On the equivariant stability of harmonic self-maps of cohomogeneity one manifolds

The systematic study of harmonic self-maps on cohomogeneity one manifolds has recently been initiated by P\"uttmann and the second named author in \cite{MR4000241}. In this article we investigate the corresponding Jacobi equation describing the equivariant stability of such harmonic self-maps. Besides several general statements concerning their equivariant stability we explicitly solve the Jacobi equation for some harmonic self-maps in the cases of spheres, special orthogonal groups and $\SU(3)$. In particular, we show by an explicit calculation that for specific cohomogeneity one actions on the sphere the identity map is equivariantly stable.


Introduction and results
Harmonic maps are solutions to one of the most studied geometric variational problems and have many applications in geometry, analysis and mathematical physics. They are defined as critical points of the energy of smooth maps φ between two Riemannian manifolds (M m , g) and (N n , h) which is defined by (1.1) Its critical points are characterized by the vanishing of the so-called tension field 0 = τ (φ) := Tr g ∇dφ, τ (φ) ∈ Γ(φ * T N), (1.2) which are precisely harmonic maps. The harmonic map equation (1.2) is a second order semilinear elliptic partial differential equation. Due to its non-linear nature it is a challenging mathematical endeavor to prove the existence of non-trivial solutions. In the case of both M, N being closed and N having non-positive curvature Eells and Sampson were able to establish their famous existence theorem which ensures that every homotopy class of maps contains a harmonic map under the aforementioned assumptions [8].
If the target manifold has positive curvature it is substantially more difficult to prove the existence of harmonic maps. To approach this problem it is natural to study the existence of harmonic maps in the case that both manifolds have a sufficient amount of symmetry, as described in the books [2,7]. Aiming in this direction harmonic maps between cohomogeneity one manifolds were first studied by Urakawa [28]. This work has later been extended by Püttmann and the second author in [19]. The problem of finding a harmonic map between cohomogeneity one manifolds reduces to solving a singular boundary value problem for an ordinary differential equation of second order. This ordinary differential equation simplifies further if one assumes that both domain and target manifold are the same leading to the notion of harmonic self-maps.
The study of harmonic self-maps of spheres with the round metric was initiated by Bizoń and Chmaj [3,4], see also the subsequent work of Corlette and Wald [5]. Extending the results for harmonic map between cohomogeneity one manifolds the second author constructed families of harmonic self-maps of spheres [20] and of SU(3) in [21].
One important property that characterizes the qualitative behavior of a given harmonic map is its stability. To understand the latter one has to calculate the second variation of the energy (1.1) and evaluate the resulting expression at a critical point. If a given harmonic map is stable, then there does not exist a second harmonic map "nearby", meaning that the critical points of (1.1) are isolated. Soon after Eells and Sampson established their existence result for harmonic maps assuming that the target manifold has negative curvature, Hartman showed that the condition of negative curvature ensures that such harmonic maps are stable [9]. On the other hand, in the case of a spherical domain equipped with the round metric and m ≥ 3, Xin showed that harmonic maps are unstable [29]. Soon thereafter Leung [11] proved that harmonic maps to spheres are unstable in general as well. In the same spirit of ideas Ohnita was able to extend these results to the case of harmonic maps from and to symmetric spaces [15].
In this article we study the equivariant stability of harmonic self-maps on cohomogeneity one manifolds. Here, equivariant stability refers to the fact that we only consider variations which are invariant under the cohomogeneity one action. We will often see the phenomenon that a harmonic map will be unstable in general but can be equivariantly stable as we are only allowing for a special class of directions in the formula for the second variation of the energy (1.1). Similar results have already been obtained in the case of biharmonic maps which represent a fourth order generalization of harmonic maps, see [17] for an introduction to this topic. The equivariant stability of rotationally biharmonic maps between models was investigated in [13,Section 5]. The normal stability of biharmonic hypersurfaces in spheres has been studied in [16]. In this article we derive the Jacobi operator associated to harmonic self-maps of compact cohomogeneity one manifolds describing their equivariant stability. Throughout this manuscript we assume that the orbit space of the cohomogeneity one manifold is isometric to [0, L], see Section 2 for more details. Using the Jacobi operator's explicit form, we prove that its eigenvalues λ j satisfy the asymptotic behavior Further, we will explicitly solve the Jacobi equation for some harmonic self-maps in the cases of spheres, special orthogonal groups and SU (3). In particular, we show by an explicit calculation that for specific cohomogeneity one actions on the sphere and on special orthogonal groups the identity map is equivariantly stable. For example, we will prove the following theorem: Theorem 1.1. Let (g, m 0 , m 1 ) be one of the following pairs (2, m 0 , m 1 ), (3, 1, 1), (3,2,2), (3,4,4), (3,8,8), (4, m 0 , 1), (4, 2, 2), (6, 1, 1), (6, 2, 2).
