors to f (M ) in N n .We define the tube of radius t > 0 over M , denoted by M t , as the image of the map f t : BM → N n defined by Note that given any p ∈ M , there is always a neighborhood U of p in M such that for all t > 0 sufficiently small, the restriction of f t to BU is an immersion onto an (2n − 1)-dimensional manifold.
Let now M ⊂ N n be a hypersurface and ξ a local choice of unit normal vector.We say that M is a Hopf hypersurface if W = −Jξ is a principal vector for the shape operator S, that is, S W = αW where α ∈ R is called the Hopf principal curvature.We will refer to W as the structure vector.In a Hopf hypersurface M of CH n or CP n , the Hopf principal curvature remains constant [6, Theorem 6.16].
In particular, Hopf hypersurfaces for which all their principal curvatures are constant have been classified by Kimura [2] in CH n and by Berndt [13] in CP n .They are all tubes of a certain radius, as shown in Table 1 for CH n and Table 2 for CP n .
The following property turned out to be useful for our purposes.
Lemma 2.2.[6, Lemma 8.1] Let M be a Hopf hypersurface with constant principal curvatures in CH n or CP n .For all eigenvalues λ, µ which are not the Hopf principal curvature, we have where T λ , T µ are the corresponding eigendistributions and ∇ is the Levi-Civita connection on M .

Type Focal submanifold Principal curvatures Multiplicities
A 0 Horosphere in Table 1.Hopf hypersurfaces with constant principal curvatures in CH n (−4) Type Focal submanifold Principal curvatures Multiplicities

E
Half spin embedding: In what follows we write "Hopf hypersurface" instead of "Hopf hypersurface with constant principal curvatures" since we will not deal with the general case.
2.3.Quartic polynomials.Since we will reduce our problem to the existence of solutions for some quartic equations, we found it convenient to briefly recall here some general properties of polynomials and their roots.A general quartic equation over R is any equation of the form where a 4 , a 3 , a 2 , a 1 , a 0 ∈ R and a 4 = 0.If we divide every term by a 4 and apply the change of variable y = x − a 3 4a 4 , then (2.5) reads y 4 + p 2 y 2 + p 1 y + p 0 = 0 where In particular, if p 1 = 0 then (2.5) can be reduced to a biquadratic equation and solved by In general, a quartic equation can be solved by radicals, and explicit formulas for such solutions are available.Thus, one may be tempted to conclude that once we have reduced our problem to finding the roots of a fourth-degree polynomial, we are done.This is not the case since the quartic equations we obtain depend on one, two, or even three different real parameters, converting the discriminant of the polynomial into an unmanageable expression.Therefore, to attack the problem we will make use of the so-called Cauchy bound for the roots of a polynomial (see, for example [16, Chapter VII]), which states that an upper bound of the absolute value on the roots of (2.5) is given by

Polyharmonic equation
The goal of this section is to obtain the r-harmonic equation for the inclusion of a Hopf hypersurface into N n (c).Here N n (c) denotes CP n (c) or CH n (c), depending if c > 0 or c < 0, respectively.We write i t : M t ֒→ N n (c) for the inclusion of the hypersurface M t arising as the tube of radius t over the corresponding complex submanifold.We use e 0 for the local unit normal vector field to M t in N n (c) such that if we flow in the direction of e 0 , the radius of the tube increases.Every object in M t is written with the subscript t, such as the shape operator S t or the Hopf principal curvature α t .
In order to prove the theorem above, we first state a couple of auxiliary lemmas.
Lemma 3.2.For every m ∈ N, the following identity holds: m ]e 0 .Proof.We proceed by induction on m.Take an orthonormal basis {e i } 2n−1 i=1 of T p M t and extend it to a local orthonormal frame in a neighborhood of p. Since ∆ 0 τ ≡ τ , by definition of the tension field we get: [ ∇ e i di t e i , di t e j − ∇ e i e i , e j ] di t e j = Tr S t e 0 , where in the last identity we used the fact that i t is an isometric embedding and applied the Gauss formula.Assume that the property holds for an arbitrary m, then where, in order to simplify the notation, we write β m (t) = Tr S t (Tr S 2 t ) m .Note that since the principal curvatures are constant on M t , then β m is constant on M t .By definition of the rough Laplacian, we obtain: (∇ e i S t (di t e i ) − ∇ e i e i , e j S t (di t e j )) tan .(3.6)This expression does not depend on the choice of the basis, since the normal part does neither.Take then an orthonormal basis where we used the property that the eigenvalues are constant on i t (M t ).Equivalently, by the Gauss formula.

