CONFORMAL GEOMETRY, CONTACT GEOMETRY, …...CONFORMAL GEOMETRY, CONTACT GEOMETRY, AND THE CALCULUS OF VARIATIONS JEFF A. VIACLOVSKY 1. Introduction. In the following we let.N;g0/denote

Vol. 101, No. 2 DUKE MATHEMATICAL JOURNAL © 2000

CONFORMAL GEOMETRY, CONTACT GEOMETRY,AND THE CALCULUS OF VARIATIONS

JEFF A. VIACLOVSKY

1. Introduction. In the following we let(N,g0) denote a compact, connectedsmooth Riemannian manifold of dimensionn ≥ 3. We denote the Ricci tensor andscalar curvature byRic andR, respectively. In this paper we examine the nonlinearcurvature equations

σk

(Ricg− Rg

2(n−1)·g)= constant (1)

for metricsg in the conformal class ofg0, where we use the metricg to view thetensor as an endomorphism of the tangent bundle and whereσk denotes the traceof the induced map on thekth exterior power; that is,σk is the kth elementarysymmetric function of the eigenvalues. The casek = 1,R = constant is known as theYamabe problem, and it has been studied in great depth (see [11] and [17]). We letM1 denote the set of unit volume metrics in the conformal class[g0]. We show thatthese equations have the following variational properties.

Theorem 1. If k 6= n/2 and (N, [g0]) is locally conformally flat, then a metricg ∈M1 is a critical point of the functional

Fk : g 7→∫N

σk

(Ricg− Rg

2(n−1)·g)

dvolg

restricted toM1 if and only if

σk

(Ricg− Rg

2(n−1)·g)= Ck

for some constantCk. If N is not locally conformally flat, then the statement is truefor k = 1 andk = 2.

We compute the second variation of the above functionals and use this to examinethe behavior of the functionals near a critical point. In particular, we show that theyare elliptic when the eigenvalues are restricted to lie in a certain cone (see Section 6).Following [4], we call such a solution admissible. We prove Theorem 2(k 6= n/2).

Received 2 April 1999.1991Mathematics Subject Classification.Primary 53A30; Secondary 35J20, 35J60.Author’s work supported by a National Science Foundation Graduate Fellowship and a Sloan

Dissertation Fellowship.

283

284 JEFF A. VIACLOVSKY

Theorem 2. If (N, [g0]) is locally conformally flat, then for odd (even)k, a neg-ative k-admissible critical point of the functionalFk|M1 is a strict local minimum(maximum). For allk, a positive scalar curvature Einstein metric is a strict localminimum unless(N, [g]) is conformally equivalent to(Sn,g0), in which case there isan (n+1)-parameter family of local minima. If(N, [g0]) is not locally conformallyflat, then the statement holds fork = 1 andk = 2.

For k = 1, it is known that the local extrema in the above theorem are actuallyglobal extrema (see [17]). We conjecture that this is also true fork ≥ 2, provided werestrict the functionals tok-admissible unit volume metrics.

We also do an ordinary differential equation (ODE) analysis for rotationally sym-metric solutions onSn−{p1,p2}, wherep1 andp2 are antipodal points. Fork < n/2,we find periodic orbits, thus giving solutions which descend toS1×Sn−1. Fork = 1,these are known as Delaunay metrics (see [12] and [17]). We also prove the followinguniqueness result.

Theorem 3. Suppose(N,g0) is of unit volume and has constant sectional curva-tureK 6= 0. Then for anyk ∈ {1, . . . ,n−1}, g0 is the unique unit volume solutionin its conformal class of (1) unlessN is isometric toSn with the standard metric. Inthis case, we have an(n+1)-parameter family of solutions that are the images of thestandard metric under the conformal diffeomorphisms ofSn.

Fork = 1, the constant scalar curvature case, the theorem holds just assumingN isEinstein. This is a well-known theorem of Obata (see [14]). To prove this, we use theconformal frame bundle of Cartan to show that the manifold of 1-jets of sections of adensity bundle is naturally isomorphic to a quotient of the total space of the principalbundle by the subgroupSO(n). The connection forms then descend to give nonlinearsecond-order equations on densities. These densities are just metrics in the conformalclass, so we arrive at the above nonlinear curvature equations. Using the connectionforms, we then derive an integral formula, which is the main part of the proof, andthe theorem follows ifk < n.

The negative curvature case can be handled by a maximum principle argument. Wewill prove the following theorem.

Theorem 4. Suppose(N,g0) is a compact Einstein manifold of unit volume withR < 0. Then fork ∈ {1, . . . ,n}, g0 is the unique unit volume solution in its conformalclass of (1).

The only case left isk = n andK > 0. The techniques here do not work in this case,but for a partial differential equation (PDE) proof using the moving planes method(see [18]).

The paper is organized as follows. We begin with a review of conformal geometry.In Section 3 we discuss the isomorphism between 1-jets and the quotient mani-fold. In Section 4 we introduce the curvature equations and discuss their variational

CONFORMAL GEOMETRY 285

properties. Section 5 is concerned with examples. We calculate the second variation inSection 6, examine the behavior of the functionals near a critical point, and show thatthe equations are elliptic at an admissible solution. Section 7 is devoted to provingthe uniqueness theorems.

Acknowledgements.This work is part of the author’s doctoral dissertation atPrinceton University under the guidance of Phillip A. Griffiths, without whom noneof this would have been possible. The author would also like to thank Robert Bryant,Jonathan Pakianathan, Dan Pollack, and Karen Uhlenbeck for their interest, time, andmany helpful suggestions.

2. Review of conformal geometry. In this section we review the relevant no-tions from conformal geometry (see [5] and [10]). We use the Einstein summationconvention.

2.1. Cartan’s principal conformal frame bundle.We begin with some definitions.Let Q denote the matrix 0 0 −1

0 In 0−1 0 0

.We view SO(n+ 1,1) ⊂ GL(n+ 2,R) as the subgroup of matricesA satisfyingAtQA=Q and det(A)= 1. We letSOo(n+1,1) denote the subgroup ofSO(n+1,1)preserving time orientation, that is,A0

0+An+10 > 0.

We letG denote the maximal parabolic subgroup ofSOo(n+1,1), consisting ofelements of the form r2 r2vtB (r2/2)vtv

0 B v

0 0 r−2

,whereB is in SO(n), v ∈ Rn, and r2 ∈ R+. We denote bygo the Lie algebra ofSOo(n+1,1), consisting of matrices of the form2s yt 0

x z y

0 xt −2s

,wheres ∈ R, y andx are inRn, andz is ann-by-n skew-symmetric matrix.

Cartan [5] shows that given an oriented manifoldN with dim(N) ≥ 3 and anequivalence class[g] of conformal Riemannian metrics onN , there exists a rightprincipal bundleP → N with structure groupG. Moreover, there is ago-valued


1-form onP ,

ψ =2ρ βj 0ωi αij βi

0 ωj −2ρ

,with the following properties:

(i) the formsωi , αij (i > j ), βi , andρ are linearly independent;

(ii) the ωi are semibasic, that is,vyωi = 0 for vertical vectorsv;(iii) π∗[g] = [ω2

1+·· ·+ω2n];

(iv) R∗g(ψ)= g−1ψg for all g ∈G;(v) restricted to a fiber,ψ is the left-invariant Maurer-Cartan form ofG.

This bundle can be considered a solution to the equivalence problem for confor-mal structures in the sense that any conformal automorphism ofN lifts to a uniqueautomorphism ofP , which preservesψ , and vice versa.

Furthermore, there exist real-valued functionsWijkl (the Weyl curvature) andBijk

(the Cotten tensor) onP with the following symmetries:

Wijkl =−Wj

ikl =−Wijlk,

0=Wijil,

0=Wijkl+Wi

klj +Wiljk,

Bijk =−Bikj ,0= Bijk+Bjki+Bkij .