Then the identity map of S g(m 0 +m 1 ) 2 +1 is a stable harmonic map with respect to equivariant variations.
Throughout this article we make use of the summation convention, i.e. we tacitly sum over repeated indices.
Organization: This manuscript is structured as follows. In Section 2 we provide preliminaries on harmonic maps between cohomogeneity one manifolds and on the stability of harmonic maps. The equivariant stability of harmonic maps between cohomogeneity one manifolds is discussed in Section 3, where we in particular provide Theorem 1.1. In the last section, Section 4, we study the equivariant stability of some explicitly given equivariant harmonic self-maps of cohomogeneity one manifolds and explicitly determine the spectra of the corresponding Jacobi operators.

Preliminaries
In Subsection 2.1 we give a brief introduction to harmonic maps between cohomogeneity one manifolds. Afterwards, in Subsection 2.2, we recall various facts concerning the stability of harmonic maps which are related to the results of this article.

2.1.
Harmonic maps between cohomogeneity one manifolds. In this subsection we introduce relevant notation and results from the theory of equivariant harmonic selfmaps of compact cohomogeneity one manifolds. The main source is [19]. In what follows let M be a Riemannian manifold endowed with an isometric action G × M → M of a compact Lie group G. We further assume that the orbit space M/G is isometric to a closed interval [0, L] and that the Weyl group W of the action is finite, i.e. we are considering specific cohomogeneity one actions on M. In this setting Püttmann and the second named author studied when equivariant (k, r)-maps, i.e., maps of the form g · γ(t) → g · γ(r(t)) (2.1) where r : [0, L] → R is a smooth function with r(0) = 0 and r(L) = kL, are harmonic [19]. Here, γ denotes a fixed unit speed normal geodesic with γ(0) being contained in one of the non-principal orbits. Further, the integer k is of the form j|W |/2 + 1 with j ∈ 2Z. We want to mention that for specific group actions also odd integers j might be allowed. The Brouwer degree of a (k, r)-map is given by k if the codimensions of both non-principal orbits are odd, and by 1 otherwise, see [18] for more details.
In the following, let Q be a fixed biinvariant metric on G and denote the orthonormal complement of the Lie algebra h of the principal isotropy group H in g by n. Then, the metric endomorphism P t : n → n is defined by where Z * denotes the action field of Z ∈ n. In addition, we define B + : n × n → g as follows B r(t) The tension field of a (k, r)-map as defined in (2.1) is given by the following expression τ |γ(t) =r(t) + 1 2ṙ (t) Tr P −1 tṖ t − 1 2 Tr P −1 tṖ r(t) .
Hence, a (k, r)-map is harmonic if and only if τ |γ(t) = τ nor |γ(t) + τ tan |γ(t) = 0. (2.2) We realize that both the tangent and the normal part of the tension field have to vanish. For the specific actions we deal with in this paper, the tangential component of the tension field vanishes such that the construction of harmonic maps is reduced to the construction of solutions to the ordinary differential equation Below we give further details on the particular cases of cohomogeneity one actions on spheres, on special orthogonal groups and on SU(3).