Existence of proper polyharmonic hypersurfaces in CP n
Let M t be a Hopf hypersurface in CP n (c).Note that the eigenvalues of the shape operator scale with a factor of √ c, so equation (3.1) is invariant with respect to the choice of the parameter c.That is to say, even if we want to "flatten" our space by choosing a smaller c, or we want to "curve" it by taking a bigger c, the r-harmonicity of the hypersurfaces is not affected.We will then take c = 4 to be consistent with Table 2, so the equation obtained in the last section reads: With Table 2 one gets explicit expressions for Tr S t , Tr S 2 t , and α t in each case.Then, by Theorem (3.1), a lengthy computation shows that there is a one-to-one correspondence between r-harmonic Hopf hypersurfaces and roots of the polynomial where the coefficients a 4 , a 3 , a 2 , a 1 , a 0 and the variable x are given in Table 3 for each case.

Type Variable
Coefficients a 2 = 2((4k + 5)n + 2k 2 + 11k + 5)r +2((2k − 3)n + 4k 2 + 4k + 5) For the cases of type C, D, and E, it is convenient to write everything in terms of 2t instead of t.To this end, we found the following trigonometric identities useful: Note that if n = 1, and we plug the Hopf principal curvature of a type A 1 Hopf hypersurface (α t = 2 cot 2t) in equation ( 4.1), one easily sees that the tube around a point of radius t is a proper r-harmonic curve if and only if Since CP 1 is just the Riemann sphere, this agrees with [19,Theorem 1.1].From now on, unless otherwise indicated, we will assume n ≥ 2.
We point out that, in general, there are two values of t which will be of special interest since they make the left-hand side of (4.1) independent of r: these are the solutions for Tr S t = 0, which yields minimal hypersurfaces, and Tr S t +3α t = 0.
Theorem 4.2.Let M t , 0 < t < π 2 , be a family of type A 1 Hopf hypersurfaces in CP n , n ≥ 2. Then: (1) The family M t contains at least two proper r-harmonic hypersurfaces for r ≥ 2.
Fix now k = n−1 2 .Note that P A 2 (0) = (2k + 1) 2 and P A 2 (1) = (2k − 2n + 1) 2 , so if we find x 1 ∈ (0, 1) such that P A 2 (x 1 ) < 0, we will have ensured the existence of at least two r-harmonic hypersurfaces.The equation Tr S t = 0 is solved by from where 1 follows.Assume now that k < k 1 , where k 1 is defined as stated in the theorem, and define A direct computation shows that as n, r → ∞, where On the other hand, if k > k 2 , take as n, r → ∞, where In this case the Cauchy bound leads to unmanageable expressions.We avoid this by using a more direct method, paying the price of a worse bound.Since the strategy is similar for each condition, we only write here the argument to ensure condition P Since, in any case, 3(2n − 2k − 1) > 1, a very conservative bound can be given by Q(r) < −r 4 + (2k + 13)nr 3 + 4(2k 2 + 5k + 4)r 2 + 8n(k + 4)r < −r 4 + (18n 2 + 65n + 16)r 3 since r ≥ 2 and k < n.Hence, it suffices to take r > 18n 2 + 65n + 16.
A similar computation holds for the rest of the conditions.and by (2.4) ∇ ξ e i , e i R N (ξ, e i , e i , ξ) = 0.
Then, since Tr R N (ξ, S t (•), •, ξ) = Tr S t +3α t and Tr S t is constant on M t , equation (5.1) implies We will adopt an approach similar to [21,Theorem 3.1].Since in our case we have limited knowledge about the spectrum of the Laplacian, we make use of the following bound for the first eigenvalue: µ 1 ≥ (n + 1) − 1 2 | Tr S t |.This is just an application for our particular case of Theorem 2.1.Hence, if we denote by f 1 the eigenfunction corresponding to the eigenvalue µ 1 , we have ), so to check if Q(f ℓ ) > 0 for any ℓ ∈ N, it is enough to ensure (5.2).It only remains to show that the positivity of Q for eigenfunctions of the Laplacian implies that Q is positive in general, but since M t is compact, we have the following Sturm-Liouville's decomposition: C ∞ (M t ) = ⊕ ∞ i=0 E µ i , where E µ i denotes the eigenspace of the Laplacian on M t with respect to the eigenvalue µ i .For a detailed discussion on the orthogonal decomposition of eigenspaces of Laplacian on compact manifolds see [7, Theorem III.9.1].
Corollary 5.3.There exists n ′ such that if n > n ′ , the biharmonic tube of radius t + over a totally geodesic CP n−1 is normally stable.

Theorem 2 . 1 .
[12, Theorem 2.1] Suppose that M is a compact orientable hypersurface embedded in a compact n-dimensional Riemannian manifold N .If the Ricci curvature of N is bounded below by a positive constant k, then 2µ 1 > k − max M | Tr S | where µ 1 is the first eigenvalue of the Laplacian of M .

Table 2 .
Hopf hypersurfaces with constant principal curvatures in CP n (4)