We have the structure equations

d(2ρ)=−βi∧ωi, (2)

dωi = 2ρ∧ωi−αij ∧ωj , (3)

dβi =−2ρ∧βi+αji ∧βj +1

2Bijkω

j ∧ωk, (4)

dαij =−αik∧αkj −βi∧ωj +βj ∧ωi+1

2Wijklω

k∧ωl. (5)

A necessary and sufficient condition for the conformal structure to be locally con-formally flat is the vanishing of theWi

jkl and theBijk. Forn > 3, if theWijkl ≡ 0, then

theBijk are also identically zero. So, in this case theWijkl are the only obstruction to

local conformal flatness. To see this, we differentiate the structure equation (5), applyCartan’s lemma, and find that

dWijkl =−4Wi

jklρ−Wmjklα

im+Wi

mklαmj +Wi

jmlαmk +Wi

jkmαml +Wi

jkl,mωm

for some functionsWijkl,m satisfying


Wijkl,m+Wi

jlm,k+Wijmk,l

=−Bjklδim+Biklδjm−Bjlmδik+Bilmδjk−Bjmkδil+Bimkδjl,(6)

whereδij is the Kronecker delta symbol. Tracing (6) oni andm, we find that

Wijkl,i = (3−n)Bjkl−Biilδjk+Biikδjl . (7)

Tracing (7) onj andk, we have

0= (3−n)Biil−nBiil+Biil = 2(2−n)Biil .Therefore we have the conformal Bianchi identity

Biik ≡ 0. (8)

We see that if theWijkl vanish andn > 3, thenBijk ≡ 0. Forn = 3, the symmetry

relations imply thatWijkl ≡ 0, so in this case theBijk are the only obstruction to local

conformal flatness.

2.2. Density bundles and metrics.There are some natural bundles that we canassociate to the principal bundleπ : P → N . Let h :G→ R be the homomorphismsendingg ∈G to r2. We denote byDs/n the line bundle onN associated toP by thehomomorphismhs ; that is,Ds/n = P ×R/ ∼, where∼ is the equivalence relation(p, t) ∼ (Rgp,r−2s t). A section of the bundleDs/n is represented by a functionu : P → R satisfying the condition

u(p ·g)= (h(p))−su(p).Differentiating this relation, using the structure equations, and using Cartan’s lemma,we see that there existui : P → R so thatu satisfies the relation

du=−2sρu+uiωi. (9)

Differentiating again, we see that there existuij = uji such that

dui =−suβi−2(s+1)uiρ+ujαji +uijωj . (10)

Differentiating one last time, we obtain

duij =−(s+1)(uiβj +ujβi

)+δij (ukβk)−2(s+2)uijρ+uikαkj +ukjαki +uijkωk,(11)

where theuijk satisfyuijk = ujik anduijk−uikj =−sBijk+Wlijkul .

One of the nicest features of the Cartan approach to conformal geometry is thatthe bundleP → N contains the Riemannian frame bundle of each of the metrics in


the conformal class[g], and positive sections of the density bundles are, in effect,metrics in the conformal class.

We now restrict to the cases = −1. Given a densityu ∈ 0(D−1/n), relations (9)through (11) simplify to become

du= 2ρu+uiωi, (12)

dui = uβi+ujαji +uijωj , (13)

duij = δij(ukβk

)−2uijρ+uikαkj +ukjαki +uijkωk. (14)

If u > 0, we define the subset ofP ,

Pu ={p ∈ P, u(p)= 1, ui(p)= 0

}.

We let an overbar denote restriction toPu ⊂ P . From (12), we see that

ρ = 0. (15)

From the structure equation (3), we have that

dωi =−αij ∧ ωj .This tells us thatPu → N is a principalSO(n) bundle. It is easy to verify that thesymmetric form onP ,

gu = u−2((ω1)2+·· ·+(ωn)2),

descends to a metricgu ∈ [g], and it follows thatPu→ N is the orthonormal framebundle of the metricgu, the ωi are the canonical 1-forms, and theαij are the Levi-Civita connection forms.

Equation (13) becomes

βi =−uij ωj . (16)

Using the structure equations (5), we have

dαij + αik∧ αkj =−βi∧ ωj + βj ∧ ωi+1

2W ijklω

k∧ ωl = 1

2Rijklω

k∧ ωl, (17)

where theRijkl are the components of the Riemann curvature tensor of the metricgu.Substituting (16) into this, we get

Rijkl =−(δil ujk−δjl uik−δikuj l+δjkuil

)+W ijkl .

Thus the Ricci curvature ofgu is

Rij = Rlilj = (n−2)uij +δij ull , (18)


and the scalar curvature ofgu is

R = Rll = 2(n−1)ull . (19)

Solving for uij , we get

uij = 1

n−2

(Rij − R

2(n−1)δij

). (20)

Given another densityv ∈ 0(D−1/n), if we restrict the relations

dv = 2ρv+viωi, (21)

dvi = vβi+vjαji +vijωj (22)

to Pu, then we get

dv = vi ωi ,dvi = vβi+ vj αji + vij ωj= vj αji +(vij − vuij )ωj .

We see thatv descends to a function onN , and we have the formulas

(∇v)i = vi , (23)(∇2v)ij= vij − vuij = vij − v

n−2

(Rij − R

2(n−1)δij

), (24)

where the gradient and Hessian are taken with respect to the metricgu. Sincegv =v−2ω, we have, still letting overbars denote restriction toPu,

gv = v−2ω = v−2gu.

We can think ofv−2 as the conformal factor taking us from the metricgu to themetricgv.

3. Contact geometry. There are two contact manifolds that arise naturally in thisview of conformal geometry. One isJ 1(N,D

−1/n+ ), the bundle of 1-jets of positive

sections of the density bundleD−1/n, and the other isM = P/SO(n), the quotientof the total space ofP by the subgroupSO(n) ⊂ G. In this section, we show thatthese two manifolds are isomorphic asR+ ×Rn-bundles overN and are, in fact,isomorphic as contact manifolds.

3.1. P/SO(n). We look at the set of left cosetsG/SO(n). Giveng ∈G, the cosetgSO(n) looks like

290 JEFF A. VIACLOVSKYr2 r2vtB(r2/2

)vtv

0 B v

0 0 r−2

1 0 00 SO(n) 00 0 1

=r2 ∗ ∗

0 B ·SO(n) v

0 0 r−2

. (25)

From property (v) of the connection, we see immediately that the fibers ofP →P/SO(n) = M are given by{ρ,ωi,βi} = 0. From the structure equation (2), 2ρ issemibasic and 2dρ is semibasic; therefore 2ρ descends toM. Clearly 2ρ∧(2dρ)n 6= 0,so we have thatM is a contact manifold with 2ρ as a global contact form.

From (25), it follows thatG/SO(n) is parameterized by(r2,v) ∈ R+ ×Rn. Wecompute the left action ofG onG/SO(n),s2 s2wtB

(s2/2

)wtw

0 C w

0 0 s−2

r2 ∗ ∗0 B ·SO(n) v

0 0 r−2

=s2r2 ∗ ∗

0 CB ·SO(n) Cv+wr−2

0 0 s−2r−2

.Therefore the action ofG onG/SO(n) = R+×Rn, which we denote byρM :G→Aut(R+×Rn), is given by

ρM(g) :(r2,v

)−→ (s2r2,Cv+r−2w

). (26)

Note thatM, as the quotient space ofP by the subgroupSO(n), is theR+×Rn-bundleoverN associated toP via the representationρM ; that is,M = P ×(R+×Rn)/∼M ,where∼M is the equivalence relation(p,(r2,v)) ∼M (Rgp,ρM(g

−1)(r2,v)). Wewrite the equivalence class of(p,(r2,v)) as[p,(r2,v)]M .

3.2. J 1(N,D−1/n+ ). If we takeu ∈ 0(D−1/n

+ ), the differential relations thatu andui must satisfy are

du= 2ρu+uiωi, (27)

dui = uβi+ujαji +uijωj . (28)

We see that, taken together,(u,ui) is a section of a bundle overN associated toPby a representationρJ :G→ Aut(R+×Rn). SinceG is connected, we can read offthe representation from (27) and (28); in fact,

R∗g(u,ui)= ρJ(g−1)(u,ui),

where

ρJ (g) :(r2,v

)−→ (s−2r2,Cv−r2w

). (29)


To see this, givenv ∈ g, the Lie algebra ofG, we let g(t) be a curve inG, withg(0)= I andg(0)= v. DifferentiatingR∗g(t)(u,ui)= ρJ (g(t)−1)(u,ui) with respectto t , we get

LV (u,ui)=−ρJ∗(v)(u,ui),where the left-hand side is the Lie derivative with respect toV , the fundamentalvertical vector field onP corresponding tov. From this, (27) and (28) follow.

We now defineJ 1(N,D−1/n+ )= P×(R+×Rn)/∼J , where∼J is the equivalence

relation (p,(r2,v)) ∼J (Rgp,ρJ (g−1)(r2,v)). We write the equivalence class of(p,(r2,v)) as[p,(r2,v)]J . We define the 1-jet ofu, denoted byj1(u), as the section(u,ui) of J 1(N,D−1/n). Note that givenu ∈ 0(D−1/n

+ ), λ a local section ofP →U ⊂N , andq ∈ U , the 1-jet ofu is given locally as

q −→ [λ(q),

(u◦λ(q),ui ◦λ(q)

)]J.

