Infima, Factorization and Stability for Natural Energy Functionals

Stefan Bechtluft-Sachs

July 28, 2003

Abstract

We consider the infima Ẽ(f) on homotopy classes of energy functionals E defined on smooth maps f: M^{m} → X^{k} between compact connected Riemannian manifolds. If M contains a submanifold L of codimension greater than the degree of E then Ẽ(f) is determined by the homotopy class of the restriction of f to M \ L. For functionals E estimating the conformal k-Jacobian we show that a map f with Ẽ(f) = 0 induces 0 in (co)homology of degree ≥ k and factors rationally through the (k − 1)-skeleton of the target manifold. Finally we derive variational formulas of natural first order functionals and obtain criteria for stability in particular at Riemannian subimmersions.

AMS Subject Classification: 58E20 (53C43 57D99)

1 Introduction

For functionals E_{ϕ}: C^{∞}(M, X) → ℝ_{0}^{+} on smooth maps of compact Riemannian manifolds we consider the infimum Ẽ: [M, X] → ℝ_{0}^{+} on homotopy classes, i.e. the invariant

Ẽ(f) := inf{E(g)|g: M → X, g ≃ f} .

We are interested in the information detected by Ẽ(f) for the problem of factorizing the map f: M → X in homotopy through a subskeleton X^{l} of a triangulation of X.

The energy functionals we will consider are universally defined locally computable functionals. They are given by an integral ∫_{M} ϕ(f) of a differential operator ϕ(f) = ϕ(f, ∇f, …, ∇^{r}f). Examples of such functionals are the classical energy, the p-energy, the volume, Jacobians, exponential energy, Willmore energy (see section 2). The dilatation [14], the p-dilatation [18] nor the L^{p}-norms of the differential are locally computable.

We show that under a degree condition on ϕ the value of Ẽ_{ϕ}(f) does not change if a submanifold of sufficiently large codimension is cut out from M. By a theorem of White, [24], Pluzhnikov [19], the infimum of a first order functional depends only on the homotopy class of the restriction of f to the d-skeleton M^{d} of a triangulation of M, where d is the degree of ϕ. Theorem A contains an analogous statement for functionals involving higher derivatives of f.

Theorem A Let ϕ have degree d, i.e.

|ϕ(A_{1}, …, A_{r}, …)| ≤ C(1 + ∥A_{1}∥^{d} + ∥A_{2}∥^{d/2} + ⋯ + ∥A_{r}∥^{d/r} + ⋯)

for A_{r} ∈ Hom(T M^{⊗r}, T X) → M × X in the exterior Hom-bundle. Let L ⊂ M be a submanifold of codimension q with q > d. If f, g: M → X are homotopic on M \ L then Ẽ_{ϕ}(f) = Ẽ_{ϕ}(g).

For instance in order to compute the infimum W̃(f) of the Willmore energy W (f) = ∫_{M} ∥∇df∥^{2}, we may modify f in a tubular neighbourhood of a submanifold L of codimension at least 5. Even more specifically W̃(f) = 0 for any map S^{n} → X if n ≥ 5.

We now discuss restrictions imposed by small values of Ẽ_{ϕ}(f) on the homotopy class of f. In [13], [14] the number of homotopy classes of maps is bounded in terms of the dilatation or other functionals estimating a norm of the differential. Concerning consequences of Ẽ_{ϕ}(f) = 0, a converse to Theorem A is obtained in [24] for functionals whose density satisfies ϕ(f) ≥ |df|^{n}, if n = dim M. It is proved in [24] that for such ϕ, Ẽ_{ϕ}(f) = 0 implies that f ≃ ∗ = X^{0} is nullhomotopic. If ϕ(f) ≥ |df|^{n} with n > dim M this is immediate from the Sobolev inequality since then E_{ϕ} directly gives an upper bound for the diameter of the image of f. For more general functionals we can at least show that f behaves homologically and rationally as if it factored over a subskeleton of X if ϕ estimates the corresponding Jacobian. In general, however, and even for functionals of higher than conformal degree, f need not factor in homotopy.

The probably most important case is that of gauge invariant first order functionals E_{ϕ}. Their density ϕ = ϕ(df) depends explicitely only on the differential df and is induced by a nonnegative function on (m × k)-matrices which is invariant under the action of O(m) × O(k). Thus ϕ(df) = Φ(df^{∗}df) for some symmetric function Φ in the eigenvalues of df^{∗}df. Fundamental are the Jacobians, i.e. the conformal powers ϕ(df) = σ_{l}^{n/2l}(df^{∗}df) of the elementary symmetric polynomials σ_{l} in the eigenvalues of df^{∗}df. There is a hierarchy (Proposition 2.3)

\begin{matrix}σ_{1}^{n/2} >> σ_{2}^{n/4} >> ⋯ σ_{l}^{n/2l} >> ⋯ >> σ_{n}^{1/2} , \\ ϕ^{q} >> ϕ if q ≥ 1 \end{matrix}  (1.1)

where ϕ >> ψ indicates that \xymatrix{E_{ϕ}(f_{ν}) \ar[r]^{ν→∞} & 0} implies \xymatrix{E_{ψ}(f_{ν}) \ar[r]^{ν→∞} & 0} for sequences (f_{ν})_{ν}.

With respect to >> the n-norm ∥df∥^{n} = Tr((df^{∗}df)^{n/2}) is equivalent to σ_{1}^{n/2}(df^{∗}df) = (Tr(df^{∗}df))^{n/2}. At the other extreme, l = n, the density ϕ(df) = det(df^{∗}df)^{1/2} is the volume and E_{ϕ}(f) ≤ ϵ clearly implies that the image of f cannot contain any n-cell of X if ϵ is sufficiently small. Hence f is homotopic to a map f̃: M → X^{n−1} into the n − 1-skeleton of X. One might conjecture the analogue for the intermediate steps in (1.1), i.e

Hypothesis: Let ϕ >> σ_{l}^{q/2l} and assume Ẽ_{ϕ}(f) = 0 Then f is homotopic to a map f̃: M → X^{l−1} ⊂ X into the (l − 1)-skeleton of a triangulation of X.

If this were true with q ≥ dim M then any energy E_{Φ} with monomial Φ of degree q would be equivalent to some Jacobian σ_{l}^{n/2l} with regard to the implications of Ẽ_{ϕ}(f) = 0.

Homologically the hypothesis holds:

Theorem B Assume that ϕ >> σ_{l}^{n/2l} and that Ẽ_{ϕ}(f) = 0. Then f induces 0 in homology and cohomology groups of degree ≥ l.

