We derive variational formulae for natural first order energy function-

als and obtain criteria for the stability of isometric immersions. This

generalizes known results for the classical energy, the p-energy and the

exponential energ

Bechtluft-Sachs, Stefan

English

MURAL - Maynooth University Research Archive Library

Tension field and Index form of Energy-TypeFunctionalsStefan Bechtluft-SachsJuly 22, 2002AbstractWe derive variational formulae for natural first order energy function-als and obtain criteria for the stability of isometric immersions. Thisgeneralizes known results for the classical energy, the p-energy and theexponential energyAMS Subject Classification: 58E20 (53C43)1 IntroductionBy an energy-type functional defined on smooth maps f : (Mn, g) → (V k, h)of compact Riemannian manifolds we mean a functional obtained by integrationof a first order differential operator φ(df) where df ∈ Γ(T ∗M ⊗ f ∗TV ) denotesthe differential of f and φ : M(R, n × k) → R+0 is invariant under the actionof O(n) × O(k). Especially φ yields a parallel function T ∗M ⊗ f ∗TV → R+0 .We can rewrite φ(df) = Φ(df ∗df) for some function Φ : M(R, n × n)+ → R onnonnegative symmetric matrices which is invariant under conjugation by O(n).The functionals in question take the formEΦ(f) :=∫MΦ(df ∗df)dvolg ,where we have used the Riemannian metrics to identify T ∗M = TM and T ∗V =TV to get the endomorphism df ∗df of TM .Famous examples of this construction are the classical energy, Φ(A) = TrA,the exponential energy, Φ(A) = exp(TrA) as in [EL3], the p-energy, Φ(A) =(TrA)p but also the volume, where Φ(A) = (detA)1/2. Results similiar to ours inthe case where Φ is a function of the Trace, Φ(A) = F (TrA), have been obtainedin [A]. In particular the exponential energy was treated in [C-L] and the p-energyin [C-L2]. There is a vast literature for the classical energy, see e.g. the survey1papers [EL1], [EL2]. For a discussion of stability results in this case we refer to[X] and the references there.Here we will derive the first and second variational formulae for the Φ-energyfunctional. The Bochner formula for vector fields then implies that isometriesare Φ-stable under certain conditions on the first and second derivative of Φ. Asin the classical case, (see [EL], [X]) there is also a range of maps Φ such that theidentity on the sphere Sn is unstable for the Φ-energy.2 Variation formulae for the Φ-EnergyIn order to derive variational formulae we will restrict ourselves to functionalswhich can be expressed with smooth Φ, i.e we work with Φ rather than φ. Thishas the advantage that the domain TM∗ ⊗ TM of Φ is independent of f . Forpolynomial (or even analytic) φ this is no loss of generality by the remark at theend of this section. In the sequel we will always assume M compact or at leastthat the variations are compactly supported. Consider a 2-parameter variationof f , i.e. a mapF : I × J ×M → V (s, t,m) 7→ fs,t(m)where I, J are intervalls around 0. Denote by ∇ the Riemannian connectionson the bundles TM , F ∗TV and f ∗TV and let v := dF(∂∂t)= ∂∂tfs,t(m), w :=dF(∂∂s)= ∂∂sfs,t(m) be the variation vector fields along f = f0 = f0,0, ft =f0,t. We compute the variation at a point p ∈ M . Let e1, . . . , en be a localorthonormal framing of TM in a vicinity of p with ∇eiej = 0 at p. Note thatfor the commutators we have [ei,∂∂s] = 0, [ei,∂∂t] = 0 and [ei, ej](p) = 0. We alsowrite ∂¯i,jΦ := ∂i,jΦ + ∂j,iΦ. In the subsequent calculations summation over theindices i, j, k, l is tacitely assumed. For the first variation of the Φ-energy densitywe obtainddtΦ(df ∗t dft) = dΦ(∇df ⊗ df + df ⊗∇df)= ∂¯i,jΦ(df∗df)〈∇ ∂∂tdFei | dFej〉= ∂¯i,jΦ(df∗df)〈∇eiv | dfej〉= ei(∂¯i,jΦ(df∗df)〈v | dfej〉)− 〈v | ∇ei(∂¯i,jΦ(df∗df)dfej)〉= div((∂¯i,jΦ(df∗df)〈v | dfej〉)ei)− 〈v | τΦ(f)〉 .We thus get theProposition 2.1 Define the Φ-tension of a smooth map f : M → V of compactRiemannian manifolds to be the vector field along fτΦ(f) := ∇ei(∂¯i,jΦ(df∗df)dfej)= ∂¯k,l∂¯i,jΦ(df∗df)〈∇eidfek | dfel〉dfej (2.2)+ ∂¯i,jΦ(df∗df)∇eidfej2Then f is Φ-harmonic, i.e. critical for the Φ-energy, if and only if τΦ(f) = 0.For the second variation we get up to divergenced2dsdtΦ(df ∗s,tdfs,t)= −dds〈v | τΦ(fs)〉= − 〈∇ ∂∂sv | τΦ(f)〉 − 〈v | ∇ ∂∂sτΦ(fs)〉= − 〈∇ ∂∂sv | τΦ(f)〉 − 〈v | ∇ ∂∂s∇ei(∂¯i,jΦ(df∗s dfs)dFej)〉= − 〈∇ ∂∂sv | τΦ(f)〉 − 〈v | Rw,dfei(∂¯i,jΦ(df∗df)dfej)〉− 〈v | ∇ei∇ ∂∂s(∂¯i,jΦ(df∗s dfs)dFej)〉where R denotes the curvature tensor of V . The last term is= −〈v | ∇ei∇ ∂∂s(∂¯i,jΦ(df∗s dfs)dFej)〉= −〈v | ∇ei(d∂¯i,jΦ(df∗s dfs)dsdfej + ∂¯i,jΦ(df∗df)∇ ∂∂sdFej)〉= −〈v | ∇ei(∂¯k,l∂¯i,jΦ(df∗df)〈∇ ∂∂sdFek | dfel〉dfej +(∂¯i,jΦ(df∗df)∇ejw))〉= −〈v | ∇ei(∂¯k,l∂¯i,jΦ(df∗df)〈∇ekw | dfel〉dfej +(∂¯i,jΦ(df∗df)∇ejw))〉= + ∂¯k,l∂¯i,jΦ(df∗df)〈∇eiv | dfej〉〈∇ekw | dfel〉+ ∂¯i,jΦ(df∗df)〈∇eiv | ∇ejw〉where the last identity holds only up to divergence.Proposition 2.3 The second variation of the Φ-energy at a Φ-harmonic map fis the integral overIΦ(f)(v, w) = − 〈v | Rw,dfei(∂¯i,jΦ(df∗df)dfej)〉+ ∂¯k,l∂¯i,jΦ(df∗df)〈∇eiv | dfej〉〈∇ekw | dfel〉+ ∂¯i,jΦ(df∗df)〈∇eiv | ∇ejw〉for any vector fields v, w along f .We finally compute the leading symbol of the second variation. We haved2dsdtEΦ (fs,t) =∫M〈v | Pw〉dvolg (2.4)with a symmetric second order partial differential operator P acting on vectorfields along f , i.e on sections v, w of f ∗TV → M . The restriction P⊥f of P (orof the bilinear form given by (2.4)) to the orthogonal complement of the imageof df : TM → f ∗TV will be called second variation perpendicular to f . Theleading symbol of P is determined by the highest order term−〈v | ∂¯k,l∂¯i,jΦ(df∗df)〈∇ei∇ekw | dfel〉dfej + ∂¯i,jΦ(df∗df)∇ei∇ejw〉in Proposition 2.3. Hence we get3Proposition 2.5 The leading symbol of the second variation of the Φ-energy isσ(ξ) = ∂¯k,l∂¯i,jΦ(df∗df)ξiξk dfel ⊗ dfej + ∂¯i,jΦ(df∗df)ξiξj , (2.6)for ξ =∑i ξiei. Thus〈σ(ξ)w | w〉 = ∂¯k,l∂¯i,jΦ(df∗df)ξiξk〈w | dfel〉〈w | dfej〉+ ∂¯i,jΦ(df∗df)ξiξj||w||2for ξ ∈ TpM∗ and w ∈ (f ∗TV )p.Remark: Let φ : M(n × k) → R+0 be a polynomial function, invariant underthe action of O(n) × O(k), i.e. such that φ(BXA) = φ(X) for all B ∈ O(k),A ∈ O(n) and X ∈ M(n × k). For any X ∈ M(n × k) we can diagonalize X∗Xand find othogonal matrices B and A as before such