UsuallyJ 1(N,D−1/n+ ) is defined as the union over allp ∈N of equivalence classes

of sections ofD−1/n that agree to the first order atp. This way,J 1(N,D−1/n+ ) embeds

naturally intoGn(TD−1/n+ ), the Grassmanian ofn-planes of the tangent bundle of

D−1/n+ (see [7]). This induces a natural contact structure such that graphs of 1-jets

of sections ofD−1/n+ are Legendre submanifolds. We note that there is a natural

isomorphism between these two definitions such that graphs of 1-jets map to graphsof 1-jets. Therefore,J 1(N,D

−1/n+ ) (using our original definition) has a natural contact

structure.

3.3. J 1(N,D−1/n+ ) is isomorphic toM. DefineF : R+×Rn→ R+×Rn by

F((r2,v

))= (r−2,−v).We have(

ρJ (g)◦F)(r2,v

)= ρJ (g)(r−2,−v)= (s−2r−2,−Cv−r−2w),(

F ◦ρM(g))(r2,v)= F (s2r2,Cv+r−2w

)= (s−2r−2,−Cv−r−2w).

This shows that the representationsρM andρJ are isomorphic; thereforeF inducesan isomorphism between the bundlesJ 1(N,D

−1/n+ ) andM.

Givenu ∈ 0(D−1/n+ ), we denote byFu the mapF(j1(u)) :N ↪→M. LetπM denote

the projectionP →M. Throughout the rest of the paper, we use the important factthat if λ is a local section of the bundlePu = {p ∈ P, u(p) = 1, ui(p) = 0} → N ,thenπM ◦λ = Fu. That is,λ is a local lifting of the mapFu. To see this, first notethatπM sendsp ∈ P to the equivalence class[p,(1,0)]M . Forq ∈N , we have

πM ◦λ(q)=[λ(q),(1,0)

]M,


Fu(q)= F ◦j1(u)(q)= F([λ(q),

(u◦λ(q),ui ◦λ(q)

)]J

)= [λ(q),F ((1,0))]

M= [λ(q),(1,0)]

M.

Forλ a section ofPu, we thus have from (15) that

j1(u)∗F ∗ρ = F ∗u ρ =(πM ◦λ

)∗ρ = λ∗ρ = 0. (30)

Theorem 5. The mapF : J 1(N,D−1/n+ ) → M is an isomorphism of contact

manifolds.

Proof. We already know that the map is a diffeomorphism. To show it is a contactmap, we need to verify thatF ∗IM = IJ , where these are the respective contact linesubbundles of the cotangent bundles. This is equivalent to showing thatF∗(I⊥J )= I⊥M ,where the notation⊥ means the rank-2n subbundle of the tangent bundle that isannihilated by the contact ideal. From (30), we know thatF∗(V ) ⊂ I⊥M for V ann-plane tangent to graph of a 1-jet; that is,V is of the formj1(u)∗(TpN). Sincen-planes of this form are dense inI⊥J , the theorem follows.

4. Calculus of variations. In this section, we discuss how the above curvatureequations arise, and we prove Theorem 1.

4.1.M. We first define then-forms onP

Ek = (−1)k∑|I |=k

βI ∧ω[I ], (31)

whereI = i1 < · · · < ik is an increasing multi-index of lengthk. The notationβI isshort forβi1∧·· ·∧βik , andω[I ] stands forωIc (I c are the indices not appearing inI ), with sign determined by (no sum onik)

ωik ∧ω[i1···ik] = ω[i1···ik−1].

We extend the definition to all strings of indices by skew-symmetry. For example,

ωi∧ω[j ] = δijω1∧·· ·∧ωn,

ωl∧ω[ij ] = δljω[i] −δliω[j ],and so on. We thus have

ωi1∧·· ·∧ωik ∧ω[j1···jk] = δi1···ikj1···jk ,

the generalized Kronecker delta symbol. We haveE0 = ω1∧·· ·∧ωn. We also referto ω1∧·· ·∧ωn asω.


Proposition 6. If N is locally conformally flat, then we have the formulas for0≤ k ≤ n,

dEk = (n−2k)(2ρ)∧Ek. (32)

If N is not locally conformally flat, the above equation still holds for0≤ k ≤ 2.

Proof. We first note that the following formulas hold forI = i1< · · ·< ik:dω[I ] = 2(n−k)ρ∧ω[I ] +αli1ω[li2···ik] +αli2ω[i1l···ik] +· · ·+αlikω[i1i2···l], (33)

and in the conformally flat case, sinceBijk = 0,

d(βI )=−2kρ∧βI +αli1βl∧βi2∧·· ·∧βik+αli2βi1∧βl∧·· ·∧βik+·· ·+αlikβi1∧βi2∧·· ·∧βl.

(34)

Therefore we have fork = 0, . . . ,n,

(−1)k dEk = d∑|I |=k

βI ∧ω[I ]= dβI ∧ω[I ] +(−1)kβI ∧ dω[I ].

Substituting (33) and (34) into this equation, we get

(−1)k dEk =(−2kρ∧βI +αli1βl∧·· ·∧βik+·· ·+αlikβi1∧·· ·∧βl

)∧ω[I ]

+(−1)kβI ∧(2(n−k)ρ∧ω[I ] +αli1ω[li2···ik] +· · ·+αlikω[i1i2···l]

).

By reindexing, we see that the alpha terms cancel, and we are left with (32). Nowif Bijk 6= 0, then fork = 0 the statement is obvious since there are noβi terms. Fork = 1, we have

−d(E1)= d(βi∧ω[i])= d(βi)∧ω[i] −βi∧ dω[i]=(−2ρ∧βi+αji ∧βj +

1

2Bijkω

j ∧ωk)∧ω[i] −βi∧ dω[i].

TheBijk terms vanish sinceωj ∧ωk∧ω[i] = 0, so the computation is the same as inthe conformally flat case. Fork = 2, we have (the 2 is in front because we sum on alli andj )

2d(E2)= d(βi∧βj ∧ω[ij ])= 2d

(βi)∧βj ∧ω[ij ] +βi∧βj ∧ dω[ij ]

= 2(−2ρ∧βi+αji ∧βj +

1

2Biklω

k∧ωl)∧βj ∧ω[ij ] +βi∧βj ∧ dω[ij ].

We see that the only term different from the conformally flat computation is

Bikl∧βj ∧ωk∧ωl∧ω[ij ] =(Biij −Biji

)βj ∧ω = 2Biijβj ∧ω = 0

by the conformal Bianchi identity (8).


From the proposition,Ek anddEk are both semibasic forP → M; therefore theEk descend toM.

SinceM is a contact manifold, we can consider the calculus of variations onM. Thegeneral reference for this theory and the following computations is [3]. We considerthe functional

Fk(N)= 1

(n−2k)

∫N

Ek

defined forN a Legendre submanifold ofM. Whenn is even, we do not defineFk

for k = n/2.Let H : (−ε,ε)×N →M be a smooth variation of Legendre submanifolds. That

is, for eacht ∈ (−ε,ε),Ht :N→M given byHt =H|{t}×N is an immersed Legendresubmanifold with imageNt . Then we have

(n−2k)F′(N)= d

dt

(∫Nt

Ek

)=∫Nt

L∂/∂tH∗Ek

=∫Nt

∂

∂tyH ∗ dEk+d

(∂

∂tyH ∗Ek

).

So by the proposition, we have in the locally conformally flat case (drop theH ∗notation)

F′(N)=∫Nt

∂

∂ty(2ρ∧Ek)=

∫Nt

hEk,

whereh(t,p) is a smooth function on(−ε,ε)×N defined byH ∗(2ρ)= hdt . We canthink of h(0,p) : N → R as the “tangent vector” to the variation since it uniquelydetermines the normal vector field∂/∂t . For t small, the functionh can be chosenarbitrarily (see [3]), so we have the following theorem (k 6= n/2).

Theorem 7. H0 : N →M is a critical point for the functionalFk if and only ifH ∗0 Ek = 0. This holds for allk in the locally conformally flat case and for0≤ k ≤ 2in the general case.

Note that critical points ofFk restricted to the set of Legendre submanifolds suchthatn·F0= 1 correspond to critical points ofFk−CkF0 for some constantCk, by thetheory of Lagrange multipliers. These are then Legendre submanifoldsH0 :N ↪→M

such thatH ∗0 (Ek−CkE0)= 0.

