Infima of Universal Energy Functionals on Homotopy Classes

Stefan Bechtluft-Sachs

February 25, 2003

Abstract

We consider the infima Ẽ(f) on homotopy classes of energy functionals E defined on smooth maps f: M^{n} → V^{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. Conversely if the infimum on a homotopy class of a functional of at least conformal degree vanishes then the map is trivial in homology of high degrees.

AMS Subject Classification: 58E15

1 Introduction

The main objective of this paper is a correspondence between universal energy functionals on one side and factorization over subskeleta up to homotopy on the other. By an energy functional E_{ϕ}: C^{∞}(M, V) → ℝ^{+} we mean an integral ∫_{M} ϕ(∇f, …, ∇^{r}f) of a constant coefficient differential operator ϕ over M, where M and V are smooth compact connected Riemannian manifolds, see section 2.

By a theorem of White, [6], the infimum

Ẽ_{Φ}(f) := inf {E_{Φ}(g)|g: M → V, g ≃ f}

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 3.1 contains an analogous statement for functionals involving higher derivatives of f.

For first order functionals we consider a kind of converse, namely the implications of Ẽ_{ϕ}(f) = 0, restricting our considerations to functionals of (at least) conformal degree d = n = dim M. For instance if ϕ(df) = det (df^{∗}df)^{1/2} and Ẽ_{ϕ}(f) = 0 then f is homotopic to a map f̃: M → V^{n−1} into the n − 1-skeleton of a triangulation of V. For the n-norm ϕ(df) = ∥df∥^{n} = Tr ((df^{∗}df)^{n/2}) it is proved in [6] that Ẽ_{ϕ}(f) = 0 implies that f ≃ ∗ = V^{0} is nullhomotopic.

Theorem 3.2 interpolates homologically between these facts. Since ϕ depends explicitely only on the differential df, the functional E_{ϕ} corresponds biuniquely to a symmetric function in n variables, see section 2. 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. Proposition 2.1 establishes a hierarchy σ_{1}^{n/2} >> σ_{2}^{n/4} >> ⋯ σ_{l}^{n/2l} >> ⋯ >> σ_{n}^{1/2}, where ϕ >> ψ indicates that we can estimate the energies CE_{ϕ}(f) ≥ E_{ψ}(f) with a constant C independent of f. 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}. If ϕ >> σ_{l}^{n/2l} then Ẽ_{ϕ}(f) = 0 implies that f induces 0 in (co)homology of degree ≥ l with arbitrary coefficients, just like a map which factors over the (l − 1)-skeleton V^{l−1} ↪ V of some triangulation of V^{0} ⊂ V^{1} ⊂ ⋯ ⊂ V^{k−1} ⊂ V.

If f is homotopic to a map f̃: M → V^{l−1} ⊂ V 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. Theorem 3.2 only yields the vanishing of the primary obstructions in this factorization problem and this needs at least conformal degree r ≥ n/2l. Proposition 3.3 shows that a general converse 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 [2],[5] for instance. The energy functionals behave like an absolute value thereof, similiar to the case of mapping degree and volume for maps between manifolds of the same dimension n = k. Consider as an example the 2-Jacobian ϕ(A) = σ_{2}^{3/4}(A^{∗}A) for maps M^{3} → V^{2}, i.e. E_{ϕ}(f) = ∫_{M} |f^{∗}ω|^{3/2} where ω is the volume form of V. In [4] it is shown that the Hopf invariant of a map M = S^{3} → S^{2} = V 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.

2 Energy functionals

For smooth maps f: M^{n} → V^{k} of compact Riemannian manifolds we consider functionals

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

of order r parametrized by functions

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

These functions are assumed invariant under the (diagonal) action of O(n)×O(k). They can therefore be evaluated on the derivatives ∇^{j}f ∈ Γ(T M^{∗}^{⊗j} ⊗ f^{∗}T V) to yield well defined energy densities ϕ(∇f, ∇^{2}f, …, ∇^{r}f). We will say that ϕ (or E_{ϕ}) has degree ≤ d if it satisfies an estimate

|ϕ(A_{1}, …, A_{r})| ≤ C(1 + ∥A_{1}∥^{d} + ∥A_{2}∥^{d/2} + ⋯ + ∥A_{r}∥^{d/r}) , A_{j} ∈ (ℝ^{n})^{⊗j} ⊗ ℝ^{k} .