If f is homotopic to a map f̃: M → X^{l−1} ⊂ X then clearly Ẽ_{σ_{l}^{r}}(f) = 0 for all r. Actually we have Ẽ_{Φ}(f) = 0 for all polynomials Φ of length ≥ l. I do not know in how far a converse holds. More specifically torsion classes in the homotopy groups might be detected only by functionals dominating the m-norm of the differential. Theorem B only yields the vanishing of the primary obstructions in this factorization problem and this needs at least conformal degree q ≥ n/2l. Proposition 6.1 shows that the hypothesis as stated above in general requires to assume even higher than conformal degree.

Rationally these obstructions can be expressed by integrals over M of certain forms constructed from minimal models, as in [13], [21], [10], [11]. The smallest example not covered by the above is that of the 2-Jacobian ϕ(A) = σ_{2}^{3/4}(A^{∗}A) for maps M^{3} → X^{2}, i.e. E_{ϕ}(f) = ∫_{M} |f^{∗}ω|^{3/2} where ω is the volume form of X. In [20] it is shown that the Hopf invariant of a map M = S^{3} → S^{2} = X can be estimated by E_{σ_{2}^{3/4}}(f). Thus if Ẽ_{σ_{2}^{3/4}}(f) = 0 then f is nullhomotopic. More generally the infinite higher homotopy groups of spheres are π_{4k−1}(S^{2k}) = ℤ ⊕ G, G finite, and the Hopf invariant h_{k} : π_{4k−1}(S^{2k}) → ℤ detects the free part. As before one gets that h_{k}(f) = 0 if Ẽ_{σ_{2k}^{(4k−1)/2k}}(f) = 0 but if k > 1 this does not imply that f is nullhomotopic. I am grateful to T. Rivière for this remark.

More generally we show that the above hypothesis is true in rational homotopy.

Theorem C Assume that π_{1}(X^{l−1}) = 0 and ϕ >> σ_{l}^{m/2l}, Ẽ_{ϕ}(f) = 0 as in Theorem B. Then the rationalization f_{ℚ}: M → X_{ℚ} is homotopic to a map f_{ℚ}: M → X_{ℚ}^{l−1} into the rationalization of the (l − 1)-skeleton of X.

Infima of energy functionals on a homotopy class are usually hard to compute. In [6] for instance an explicit formula for the infimum of the 2-energy ∫_{M} |df|^{2} for maps f from punctured regions in ℝ^{3} to the sphere S^{2} is obtained. The proofs of Theorem B and Theorem C yield a (much cruder) procedure to obtain lower bounds for functionals estimating the Jacobians.

In the final part we consider stability of critical points of natural first order functionals. It has been shown in [22], [25], [16] that there are usually few stably critical maps of the 2-energy (i.e. with nonnegative second variation). In particular this holds if the 2-skeleton of one of the spaces involved is trivial. The vector field whose second variation is negative belongs to a deformation similiar to that used in [24] or in the proof of Theorem A. A typical result states that a stably harmonic map S^{m} → X or X → S^{k} is constant if m, k > 2. In these assertions the sphere may be replaced by certain Riemannian symmetric spaces, see [17], [12]. In [2], [7], [8] stability criteria 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. Extending the above to natural first order functionals E_{Φ} we show that stability at a Riemannian subimmersion can be read off a few parameters determined by Φ. We show

Theorem D Let λ_{k}, λ′, μ_{k}, ν_{k} ∈ ℝ be the Parameters for Φ defined by d_{𝔭_{k}}ΦA = λ_{k} Tr A + λ′_{k} Tr A_{V} as in (7.11) and d_{𝔭_{k}}^{2}Φ(A) = μ_{k} Tr(A_{H}^{2}) + ν_{k}(Tr A)^{2} + … as in (7.12), where 𝔭_{k} denotes the orthogonal projection in ℝ^{m} on a k-dimensional subspace ℝ^{k}.

If λ_{k} ≠ 0 then the Φ-energy has the same critical Riemannian subimmersions as the 2-energy, i.e. the minimal ones. If λ_{k} = 0 then any Riemannian subimmersion is critical (“Φ-harmonic”, Proposition 7.13).

If λ_{k}, λ_{k} + λ′_{k}, λ_{k} + 2μ_{k} + 2ν_{k}, λ_{k} + μ_{k} are all positive then the second variation of the Φ-energy has finite index at any minimal Riemannian subimmersion (Proposition 8.4).

Any Riemannian submersion f: M → X of rank k with totally geodesic fibres is Φ-harmonic. For vector fields v along f the second variation of E_{Φ} at f is given by (Proposition 8.9)

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

+ (2μ_{k} + 4ν_{k})(div v)^{2} + 2μ_{k} Tr (∇v ∘ A(v) + A(v)^{∗}(∇v)^{∗}) .

In section 2 we specify the type of functionals we consider. Section 3 proves that the infimum of a degree-d-functional does not change, when cutting out a submanifold of codimension more than d. The homological statement of Theorem B is proved in section 4 and the lifting in rational homotopy of Theorem C in section 5. In section 6 we present some counterexamples to the general hypothesis in conformal degree. The last two section 7 and 8 deal with the variational formulas for first order functionals and derive criteria for stability of critical Riemannian subimmersions.

2 Energy Functionals

2.1 Natural local functionals

Our main concern is with locally computable functionals defined on smooth maps f: M → X between Riemannian manifolds.

Definition 2.1 An energy functional assigns to any pair (M, X) of Riemannian manifolds and each smooth map f: M → X a nonnegative real number E(f) ∈ ℝ_{0}^{+} ∪ {∞} such that the following holds: For fixed f ∈ C^{∞}(M, X) and any open submanifold U ⊂ M let μ(U) := E(f|_{U}). Then μ defines a continuous measure on M whose density ϕ(f) with respect to the Lebesgue measure on M is a smooth function.

By the nonlinear version of Peetre’s Theorem, see [15], 19.7., the density is given by a differential operator ϕ(f) = ϕ(j(f)) of order r, where j(f) denotes the jet of f. In fact, since we require ϕ(f) to be smooth we obtain that ϕ depends on jets j^{r}(f) of some finite degree r = r(f) only. Using the Levi-Civita connections on M and X, we may identify the jet bundle

J^{r}(M, X) ⟶^{∇} ∏_{l=1}^{r} Hom(T M^{⊗l}, T X)

with the product of the (exterior) Hom-bundles over M × X. We will write j^{r}(f)(p) = (f(p), ∇f, … ∇^{r}f), with sections ∇f = df ∈ Γ (T M^{∗} ⊗ f^{∗}T X),

∇^{l}f ∈ Γ (T M^{∗}^{⊗l} ⊗ f^{∗}T X)

for the r-jet of f at p ∈ M. An energy

E_{ϕ}(f) := ∫_{M} ϕ(∇f, ∇^{2}f, …, ∇^{r}f, …)

is thus given by a function

ϕ: ∏_{l=1}^{∞} Hom(T M^{⊗l}, T X) → ℝ_{0}^{+} .  (2.2)

A natural energy functional additionally has the property that for all isometries χ: M → M′ and ψ: X → X′ we have that E(f) = E(ψ ∘ f ∘ χ^{−1}), or, equivalently, ϕ(f) = ϕ(ψ ∘f ∘χ^{−1})∘χ. In this case ϕ is some universal expression in the jet of f and the Riemannian curvature tensor, see [15].