4.2. J 1(N,D−1/n+ ). Using the isomorphism between the contact manifolds, we

can transfer everything above to the jet space. If we takeu ∈ 0(D−1/n+ ) and letλ be

a local section ofPu on an open setU ∈ N , then we haveπM ◦λ = Fu. ThereforesinceEk descends toM, it follows from (16) that

F ∗uEk = F ∗u((−1)kβI ∧ω[I ]

)= λ∗((−1)kβI ∧ω[I ])= λ∗(σk(uij )ω). (35)

Note thatF ∗uE0 is the volume form of the metricgu onN . By Theorem 7, the remarkabove on Lagrange multipliers, and formula (20), we have proved Theorem 1.


4.3. The casek = n/2. Notice that in the conformally flat case, fork = n/2,Proposition 6 says thatdEk = 0. Therefore

∫N

Ek is a constant, independent of thechoice of metric in the conformal class. This is no accident. In this case, it turns outthat the integrand is a multiple of the Pfaffian of the curvature matrix, so we get theEuler characteristic by the generalized Chern-Gauss-Bonnet theorem. To see this, weargue as follows. Letting� denote the Kulkarni-Nomizu product (see [2]), we candecompose the full curvature as

Riem=Weyl+ 1

n−2

(Ric− R

2(n−1)g

)�g.

We recall the definition of the Chern-Gauss-Bonnet integrand. We let{e1, . . . ,en} bea local orthonormal frame and{e∗1, . . . ,e∗n} be the dual frame. We view the curvaturetensor as a skew-symmetric matrix of 2-forms

�ij = Rijkle∗k ∧e∗l .Then we define

Pfaffian(Riem)≡ Pfaffian(�ij

)= 1

2n/2 ·(n/2)! δ1 2···ni1i2···in�i1i2∧�i3i4∧·· ·∧�in−1in ,

which is ann-form and is the Chern-Gauss-Bonnet integrand.

Proposition 8. For a symmetric(0,2)-tensorC,

Pfaffian(C�g)= 2n/2(n/2)! ·σn/2(C) dvol.

Proof. Let

�ij = (C�g)ijkle∗k ∧e∗l= (Cikδjl−Cjkδil+Cjlδik−Cilδjk)e∗k ∧e∗l=−2Cike

∗j ∧e∗k+2Cjke

∗i ∧e∗k .

We may assume thatC is diagonalized,Cij = λiδij , and we have

�ij = 2(λi+λj

)e∗i ∧e∗j .

We then have

2n/2 ·(n/2)! ·Pfaffian(�ij

)= δ1 2···ni1i2···in�i1i2∧�i3i4∧·· ·∧�in−1in

= 2n/2δ1 2···ni1i2···in

(λi1+λi2

)(· · ·)(λin−1+λin

)e∗i1e∗i2· · ·e∗in

= 2n/2∑Sn

(λi1+λi2

)(· · ·)(λin−1+λin

)dvol

= 2n∑Sn

λi1λi2 · · ·λin/2 dvol

= 2n((n/2)!)2 ·σn/2(C) dvol.


Therefore, in the locally conformally flat case, we have

Pfaffian(Riem)= 2n/2(n/2)! ·σn/2(

Ric− R

2(n−1)g

)dvol.

4.4. Conformal factors.Fixing a background metric in the conformal class, wewant to write out the above functionals and Euler-Lagrange equations with respect toa conformal factor. We letπ denote the projection fromP toN .

Claim 9. The following formula holds onP :

π∗F ∗uEk = 1

unσk

(u ·uij −

∑(ul)

2

2δij

)ω. (36)

Proof. In order to prove this, we first need a few lemmas.

Lemma 10. OnP , we have

d

(σk

(u ·uij −6(uj )

2

2δij

))≡ 0 modωi.

Proof. Let vij = u ·uij −(6(uj )2/2)δij . A computation shows that

dvij = 0+vikαkj +vkjαki +vijkωk.

Moduloω terms, this is just the differential of conjugation in the fiber by the orthog-onal part of a group element. This tells us that under the group action,vij transformsby conjugation of the orthogonal part of a group element, that is,

R∗gvij = B−1vijB.

Takingσk of both sides, we see thatσk(vij ) is a zero-density.

Lemma 11. If u is ans/n-density andv is anr/n-density, thenuv is an(r+s)/n-density.

Proof. We have

d(uv)= v du+udv ≡ v(−2suρ)+(−2rvρ)≡−2(r+s)uvρ modωi.

Lemma 12. If f ∈ C∞(P,R), thenfω = π∗α for an n-form α onN if and onlyif f is a 1-density.

Proof. The identity dα = 0 pulls up to(df + 2nfρ) ∧ ω = 0, so thatdf =−2nfρ+fiωi . Conversely, given a functionf satisfying this density equation, thereis a uniquen-form α onN such thatπ∗α = fω.


Applying the above lemmas, we see that the expression

1

un

{σk

(u ·uij −6(uj )

2

2δij

)}is an(n+0)/n= 1-density. Therefore

1

un

{σk

(u ·uij −6(uj )

2

2δij

)}ω = π∗(α)

with α ann-form onN . To prove the claim, we now need to verify thatπ∗F ∗uEk =π∗α. If we pull back both sides of this by the sectionλ, we get equation (35), so weare done.

Previously, we requiredλ to be a local section ofPu ⊂ P . Using the claim, we cannow write down the general formula forF ∗uEk, whenλ is any local section. Pullingback (36) byλ, we have

λ∗π∗F ∗uEk = (π ◦λ)∗F ∗uEk = F ∗uEk = λ∗{

1

unσk

(u ·uij −

∑(ul)

2

2δij

)ω

}.

We now fix au0 ∈ 0(D−1/n) and denote byg0 the associated metric onN . Wecan now restate Theorem 1 as an equation on conformal factors. Giveng ∈ M1, wewrite g = v−2g0, wherev ∈ C∞(N,R+). Using equations (23) and (24), we get thefollowing theorem (k 6= n/2).

Theorem 13. If (N, [g0]) is locally conformally flat, a metricgw = w−2g0 is acritical point of the functional

Fk : v 7→∫N

1

vnσk

(v

(∇2v+ v

n−2

(Ric0− R0

2(n−1)g0

))− |∇v|

2

2g0

)dvolg0

(37)

restricted toM1 if and only if

σk

(w∇2w+ w2

n−2

(Ric0− R0

2(n−1)g0

)− |∇w|

2

2g0

)= Ck (38)

for some constantCk. Hereσk,∇2w, and∇w are taken with respect tog0. If (N, [g0])is not locally conformally flat, the statement is true fork = 1 andk = 2.

We summarize the results of this section in Table 1.

5. Examples. The scalar curvature casek = 1 is known as the Yamabe problemand has been completely solved. That is, for any compact Riemannian manifold


Table 1

M J 1(N,D

−1/n+

)Fk(N) Fk(j

1(u))

= (−1)k

(n−2k)

∫N

βI ∧ω[I ] = 1

(n−2k)

∫N

σk

(1

(n−2)

(Ricu− Ru

2(n−1)·gu))

volgu

Critical point ofFk Critical point ofFk

restricted ton ·F0= 1 restricted toM1

Critical point ofFk−CkF0 Critical point ofFk−(Ck/n)Vol

(−1)kβI ∧ω[I ] = Ckω σk

(Ric− R

2(n−1)·g)= (n−2)kCk

(N,g), there is a constant scalar curvature metric in the conformal class ofg (see[11] and [17]). For an Einstein metric in[g] represented by a densityu, we note that

uij = 1

n−2

(R

nδij − R

2(n−1)δij

)= R

2n(n−1)δij .

So when viewed as a submanifold ofM under the map

Fu = F ◦j1(u) :N ↪→M,

a metric is Einstein if and only ifβi = −(C1/n)ωi , when restricted toFu(N) ⊂

M. Therefore an Einstein metric solves all of the equationsEk = CkE0, and thennecessarily,

Ck = σk(

R

2n(n−1)δij

)=(

R

2n(n−1)

)k(n

k

). (39)

For k = 2, we do not require local conformal flatness; therefore, Einstein metrics arecritical points as well as products of Einstein manifolds. A constant curvature manifold(space form) is exactly a locally conformally flat Einstein manifold. Therefore fork > 2, space forms are critical points. The product of space form with sectionalcurvature 1 and a space form with sectional curvature−1 is locally conformally flatand is a critical point. For any space formX, the productN = S1×X is locallyconformally flat and is a critical point.

An important example is(Sn, [g0]), whereg0 is the standard metric. We have thefollowing theorem due to Obata (see [14]).


Theorem 14. Any constant scalar curvature metric in the conformal class ofg0

is necessarily Einstein and, moreover, is the image of the standard metric under aconformal diffeomorphism ofSn.

