STABILITY OF NATURAL ENERGY FUNCTIONALS AT RIEMANNIAN SUBIMMERSIONS

STEFAN BECHTLUFT-SACHS, ANDREAS DÖRING

Abstract. We derive variational formulas of natural first order functionals and obtain criteria for stability in particular at Riemannian subimmersions.

1. Introduction

Any function Φ: Sym^{+}(m) → ℝ_{0}^{+} on symmetric (m×m) matrices which is invariant under conjugation by O(m) gives rise to a functional E_{Φ} : C^{∞}(M^{m}, X^{k}) → ℝ_{0}^{+}. Here (M^{m}, g) and (X^{k}, h) denote compact Riemannian manifolds of dimensions m and k, carrying Riemannian metrics g and h, respectively. We denote by df^{∗} ∈ Γ Hom(f^{∗}T X, T M) the adjoint of the differential df ∈ Γ(Hom(T M, f^{∗}T X)) with respect to the Riemannian metrics on M and X. More explicitely it is defined by the condition that h(df v, x) = g(v, df^{∗}x) holds for any p ∈ M and v ∈ T_{p}M, x ∈ T_{f(p)}X. The Φ-energy of a smooth map f: M → X is the integral

E_{Φ}(f) = ∫_{M} Φ(df^{∗}df) .

Essentially the Φ-energies are the natural locally computable functionals whose density depends explicitely only on the first derivatives of f and does not involve (derivatives of) the Riemannian curvature tensors (cf. [2], [10]).

Examples of such functionals are the 2-energy, Φ(A) = Tr(A), the p-energies, Φ(A) = (Tr(A))^{p/2} (cf. [6]), the exponential energy, Φ(A) = e^{Tr(A)} (cf. [7], [5]), but also Φ(A) = Tr(A^{p}), Φ(A) = Tr(e^{A}), the volume and the Jacobians, Φ(A) = (σ_{l}(A))^{q}, where σ_{l}(A) denotes the l-th elementary symmetric polynomial in the eigenvalues of A. By a theorem of Glaeser, [8], all these functions Φ may be written as Φ(A) = Φ^{s}(Tr(A), Tr(A)^{2}, …, Tr(A)^{m}) with some smooth function Φ^{s}: ℝ^{m} → ℝ. Functionals with Φ(A) = F (Tr(A)) have been studied by Ara in [1].

In this paper we will derive (un)stability criteria for the Φ-energies at Riemannian subimmersions. These are maps f such that df^{∗}df is an orthogonal projection of constant rank. Examples of such maps are immersions (cf. [5], [6], [1], [3]) and Riemannian submersions like G-maps f: G/H → G/K, H ⊂ K ⊂ G, between compact normal homogeneous spaces.

2000 Mathematics Subject Classification. Primary 58E20; Secondary 53C43.

Key words and phrases. Φ-energy, harmonic maps, stability

We gratefully acknowledge the referee’s valuable suggestions

2 STEFAN BECHTLUFT-SACHS, ANDREAS DÖRING

By a Φ-harmonic map we mean a critical point for E_{Φ}. Such a map f is called stably Φ-harmonic if the index form, i.e., the second variation, of E_{Φ} is nonnegative at f. At a Riemannian subimmersion f of rank k these properties depend on a few parameters determined by Φ: Let λ_{k}(Φ), λ′_{k}(Φ), μ_{k}(Φ), ν_{k}(Φ) ∈ ℝ be the parameters for Φ defined by expanding the first and second derivatives of Φ at the orthogonal projection 𝔭_{k}: ℝ^{m} → ℝ^{k} in terms of traces (cf. (2.8) and (2.9))

d_{𝔭_{k}}ΦA = λ_{k} Tr A + λ′_{k} Tr A_{V} and d_{𝔭_{k}}^{2}Φ(A) = μ_{k} Tr(A_{H}^{2}) + ν_{k}(Tr A)^{2} + ….

In Proposition 3.2 we show that for all Φ with λ_{k}(Φ) ≠ 0 the Φ-harmonic Riemannian subimmersions are those with minimal image and fibres. If λ_{k}(Φ) = 0, then any Riemannian subimmersion is Φ-harmonic.

The second variation of the 2-energy E_{Tr} has always finite index at a harmonic map but usually there are few stably critical maps of the 2-energy. For instance, the identity map on the standard k-sphere S^{k} is unstable for E_{Tr} if k ≥ 3, cf. [13]. More generally, a stable harmonic map S^{m} → X or X → S^{k} is constant if m, k > 2, cf. [16], [11]. In these assertions the sphere may be replaced by certain Riemannian symmetric spaces, see [12], [9]. In [1], [5], [6] stability criteria for isometries have been derived for functionals of the type ∫_{M} F (∥df∥), F: ℝ_{0}^{+} → ℝ_{0}^{+}, such as the exponential energy or the p-energy. For Riemannian submersions in this case, see [14].