thatBXA =λ1 0. . .0 λq0 · · · 0 orλ1 0 0. . ....0 λq 0as q := min{n, k} = n or q = k. Hence φ(X) = φ(λ1, . . . , λq) is a symmetricpolynomial and since φ(±λ1, . . . ,±λq) = φ(λ1, . . . , λq) this does not involve oddpowers of the λi. Thus we find a symmetric polynomial Φ in n variables suchthat φ(λ1, . . . , λq) = Φ(λ21, . . . , λ2q, 0, . . . , 0). This extends to a polynomial Φ :M(n × n)+ → R+0 such that φ(X) = Φ(X∗X). For analytic φ this constructionyields an analytic function Φ.Note that if φ is differentiable we do not necessarily get a differentiable func-tion Φ with the above properties. In general Φ is only differentiable on the set ofmatrices of full rank q. For instance φ(X) := det (X∗X)3/4 is differentiable butΦ(A) := det (A)3/4 is not.For polynomial φ there are polynomials Φs and Φσ such thatφ(X) = Φ(X∗X) = Φs(s1, . . . , sq) = Φσ(σ1, . . . , σq)where σl is the lth elementary symmetric polynomial in the eigenvalues λ21, . . . , λ2qof X∗X determined byn∑l=0σl(X∗X)tl = det (1 + tX∗X)andsk =n∑l=0λ2ql = Tr((X∗X)l).In the analytic case one can use a theorem of Glaeser, [Gl], to get analytic func-tions Φs and Φσ.43 Applications3.1 Isometric ImmersionsFor isometric immersions the preceeding formulae simplify substantially. By in-variance dΦ(id) must be some multiple λTr of the trace. We have the followingTheorem 1 Let f :M → V be an isometric immersion and assume that dΦ(id) 6=0. Then1. f is Φ-harmonic if and only if it is harmonic.2. If λ > 0 then the leading symbol of P⊥f is positive definite, hence the secondvariation perpendicular to f has finite index.Proof: (1) For an isometric immersion or a Riemannian submersion the firstterm in (2.2) vanishes. Since an isometric immersion f has df ∗df = id we getτΦ(f) = ∂¯i,jΦ(id)∇eidfej = 2λTr∇df = 2λτ(f) .(2) On vector fields w normal to f , i.e perpendicular to the dfel in (2.6), thefirst summand in (2.6) vanishes. As before the second summand is some multipleof the trace which shows thatσ(ξ) = ∂¯i,jΦ(df∗df)ξiξj = 2λ||ξ||2 > 0for ξ 6= 0. Thus the restriction of P to (Image(df))⊥ ⊂ f ∗TV is elliptic withpositive definite leading symbol and therefore has only finitely many negativeeigenvalues. •3.2 Stability of IsometriesBy invariance, the second derivative d2Φ(id) is a homogeneous polynomial ofdegree 2. Therefore there are µ, ν ∈ R such thatd2Φ(id)(H) = µTr(H2)+ ν(TrH)2The second variation formula in Proposition 2.3 simplifies toIΦ(f)(v, v) = −〈v | Rv,ei(∂¯i,jΦ(id)ej)〉+ ∂¯k,l∂¯i,jΦ(id)〈∇eiv | ej〉〈∇ekv | el〉+ ∂¯i,jΦ(id)〈∇eiv | ∇ejv〉= − 2λRic(v)+ µ(〈∇eiv | ej〉+ 〈∇ejv | ei〉)2+ 4ν〈∇eiv | ei〉〈∇ekv | ek〉+ 2λ〈∇eiv | ∇eiv〉5= − 2λRic(v)+ 2µ(||∇v||2 + Tr((∇v)2))+ 4ν (div(v))2 + 2λ||∇v||2= − 2λRic(v) + 2(µ+ λ)||∇v||2 + 2µTr((∇v)2)+ 4ν (div(v))2= − 2λRic(v) + µ||Lvg||2 + 4ν(div(v))2 + 2λ||∇v||2since Tr(∇v) = div(v). Comparing this with the Bochner formula (see e.g. [Y]):∫M−Ric(v)−12||Lvg||2 + (div(v))2 + ||∇v||2 = 0 (3.1)we obtainTheorem 2 Assume that µ ≥ −λ and that 2ν ≥ λ. Then any isometry of M isΦ-stable.We now derive a sufficient criterion for the identity map on a sphere to be unsta-ble. To that end