This theorem tells us that onSn we have an (n+1)-parameter family of unit volumeEinstein metrics. Therefore we have an (n+1)-parameter family of critical points ofthe functionalsFk|M1 (k 6= n/2). In stereographic coordinates, fixing the flat metric,equation (38) becomes

σk

(u · ∂2u

∂xi∂xj− |∇u|

2

2δij

)= Ck.

If we write g = u−2gflat, then these critical metrics are given by

u(x)= a|x|2+bixi+cfor some constantsa,bi , andc. The standard metric is represented by 1+|x|2.

We now consider symmetric solutions onS1× Sn−1. To do this, we pass to theuniversal coverN = R×Sn−1, the cylinder. We letgc denote the product metric. TheRicci tensor ofN looks like (

0 00 (n−2)In−1

)andR = (n−1)(n−2). Therefore,

Ric− R

2(n−1)δij =

(0 00 (n−2)In−1

)−((n−2)/2 0

0 −((n−2)/2)In−1

)

=(−(n−2)/2 0

0 ((n−2)/2)In−1

).

If we assume that the conformal factoru is independent ofSn−1, then the equationreduces to an ODE, andu= u(t), wheret is the coordinate onR. We have

∇2(u)=(u′′ 00 0

)and|∇u|2= (u′)2. Substituting this into equation (38), the PDE becomes the ODE

σk

(uu′′ −(u2/2

)−((u′)2/2) 00

((u2/2

)−((u′)2/2))In−1

)= Ck. (40)

Note thatu(t)= cosh(t) is a solution for allk.We briefly describe cylindrical coordinates onSn. Let p1 andp2 denote the north

and south poles onSn, and letf : Rn → Sn − {p2} denote inverse stereographicprojection. Cylindrical coordinates are then given by

F : (t,v)−→ etvf−→ Sn−{p1,p2}.


It is easily verified that 4F ∗g0 = (cosh(t))−2gc, so cosh(t) represents the sphericalmetric. We now fixCk corresponding to this solution.

SinceO(n−1) is compact, by the principle of symmetric criticality (see [15]), it fol-lows that the ODE above is still variational for the functional restricted to symmetricfunctions. We observe that the Lagrangian does not depend explicitly ont ; therefore,by Noether’s theorem, we have a conserved quantity associated to the time-translationsymmetry. For any one-dimensional functional of the form∫

L(u,u′,u′′

)dt,

then (see [8]) we have the first integral

L−u′(Lu′ − d

dtLu′′

)−u′′Lu′′ = constant.

With the Lagrange multiplier, our Lagrangian is (see Table 1)

L= 1

un

(1

(n−2k)σk

(uu′′ −(u2/2

)−((u′)2/2) 00

((u2/2

)−((u′)2/2))In−1

)− Ckn

).

We then compute the conservation law, and since we are at a solution, we use (40)to solve foru′′ and substitute this in to get a first-order Hamiltonian involving onlyu

andu′. After a tedious computation, we find that the conservation law takes the form

1−(u2−(u′)2)k =Dk,nun,whereDk,n is a constant parameterizing the solutions. Instead of computing it thisway, we can just show directly that this is indeed a conservation law by substitution.This verification may be found in [18].

If we look in the phase plane(u,u′), the spherical metric is represented by thehyperbola(cosh(t),sinh(t)). Note that there is a constant solutionu0, correspondingto the cylindrical metric. The orbits in the region to the right of the hyperbola stayinside of this region. In this region, fork < n/2 theun term dominates, and it isnot difficult to see that all of the orbits are closed. These solutions, parameterized byDk,n, orbit around the constant solution. Any one of these periodic solutions descendsto give a solution onS1(T )×Sn−1, whereT is the radius ofS1 and 2πT is somemultiple of the period of the solution. The analysis here is similar to thek = 1 case(Delaunay metrics), so we refer the reader to [17] for details.

6. The second variation. In this section we prove Theorem 2. We begin withsome important definitions.


Definition 1. Let (λ1, . . . ,λn) ∈ Rn. We view the elementary symmetric functionsas functions onRn,

σk(λ1, . . . ,λn)=∑

i1<···<ikλi1 · · ·λik ,

and we let0+k equal the component of{σk > 0} containing the positive cone. Forkeven (odd),0−k equals the component of{σk > (<) 0} containing the negative cone.

For a symmetric linear transformationA : V → V , whereV is ann-dimensionalinner product space, the notationA ∈ 0±k means that the eigenvalues ofA lie in thecorresponding set.

Definition 2. Let A : V → V be as above. For 0≤ q ≤ n, theqth Newton trans-formationassociated withA is

Tq(A)= σq(A) ·I−σq−1(A) ·A+·· ·+(−1)qAq.

It is proved in [16] that ifAij are the components ofA with respect to some basisof V , then

Tijq (A)= 1

q! δi1···iq ij1···jqjAi1j1 · · ·Aiqjq , (41)

whereδi1···iq ij1···jqj is the generalized Kronecker delta symbol.

The following proposition describes some important properties of the sets0±k .

Proposition 15. Each set0+(−)k is an open convex cone with vertex at the origin,and we have the following sequences of inclusions:

0+(−)n ⊂ 0+(−)n−1 ⊂ ·· · ⊂ 0+(−)1 .

Furthermore, for symmetric linear transformationsA ∈ 0+(−)k , B ∈ 0+(−)k , we have

tA+(1− t)B ∈ 0+(−)k for t ∈ [0,1]. If A ∈ 0+k , thenTk−1(A) is positive definite, andif k is odd (even) andA ∈ 0−k , thenTk−1(A) is positive (negative) definite.

The proof of this proposition is standard and may be found in [4], [6], and [19].

Definition 3. A metric g is positive k-admissibleor negative k-admissibleif

Ricg− Rg

2(n−1)·g

is everywhere in0+k or 0−k , respectively.

Note that we are using the metricg to view this tensor as an endomorphism of thetangent space at any point, and we take the eigenvalues of this map. Since the Riccitensor is symmetric, these eigenvalues are real.


When is a solutiong of the equation

σk

(Ricg− Rg

2(n−1)·g)= Ck

admissible? Fixing a metricg0 in the conformal class and writingg = w−2g0, wehave

w2

n−2

(Ricg− Rg

2(n−1)g

)= w∇2w+ w2

n−2

(Ric0− R0

2(n−1)g0

)− |∇w|

2

2g0.

If the fixed metricg0 hasRic0−(R0/2(n−1))g0 ∈ 0+(−)k , then looking at a minimum

(maximum), we see, at the point, thatRicg−(Rg/2(n−1))g ∈ 0+(−)k with respect tog, sinceg andg0 are conformal metrics. Therefore, since the cones are connected,by continuity we haveRicg−(Rg/2(n−1))g ∈ 0+(−)k everywhere.

Returning to the second variation, from the theory of Lagrange multipliers, at acritical point of Fk restricted toC = {N ⊂ M : n ·F0(N) = 1}, we know the firstvariation is given by a Lagrange multiplier. Furthermore, we have at the critical pointthat

{F|C}′′ ={F′′k−CkF′′0

}|C.

That is, we can compute the second variation at a critical point using the Lagrangemultiplier (see [1, page 125]).

We letH : (−ε,ε)× (−ε,ε)×N →M be a two-parameter Legendre variation ofthe Legendre submanifoldH(0,0) :N→M, satisfying the constraintn·F0(H(s,t))= 1and such thatH ∗(0,0)(Ek−CkE0) = 0. That is, we are at a critical point. SinceH isa Legendre variation, we haveH ∗(2ρ) = h1ds + h2dt whereh1,h2 : (−ε,ε)×(−ε,ε)×N → R. We can think of the real-valued functions onN , h1(0,0,p) andh2(0,0,p) as “tangent vectors” to the two-parameter variation, since these uniquelydetermine the normal vector fields∂/∂t and∂/∂s.

In this entire section, we make the following restriction to allowable variations.SinceM is a bundle overN , we require that a variation move only in the fiber; that is,it is a variation of sections of the bundle. Note that for such variations, in the followingcomputations all terms of the form ((∂/∂t)yωi) are zero, since theωi are semibasicfor P → N and therefore vanish on vertical vectors. Also note that in the followingcomputations, the forms in the integrand are defined onP , but they descend toM.To be technically correct, we should be pulling back by a local section ofP →M,but we take this as understood.

At a critical point, we have the computation of the second variation (we omit theH ∗ notation and abbreviateN =N(0,0))


(Fk|C)′′(N)(h1,h2)=(F′′k−CkF′′0

)|C(N)(h1,h2)

= d

dt

(d

ds

(∫N(s,t)

1

(n−2k)Ek− Ck

nE0

))∣∣(0,0)

= d

dt

(∫N(0,t)

h1(Ek−CkE0)

)∣∣t=0

=∫N

h1L∂/∂t (Ek−CkE0)

=∫N

h1

(∂

∂ty dEk+d

(∂

∂tyEk

))−∫N

nCkh1h2ω.