For the Φ-energy at a Riemannian subimmersion we compute the leading symbol of the second variation. By Proposition 3.8 the second variation d_{f}^{2}E_{Φ} of the Φ-energy at a minimal Riemannian subimmersion f has finite index if λ_{k}(Φ), λ_{k}(Φ) + λ′_{k}(Φ), λ_{k}(Φ) + 2μ_{k}(Φ) + 2ν_{k}(Φ) and λ_{k}(Φ) + μ_{k}(Φ) are all positive.

Any Riemannian submersion f: M → X of rank k with totally geodesic fibres is Φharmonic. For vector fields v along f the second variation of E_{Φ} at f is given in Theorem 3.10. Some results of Urakawa on the 2-energy, cf. [15], immediately carry over to these more general functionals. For instance, if the identity map on X is Φ-unstable, then the same holds for f (cf. (3.13)). Conversely, if the identity map on X is stably Φ-harmonic, we can make f stably Φ-harmonic by shrinking its fibres. For equivariant maps among normal homogeneous spaces one can explicitely compute the index of the second variation by translating the Jacobi operator into Lie theory and computing its small eigenvalues, analogous to [15]. This will be pursued elsewhere.

2. Preliminaries

2.1. Variational formulas for the Φ-Energy

We first recall the general variational formulas for the Φ-energy from [3]. We may always extend Φ to a function Φ: M(m) → ℝ on all real (m × m)-matrices which remains invariant under conjugation by orthogonal matrices. A straightforward calculation yields

STABILITY OF NATURAL ENERGY FUNCTIONALS AT RIEMANNIAN SUBIMMERSIONS 3

the following formulas for the first and second variations. Denote by ∇^{M}, ∇^{X} the Levi-Civita connections on M, X, and by ∇ the induced connection on f^{∗}T X or T M^{∗}⊗f^{∗}T X. A smooth map f: M → X is Φ-harmonic, i.e., a critical point of the Φ-energy if

(2.1)  d_{f}E(v) = \frac{d}{dt}E_{Φ}(f_{t}) = ∫_{M} d_{df^{∗}df}Φ((∇v)^{∗}df + df^{∗}∇v) = 0

for any smooth one parameter variation f_{t} = F (t, ·), F: (−ϵ, ϵ) × M → X with v = (d/dt) f_{t}. In [3] the tension field τ_{Φ}(f) ∈ Γf^{∗}T X defined by (d/dt) E_{Φ}(f_{t}) = ⟨τ_{Φ}(f) | w⟩ is computed. For the 2-energy, Φ = dΦ = Tr, d^{2}Φ = 0, we have

τ(f) = τ_{Tr}(f) = Tr ∇df .

In [3, Proposition 2.3] the index form of the Φ-energy at a Φ-harmonic map f: M → X is computed. It is given by

d_{f}^{2}E(v, w) = \frac{d^{2}}{dtds}E_{Φ}(f_{t,s}) = ∫_{M} I_{Φ}(f)(v, w) , where
I_{Φ}(f)(v, w) = d_{df^{∗}df}Φ (−df^{∗}R^{X}(·, v)w − df^{∗}R^{X}(·, w)v)
+ d_{df^{∗}df}Φ ((∇v)^{∗}(∇w) + (∇w)^{∗}(∇v))
+ d_{df^{∗}df}^{2}Φ (df^{∗}∇v + (∇v)^{∗}df, df^{∗}∇w + (∇w)^{∗}df) .

(2.2)

The vector fields v = (d/dt) f_{t,s}, w = (d/ds) f_{t,s} along f are variation fields of a smooth 2-parameter family f_{t,s} = F (t, s, ·), F: (−ϵ, ϵ) × (−ϵ, ϵ) × M → X. By R^{X} we denote the Riemannian curvature tensor of X and by df^{∗}R^{X}(·, w)v the homomorphism R_{v,w}: T M → T M defined by ⟨R_{v,w}x | y⟩ = ⟨R^{X}(dfx, w)v | dfy⟩.

The second variation of E_{Φ} can be written in the form d_{f}^{2}E_{Φ}(v, w) = ⟨J_{Φ}(f)v | w⟩ with a second order differential operator, the Jacobi operator of Φ, acting on vector fields v, w along f. Its leading symbol is calculated from (2.2).