The fundamental examples considered here have gauge invariant densities. Identifying ℝ^{m} and its dual via the standard scalar product we may write

∏_{l=1}^{∞} Hom(T M^{⊗l}, T X) = P_{O(m)×O(k)}(M × X) ×_{O(m)×O(k)} ∏_{l=1}^{∞} Hom(ℝ^{m}^{⊗l}, ℝ^{k}) .

Thus gauge invariant energies are of finite order and parametrized by functions

ϕ: ∏_{j=1}^{r} ((ℝ^{m})^{⊗j} ⊗ ℝ^{k}) → ℝ_{0}^{+} .

which are invariant under the (diagonal) action of O(m) × O(k).

2.2 Examples

Using the Riemannian metrics to identify the dual tangent bundles T M^{∗} ≅ T M and T X^{∗} ≅ T X first order functionals take the form

E_{ϕ}(f) = E_{Φ}(f) := ∫_{M} Φ(df^{∗}df))d vol_{g}

where df^{∗} denotes the adjoint of df. Famous examples of such functionals are the classical (2-)energy, or more general p-energy ∫_{M} (Tr(df^{∗}df))^{p/2} of degree p, the exponential energies ∫_{M} e^{Tr(df^{∗}df)}, ∫_{M} Tr e^{df^{∗}df} of infinite degree, F -energies ∫_{M} F (Tr(df^{∗}df)) considered by Ara [2], where Φ(A) = F (Tr A) and the Jacobians ∫_{M} σ_{p}(df^{∗}df)^{q} with σ_{p} the pth elementary symmetric polynomial. The Willmore energies W (f) := ∫_{M} ∥∇df∥^{2} of degree 4 and W_{0}(f) := ∫_{M} ∥∇df(df^{∗}df)^{−1}∥^{2} are gauge invariant.

We need the following estimates between polynomial energies. An invariant function Φ on symmetric matrices is essentially the same as a symmetric function in the eigenvalues. For an arbitrary symmetric function Φ in n variables denote by

l(Φ) := min{r | Φ(x_{1}, …, x_{r}, x_{r+1} = 0, …, x_{n} = 0) ≠ 0} .

the length of Φ. If Φ is polynomial (eventually with real exponents) then l(Φ) is the length of the shortest monomial ocurring in Φ. By ((x_{1}^{i_{1}} ⋯ x_{s}^{i_{s}})), for i_{1} ≥ i_{2} ≥ ⋯ ≥ i_{s}, i_{j} ∈ ℝ_{0}^{+}, we denote the symmetrization of x_{1}^{i_{1}} ⋯ x_{s}^{i_{s}}, i.e. the function

∑_{π∈S_{n}} x_{π1}^{i_{1}} ⋯ x_{πs}^{i_{s}} .

We have a simple hierarchy of the functionals E_{((x_{1}^{i_{1}}···x_{s}^{is}))}:

Proposition 2.3 1. If 1 ≤ r < s ≤ l ≤ n and i_{s} ≥ ϵ ≥ 0 then

E_{((x_{1}^{i_{1}}···x_{l}^{i_{l}}))}(f) ≤ n!E_{((x_{1}^{i_{1}}···x_{r}^{ir+ϵ}···x_{s}^{is−ϵ}···x_{l}^{i_{l}}))}(f) .

2. For any μ > 0, r ≥ 1 we have

E_{Φ} ≤ \frac{1}{μ^{r−1}} E_{Φ^{r}} + μ vol M .

Proof: Let a_{1}(x) ≥ a_{2}(x) ≥ … ≥ a_{n}(x) be the eigenvalues of d_{x}f^{∗}d_{x}f, x ∈ M. Now

E_{((x_{1}^{i_{1}}···x_{l}^{i_{l}}))}(f) = ∫_{M} ∑_{π∈S_{n}} a_{π1}^{i_{1}} · · · · · · a_{πr}^{i_{r}} ⋯ a_{πs}^{i_{s}} ⋯ a_{πl}^{i_{l}}

≤ n! ∑_{π∈S_{n}} ∫_{a_{π1}≥...≥a_{πl}} a_{π1}^{i_{1}} · · · · · · a_{πr}^{i_{r}} ⋯ a_{πs}^{i_{s}} ⋯ a_{πl}^{i_{l}}

≤ n! ∑_{π∈S_{n}} ∫_{a_{π1}≥...≥a_{πl}} a_{π1}^{i_{1}} · · · · · · a_{πr}^{i_{r}+ϵ} ⋯ a_{πs}^{i_{s}−ϵ} ⋯ a_{πl}^{i_{l}}
≤ n!E_{((x_{1}^{i_{1}}···x_{r}^{ir+ϵ}···x_{s}^{is−ϵ}···x_{l}^{i_{l}}))}(f)

For the second claim we decompose the domain of integration to estimate

∫_{M} Φ(df^{∗}df) ≤ ∫_{\{x|Φ(d_{x}f^{∗}d_{x}f)>μ\}} \frac{Φ(df^{∗}df)^{r}}{μ^{r−1}} +  ∫_{\{x|Φ(d_{x}f^{∗}d_{x}f)≤μ\}} μ

≤ \frac{1}{μ^{r−1}} E_{Φ^{r}} + μ vol M .
•

We denote by σ_{l}(A) the lth elementary symmetric polynomial in in the eigenvalues of a matrix A. It is determined by the identity det(1 + λA) = ∑_{l} σ_{l}(A)λ^{l}. If Φ and Ψ are homogeneous polynomials of the same degree with l(Ψ) ≤ l(Φ) = l then there are constants C̃ = C̃(Ψ, Φ, n) and C = C(Φ, n, l) such that

C̃CE_{Ψ}(f) ≥ CE_{Φ}(f) ≥ E_{σ_{l}^{n/2l}}(f)

for all f.

3 Removing submanifolds

We prove Theorem A. Let ϕ have degree ≥ d and let L ⊂ M be a submanifold of codimension q with q > d. Choose R small enough such that the exponential map restricted to the R-disc bundle of the normal bundle of L is a diffeomorphism, i.e. the map

RDν(L) → U_{R}(L) := {x ∈ M | d(x, L) ≤ R}
v ∈ ν_{p}(L) ↦ exp_{p}(v) .