Denoting by df^{∗} the adjoint of df and using the Riemannian metrics to identify the dual tangent bundles we get a bundle endomorphism df^{∗}df of T M. Examples of functionals as above are the classical (2-)energy, or more general p-energy ∫_{M} (Tr (df^{∗}df))^{p/2} of degree p, the exponential energy ∫_{M} e^{Tr (df^{∗}df)} of infinite degree, the Willmore energies W (f) := ∫_{M} ∥∇df∥^{2} of degree 4, W_{0}(f) := ∫_{M} ∥∇df(df^{∗}df)^{−1}∥^{2}. For an immersion W_{0} is the L^{2}-norm of the mean curvature.

Functionals of first order, i.e. involving only first order derivatives, are conveniently described by nonnegative functions Φ: M(n × n, ℝ)^{+} → ℝ_{0}^{+} defined on nonnegative symmetric matrices and invariant under conjugation by O(n). We have Φ(A^{∗}A) = ϕ(A) and

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

We describe the estimates between polynomial energy functionals corresponding to the successive lifting problem for a map f: M → V over the skeleta of V.

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 2.1 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 volM .

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}} + μvolM .
•

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 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 Statement of Results

Theorem 3.1 Let ϕ have degree d and let L ⊂ M be a submanifold of codimension q with q > d. If f, g: M → V are homotopic on M \ L then Ẽ_{ϕ}(f) = Ẽ_{ϕ}(g).

For instance in order to compute W̃(f) 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} → V if n ≥ 5. It was proved by B. White in [6], see also Pluzhnikov [3], that for functionals of order 1 the same holds if f and g are homotopic on a (q − 1)-skeleton of M with q > d, i.e. outside a skeleton of codimension q > d.

Theorem 3.2 Assume that ϕ(A) ≥ σ_{l}(A^{∗}A)^{n/2l} and that Ẽ_{ϕ}(f) = 0. Then f induces 0 in (co)homology groups of degree ≥ l.

The following proposition shows that the assumptions of theorem 3.2 do not in general imply that f is homotopic to a map to f̃: M → V^{l−1} ⊂ V.

Proposition 3.3 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 [6] or theorem 3.1, 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.

4 Proof of Theorem 3.1

Let R be 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 → V 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)
≤ volU_{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. •

5 Proof of Theorem 3.2

Let M^{q}, V^{q} denote the q-skeleta of smooth triangulations of M and V respectively. By the Deformation Theorem of geometric measure theory, [1], we can deform the restriction f|_{M^{q}} of f to a Lipschitz map f̃ mapping M^{q} to V^{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.

It suffices to consider the case Φ = σ_{l}^{n/2l}. 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 .  (5.1)

Assume that E_{Φ}(f) ≤ δ for some δ. By (5.1) we find v ∈ D^{s} such that

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

From proposition 2.1 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}} + μvolM )
≤ C_{4} (\frac{1}{μ^{n/q−1}} δ/C_{3} + μvolM )

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 V^{q}. Thus we get a map f̃_{v} homotopic to f which maps M^{q} to V^{q} and misses a point in every q-cell of V^{q}. Therefore it is homotopic to a map M^{q} → V^{q−1}. •

References

[1] Federer, H. Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften 153, Springer-Verlag, 1969.

[2] Gromov, M. Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152, Birkhuser, 1999

[3] Pluzhnikov, A. I. Topological aspects of the problem of minimizing the Dirichlet functional Proc. Steklov Inst. Math. 1993, no. 3 (193), 167–171

[4] Rivière, T. Minimizing fibrations and p-harmonic maps in homotopy classes from S^{3} into S^{2}, Comm. Anal. Geom. 6 (1998), no. 3, 427–483.

[5] Sullivan, D. Infinitesimal computations in topology, Inst. Hautes tudes Sci. Publ. Math. No. 47 (1977), 269–331.

[6] White, B. Infima of energy functionals in homotopy classes of mappings, J. Differential Geom. 23 (1986), no. 2, 127–142.