Corollary 2.3 (see [3, Proposition 2.5]). The leading symbol of the second variation of E_{Φ} is given by

\begin{aligned}⟨σ(ξ)v | w⟩ & = d^{2}Φ (df^{∗}v ⊗ ξ + ξ ⊗ df^{∗}v, df^{∗}w ⊗ ξ + ξ ⊗ df^{∗}w) \\ & + 2⟨v | w⟩d_{df^{∗}df}Φ(ξ ⊗ ξ) , \end{aligned}

where ξ ∈ T M = T M^{∗} by the Riemannian metric, v, w ∈ f^{∗}T X. If σ(ξ) > 0 for all ξ ≠ 0, then J_{Φ}(f) is elliptic and the second variation of E_{Φ} has finite index.

2.2. A Bochner Formula

In Section 3 some stability criteria of Riemannian submersions will be derived by comparing (2.2) with the following Bochner formula. With ∥A∥^{2} = Tr A^{∗}A and B = ∇df, B(v), B(df^{∗}v) defined by ⟨B(v)x | y⟩ = ⟨v | (∇_{x}df)y⟩, resp. B(df^{∗}v) = ∇_{df^{∗}v}df, a straightforward calculation gives for a closed manifold M that

\frac{1}{2} ∫_{M} ∥df^{∗}∇v + (∇v)^{∗}df∥^{2} = ∫_{M}∥df^{∗}∇v∥^{2} − Tr (R^{X}(·, df^{∗}v)v) + (Tr(df^{∗}∇v))^{2}
− ⟨∇_{df^{∗}v}v | τ(f)⟩ − Tr (df^{∗}∇vB(v))
+ ⟨τ(f) | v⟩ Tr (df^{∗}∇v) + Tr ((∇v)^{∗}B(df^{∗}v)) .

(2.4)

2.3. Derivatives of Φ at a projection

The endomorphism df^{∗}df of a Riemannian subimmersion f is the projection onto the horizontal distribution. We therefore need the derivatives of Φ at a projection. Decomposing ℝ^{m} = ℝ^{k} ⊕ ℝ^{m−k} and accordingly

(2.5)  Sym(m) = Sym(k) ⊕ Sym(m − k) ⊕ M(k × (m − k)) ,

we determine the components of dΦ and d^{2}Φ. Since the projection 𝔭_{k} onto the first factor ℝ^{k} is fixed by conjugation with O(k) × O(m − k), the first and second derivatives of Φ at 𝔭_{k} can be expressed in terms of traces and involve only few parameters. In order to derive this expansion we choose a curve X(s) ∈ Sym(m) through X(0) = 𝔭_{k} and a skew symmetric matrix

v = ( \begin{matrix}P & Z \\ −Z^{t} & Q \end{matrix} ) ∈ 𝔰𝔬(m)

with P ∈ 𝔰𝔬(k), Q ∈ 𝔰𝔬(m − k) and Z ∈ M(k × (m − k)). Since Φ is invariant under conjugation by O(m), we have

(2.6)  \frac{d}{dt}|_{t=0} Φ(e^{−tv}X(s)e^{tv}) = d_{X(s)}Φ[X(s), v] = 0 and
(2.7)  \frac{d^{2}}{dsdt}|_{s=0,t=0} Φ(e^{−tv}X(s)e^{tv}) = d_{𝔭_{k}}^{2}Φ(Ẋ ⊗ [𝔭_{k}, v]) + d_{𝔭_{k}}Φ[Ẋ, v] = 0 .

The only linear invariants on the first two summands Sym(k) and Sym(m − k) in (2.5) are multiples of traces. Since

[𝔭_{k}, v] = ( \begin{matrix}0 & Z \\ Z^{t} & 0 \end{matrix} ) ,

it follows from (2.6) that d_{𝔭_{k}}Φ vanishes on the third summand M(k × (m − k)) in (2.5). Hence there are λ_{k} = λ_{k}(Φ), λ′_{k} = λ′_{k}(Φ) ∈ ℝ such that

(2.8)  d_{𝔭_{k}}ΦA = λ_{k} Tr A + λ′_{k} Tr A_{V}

STABILITY OF NATURAL ENERGY FUNCTIONALS AT RIEMANNIAN SUBIMMERSIONS 5

for A ∈ Sym(m) and A_{H} ∈ Sym(k), A_{V} ∈ Sym(m − k) and A_{VH} ∈ M(k × (m − k)) such that

A = ( \begin{matrix}A_{H} & A_{VH} \\ A_{VH}^{t} & A_{V} \end{matrix} ) .