We will write ρ(x) = d(x, L) for the distance from L and μ(r, x) = exp_{p}(rv) for x = exp_{p}(v) ∈ U_{R}(L) and r ≤ R/ρ(x). Let 0 < α < β = 2α < ϵ = 3α < R and choose smooth functions χ: [0, R] → [0, 1] and ψ: [0, R] → [0, R/α] such that χ(x) = 0 for x ∈ [0, α], χ(x) = 1 for x ∈ [β, R], χ′ ≥ 0 and ψ(x) = R/α for x ∈ [0, α], ψ(x) = 1 for x ∈ [ϵ, R], ψ′ ≤ 0 and R/x ≥ ψ(x) ≥ R/2x for x ∈ [α, β]. If the functional in question has order r we also require that for the derivatives χ^{(j)} and ψ^{(j)}, j = 0 … r, we have that |χ^{(j)}(x)| ≤ C/α^{j} and |ψ^{(j)}| ≤ C/α^{j+1}. Here and in the sequel C denotes a suitable constant independent of α. Denote by π the map M → M × [0, 1] given by

π(x) := (μ(ψ(ρ(x)), x), χ(ρ(x))) .

Note that the image of π is contained in the compact set

K := M × {1} ∪ M \ U_{R/2}(L) × [0, 1] ∪ M × {0} .

Let H: K → X be a smooth homotopy between f = H|_{M×{0}} and g|_{M×{1}} outside L. The map h defined by h(x) := H(π(x)) is homotopic to f and coincides with g on M \ U_{ϵ}(L). Thus

Ẽ_{ϕ}(f) − Ẽ_{ϕ}(g) = Ẽ_{ϕ}(h) − Ẽ_{ϕ}(g) = ∫_{U_{ϵ}} ϕ(∇h, …, ∇^{r}h) − ∫_{U_{ϵ}} ϕ(∇g, …, ∇^{r}g) .

For the proof of the theorem it suffices to show that the energy of h on U_{ϵ}(L) can be made arbitrarily small by choosing α appropriately. So we need to estimate the (covariant) derivatives of order ≤ r of h. Up to some constants depending on the derivatives of H on the compact set K we have to estimate ∇^{j}π, j = 1 … r. At p ∈ L we use a chart ξ of U_{R}(L) mapping L to ℝ^{n−q} × {0}, preserving ρ, i.e. ξ(z) = (x, y) with ρ(z) = |x|, and such that ξ(μ(r, z)) = (rx, y). Up to constants depending on derivatives of ξ we have to estimate the derivatives of

(x, y) ↦ (ψ(|x|)x, y, χ(|x|)) .

A straightforward calculation shows that

∇^{j}χ(|x|) = ∑_{i=1}^{j} χ^{(i)}(|x|)O(1/|x|^{j−i}) ,

∇^{j}ψ(|x|)x = ∑_{i=1}^{j} ψ^{(i)}(|x|)O(1/|x|^{j−i−1}) .

Hence

||∇^{j}χ(|x|)|| ≤ Cα^{−j} and ||∇^{j} (ψ(|x|)x) || ≤ Cα^{−j} .

Therefore we can estimate ∇^{j}π by α^{−j} and the energy integral by

E_{ϕ}(h|_{U_{ϵ}(L)}) = ∫_{U_{ϵ}(L)} ϕ(∇h, …, ∇^{r}h)
≤ vol U_{ϵ}(L)C(1 + |∇h|^{d} + ⋯ + |∇^{r}h|^{d/r})
≤ Cϵ^{q}α^{−d} = C3^{q}α^{q−d}

If q > d then this tends to 0 for α → 0. •

4 Vanishing in homology

In order to prove Theorem B assume ϕ(A) = Φ(A^{∗}A) = σ_{l}(A^{∗}A)^{n/2l}. Let f: M → X have Ẽ_{ϕ}(f) = 0. Let M^{q}, X^{q} denote the q-skeleta of smooth triangulations of M and X respectively. We will show that the restriction of f|_{M^{q}} is homotopic to a map into X^{q−1} for l ≤ q.

By the Deformation Theorem of geometric measure theory, [3], we can deform the restriction f|_{M^{q}} of f to a Lipschitz map f̃ mapping M^{q} to X^{q} such that for the q-dimensional Hausdorff measures H we have

H^{q}(f̃(M^{q})) ≤ C_{1}H^{q}(f(M^{q}))

where C_{1} does not depend on f.

We choose an embedding of the tangent bundle T M ↪ M × ℝ^{s} into a trivial bundle of rank s. Let π be the orthogonal projection M × ℝ^{s} → T M. Chose ϵ small enough so that Ψ_{v}: m ↦ exp_{m}(πv) is a diffeomorphism of M for all v ∈ D^{s} = {v ∈ ℝ^{s} | |v| ≤ ϵ}. Also let U_{ϵ}M^{q} denote the image of M^{q} × D^{s} under the map Ψ: (m, v) ↦ exp_{m}(πv). Let f_{v}(m) = f(exp_{m}(πv)). We denote by e_{Φ}(f) = Φ(df^{∗}df) the energy density and apply the transformation formula to the submersion Ψ: M^{q} × D^{s} → U_{ϵ}M^{q}. Let J_{n}Ψ be the n-Jacobian of this map,

C_{2} = \frac{1}{max_{z∈U_{ϵ}M^{q}} H^{s+q−n}(Ψ^{−1}(z))}

and

C_{3} = \frac{min_{M^{q}×D^{s}} J_{n}Ψ}{max_{z∈U_{ϵ}M^{q}} H^{s+q−n}(Ψ^{−1}(z)) max_{m,v} |d_{m}ψ_{v}^{∗}d_{m}ψ_{v}|^{n/2}} ,

where |d_{m}ψ_{v}^{∗}d_{m}ψ_{v}| denotes the operator norm of d_{m}ψ_{v}^{∗}d_{m}ψ_{v} for m ∈ M^{q}, v ∈ D^{s}, i.e. its largest eigenvalue. Then

E_{Φ}(f) ≥ ∫_{U_{ϵ}M^{q}} e_{Φ}(f)
≥ C_{2} ∫_{U_{ϵ}M^{q}} e_{Φ}(f)H^{s+q−n}(Ψ^{−1}(z))dz
= C_{2} ∫_{M^{q}×D^{s}} e_{Φ}(f)(Ψ(y))J_{n}Ψ(y)dy
≥ C_{2} min_{M^{q}×D^{s}} J_{n}Ψ ∫_{M^{q}×D^{s}} e_{Φ}(f)(Ψ(y))dy
= C_{2} min_{M^{q}×D^{s}} J_{n}Ψ ∫_{D^{s}} ∫_{M^{q}} e_{Φ}(f)(Ψ_{v}(m))dmdv
≥ C_{3} ∫_{D^{s}} ∫_{M^{q}} e_{Φ}(f ∘ Ψ_{v})dmdv .  (4.1)

Assume that E_{Φ}(f) ≤ δ for some δ. By (4.1) we find v ∈ D^{s} such that

∫_{M^{q}} e_{Φ}(f_{v}|_{M^{q}})(m)dm ≤ δ/C_{3} .