We letHt =H(0,t) : (−ε,ε)×N→M, and we have that

H ∗t βi = sijH ∗t ωj +Pi dt, (42)

H ∗t 2ρ = h2(0, t)dt (43)

for some functionsPi andsij . Note that for fixedt , since we are on Legendre subman-ifolds, d(2ρ) = −βi ∧ωi restricts to zero. Therefore by Cartan’s lemma,βi = sijωjwith sij = sji .

6.1. k = 1. In this case,H is a two-parameter variation of Legendre submanifoldssatisfying the constraintn ·F0(H(s,t))= 1 and such thatH ∗(0,0)(E1−C1E0)= 0, thatis, βi∧ω[i] = −C1ω. We then have

(F1|C)′′(N)(h1,h2)

=∫N

h1

(∂

∂tydE1+d

(∂

∂tyE1

))−∫N

nC1h1h2ω

=−∫N

h1

(∂

∂ty2(n−2)ρ∧βi∧ω[i] +d

(∂

∂ty(βi∧ω[i]

)))−∫N

nC1h1h2ω.

The first term in the integral is

(n−2)h1(h2βi∧ω[i] −2Piρ∧ω[i]

).

Theρ term vanishes since we are on a Legendre submanifold. For the second term,we have

d

(∂

∂ty(βi∧ω[i]

))= d(Piω[i])= dPiω[i] +Pi

(2(n−1)ρ∧ω[i] +αliω[l]

)= (dPi−Plαli)ω[i].


We define the functionsPij by

dPi−Plαli = Pijωj .Therefore

(F1|C)′′(N)(h1,h2)=−∫N

h1Piiω+(n−2)h1h2βi∧ω[i] +nC1h1h2ω

=−∫N

h1Piiω+2C1h1h2ω.

Now we pull back these computations to the jet space. If we takeN to be of the formF(j1(u)), then we chooseλ to be a section ofPu. Since thisλ is a lifting of Fu, wejust need to pull back the above computations underλ. We first identify thePii . Wehave

d(2ρ)= d(h2dt)= d(h2)∧ dt =−βi∧ωi=−(Pi dt∧ωi+sijωj ∧ωi)= Piωi∧ dt,

where thesij terms vanish from symmetry. Therefore from Cartan’s lemma we getthat

dh2≡ Piωi mod{dt}.So now when we pull back by the sectionλ, this tells us that

Pi = (∇h2)i, (44)

and by the definition of thePij , we have

Pij =(∇2h2

)ij. (45)

Noting thatC1 = (R/2(n−1)), we then substitute this and (45) into the secondvariation to get

(F1|M1

)′′(N)(h1,h2)=−

∫N

h1Piiω+2C1h1h2ω

=−∫N

h1

(1h2+ R

n−1h2

)dvolgu .

Therefore the Jacobi operator ofF1|M1 is

J1(h)=−(1h+ R

n−1h

).


Applying the Rayleigh-Ritz characterization of the first eigenvalue of the Laplacian,we have

F′′1|M1(N)(h,h)=−∫N

h

(1h+ R

n−1h

)dvolgu

≥(λ1(−1)− R

n−1

)∫N

h2 dvolgu .

We see immediately that ifR ≤ 0, then a critical point is a strict local minimum forF1|M1. To analyze the caseR > 0, we state the following theorem of Obata [13].

Theorem 16. LetN be a compact Einstein manifold withR > 0. Then

λ1(−1)≥ R

n−1

with equality if and only ifN is isometric toSn with the standard metric.

Using this theorem, we see that an Einstein metric is a strict local minimum forF1|M1 unlessN is isometric toSn with the standard metric.

6.2. k = 2. We restrictH to be a two-parameter variation of Legendre submani-folds satisfying the constraintn ·F0(H(s,t))= 1 and such thatH ∗(0,0)(E2−C2E0)= 0,that is,βi∧βj ∧ω[ij ] = 2C2ω. We compute

(F2|C)′′(N)(h1,h2)=∫N

h1

(∂

∂tydE2+d

(∂

∂tyE2

))−∫N

nC2h1h2ω

= 1

2

∫N

h1

(∂

∂ty(n−4)2ρ∧βi∧βj ∧ω[ij ]

+d(∂

∂ty(βi∧βj ∧ω[ij ]

)))−∫N

nC2h1h2ω.

The 1/2 is in front because we are summing on alli,j . For the first term in theintegral, we have

∂

∂ty2ρ∧βi∧βj ∧ω[ij ] = h2βi∧βj ∧ω[ij ] −2Pi(2ρ)∧βj ∧ω[ij ]

= h2βi∧βj ∧ω[ij ]since we are on a Legendre submanifold. For the second term we have

d

(∂

∂ty(βi∧βj ∧ω[ij ]

))= d(2Pi∧βj ∧ω[ij ])= 2(dPi)βj ∧ω[ij ] +2Pi dβj ∧ω[ij ] −2Pi∧βj ∧dω[ij ].

(46)


From now on, since theρ terms vanish, we leave them out. The middle term of this is

2Pi dβj ∧ω[ij ] = 2Pi

(αlj ∧βl+

1

2Bilmω

l∧ωm)∧ω[ij ]

= 2Piαlj ∧βl∧ω[ij ] +Pi

(Bjij −Bjji

)ω

= 2Piαlj ∧βl∧ω[ij ] −2Pi

(Bjji

)ω

= 2Piαlj ∧βl∧ω[ij ]

(47)

by the conformal Bianchi identity (8). The last term in (46) is

−2Piβj ∧ dω[ij ] = −2Piβj ∧(αli ∧ω[lj ] +αlj ∧ω[il]

)=−2Plα

li ∧βj ∧ω[ij ] −2Piα

lj ∧βl∧ω[ij ].

(48)

The last term here cancels out with the term in (47), so we have(F2|C

)′′(N)(h1,h2)=−4

∫N

h1h2C2ω+∫N

h1Pilωl∧βj ∧ω[ij ].

Pulling back to the jet space, we haveβj =−ujmωm, and using Definition 2, we get(F2|M1

)′′(h1,h2)=−4

∫N

h1h2C2 dvolgu−∫N

h1(∇2h2

)ilT il1 (u∗∗) dvolgu .

Therefore the Jacobi operator ofF2|M1 is

J2(h)=−T ij1 (u∗∗)(∇2h

)ij−4C2h.

Note that this is just the expression in a local orthonormal basis of a globally definedoperator. Using Proposition 15, we see that if we are at a two-admissible criticalmetric, then the Jacobi operator is elliptic. The Jacobi operator is just the linearizationof the Euler-Lagrange equations at a solution, so it follows that the Euler-Lagrangeequations are elliptic at a two-admissible solution.

It is easy to see from the derivation that the Jacobi operator must be of divergenceform, that is,T ij1 (u∗∗),j = 0, so we see immediately from the second variation thata negative two-admissible solution is a strict local maximum forF2|M1. For thepositive curvature case, assume we are at an Einstein metric. Then we have thatC2= (R2/8n(n−1)), uil = (R/2n(n−1))δil , and

T il1 (u∗∗)= δi1il1l ui1l1 =R

2n(n−1)δi1l1δ

i1il1l= R

2nδil .

The second variation becomes(F2|M1

)′′(h1,h2)=− R

2n

∫N

h1

(1h2+ R

(n−1)h2

)dvolgu .


Up to the factor(R/2n), we see that the linearization is exactly the same as that forthe k = 1 case. Using Obata’s theorem, as in the section above, we conclude that apositive scalar curvature Einstein metric is a strict local minimum forF2|M1, unlessN is isometric toSn with the standard metric.

6.3. k > 2. In this case, we requireN to be locally conformally flat. A computationsimilar to that of thek = 2 case shows that at a critical point we have

(Fk|C

)′′(h1,h2)=−2k

∫N

h1h2Ckω+ (−1)k

(k−1)!∫N

h1Pi1lωlβi2 · · ·βikω[i1···ik].

Pulling back underλ a section ofPu, we get

=−2k∫N

h1h2Ckω−∫N

h1(∇2h2

)ijTij

k−1(u∗∗) dvolgu .

Therefore the Jacobi operator ofFk|M1 is

Jk(h)=−T ijk−1(u∗∗)(∇2h

)ij−2kCkh.