For the second derivative we get from (2.7) with Ẋ = ( \begin{matrix}0 & Y \\ Y^{t} & 0 \end{matrix} ), resp. Ẋ =

( \begin{matrix}R & 0 \\ 0 & S \end{matrix} ), that

\begin{aligned}d_{𝔭_{k}}^{2}Φ(( \begin{matrix}0 & Y \\ Y^{t} & 0 \end{matrix} ) ⊗ ( \begin{matrix}0 & Z \\ Z^{t} & 0 \end{matrix} )) & = −d_{𝔭_{k}}Φ [( \begin{matrix}0 & Y \\ Y^{t} & 0 \end{matrix} ) , ( \begin{matrix}P & Z \\ −Z^{t} & Q \end{matrix} )] \\ & = −2λ′_{k} Tr(Y^{t}Z) \end{aligned}

and

\begin{aligned}d_{𝔭_{k}}^{2}Φ(( \begin{matrix}R & 0 \\ 0 & S \end{matrix} ) ⊗ ( \begin{matrix}0 & Z \\ Z^{t} & 0 \end{matrix} )) & = −d_{𝔭_{k}}Φ [( \begin{matrix}R & 0 \\ 0 & S \end{matrix} ) , ( \begin{matrix}P & Z \\ −Z^{t} & Q \end{matrix} )] \\ & = −λ′_{k} Tr(SQ) = 0 , \end{aligned}

since S is symmetric and Q is skew symmetric. Finally, since 𝔭_{k} is fixed under conjugation with O(k) × O(m − k) and since the only quadratic O(q)-invariants on B ∈ Sym(ℝ^{q}) are linear combinations of Tr(B^{2}) and (Tr(B))^{2}, we get that

d_{𝔭_{k}}^{2}Φ(( \begin{matrix}R & 0 \\ 0 & S \end{matrix} ) ⊗ ( \begin{matrix}U & 0 \\ 0 & V \end{matrix} ))

is a linear combination of

Tr(RU) , Tr(SV) , Tr(R) Tr(U) , Tr(S) Tr(V) , Tr(S) Tr(U) + Tr(R) Tr(V) .

The second derivative may therefore be put into the form

d_{𝔭_{k}}^{2}Φ(A) = d_{𝔭_{k}}^{2}Φ(A ⊗ A)
= μ_{k} Tr(A_{H}^{2}) + ν_{k}(Tr A)^{2} − 2λ′_{k} Tr(A_{VH}^{t}A_{VH})
+ μ′_{k} Tr(A_{V}^{2}) + ν′_{k}(Tr A_{V})^{2}
+ κ_{k} Tr A_{H} Tr A_{V}

(2.9)

with coefficients μ_{k} = μ_{k}(Φ), μ′_{k} = μ′_{k}(Φ), … ∈ ℝ determined by Φ. Only the coefficients λ_{k}, λ′_{k}, μ_{k}, ν_{k} will show up in the index form. We list these for some examples in Table 1. Here σ_{p}(A) denotes the p-th elementary symmetric polynomial in the eigenvalues. It is determined by the relation

det(1 + tA) = ∑_{p=0}^{m} σ_{p}(A)t^{p} .

Φ(A) Tr A Tr(A^{p}), p > 1 (Tr A)^{p} e^{Tr A} Tr e^{A} σ_{p}(A)
λ_{k}  1  p  pk^{p−1}  e^{k}  e  \binom{k−1}{p−1}
λ′_{k}  0  −p  0  0 1 − e \binom{k−1}{p−2}
μ_{k}  0  p(p − 1)  0  0  e −\binom{k−2}{p−2}
ν_{k}  0  0  p(p − 1)k^{p−2} e^{k}  0  \binom{k−2}{p−2}

Table 1. Coefficients for some functionals

3. Riemannian subimmersions

A Riemannian, or metric, subimmersion of rank k is a map f: M → X such that for every point p ∈ M there exist a neighborhood U of p, a k-dimensional manifold Y, a metric submersion π: U → Y and an isometric immersion ı: Y → X such that f|_{U} = ı ∘ π. We denote by V = ker df and H = (ker df)^{⊥} ⊂ T M its vertical and horizontal distributions and by N = (im df)^{⊥} ⊂ f^{∗}T X the pull back of the normal bundle of M in X. We denote by X_{V} = 𝔭_{V} X, X_{H} = 𝔭_{H} X and X^{⊥} = 𝔭_{⊥} X the orthogonal projections onto V, H and N, respectively. Let T^{ker} ∈ Γ Hom(V ⊗ V, H) and T^{im} ∈ Γ Hom(H ⊗ H, N) be the second fundamental forms of the fibres and the image of f, respectively. Let A ∈ Γ Hom(H ⊗ H, V) be the tensor field A(h, h′) = 𝔭_{V} ∇_{h}h′. We will denote the various adjoints by the same letters and write