From proposition 2.3 there is a constant C_{4} such that

H^{q}(f_{v}(M^{q})) = E_{σ_{q}^{1/2}} ≤ C_{4}E_{σ_{l}^{q/2l}} ≤ C_{4} (\frac{1}{μ^{n/q−1}} E_{σ_{l}^{n/2l}} + μ vol M )
≤ C_{4} (\frac{1}{μ^{n/q−1}} δ/C_{3} + μ vol M )

for any μ > 0 provided q ≥ l. Choosing δ and μ appropriately we can achieve that this is strictly less than the volume of the smallest q-cell of X^{q}. Thus we get a map f̃_{v} homotopic to f which maps M^{q} to X^{q} and misses a point in every q-cell of X^{q}. Therefore it is homotopic to a map M^{q} → X^{q−1}. •

5 Factorization in Rational Homotopy

In this section we show Theorem C, i.e. that rationally, the vanishing of the infimal l-Jacobian suffices to obtain factorization over the (l − 1)-skeleton. As in the previous section let ϕ(A) = Φ(A^{∗}A) = σ_{l}(A^{∗}A)^{n/2l} assume Ẽ_{ϕ}(f) = 0. Denote the rationalization of f: X → M by f_{ℚ}: M → X_{Q} and let X^{l−1} ↪ X_{ℚ}^{l−1} be the rationalization of the of the (l − 1)-skeleton of X.

The proof is a computation in suitable minimal models similiar to [13]. By the functorial properties of minimal models the rationalization f_{ℚ}: M → X_{ℚ} is homotopic to a map F: M → X_{ℚ}^{k} ⊂ X_{ℚ} if there is a relative Sullivan algebra S = Ω(X) ⊗ ΛV ≃ Ω(X^{k}) and an extension F: Ω(X) ⊗ ΛV → Ω(M) of f^{∗}: Ω(X) → Ω(M), see [10], [11]. We will recall the construction of S and use the estimates on f to define F.

5.1 Sullivan Algebras

Let j: X → Y (X := Ω(X), Y := Ω(X^{k}) in our case) be a morphism of commutative cochain algebras. Up to homotopy j can be replaced by a morphism j̃ into a relative Sullivan algebra homotopy equivalent to Y such that the triangle

\xymatrix{X \ar[r]^{j̃} \ar[rr]_{j} & S = X ⊗_{d} ΛV \ar[r]^{m}_{≃} & Y}

commutes up to homotopy. By definition S is called a relative Sullivan allgebra if

V = ⨁_{i>0} V_{i} = ⋃_{i>0} V (i) , V (i) = V (i − 1) ⊕ V_{i}

with graded vector spaces V_{i}, V (i). The differential is required to satisfy the nilpotence condition

dV (i) ⊂ S(i − 1) = X ⊗_{d} ΛV (i − 1) where S(−1) = X .

5.2 Construction of S

The algebras S(q) are constructed together with morphisms m_{q}: S(q) → Y which are quasiisomorphisms in degrees increasing with q. To begin, let V (0) ↪^{ι} ker d ⊂ Y be a vectorspace of cycles in Y such that V (0) ⊕ im Hj generates H(Y) as an algebra. Define

S(0) := X ⊗ ΛV (0) with d|_{V (0)} = 0

and

\xymatrix{m_{0} : X ⊗ ΛV (0) \ar[r]^{j⊗ι} & Y .}

By construction m_{0} induces an epimorphism in homology.

Proceeding inductively assume

\xymatrix{S(q) := X ⊗_{d} ΛV (q) \ar[r]^{m_{q}} & Y}

has already been constructed and Hm_{q} is surjective. Let V_{q+1} = ker Hm_{q} as a vector space but with degrees lowered by 1. The differential d: V_{q+1} → S(q) is defined to be the inclusion with this degree shift. The map m_{q+1} : V_{q+1} → Y is the lift of m_{q} over d:

\xymatrix{ker Hm_{q} \ar@{^{(}->}[r] & HS(q) \ar[r]^{Hm_{q}} \ar@{^{(}->}[d] & HY \ar@{^{(}->}[d] \\ V_{q+1} \ar[drr]^{m_{q+1}} \ar@{^{(}->}[r]^{d} \ar[u]^{=} & S(q) \ar[r]^{m_{q}} & Y \\  &  & Y \ar[u]^{d}}  (5.1)

Extending m_{q+1} to ΛV_{q+1} ⊕ V (q) we obtain m_{q+1} : S(q + 1) := X ⊗ V (q + 1) → Y which again induces an epimorphism in homology.

5.3 Extension of f

Let f: X = Ω(X) → M = Ω(M) and j: X = Ω(X) → Y = Ω(X^{k}) be the morphisms of commutative cochain algebras induced by f: M → X and the inclusion j: X^{k} ↪ X respectively. The homomorphism induced by j in cohomology is an isomorphism in degrees < k, injective in degree k and 0 in degrees > k. Hence we may choose V (0) ⊂ H^{k}(Y) such that

H(Y) = V (0) ⊕ im Hj

and Hm_{0} is isomorphic in degrees ≤ k. Denote by 𝔦(q) the ideal generated by V (0) in S(q). In the diagram (5.1) we may split

V_{q+1} = V′_{q+1} ⊕ d^{−1}𝔦(q)

where V′_{q+1} lies in degrees > k.

We will inductively extend f to maps f_{q}: S(q) → M such that the f_{q} vanish on 𝔦(q). To this end set

f_{−1} := f: S(−1) = X → M

and assume that we already have constructed the extension

f_{q}: S(q) → M

satisfying estimates

∥f_{q}ω∥_{n/r} ≤ ϵ∥ω∥_{n/r}  (5.2)

for all ω ∈ S(q) of degree r, n ≥ r > k. Changing f by a homotopy we can have (5.2) with arbitrarily small value of ϵ. If dω = 0 then by proposition 2.3 an estimate

|∫_{σ} f_{q}ω| ≤ ∫_{σ} |f_{q}ω| ≤ μ^{1−n/r} ∫_{σ} |f_{q}ω|^{n/r} + μ vol σ ≤ μ^{1−n/r}ϵ^{n/r} + μ vol σ

holds for any r-cycle σ in M and any ϵ, μ > 0. In particular f_{q}ω is exact.

Thus f_{q}(dV′_{q+1}) ⊂ dM. Let {ω_{j}}_{j} be a basis for V′_{q+1}. We define f_{q+1} on V_{q+1} by lifting f_{q}. More precisely, choose for each ω_{j} some α_{j} ∈ M satisfying dα_{j} = f_{q}(dω_{j}) and the estimate of Lemma 5.3. Define f_{q+1}|_{𝔦}(q) := 0 and f_{q+1}(ω_{j}) := α_{j}.