2.1.1. Cohomogeneity one actions on spheres. By proving that each cohomogeneity one action on a sphere is orbit equivalent to the isotropy representation of a Riemannian symmetric space of rank 2, Hsiang and Lawson [10] classified such actions. The orbits of any isometric cohomogeneity one action G×S n+1 → S n+1 of a compact Lie group G on the sphere S n+1 yield an isoparametric foliation of the sphere. The latter is a family of parallel hypersurfaces with constant principal curvatures together with two focal submanifolds which together foliate the sphere. Takagi  Therefore, we have n = m 0 +m 1 2 g. Later, Münzner [14] showed that this holds true for all isoparametric foliations of spheres, i.e. also for those not stemming from group actions. Up to ordering of m 0 and m 1 there exist only actions with the following (g, m 0 , m 1 ): (1, m, m), (2, m 0 , m 1 ), (3, 1, 1), (3, 2, 2), (3,4,4), (3,8,8), (4, m 0 , 1), (4, 2, 2), (4, 2, 2ℓ + 1), (4, 4, 4ℓ + 3), (4,4,5), (4,6,9), (6, 1, 1), (6, 2, 2), where ℓ ∈ N + . Harmonic self-maps of S n+1 can be characterized as critical points of where c denotes a positive constant. The critical points are those which are solutions to the ordinary differential equation where k ∈ Z. The above boundary value problem (2.4), (2.5) will be referred to as (g, m 0 , m 1 , k)-boundary value problem. When the multiplicities m 0 and m 1 coincide, we will refer to this problem as (g, m, k)-boundary value problem. It can be easily seen that r(t) = t is an explicit solution of the (g, m 0 , m 1 , 1)-boundary value problem. It is clear that this solution corresponds to the identity map of the Riemannian manifold M. Further, the function r(t) = (1 − g)t is an explicit solution of the (g, m, 1 − g)boundary value problem. In the following we will refer to these particular solutions as linear solutions. It has been shown in [19] that the (g, m 0 , m 1 , k)-boundary value problem admits linear solutions for k = 1 and k = 1 − g, m 0 = m 1 only and that the above mentioned solutions exhaust all linear solutions.
In [4] Bizoń and Chmaj proved that each of the (1, m, 0)-boundary value problems and the (1, m, 1)-boundary value problems admits infinitely many solutions provided that 2 ≤ m ≤ 5. For each choice of m with 2 ≤ m ≤ 5, we subsume the solutions with k = 0 and k = 1 in a countable family r n , n ∈ N, of solutions which are labeled by a nodal number n, namely by the number of intersections with π 2 . Bizoń and Chmaj also proved that these solutions (more precisely, the solutions different from the identity map or its negative) do no longer exist for m ≥ 6.
Remark 2.1. Note that Bizoń and Chmaj [4] use a different notation than we do: the role of m is played by what they call k, which satisfies k = m + 1. Further, instead of r(t) they deal with the shifted function h(t) = r(t) − π 2 . In [20] the second named author provided an analogous result for g = 2, namely she proved that each of the (2, m 0 , m 1 , 1)-boundary value problems admits infinitely many solutions assuming that 2 ≤ m 0 ≤ 5. For each choice of m 0 with 2 ≤ m 0 ≤ 5, we thus obtain a countable family r n , n ∈ N, of solutions which are labeled by a nodal number n, namely by the number of intersections with π 2 . It was also shown in [20] that these solutions (more precisely, the solutions different from the identity map or its negative) do no longer exist for m 0 ≥ 6.

Cohomogeneity one actions on special orthogonal groups.
Any isometric cohomogeneity one action G × S n+1 → S n+1 on a sphere can be lifted to an isometric cohomogeneity one action of G × SO(n + 1) on SO(n + 2). If we denote the data of the cohomogeneity one action on the sphere by (g, m 0 , m 1 ), then we have n = (m 0 + m 1 )g/2. It was proved in [19] that a solution of the (2g, m 0 , m 1 , k)-boundary value problem yields a harmonic self-map of SO(n + 2). We can conclude that harmonic self-maps of SO(n + 2) are associated with the solutions of the ordinary differential equation where k ∈ Z. Clearly, r(t) = t is a solution to this boundary value problem with k = 1. Further, if m = m 0 = m 1 , then r(t) = (1 − 2g)t is a solution to this boundary value problem with k = 1 − 2g.
2.2. Stability of harmonic maps. In this subsection we recall various facts concerning the stability of harmonic maps where we follow [27,Chapter 5]. Thus, let φ : M → N be a smooth harmonic map. The second variation of the energy of a map (1.1) evaluated at a critical point is given by Here, J φ denotes the Jacobi operator which is defined by In the above formula R N represents the Riemannian curvature tensor of the manifold N.