⟨A(h, h′) | u⟩ = ⟨∇_{h}^{M}h′ | u⟩ = 1/2 ⟨u | [h, h′]⟩ = −⟨h′ | ∇_{h}^{M}u⟩ = −⟨A(u)h | h′⟩
= −⟨h′ | A(h, u)⟩ = −⟨A(h)h′ | u⟩ = −⟨∇_{h′}^{M}h | u⟩ ,
⟨h | T^{ker}(u, u′)⟩ = ⟨h | ∇_{u}^{M}u′⟩ = −⟨∇_{u}^{M}h | u′⟩ = −⟨T^{ker}(h)u | u′⟩ ,
⟨r | T^{im}(h, h′)⟩ = ⟨r | T^{im}(h)h′⟩ = ⟨r | ∇_{h}^{X}h′⟩ = −⟨∇_{h}^{X}r | h′⟩ = −⟨T^{im}(r)h | h′⟩

for vector (fields) h, h′ ∈ H, u, u′ ∈ V, r ∈ N, see [4]. The second fundamental form ∇df may be expressed in terms of the second fundamental forms T^{im}, T^{ker} and A,

(3.1)  (∇_{X}df)Y = T^{im}(X_{H}, Y_{H}) − T^{ker}(X_{V}, Y_{V}) − A(X_{H}, Y_{V}) − A(Y_{H}, X_{V}) .

In the sequel (3.1) will be used to simplify the expressions (2.1) and (2.2) for the first and second variation at a Riemannian subimmersion of rank k.

3.1. The tension field at Riemannian subimmersions

For the tension field, (2.1) and (2.8) yield the following

Proposition 3.2. If f is a metric subimmersion of rank k, then we have

(3.3)  τ(f) = Tr T^{im} − Tr T^{ker}

and

(3.4)  τ_{Φ}(f) = λ_{k}(Φ)τ(f) .

In particular, all Φ with λ_{k}(Φ) ≠ 0 have the same critical subimmersions of rank k as the 2-energy. A Riemannian subimmersion is harmonic for such Φ if and only if both the image and the fibres are minimal.

Proof. The first formula (3.3) is immediate from (3.1) since the terms involving the Atensor do not contribute to the trace τ(f) = Tr ∇df.

For the second formula (3.4) we identify T_{p}M ≅ ℝ^{m} = ℝ^{k} ⊕ ℝ^{m−k} ≅ H_{p} ⊕ V_{p}, 𝔭_{k} = 𝔭_{H} and 𝔭_{n−k} = 𝔭_{V}, and apply (2.8). The integrand in the first variational formula (2.1) thus becomes

d_{df^{∗}df}Φ((∇v)^{∗}df + df^{∗}∇v) = λ_{k} Tr((∇v)^{∗}df + df^{∗}∇v) + λ′_{k} Tr((∇v)^{∗}df + df^{∗}∇v)_{V} .

Since df ∘ 𝔭_{V} = 0, the second term on the right hand side vanishes and we finally obtain

d_{df^{∗}df}Φ((∇v)^{∗}df + df^{∗}∇v) = λ_{k} Tr((∇v)^{∗}df + df^{∗}∇v) .

But this is λ_{k} times the integrand in the first variational formula for the 2-energy. Therefore τ_{Φ}(f) = λ_{k}(Φ)τ(f).

Finally if τ(f) vanishes then both Tr T^{im} ∈ ΓN and Tr T^{ker} ∈ ΓH vanish.

□

3.2. The index form at Riemannian subimmersions

The first and second derivative of Φ at 𝔭_{H} = df^{∗}df are given by (2.8) and (2.9). As before we will identify v ∈ f^{∗}T X ≅ H ⊕ N ⊂ T M ⊕ N and ∇v ∈ T M^{∗} ⊗ f^{∗}T X ≅ T M^{∗} ⊗ (H ⊕ N) ⊂ T M^{∗} ⊗ (T M ⊕ N).

In order to evaluate the index form (2.2) on a vector field v ∈ Γf^{∗}T X = Γ(H ⊕ N) we need to compute (2.8) for A_{1} = df^{∗}R^{X}(v, df·)v + (∇v)^{∗}(∇v) for the terms involving the first derivative of Φ and (2.9) with

A_{2} = df^{∗}∇v + (∇v)^{∗}df = ( \begin{matrix}𝔭_{H} ((∇v)^{∗} + ∇v) 𝔭_{H} & 𝔭_{H} ∇v 𝔭_{V} \\ 𝔭_{V}(∇v)^{∗} 𝔭_{H} & 0 \end{matrix} )

for the second order terms.