We note that this is just the expression in a local orthonormal basis of a globallydefined operator. From Proposition 15, we see that the equations are elliptic at ak-admissible solution. Again we see from the derivation that the Jacobi operator must beof divergence form, that is,T ijk−1(u∗∗),j = 0, so we see immediately from the secondvariation that fork odd, a negativek-admissible critical point is a strict local minimumfor Fk|M1. Fork even, a negativek-admissible critical point is a strict local maximum,just as in thek = 1 andk = 2 cases. At a positive scalar curvature Einstein metric(which is a constant curvature metric since we are assumingN is locally conformallyflat), we haveβi =−(C1/n)ω

i . Therefore we have

(Fk|M1

)′′(h1,h2)=−

(C1

n

)k−1(n−1

k−1

)∫N

h1

(1h2+ R

n−1h2

)dvolgu .

Up to a positive factor, we see the linearization is exactly the same as for the casek = 1. Therefore we conclude that a positive constant sectional curvature metric is astrict local minimum forFk|M1, unlessN is isometric toSn with the standard metric.

7. Uniqueness. In this section we prove Theorems 3 and 4.

7.1. Einstein metrics.Given a densityv ∈ 0(D−1/n+ ), we recall the formula (14):

dvij = (vlβl)δij −2vijρ+vilαlj +vljαli+vij lωl. (49)

We let◦vij denote the tracelessvij , that is,

◦vij = vij − (vll/n)δij . We then have the

following proposition.


Proposition 17. We have

d(◦vij)=−2

◦vijρ+ ◦vilαlj +

◦vljα

li modωi. (50)

Proof. Since the statement is modulo theωi , we ignore these terms in the followingcomputations. We trace formula (49) to get

d(vii)= n(vlβl)−2viiρ+vilαli+vliαli= n(vlβl)−2viiρ,

since thevil are symmetric and theαli are skew. We thus have

d◦vij = dvij −

(1

n

)dvllδij

= (vlβl)δij −2vijρ+vilαlj +vljαli−vlβlδij +(

2

n

)vllρδij

=−2◦vijρ+vilαlj +vljαli−

(vkkn

)(αij +αji

)=−2

◦vijρ+vilαlj −

(vkkn

)(δilα

lj

)+vljαli−(vkkn )(δjlαli)=−2

◦vijρ+ ◦vilαlj +

◦vljα

li .

This proposition tells us that◦vij scales and conjugates when we move in the fiber;

that is, it is a section of a vector bundle. If we restrict to

Pv ={p ∈ P, v(p)= 1, vi(p)= 0

},

then we have

◦vij = vij −

(vll

n

)δij

= 1

n−2

(Rij − R

2(n−1)δij

)− R

2n(n−1)δij

= 1

n−2

(Rij −

(R

n

)δij

)= 1

n−2

◦Rij .

So we have the fact that◦vij restricts to be the traceless Ricci tensor of the metricgv.

If gv should happen to be Einstein, then the traceless Ricci vanishes. Since◦vij just

scales and conjugates in the fiber ofP →N , we have the following lemma.

Lemma 18. Given a densityv ∈ 0(D−1/n+ ), if gv is an Einstein metric, then

◦vij ≡ 0

onP .

For the rest of this section we assume that(N, [g]) has an Einstein metric in theconformal class[g], and we now fixv ∈ 0(D−1/n

+ ) to be an Einstein metric.


7.2. k = 1. We state the following result of Obata (see [14]).

Theorem 19. Let (N,g) be a compact Einstein manifold. Then any constantscalar curvature metric in the conformal class ofg is also Einstein. Moreover,gis the unique constant scalar curvature metric in its conformal class (up to scalingby a constant) unless(N,g) is isometric to(Sn,g0).

If there are two distinct unit volume Einstein metrics in the conformal class, thenObata constructs an isometry to (Sn,g0) to prove the second statement. We reprovethe first statement to illustrate our methods. We assumeN is oriented, noting that thenonorientable case follows by passing to the orientable double cover. We have thefollowing proposition.

Proposition 20. On a Legendre submanifold ofM, we have

d(viβj ∧ω[ij ]

)= vβi∧βj ∧ω[ij ] + n−1

nvllβi∧ω[i]

= 2vE2− n−1

nvllE1.

Proof. We omit ρ terms because we are restricting to a Legendre submanifold.We have

d(viβj ∧ω[ij ]

)= dviβj ∧ω[ij ] +vi dβj ∧ω[ij ] −viβj ∧ dω[ij ]. (51)

Using Lemma 18, the first term on the right is

dviβj ∧ω[ij ] =(vβi+vlαli+vilωl

)βj ∧ω[ij ]

= (vβi+vlαli)∧βj ∧ω[ij ] +(1

n

)vkkδilω

lβj ∧ω[ij ]

= (vβi+vlαli)∧βj ∧ω[ij ] −(1

n

)vkkβjω

i∧ω[ij ]

= (vβi+vlαli)∧βj ∧ω[ij ] + n−1

nvkkβj ∧ω[j ].

The middle term in (51) is

vi dβj ∧ω[ij ] = vi(αlj ∧βl+

1

2Bjklω

k∧ωl)∧ω[ij ]

= vi(αlj ∧βl

)∧ω[ij ] −viBjjiω= vi

(αlj ∧βl

)∧ω[ij ]by the conformal Bianchi identity (8). The last term in (51) is

−viβj ∧ dω[ij ] = −viβj ∧(αli ∧ω[lj ] +αlj ∧ω[il]

).


Putting together all of the alpha terms, we get (omitting the wedges)

vlαliβjω[ij ] +viαljβlω[ij ] −viβjαliω[lj ] −viβjαljω[il]

= vlαliβjω[ij ] +viαljβlω[ij ] −vlαliβjω[ij ] −viαljβlω[ij ] = 0.

Now we assume we haveu ∈ 0(D−1/n+ ) of constant scalar curvature. Thenβi ∧

ω[i] = −σ1ω onF(j1(u)), whereσ1 is a constant. We now letσk denote the functiondefined byEk = σkω when restricted toF(j1(u)), that is,σk = σk(uij ). We pull backthe formula in the proposition toN and integrate to get

0=∫N

(2vσ2− n−1

nvllσ1

)dvolgu .

We have a formula forvll restricted to the setPu (see (24)), so we get

0=∫N

(2vσ2− n−1

n

(1v+ vuii

)σ1

)dvolgu

=∫N

(2vσ2− n−1

n

(1v+ vσ1

)σ1

)dvolgu

= 2∫N

v

(σ2− n−1

2nσ 2

1

)dvolgu−

∫N

n−1

nσ11v dvolgu

= 2∫N

v

(σ2− n−1

2nσ 2

1

)dvolgu .

(52)

Lemma 21. If A is a symmetricn×n matrix, then

σ2(A)≤ n−1

2nσ1(A)

2

with equality if and only ifA= λI .

Proof. If we let λi denote the eigenvalues ofA, the lemma is a consequence ofthe factorization

σ2(A)− n−1

2nσ1(A)

2=− 1

2n

∑i<j

(λi−λj

)2.

Using the lemma and the fact thatv is positive, we see that the integrand in (52)is nonpositive. Since the integral is equal to zero, we conclude that the integrand isidentically zero, and therefore, by the lemma, the metricgu is Einstein.


7.3. k > 1. In this case we assume that(N, [g]) is locally conformally flat. Weprove Theorem 3 by showing that any solution is necessarily Einstein. The result thenfollows from Theorems 14 and 19. Omitting the wedge product notation we have thefollowing.

Proposition 22. On a Legendre submanifold ofM, we have fork 6= n,

d(vi1βi2 · · ·βik+1ω[i1···ik+1]

)= vβi1 · · ·βik+1ω[i1···ik+1] +n−kn

vllβi1 · · ·βikω[i1···ik]

= (−1)k+1(k+1)!(vEk+1− n−k

(k+1)nvllEk

).

Proof. We omit ρ terms because we are restricting to a Legendre submanifold.We have

d(vi1βi2 · · ·βik+1ω[i1···ik+1]

)= dvi1βi2 · · ·βik+1ω[i1···ik+1] +kvi1 dβi2 · · ·βik+1

×ω[i1···ik+1] +(−1)kvi1βi2 · · ·βik+1 dω[i1···ik+1].(53)

Using Lemma (18), the first term on the right is

dvi1βi2 · · ·βik+1ω[i1···ik+1]= (vβi1+vlαli1+vi1lωl)βi2 · · ·βik+1ω[i1···ik+1]

= (vβi1+vlαli1)βi2 · · ·βik+1ω[i1···ik+1] +(

1

n

)vjj δi1lω

lβi2 · · ·βik+1ω[i1···ik+1]

=(vβi1+vlαli1)βi2 · · ·βik+1ω[i1···ik+1]+(−1)k(

1

n

)vjjβi2 · · ·βik+1ω

i1ω[i1···ik+1].