We use the following sign convention for the rough Laplacian where ∇ φ represents the connection on φ * T N and {e i }, i = 1, . . . , m is an orthonormal basis of T M. Note that the Jacobi operator J φ defined in (2.9) is self-adjoint when M is compact. We say that a harmonic map φ is stable if From the general spectral theory for elliptic operators over a compact Riemannian manifold we know that the eigenvalues λ of J φ satisfy The vector space is called the multiplicity of λ and we know from general elliptic theory that dim V λ (φ) < ∞.
In terms of these spectral data we define It follows directly that a harmonic map φ is stable if and only if λ j (φ) > 0 for j ∈ N + .
In the following we will cite four important results concerning the stability of harmonic maps which are closely related to the main results of this article, the first two have already been mentioned in the introduction.
Theorem 2.2 (Xin). For m ≥ 3 any stable harmonic map from S m with the round metric into any Riemannian manifold must be a constant map.
In order to prove the above result the author make use of the fact that the sphere admits conformal vector fields. In particular, this implies that for n ≥ 3 a stable harmonic map from any Riemannian manifold into S n with the round metric must be a constant map. The above results have been extended to Riemannian symmetric spaces in [15,25]. On the other hand, it was shown by Urakawa that by a suitable deformation of the metric on the domain a harmonic map to the sphere can also be stable [26,Proposition 7.4]. For further results on the stability of harmonic maps we refer to [27,Chapter 5]. Moreover, let us recall the following result concerning the stability of the identity map on S m+1 , which was obtained independently by Mazet [  Here, proper means that we exclude the eigenspace spanned by the constant function. The above result follows from calculating the spectrum of the operator acting on functions. It is well known that and each eigenvalue has multiplicity m + 2.
For the sake of completeness we also want to mention the following result providing a geometric characterization of the stability of the identity map, which is also due to Smith [22]. (2) The nullity of the identity map is given by
In the following we will study the equivariant stability of harmonic self-maps between cohomogeneity one manifolds. Such maps may be stable with respect to equivariant variations but unstable with respect to general variations. Hence, our results on the equivariant stability of some particular harmonic maps do not contradict the general theorems that have been presented in this subsection.
3. Equivariant stability of harmonic self-maps between cohomogeneity one manifolds In the present section we discuss the equivariant stability of equivariant harmonic selfmaps of compact cohomogeneity one manifolds M. We make use of the notation introduced in Subsection 2.1.
In order to investigate the equivariant stability of harmonic (k, r)-maps, which are characterized as solutions of (2.2), we consider a variation of r(t), denoted by r s (t), which satisfies d ds s=0 r s (t) = ξ(t).
In the following we will calculate the linearization of the normal and the tangential part of the tension field separately. The variation of the normal part is given by whereas for the tangential part we find Hence, we obtain the following
Proof. This is a direct consequence of the above calculations.
Note that we can write the Jacobi equation (3.3) in the following form which allows us to apply the general theory of Sturm-Liouville for one-dimensional eigenvalue problems. For more details on Sturm-Liouville theory as it is used in this article we refer to Appendix A. We get the following general result: (2) The eigenfunction ξ j (t) corresponding to the eigenvalue λ j has exactly j zeros in (0, L). Between two zeros of ξ j (t) there is exactly one zero of ξ j+1 (t). (3) For j → ∞ the eigenvalues have the following asymptotic behavior Proof. Performing the substitutions z(t) = p(t) = √ det P t and + (e j , e j ) − 1 2 Tr(P −1 tP r(t) ) det P t we conclude that the Jacobi equation (3.4) can be written in the form of a Sturm-Liouville eigenvalue problem, see Appendix A for the precise details. Moreover, it can be directly seen that the first three conditions of (SL), which are given in Appendix A in full detail, are satisfied. Regarding the fourth condition of (SL) we note that the boundary conditions (A.2) are satisfied by choosing α 1 = β 1 = 0 and α 2 , β 2 = 0 as p(t) := √ det P t satisfies p(0) = p( π 2g ) = 0 and hence the fourth condition of (SL) also holds true. The statement (3.5) on the asymptotic behavior of the eigenvalues is a direct consequence of the Weyl asymptotic (A.3).