With Tr A^{∗}A = ∥A∥^{2} and Tr A_{1,V} = ∥∇v∥_{V}^{2} = ∑_{i=k+1}^{m} |∇_{e_{i}}v|^{2}, for a local orthonormal framing {e_{k+1}, …, e_{m}} of V, we infer

\begin{aligned}I_{Φ}(f)(v, v) & = 2λ_{k} (∥∇v∥^{2} − Tr⟨R^{X}(df·, v)v | df·⟩) + 2λ′_{k}∥ 𝔭^{⊥} ∇v∥_{V}^{2} \\ & + μ_{k} Tr (A_{2,H}^{2}) + 4ν_{k}(Tr 𝔭_{H} ∇v)^{2} \end{aligned} ; (3.5)

8 STEFAN BECHTLUFT-SACHS, ANDREAS DÖRING

Let div x = Tr ∇^{M}x denote the divergence of a vector field x on M. We will need the following relations.

𝔭_{⊥} ∇v_{H} 𝔭_{V} = 𝔭_{H} ∇v^{⊥} 𝔭_{V} = 0 ,
Tr (A_{2,H}^{2}) = Tr ((𝔭_{H} ∇v)^{∗} + 𝔭_{H} ∇v)^{2})
= Tr ((𝔭_{H} ∇v_{H})^{∗} + 𝔭_{H} ∇v_{H})^{2}) + 4∥T^{im}(v^{⊥})∥^{2}
+ 8 Tr ((𝔭_{H} ∇v_{H} 𝔭_{H})T^{im}(v_{H})) ,
Tr 𝔭_{H} ∇v = Tr 𝔭_{H} ∇v_{H} + Tr 𝔭_{H} ∇v^{⊥}
= div v_{H} − Tr T^{ker}(v_{H}) + Tr T^{im}(v^{⊥}) .

(3.6)

Inserting (3.6) into (3.5) yields

I_{Φ}(f)(v, v) = 2λ_{k} (∥∇v∥^{2} − Tr⟨R^{X}(df·, v)v | df·⟩) + 2λ′_{k}∥∇^{⊥}v^{⊥}∥_{V}^{2}
+ μ_{k} [Tr ((𝔭_{H} ∇v_{H})^{∗} + 𝔭_{H} ∇v_{H})^{2}) + 4∥T^{im}(v^{⊥})∥^{2}
+8 Tr ((𝔭_{H} ∇v_{H} 𝔭_{H})T^{im}(v_{H}))]
+ 4ν_{k} (div v_{H} − Tr T^{ker}(v_{H}) + Tr T^{im}(v^{⊥}))^{2} .

(3.7)

This specializes to the formulae in [3] for isometric immersions. As in (2.3) we get

Proposition 3.8. The leading symbol of the second variation of a functional E_{Φ} at a minimal Riemannian subimmersion of rank k is

σ(ξ) = 2λ_{k}|ξ|^{2} + 2λ′_{k}|ξ_{V}|^{2} 𝔭_{⊥} +2μ_{k}ξ_{H} ⊗ ξ_{H} + 2μ_{k}|ξ_{H}|^{2} 𝔭_{H} +4ν_{k}ξ_{H} ⊗ ξ_{H} .

In particular, the p-energy, the exponential energies and the Jacobians have elliptic second variation with positive symbol and finite index at Riemannian subimmersions, cf. Table 1.

3.3. Riemannian submersions with totally geodesic fibres

We will consider in more detail the case of Riemannian submersions with totally geodesic fibres. Then T^{im} = 0, T^{ker} = 0, v = v_{H} ∈ H ⊂ T M and

Tr⟨R^{X}(v, df·)v | df·⟩ = − Ric^{X}(v, v) = − Ric^{M}(v, v) − 2∥A(v)∥^{2} .

The Bochner formula (2.4) becomes

(3.9) ∫_{M} \frac{1}{2} ∥∇v + (∇v)^{∗}∥^{2} = ∫_{M} ∥∇v∥^{2} − Ric^{X}(v, v) + (div v)^{2} − 2 Tr (∇v ∘ A(v)) .