Lemma 5.3 Let M be a compact n-dimensional Riemannian manifold and denote by ∥ · ∥_{p} the L^{p}-norm of differential forms given by the Riemannian metric. There is a constant C ∈ ℝ depending on M (but not on ω) such that for each exact r-form ω ∈ Ω^{r}(M), ω ∈ dΩ^{r−1}(M) there is α ∈ Ω^{r−1}(M) with

ω = dα and ∥α∥_{n/(r−1)} ≤ C∥ω∥_{n/r} .

Proof: From the Hodge decomposition

Ω(M) = ker Δ ⊕ im d ⊕ im d^{∗}

the Laplacian Δ is invertible on im d ⊕ im d^{∗}. Let α := d^{∗}Δ^{−1}ω. Clearly dα = ω. Extending the operator Δ^{−1} on im d⊕im d^{∗} by 0 to all of Ω^{r}(M) yields a bounded operator L^{p} = W^{0,p} → W^{2,p} into the Sobolev space W^{2,p}, [23]. Also d^{∗}: W^{2,p} → W^{1,p} is bounded, [1]. From the Sobelev-embedding ι: W^{1,p} ⊂ L^{np/(n−p)} we infer that

∥α∥_{np/(n−p)} ≤ C∥ω∥_{p}

where C := ∥ι d^{∗}Δ^{−1}∥_{op} is the operator norm. With p = n/r, np/(n − p) = n/(r − 1) the assertion follows. •

6 Further Examples

The following proposition shows that the assumptions of theorem B do not in general imply that f is homotopic to a map to f̃: M → X^{l−1} ⊂ V.

Proposition 6.1 Let n = m + m′, k = l + l′, g: S^{m} → S^{l}, g′: S^{m′} → S^{l′} and f = g × g′: S^{m} × S^{m′} → S^{l} × S^{l′}. Assume that r < m/2l or r < m′/2l′. Then Ẽ_{ϕ}(g × g′) = 0 = Ẽ_{ϕ}(g ∧ g′) for ϕ = σ_{k}^{r} where g ∧ g′: S^{m+m′} = S^{m} ∧ S^{m′} → S^{l+l′} denotes the smash product of g and g′. In particular if g: S^{n−1} → S^{k−1} and f = Sg = g ∧ id: S^{m} → S^{k} is the suspension of g then Ẽ_{ϕ}(f) = 0 for ϕ = σ_{k}^{r} if r < (n − 1)/2(k − 1).

Proof: For the cartesian product f = g × g′ we get

df^{∗}df = ( \begin{matrix}dg^{∗}dg & 0 \\ 0 & dg′^{∗}dg′ \end{matrix} ) ,

σ_{k}(df^{∗}df) = σ_{l}(dg^{∗}dg)σ_{l′}(dg′^{∗}dg′) and

Ẽ_{ϕ}(f) = Ẽ_{σ_{l}^{r}}(g)Ẽ_{σ_{l′}^{r}}(g′) .

If r < m/2l or r < m′/2l′ then one of the factors vanishes by [24] or theorem A, hence Ẽ_{σ_{k}^{r}}(f) = 0.

The assertion about the smash product follows from that for the cartesian product. We deform g and g′ to maps which are constant on some open sets U ⊂ S^{m} and U′ ⊂ S^{m′} containing the base points. The projection π: S^{m}×S^{m′} → S^{m} ∧ S^{m′} and its inverse have bounded differential over S^{m} \ U × S^{m′} \ U′. Hence we can estimate

E_{ϕ}(g ∧ g′) = ∫_{π(S^{m}\U×S^{m′}\U′)} σ_{k}^{r}(g ∧ g′) ≤ C  ∫_{S^{m}\U×S^{m′}\U′} σ_{k}^{r}(π′ ∘ (g × g′))

≤ C ∫_{S^{m}×S^{m′}} σ_{k}^{r}(g × g′) = CE_{σ_{l′}^{r}}(g × g′)

where π′: S^{l} × S^{l′} → S^{l} ∧ S^{l′}. •

The generator of π_{4}(S^{3}) ≅ ℤ/2 is the suspension of the Hopf map η: S^{3} → S^{2}. Hence Ẽ_{σ_{3}^{r}}(sη) = 0 for r < 3/4 but the conformal power is r = 2/3.

7 Variational formulae for Riemannian subimmersions

In the sequel we will be concerned with gauge invariant first order energies E_{Φ} : C^{∞}(M, N) → ℝ_{0}^{+},

E_{Φ}(f) = ∫_{M} Φ(df^{∗}df) .

with a function Φ: Sym(m) → ℝ_{0}^{+} defined on nonnegative symmetric m × m matrices invariant under conjugation by O(m). We assume that Φ extends to a smooth symmetric function Φ: M(m) → ℝ on all real (m × m)-matrices which is again invariant under conjugation by orthogonal matrices.

7.1 Tension field and index form of a Φ-harmonic map

A straightforward calculation yields the following formulas for the first and second variation, [4]. Note that ∇ denotes the induced connection on f^{∗}T X respectively T M^{∗} ⊗ f^{∗}T X. A smooth map f: M → X is Φ-harmonic, i.e. a critical point of the Φ-energy if

\frac{d}{dt}E_{Φ}(f_{t}) = ∫_{M} d_{df^{∗}df}Φ((∇v)^{∗}df + df^{∗}∇v) = 0  (7.1)

for any smooth one parameter variation f_{t} = F (t, ·), F: (−ϵ, ϵ) × M → X with v = d/dtf_{t}. If M is closed or the variation has compact support in the interior then this is equivalent to the vanishing of the Φ-tension of f defined by

τ_{Φ}(f) := d_{df^{∗}df}Φ ∇df + d_{df^{∗}df}^{2}Φ(∇(df^{∗}df) ⊗ df) .  (7.2)

In a local orthonormal frame {e_{i}} around p ∈ M this reads

τ_{Φ}(f) := ∇_{e_{i}} (d_{i,j}Φ dfe_{j})
= d_{i,j}Φ (∇_{e_{i}}dfe_{j} − df∇_{e_{i}}e_{j})
+ d_{k,l}d_{i,j}Φ⟨∇_{e_{i}}dfe_{k} − df∇_{e_{i}}e_{k} | dfe_{l}⟩dfe_{j}

where d_{i,j} = ∂/∂x_{i,j} is the partial derivative for the i, j-matrix entry at df^{∗}df. For the 2-energy, Φ = dΦ = Tr, d^{2}Φ = 0, this gives

τ(f) = τ_{Tr}(f) = Tr ∇df .