Explicit calculation of spectra
In this section we study the equivariant stability of some explicitly given equivariant harmonic self-maps of cohomogeneity one manifolds. These harmonic maps have been provided in [19,21]. We assume that we only deal with those cohomogeneity one actions for which the tangential component of the tension field vanishes trivially, in other words, for which solutions to (2.3) induce harmonic maps between the corresponding cohomogeneity one manifolds. For these kinds of maps also the tangential contribution to the Jacobi equation (3.2) vanishes. The cohomogeneity one actions on spheres, special orthogonal groups and on SU(3) discussed in Subsection 2.1 satisfy this condition.
(The constant λ in the previous equation differs from λ in (2.4) by a factor g 2 . By abuse of notation we call both constants λ.) In order to solve (4.2), we make the ansatz and obtain is a solution of (4.3). In order to find additional solutions of (4.3) and to solve the spectral problem (4.2) we perform the transformation f (x) = u(tanh(x)) which gives the equation )u(tanh(x)) = 0. The above equation is solved by the Jacobi polynomials, see Appendix B.1 for the precise details. We summarize our calculations as follows: Proposition 4.1. The spectral problem (4.2) describing the equivariant stability of the identity map, which we parametrize by r(x) = 2 g arctan(e x ), is solved by where j ∈ N + .
Then the identity map of S g(m 0 +m 1 ) 2 +1 is equivariantly stable.
(2) For the case g = 1, Proposition 4.1 has been established by Bizoń and Chmaj in [4]. (3) Note that for g = 1 the spectrum (4.4) is precisely the spectrum of the operator (2.11) which was calculated by abstract methods instead of a direct calculation. (4) Setting g = 1 and performing the change of variables t = 2 arctan(e x ), (2.4) transforms into By differentiating this identity one finds that r ′ (x) solves the eigenvalue problem (4.2) with eigenvalue λ = 1 − m. There is a geometric reason for this fact: The proof of Theorem 2.2 makes use of a conformal vector field on the sphere which is obtained by projecting a parallel vector field from R m+2 onto S m+1 . As ∂ ∂x is the generator of conformal transformations it is clear that r ′ (x) solves (4.2) with the corresponding eigenvalue. However, for g ≥ 2 the above statement does no longer hold true. (5) If we inspect the complete list of triples (g, m 0 , m 1 ) presented in Subsection 2.1.1 then we realize that the triples (4, 2, 2ℓ + 1), (4, 4, 4ℓ + 3), (4,4,5), (4,6,9), where ℓ, m 0 , m 1 ∈ N + with m 0 ≤ m 1 , do not appear in the statement of Theorem 4.2. It is currently unknown if the tangential part of the tension field (2.2) vanishes for these triples and hence we cannot make a statement on the equivarant stability of the identity map in these cases.

The equivariant stability of the linear solution r(t)=(1-g)t.
As a next step we study the equivariant stability of the linear solution r(t) = (1 − g)t of (2.4). Note that this solutions exists only for m = m 0 = m 1 . Performing the change of variables t = 2 g arctan(e x ) equation (4.1) acquires the following form Again, we make the ansatz leading to We conclude that is a solution of (4.6). Again, to obtain the additional solutions of (4.6) and to solve the spectral problem (4.2), we perform the transformation f (x) = u(tanh(x)) which results in the equation Hence, by the same reasoning as in the previous section we find that the above equation is solved by the so-called Gegenbauer polynomials, see Appendix B.2 for the precise details.
As an immediate consequence of Proposition 4.4 we obtain the following theorem. where m ∈ N + . Then the harmonic self-map of S mg+1 , which is associated with the solution r(t) = (1 − g)t of (2.4), is equivariantly stable.
(1) In the case g = 2 the eigenvalues (4.4) and (4.7) coincide as one should expect from the explicit form of the linear solutions r(t).
(2) We would like to point out that the explicit spectra (4.4), (4.7) do not contradict Theorems 2.2 and 2.3 as we are dealing with a special kind of stability, that is equivariant stability.

The equivariant stability of the linear solution r(t)=t.