The left hand side is

\frac{1}{2} ∥∇v + (∇v)^{∗}∥^{2} = \frac{1}{2} ∥ 𝔭_{H}^{∗} L_{v} g∥^{2} + ∥∇v∥_{V}^{2} ,

where 𝔭_{H}^{∗} L_{v} g denotes the horizontal component of the Lie derivative of the metric of M. From (3.7) and (3.9) we therefore get the following expressions for the second variation.

Theorem 3.10. The index form of a Riemannain submersion f: M → X with totally geodesic fibres is given by

∫_{M} I_{Φ}(f)(v, v) = ∫_{M} 2λ_{k} (∥∇v∥^{2} − Ric^{X}(v, v)) + μ_{k}∥ 𝔭_{H}^{∗} L_{v} g∥^{2} + 4ν_{k}(div v)^{2} (3.11)
= ∫_{M} 2λ_{k} (∥∇v∥_{V}^{2} + 2 Tr (∇v ∘ A(v)))
+(λ_{k} + μ_{k})∥ 𝔭_{H}^{∗} L_{v} g∥^{2} + (4ν_{k} − 2λ_{k})(div v)^{2}
(3.12)  = ∫_{M}(2λ_{k} + 2μ_{k}) (∥∇v∥^{2} − Ric^{X}(v, v)) − 2μ_{k}∥∇v∥_{V}^{2}
+(2μ_{k} + 4ν_{k})(div v)^{2} − 4μ_{k} Tr (∇v ∘ A(v)) .

3.4. Applications

There are a number of immediate consequences of these formulas. Let f: M^{m} → X^{k} as before be a Riemannian submersion of rank k with totally geodesic fibres and Φ: M(m) → ℝ be O(m)-invariant. In case λ_{k}(Φ), μ_{k}(Φ), ν_{k}(Φ) ≥ 0, it follows from (3.11) that f is stably Φ-harmonic if it is stably harmonic.

3.4.1. Positive elliptic symbol. By Proposition 3.8 the second variation of E_{Φ} has positive symbol at f if

λ_{k}(Φ), λ_{k}(Φ) + μ_{k}(Φ), λ_{k}(Φ) + 2μ_{k}(Φ) + 2ν_{k}(Φ) > 0 .

Hence in this case we always have finite index and nullity of the second variation of E_{Φ}. Some results of Urakawa, [15], for the 2-energy extend to such Φ-energies. For instance, from (3.11), the second variation of E_{Φ} at f restricted to basic vector fields coincides with the second variation of E_{Φ} at the identity map of X. Hence the index, nullity and smallest eigenvalue of the Jacobi operator J_{Φ}(f) can be compared to the corresponding quantities of the identity map on X by (cf. [15], Proposition 6.3)

index d_{f}^{2}E_{Φ} ≥ index d_{id_{X}}^{2}E_{Φ} ,
nullity d_{f}^{2}E_{Φ} ≥ nullity d_{id_{X}}^{2}E_{Φ} ,
λ_{1}(J_{Φ}(f)) ≤ λ_{1}(J_{Φ}(id_{X}) .

(3.13)

Combining this with the instability result of [3, Theorem 3] yields

Proposition 3.14. If Φ has λ_{k}(k − 2) > 2μ_{k} + 2ν_{k}k, then any Riemannian submersion f: M^{m} → S^{k} with totally geodesic fibres onto the standard k-sphere is unstable for E_{Φ}.

The canonical variation of the metric g on M is by definition the 1-parameter family g_{t} of metrics on M obtained by rescaling the metric on the fibres f^{−1}(x), x ∈ X, by a factor t ∈ ℝ^{+}, i.e., g_{t} = 𝔭_{H}^{∗} g + t^{2} 𝔭_{V}^{∗} g. The second variation may be written as

d_{f}^{2}E_{Φ} = Q^{V} + Q^{H}

10 STEFAN BECHTLUFT-SACHS, ANDREAS DÖRING

with Q^{V}(v, w) = ⟨v | J^{V}w⟩ = 2 ∫_{M} λ_{k}∥∇v∥_{V}^{2} and Q^{H}(v, w) = ⟨v | J^{H}w⟩ collecting the remaining horizontal terms in (3.11). The vertical Jacobi operatorJ^{V} coincides with that of [15] and the horizontal Jacobi operator J^{H} is the one of [15] plus the terms involving μ_{k} and ν_{k}. Let Q_{t} be the second variation with respect to the metric g_{t}. Then Q_{t} = t^{−2}Q^{V} +Q^{H} = (t^{−2} −1)Q^{V} +Q_{1}. We decompose L^{2}(f^{∗}T X) = ker J^{V} ⊕(ker J^{V})^{⊥} and identify ker J^{V} with the space of basic vector fields and (ker J^{V})^{⊥} with those vector fields whose average along the fibres of f vanishes. A straightforward calculation shows that Q^{H}(v^{0}, v^{⊥}) = 0 for v^{0} ∈ ker J^{V}, v^{⊥} ∈ (ker J^{V})^{⊥}. Furthermore, J^{V} is the Laplacian along the fibres with values in the vector bundle f^{∗}T X. Hence the restriction of J^{V} to (ker J^{V})^{⊥} has a smallest eigenvalue λ_{1}(J^{V}) > 0. Let −C be a lower bound for the spectrum of J_{Φ} (for t = 1) and choose t ≤ (C + 1)^{−1/2}. Then