The index form, i.e. the second variation, of the Φ-energy at Φ-harmonic map f: M → X is

\begin{aligned}\frac{d^{2}}{dtds}E_{Φ}(f_{t,s}) = ∫_{M} & I_{Φ}(f)(v, w) \\ = ∫_{M} & 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) \end{aligned} (7.3)

where f_{t,s} is a smooth 2-parameter variation with v = d/dtf_{t,s}, w = d/dsf_{t,s}. 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, such that ⟨R_{v,w}x | y⟩ = ⟨R^{X}(dfx, w)v | dfy⟩.

The second variation of E_{Φ} can be written in the form d^{2}E_{Φ}(v, w) = ⟨P v | w⟩ with a second order differential operator acting on vector fields v, w along f. Its leading symbol can be calculated from (7.3).

Corollary 7.4 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 P is elliptic and in particular the second variation has finite index.

In section 8 stability of Riemannian subimmersions will be derived by comparing (7.3) with the following Bochner type 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)x := (∇_{df^{∗}v})x, we have for closed manifolds M:

\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)) (7.5)

Proof: The left hand side is

\frac{1}{2}∥df^{∗}∇v + (∇v)^{∗}df∥ = ∥df^{∗}∇v∥^{2} + Tr (df^{∗}∇v(∇v)^{∗}df) .  (7.6)

In a local orthonormal framing {e_{i}}_{i} of T M with ∇e_{i} = 0 at the point in question we have

⟨∇_{e_{i}}^{M}df^{∗}v | e_{j}⟩ = ⟨(∇_{e_{i}}df^{∗})v + df^{∗}∇_{e_{i}}v | e_{j}⟩ = ⟨v | (∇_{e_{i}}df)e_{j}⟩ + ⟨∇_{e_{i}}v | dfe_{j}⟩ .

Computing modulo divergences yields

\begin{matrix}Tr (R^{X}(·, df^{∗}v)v) & = ⟨∇_{e_{i}}∇_{df^{∗}v}v | dfe_{i}⟩ − ⟨∇_{df^{∗}v}∇_{e_{i}}v | dfe_{i}⟩ − ⟨∇_{[e_{i},df^{∗}v]}v | dfe_{i}⟩ \\ = −⟨∇_{df^{∗}v}v | τ(f)⟩ − ⟨∇_{df^{∗}v}∇_{e_{i}}v | dfe_{i}⟩ − ⟨∇_{e_{j}}v | dfe_{i}⟩⟨∇_{e_{i}}^{M}df^{∗}v | e_{j}⟩ \\ = −⟨∇_{df^{∗}v}v | τ(f)⟩ − ⟨∇_{df^{∗}v}∇_{e_{i}}v | dfe_{i}⟩ − Tr (df^{∗}∇v(B(v) + (∇v)^{∗}df)) \end{matrix} ; (7.7)

and

\begin{matrix}(Tr(df^{∗}∇v))^{2} & = ⟨dfe_{i} | ∇_{e_{i}}v⟩⟨dfe_{j} | ∇_{e_{j}}v⟩ \\ = −⟨τ(f) | v⟩ Tr (df^{∗}∇v) − Tr ((∇v)^{∗}B(df^{∗}v)) − ⟨dfe_{j} | ∇_{df^{∗}v}∇_{e_{j}}v⟩ . \end{matrix} (7.8)
•

7.2 Derivatives of Φ at a projection

The endomorphism df^{∗}df of a Riemannian subimmersion f is the projection onto the horizontal distribution. Decompose ℝ^{m} = ℝ^{k} ⊕ ℝ^{m−k} and accordingly

Sym(m) = Sym(k) ⊕ Sym(m − k) ⊕ ℝ^{k} ⊗ ℝ^{m−k} .

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 determined by Φ.

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), Z ∈ M(k × m − k). Since Φ is invariant under conjugation by O(n),

\frac{d}{dt}|_{t=0} Φ(e^{−tv}X(s)e^{tv}) = d_{X(s)}Φ[X(s), v] = 0  (7.9)

and

\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 . (7.10)

Since

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

it follows from (7.9) that

d_{𝔭_{k}}ΦA = λ_{k} Tr A + λ′_{k} Tr A_{V}  (7.11)

for

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

with λ_{k} = λ_{k}(Φ), λ′_{k} = λ′_{k}(Φ) ∈ ℝ uniquely determined by Φ.

For the second derivative we get from (7.10) with Ẋ = ( \begin{matrix}0 & Y \\ Y^{t} & 0 \end{matrix} ) respec-

tively 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 skew symmetric. Finally, since 𝔭_{k} is fixed by conjugation with O(k) × O(m − k) 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}

(7.12)

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 in some examples:

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

and

ν_{k}(σ_{p}^{q}) = q(q − 1)\binom{k}{p}^{q−2}\binom{k − 1}{p − 1}^{2} + q\binom{k}{p}^{q−1}\binom{k − 2}{p − 2}

7.3 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. Let T^{ker} ∈ Γ Hom(V ⊗ V, H) and T^{im} ∈ Γ Hom(H ⊗ H, N) be the second fundamental forms of the fibres respectively the image of f and A ∈ Γ Hom(H ⊗ H, V) be the tensor field A(h, h′) = 𝔭_{V} ∇_{h}h′. We will denote the 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 [5].

The variational formulas (7.2) and (7.3) for the Φ-tension and index form simplify at a Riemannian subimmersion of rank k. Split T_{p}M ≅ ℝ^{m} = ℝ^{k} ⊕ ℝ^{m−k} ≅ H_{p} ⊕ V_{p} and denote by 𝔭_{k} = 𝔭_{H}, 𝔭_{n−k} = 𝔭_{V} the projections on the first respectively second factor. The second fundamental form ∇df may be expressed in terms of the second fundamental forms T^{im}, T^{ker} and A,

(∇_{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}) .

From (7.1) or (7.2) and (7.11) we get

Proposition 7.13 If f is a metric subimmersion of rank k, then we have

τ(f) = Tr T^{im} − Tr T^{ker}

and

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 iff both image and fibres are minimal.

8 The Index Form at Riemannian subimmersions

The first and second derivative of Φ at 𝔭_{H} = df^{∗}df are given by (7.11) and (7.12). As before we will identify v ∈ f^{∗}T X ≅ H ⊕ N ⊂ T M ⊕ N and express ∇v ∈ T M^{∗} ⊗f^{∗}T X ≅ T M^{∗} ⊗(H⊕N) ⊂ T M^{∗} ⊗(T M ⊕N) in terms of the Levi-Civita connection ∇^{M} of M, the induced connection ∇^{⊥} on N and the tensor fields A, T^{ker} and T^{im}. For v = v_{H} + v^{⊥}, v_{H} ∈ H ⊂ T M, v^{⊥} ∈ N we get

⟨z | ∇_{x}v⟩ = ⟨z | ∇_{x}^{M}v_{H}⟩ + ⟨z^{⊥} | ∇_{x}^{⊥}v^{⊥}⟩

− ⟨z_{V} | A(x_{H}, v_{H})⟩ + ⟨T^{ker}(z_{V}, x_{V}) | v_{H}⟩ + ⟨z^{⊥} | T^{im}(x_{H}, v_{H})⟩

− ⟨T^{im}(x_{H}, z_{H}) | v^{⊥}⟩ , i.e.
∇v = ∇^{M}v_{H} + ∇^{⊥}v^{⊥} − A(v_{H}) − T^{ker}(v_{H}) + T^{im}(v_{H}) + T^{im}(v^{⊥}) .