let v be the gradient vectorfield on Sn ⊂ Rn+1 of the restrictionof a linear map p : Rn+1 → R, p(x) = 〈p, x〉 for a unit vector p ∈ Rn+1 asin [X]. Then ||v(x)||2 + p(x)2 = 1 and ∇xv = −px for all x ∈ TSn, hence〈∇eiv, ej〉 = −pδi,j . Since the Ricci curvature of Sn is Ric(v) = (n− 1)||v||2, theformula for the index form yieldsIΦ(v, v) = −2λ(n− 1)||v||2 + (4µn+ 4νn2 + 2λn)p2 . (3.2)Denoting by ωn−1 the volume of the standard (n− 1)-sphere we compute∫Sn‖v‖2 = ωn−1∫ pi/2−pi/2cos(θ)n+1 dθ= ωn−1([sin(θ) cos(θ)n]pi/2−pi/2 +∫ pi/2−pi/2sin(θ)2 cos(θ)n−1 dθ)= n∫Snp2 .Inserting this into (3.2) shows the followingTheorem 3 Ifλ(n− 2) > 2µ+ 2νnthen id : Sn → Sn is Φ-unstable.3.3 ExamplesFor some of the functionals mentioned in the introduction theorems 2 and 3 give:61. For the p-energy, Φ(A) = (Tr(A))p we compute λ = pnp−1, µ = 0 andν = p(p−1)np−2. Thus idSn is unstable if n > 2p. Isometries are generally stableif n ≤ 2(p− 1).2. The exponential energy, Φ(A) = eTrA, has λ = en, µ = 0, ν = en. Thusisometries are always stable for EΦ. This is the proof of [C-L].3. For Φ(A) = Tr(Ap) we get λ = p, µ = p(p − 1) and ν = 0. Thus idSn isunstable if n > 2p.4. For Φ(A) = Tr exp(A) we get λ = e, µ = e, ν = 0. Therefore idSn is unstableif n > 4.5. For Φ(A) = det (A) we get λ = 1, µ = −1, ν = 1. Thus any isometry is stablefor Edet .6. Let α1, . . . , αn be the eigenvalues of A and if n ≥ 2 define the discriminantΦ(A) :=∏1≤i<j≤n(αi − αj)2. Then EΦ has λ = µ = ν = 0 and the secondvariation at an isometry vanishes.References[A] Ara, Mitsunori Geometry of F -harmonic maps, Kodai Math. J. 22 (1999),no. 2, 243–263.[C-L] Cheung, Leung-Fu; Leung, Pui-Fai The second variation formula for expo-nentially harmonic maps, Bull. Austral. Math. Soc. 59 (1999), no. 3, 509–514.[C-L2] Cheung, Leung-Fu; Leung, Pui Fai Some results on stable p-harmonicmaps, Glasgow Math. J. 36 (1994), no. 1, 77–80.[EL] Eells, James; Lemaire, Luc Selected topics in harmonic maps, CBMS Re-gional Conference Series in Mathematics, 50. Published for the ConferenceBoard of the Mathematical[EL1] Eells, James; Lemaire, Luc A report on harmonic maps, Bull. LondonMath. Soc. 10 (1978), no. 1, 1–68.[EL2] Eells, James; Lemaire, Luc Another report on harmonic maps Bull. LondonMath. Soc. 20 (1988), no. 5, 385–524.[EL3] Eells, James; Lemaire, Luc Some properties of exponentially harmonicmaps Partial differential equations, Part 1, 2 (Warsaw, 1990), 129–136, Ba-nach Center Publ., 27, Part 1, 2, Polish Acad. Sci., Warsaw, 1992.[Gl] Glaeser, Georges Fonctions compose´es diffrentiables, Ann. of Math. (2) 77(1963) 193–209.[X] Xin, Yuanlong Geometry of harmonic maps, Progress in Nonlinear Differen-tial Equations and their Applications, 23. Birkha¨user Boston, Inc., Boston,MA, 1996.7[Y] Yano, Kentaro Integral formulas in Riemannian geometry, Pure and AppliedMathematics, No. 1 Marcel Dekker, Inc., New York 1970.8

Tension field and Index form of Energy-Type

Functionals

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Tension field and Index form of Energy-Type Functionals

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