Q_{t}(v) = (t^{−2} − 1)Q^{V}(v^{⊥}) + Q_{1}(v^{0}) + Q_{1}(v^{⊥}) ≥ Q_{1}(v^{0}) .

Thus, if the identity map on X is stably Φ-harmonic, we always have Q_{1}(v^{0}) ≥ 0 and infer that f is stably Φ-harmonic with respect to sufficiently small t (cf. Theorem 7.3 in [15]). As an example, any Riemannian submersion with totally geodesic fibres is stably harmonic for the exponential energy if the fibres are suitable rescaled, since the identity map is always stable for the exponential energy (cf. [5]).

3.4.2. Small Ricci curvature. If the Ricci-curvature of X is small relative to the A-tensor, we use the estimate

∥∇v∥_{V}^{2} + ∥A(v)∥^{2} ≥ 2 |Tr (∇v ∘ A(v))|

together with (3.12). It follows for instance that if

Ric^{X}(v, v) ≤ −∥A(v)∥^{2} , λ_{k} ≥ |μ_{k}| , 2ν_{k} ≥ −μ_{k} ,

then a Riemannian submersion f: M → X^{k} with totally geodesic fibres is stably Φharmonic. This applies to all the functionals in Table 1.

References

[1] M. Ara, Geometry of F -harmonic maps, Kodai Math. J. 22 (1999), 243–263.

[2] S. Bechtluft-Sachs, Infima of Energy-type Functionals on Homotopy Classes, preprint.

[3] S. Bechtluft-Sachs, Tension Field and Index Form of Energy-Type Functionals, Glasg. Math. J. 45 (2003), 117–122.

[4] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb. (3), 10, Springer-Verlag, Berlin, 1987.

[5] L. F. Cheung and P. F. Leung, Second Variation Formula for Exponential Harmonic Maps, Bull. Austral. Math. Soc. 59 (1999), 509–514.

[6] L. F. Cheung and P. F. Leung, Some Results on Stable p-Harmonic Maps, Glasg. Math. J. 36 (1994), 77–80.

[7] J. Eells and L. Lemaire, Selected Topics in Harmonic Maps, CBMS Reg. Conf. Ser. Math. 50, American Mathematical Society, Providence, RI, 1983.

[8] G. Glaeser, Fonctions composées différentiables, Ann. of Math. (2) 77 (1963), 193–209.

[9] R. Howard and W. Wei, Nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of Euclidean spaces, Trans. Amer. Math. Soc. 294 (1986), 319–331.

[10] I. Kolár, P. W. Michor and J. Slovák, Natural operations in differential geometry, Springer-Verlag, Berlin, 1993.

[11] P. F. Leung, On the stability of harmonic maps, Harmonic maps (New Orleans, La., 1980), pp. 122–129, Lecture Notes in Math. 949, Springer, Berlin–New York, 1982.

[12] Y. Ohnita, Stability of harmonic maps and standard minimal immersions, Tôhoku Math. J. 38 (1986), 259–267.

[13] R. T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc. 47 (1975), 229–236.

[14] S. Montaldo, p-harmonic maps and stability of Riemannian submersions, Boll. Un. Mat. Ital. A (7) 10 (1996), 537–550.

[15] H. Urakawa, Stability of harmonic maps and eigenvalues of the Laplacian, Trans. Amer. Math. Soc. 301 (1987), 557–589.

[16] Y. L. Xin, Geometry of Harmonic Maps, Progr. Nonlinear Differential Equations Appl. 23, Birkhäuser Boston, Inc., Boston, MA, 1996.

[17] K. Yano, Integral formulas in Riemannian geometry, Pure and Applied Mathematics 1, Marcel Dekker, Inc., New York, 1970.

Stefan Bechtluft-Sachs

NWF I - Mathematik

Universität Regensburg

D-93040 Regensburg

Germany

  E-mail address: stefan.bechtluft-sachs@mathematik.uni-regensburg.de