In order to evaluate the index form (7.3) on a vectorfield v ∈ Γf^{∗}T X = Γ(H⊕N) we need to compute (7.11) for A_{1} = df^{∗}R^{X}(v, df·)v + (∇v)^{∗}(∇v) for the terms involving the first derivative of Φ and (7.12) 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}  (8.1)

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

∥∇v∥^{2} = ∥ 𝔭_{H} ∇v_{H} + T^{im}(v^{⊥})∥^{2} + ∥∇^{⊥}v^{⊥} + T^{im}(v_{H})∥^{2}
= ∥∇^{M}v_{H} + T^{im}(v^{⊥})∥^{2} + ∥∇^{⊥}v^{⊥} + T^{im}(v_{H})∥^{2}
− ∥A(v_{H})∥^{2} − ∥T^{ker}(v_{H})∥^{2} ,
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^{⊥}) .

(8.2)

Inserting (8.2) into (8.1) 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} .

(8.3)

As in (7.4) we get

Proposition 8.4 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.

We will consider the special cases of isometric immersions and Riemannian submersions.

8.1 Isometric Immersions

For an isometric immersion, T^{ker} = 0, A = 0. Thus (8.3) simplifies to

I_{Φ}(f)(v, v) = 2λ_{k} Tr⟨R^{X}(v, df·)v | df·⟩
+ 2λ_{k} (∥∇^{M}v_{H} + T^{im}(v^{⊥})∥^{2} + ∥∇^{⊥}v^{⊥} + T^{im}(v_{H})∥^{2})

+ μ_{k} (∥ L_{v_{H}} g∥^{2} + 8 Tr ((𝔭_{H} ∇^{M}v_{H} 𝔭_{H})T^{im}(v^{⊥})) + 4∥T^{im}(v^{⊥})∥^{2})

+ 4ν_{k} (div v_{H} + Tr T^{im}(v^{⊥}))^{2} .

In the case of a totally geodesic isometric immersion formula (7.5) becomes (cf. [26])

∫_{M} ∥∇^{M}v∥^{2} − Ric^{M}(v, v) = ∫_{M} \frac{1}{2}∥ L_{v} g∥^{2} + (div v)^{2}  (8.5)

where L_{v} g = ∇^{M}v + (∇^{M}v)^{∗} is the Lie derivative of the Riemannian metric g on M.

Proposition 8.6 The second variation at a totally geodesic isometric immersion f: M → X is

∫_{M} I_{Φ}(f)(v, v) = ∫_{M} μ_{k}∥ L_{v_{H}} g∥^{2} + 4ν_{k}(div v_{H})^{2}

+ 2λ_{k} (∥∇^{M}v_{H}∥^{2} − Ric^{M}(v_{H}, v_{H}) − Tr R^{X}(·, v^{⊥})v^{⊥} + ∥∇^{⊥}v^{⊥}∥^{2})

= ∫_{M} 2λ_{k} (∥∇^{⊥}v^{⊥}∥^{2} − Tr R^{X}(·, v^{⊥})v^{⊥}) + (λ_{k} + μ_{k})∥ L_{v_{H}} g∥^{2}
+ (4ν_{k} − 2λ_{k})(div v_{H})^{2}
= ∫_{M} 2λ_{k} (∥∇^{⊥}v^{⊥}∥^{2} − Tr R^{X}(·, v^{⊥})v^{⊥})
+ (2λ_{k} + 2μ_{k}) (∥∇^{M}v_{H}∥^{2} − Ric^{M}(v_{H}, v_{H}))
+ (4ν_{k} + 2μ_{k})(div v_{H})^{2}.

This has been exploited in [4] to derive stability criteria for isometries.

8.2 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 and v = v_{H} ∈ H ⊂ T M. Plugging the curvature

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

into (8.3) gives

I_{Φ}(f)(v, v) = 2λ_{k} (∥∇v∥^{2} − Ric^{X}(v, v)) + μ_{k}∥ 𝔭_{H}^{∗} L_{v} g∥^{2} + 4ν_{k}(div v)^{2}
= 2λ_{k} (∥∇^{M}v∥^{2} − Ric^{M}(v, v)) + μ_{k}∥ L_{v} g∥^{2} + 4ν_{k}(div v)^{2}
− 4μ_{k} Tr ((𝔭_{H} ∇^{M}v 𝔭_{V})A(v)) − (6λ_{k} + 2μ_{k})∥A(v)∥^{2} .

(8.7)

By Proposition 7.13 A Riemannian submersion with totally geodesic fibres is Φ-harmonic for any Φ. The Bochner formula (7.5) now becomes

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

(8.8)

and the left hand side is

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

From (8.7) and (8.8) we therefore get

Proposition 8.9 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∥_{V}^{2} − Tr (∇v ∘ A(v) + A(v)^{∗} ∘ (∇v)^{∗}))
+ (λ_{k} + μ_{k})∥ 𝔭_{H}^{∗} L_{v} g∥^{2} + (4ν_{k} − 2λ_{k})(div v)^{2}
= ∫_{M}(2λ_{k} + 2μ_{k}) (∥∇v∥^{2} − Ric^{X}(v, v)) − 2μ_{k}∥∇v∥_{V}^{2}

+ (2μ_{k} + 4ν_{k})(div v)^{2} + 2μ_{k} Tr (∇v ∘ A(v) + A(v)^{∗} ∘ (∇v)^{∗})

8.3 Examples

If X has sufficiently small Ricci curvature we use the estimate

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

together with the second equation of proposition 8.9. It follows for instance that if

Ric^{X}(v, v) ≤ −∥A(v)∥^{2}, λ_{k} ≥ |μ_{k}| ≥ 0, λ_{k} + μ_{k} ≥ |μ_{k}|, 2ν_{k} ≥ −μ_{k}

then a Riemannian submersion f: M → X^{k} with totally geodesic fibres is stably Φ-harmonic. In table 1 these are Φ(A) = Tr A, (Tr A)^{p}, e^{Tr A}, Tr (e^{A}) and σ_{p}(A) if k + 1 ≥ 2p, σ_{p}(A)^{q} for q > 1 if (k − 1)\binom{k}{p} + 2 ≥ 2p.

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Stefan Bechtluft-Sachs NWF I - Mathematik Universität Regensburg D-93040 Regensburg Germany