In this subsection we investigate the equivariant stability of the linear solution r(t) = t of (2.6). Plugging in r(t) = t and performing a change of variables t = 1 g arctan(e x ) equation (4.8) transforms into the following eigenvalue problem From the considerations in Subsection 4.1.1 we get: Proposition 4.7. The spectral problem (4.2) describing the equivariant stability of the identity map, which we parametrize by r(x) = 1 g arctan(e x ), is solved by where j ∈ N + .
As an immediate consequence of Proposition 4.7 we obtain the following theorem.
As an immediate consequence of Proposition 4.9 we obtain the following theorem. where m ∈ N + . Further, we set n = 2gm. Then the harmonic self-map of SO(n + 2) which is associated with the solution r(t) = (1 − 2g)t of (2.6), is equivariantly stable.

The second variation of the identity map of SU(3).
Harmonic self-maps of SU (3) have been investigated by the second author in [21] where the existence of a countably infinite family of harmonic self-maps of SU(3) with non-trivial Brouwer degree was established. The latter can be characterized as critical points of the energy functional . (4.13) More precisely, the critical points of (4.13) are those who satisfy The second variation of (4.13) evaluated at a critical point is given by Hence, in order to study the equivariant stability of harmonic self-maps of SU(3) we have to investigate the following eigenvalue problem Note that in our coordinates the volume element is given by 1 2 dx cosh 3 (x) which leads to the factor of 1 cosh 2 (x) . Again, the only solution of (4.14) which is known in closed form is the identity map r 1 (x) = arctan(e x ). Applying the identity cos(arctan e x ) = 1 we find that (4.16) simplifies to Now, we make the ansatz and find Hence, we can conclude that f (x) = 1 and λ = −6 is a solution of (4.18). To obtain the additional solutions we make the ansatz f (x) = u 1 2 tanh(x) which yields the equation This equation is again solved by the Gegenbauer polynomials presented in Appendix B. We summarize our calculations as follows: Proposition 4.11. The spectral problem (4.17) characterizing the equivariant stability of the identity map of SU(3) parametrized by r 1 (x) = arctan(e x ) is solved by where j ∈ N + .
Remark 4.12. Recall that dim SU(3) = 8, if we now compare the spectrum of the Jacobi operator for the identity on S m+1 in the case of m = 7 and g = 1 given by (4.4) with the spectrum of the Jacobi operator on SU(3) given by (4.19) we realize that the eigenvalues coincide, while the eigensections differ slightly.
Remark 4.13. In all explicit calculations that we have carried out in this section, characterizing the equivariant stability of the identity map, we have seen that the eigenvalues λ j have a growth rate of j 2 . This is consistent with the statement of Theorem 3.2 which describes the general equivariant stability of harmonic self-maps on cohomogeneity one manifolds.
Appendix A. Aspects of Sturm-Liouville theory In this appendix we collect a number of results from Sturm-Liouville theory for ordinary differential equations which are linear and of second order. For more details on this subject we refer to [24,Chapter 5.4]. We set J := [a, b] and consider u : J → R. We define the operator L as follows We are interested in the eigenvalue problem with α i , β i ∈ R, i = 1, 2. Note that (A.2) can be read as a linear combination of Dirichlet and Neumann boundary data for u. We say that the conditions (SL) are satisfied if the following conditions hold (1) p(s) ∈ C 1 (J), (2) q(s), z(s) ∈ C 0 (J), (3) p(s) > 0, z(s) > 0 in J, (4) α 2 1 + α 2 2 > 0 and β 2 1 + β 2 2 > 0. We are now ready to state the following powerful result from Sturm-Liouville theory which is the basis in the proof of Theorem 3.2.
Theorem A.1. Consider the eigenvalue problem (A.1). If the conditions (SL) hold, then the eigenvalue problem (A.1) has infinitely many simple eigenvalues λ 0 < λ 1 < λ 2 < . . . λ j → ∞ for j → ∞. Here, we provide some facts on the specific orthogonal polynomials which are solutions to the linear second order ordinary differential equations which appear in the study of the equivariant stability of harmonic self-maps. For more details on this subject we refer to [1,Chapter 22]  δ(1 + δ)(2 + δ)x 3 .
One possibility of obtaining the higher order Gegenbauer polynomials is the recursion formula