Next we note that

ωi1ω[i1···ik+1] = (−1)k(n−k)ω[i2···ik+1],

so we have

dvi1βi2 · · ·βik+1ω[i1···ik+1] = (vβi1+vlαli1)βi2 · · ·βik+1ω[i1···ik+1]

+ n−kn

vjjβi2 · · ·βik+1ω[i2···ik+1].

The middle term in (53) is

kvi1 dβi2 · · ·βik+1ω[i1···ik+1] = kvi1αli2βlβi3 · · ·βik+1ω[i1···ik+1].


The last term in (53) is

(−1)kvi1βi2 · · ·βik+1 dω[i1···ik+1]= (−1)kvi1βi2 · · ·βik+1

(αli1ω[l···ik+1] +· · ·+αlik+1

ω[i1···l])

= (−1)kvlβi2 · · ·βik+1αi1l ω[i1···ik+1] +(−1)kvi1βi2

· · ·βik+1

(αli2ω[i1l···ik+1] +· · ·+αlik+1

ω[i1···l])

= vlαi1l βi2 · · ·βik+1ω[i1···ik+1] +(−1)kvi1βl · · ·βik+1αi2l ω[i1i2···ik+1]

+· · ·+(−1)kvi1βi2 · · ·βlαik+1l ω[i1i2···ik+1]

= vlαi1l βi2 · · ·βik+1ω[i1···ik+1] +vi1αi2l βl · · ·βik+1ω[i1i2···ik+1]+· · ·+vi1αik+1

l βi2 · · ·βlω[i1i2···ik+1]= −vlαli1βi2 · · ·βik+1ω[i1···ik+1] −kvi1αli2βl · · ·βik+1ω[i1i2···ik+1].

We see that all of the alpha terms cancel, so we are left with

d(vi1βi2 · · ·βik+1ω[i1···ik+1]

)= vβi1 · · ·βik+1ω[i1···ik+1] +n−kn

vllβi1 · · ·βikω[i1···ik]

= (−1)k+1(k+1)!(vEk+1− n−k

(k+1)nvllEk

).

Now we assume we haveu ∈ 0(D−1/n+ ) with σk = σk(uij ) constant. We pull back

the formula in the proposition toN and integrate to get

0= (−1)k+1(k+1)!∫N

(vσk+1− n−k

(k+1)nvllσk

)dvolgu .

We have a formula forvll restricted to the setPu (see (24)), so we get

0=∫N

(vσk+1− n−k

(k+1)n

(1v+ vuii

)σk

)dvolgu

=∫N

(vσk+1− n−k

(k+1)n

(1v+ vσ1

)σk

)dvolgu

=∫N

v

(σk+1− n−k

(k+1)nσkσ1

)dvolgu−

n−k(k+1)n

∫N

σk1v dvolgu

=∫N

v

(σk+1− n−k

(k+1)nσkσ1

)dvolgu .

(54)

Lemma 23. LetA be a symmetricn×n matrix. IfA ∈ 0+k , then

σk+1(A)≤ n−k(k+1)n

σk(A)σ1(A)


with equality if and only ifA= λI . If A ∈ 0−k , then for even (odd)k

σk+1(A)≥ (≤) n−k(k+1)n

σk(A)σ1(A)

with equality if and only ifA= λI .

Proof. We let

pk =(n

k

)−1

·σk(A).The inequalities we want are thenpk+1 ≤ pkp1. For anyn× n symmetric matrix,Newton’s inequalities (see [9]) state that

pk−1pk+1 ≤ p2k .

The proof proceeds by induction. The casek = 1 follows from Lemma 21. Assumethe statement is true up tok−1. If A ∈ 0+k , then from Proposition 15, we havepj > 0for all j ≤ k. From Newton’s inequalities and sincepk > 0, we use the inductivehypothesis to get

pk−1pk+1 ≤ pkpk ≤ pkp1pk−1.

Sincepk−1> 0, we divide and arrive at the desired inequality. Assumepk+1= p1pk.Then from Newton’s inequalities we have

pk−1p1pk = pk−1pk+1 ≤ p2k .

Sincepk > 0, we havepk−1p1 ≤ pk. But by induction,pk ≤ p1pk−1; thereforepk = p1pk−1. Again by induction we conclude thatA = λI . The negative casefollows easily.

From the discussion after Definition 3, we have thatgu is k-admissible, so we canapply Lemma 23. Sincev is positive, it follows that the integrand in (54) is alwaysnonnegative or always nonpositive. Since the integral is equal to zero, we concludethat the integrand is identically zero, and therefore, by the lemma, the metricgu isEinstein.

7.4. Einstein manifolds withR < 0. We now prove Theorem 4. In the followingwe letCk denote the constant corresponding to the solutiong = w−2g0. Sinceg0 isan Einstein metric, writing out equation (1) with respect to a conformal factor, we get(see equation (38))

Ck = σk(w∇2w+w2

(R0

2n(n−1)

)g0− |∇w|

2

2g0

).

SinceN is compact, letp be a point wherew is a maximum. We then have

σk

(w(p)∇2w(p)+w(p)2

(R0

2n(n−1)

)g0

)= Ck. (55)

To expand this, we need the following lemma.


Lemma 24. For any symmetricn×n matrixB and real numbera,

σk(aI+B)=∑

j1<···<jk

(ak+ak−1σ1

(λj1, . . . ,λjk

)+·· ·+σk(λj1, . . . ,λjk )),whereI is the identity matrix andλi are the eigenvalues ofB.

Proof. We have

σk(aI+B)= σk(a+λ1, . . . ,a+λn

)= ∑j1<···<jk

(a+λj1

) · · ·(a+λjk )=

∑j1<···<jk

(ak+ak−1σ1

(λj1, . . . ,λjk

)+·· ·+σk(λj1, . . . ,λjk )).This lemma implies that

σk(aI+B)=k∑i=0

cn,k,iσk−i (B)ai

for some positive combinatorial coefficientscn,k,i . We note thatcn,k,k (the coefficientof ak) is equal to

(nk

). Equation (55) becomes

σk

(w(p)∇2w(p)+w(p)2

(C1(g0)

n

)g0

)=k−1∑i=0

w(p)k+icn,k,iσk−i(∇2w(p)

)(C1(g0)

n

)i+w(p)2kCk(g0)= Ck. (56)

7.4.1. k = 1. At the pointp, since it is a maximum forw, we have that∇2w isnegative semidefinite, so (56) becomes

w(p)1w(p)+w(p)2C1(g0)= nonpositive term+w(p)2C1(g0)= C1.

Therefore

w(p)2C1(g0)≥ C1. (57)

SinceC1(g0)= (R0/2(n−1)) < 0, this implies thatC1< 0, and we have

w(p)≤(

C1

C1(g0)

)1/2

.

Since we are considering metrics of unit volume, we must have

C1 ≤ C1(g0) (58)

with equality if and only ifw ≡ 1. The following proposition, which follows easilyfrom Hölder’s inequality, implies thatC1= C1(g0).


Proposition 25. If (N,g0) is a compact Riemannian manifold with constantscalar curvatureR0 ≤ 0, thenR0 is a global minimum for the total scalar curva-ture functional restricted toM1.

7.4.2. k = 2. In this case, at the maximum point, (56) becomes

C2= w(p)2σ2(∇2w(p)

)+(n−1)w(p)31w(p)

(C1(g0)

n

)+w(p)4C2(g0)

= nonnegative terms+w(p)4C2(g0).

Therefore

w(p)4C2(g0)≤ C2.

SinceC2(g0)= (R20/8n(n−1)) > 0, this implies thatC2 > 0, and we have

w(p)≤(

C2

C2(g0)

)1/4

.

Again, since we are considering unit volume metrics, we must have

C2 ≥ C2(g0)

with equality if and only ifw ≡ 1. From Lemma 21, we have

n−1

2nσ 2

1 ≥ σ2= C2 ≥ C2(g0)= n−1

2nC1(g0)

2.

Sinceg is negative two-admissible (Definition 3), we have thatσ1 is negative byProposition 15. Therefore

σ1 ≤ C1(g0).

Proposition 25 then tells us thatC2= C2(g0).

7.4.3. k > 2. These cases proceed exactly as in thek = 1 andk = 2 cases. Usingthe maximum principle, we can prove for oddk thatCk ≤ Ck(g0) and for evenk thatCk ≥ Ck(g0). Lemma 23 implies the following.

Lemma 26. LetA be a symmetricn×nmatrix. IfA ∈ 0−k , then for evenk,pk ≤ pk1and, for oddk, pk ≥ pk1.

Since any solution must necessarily be negativek-admissible, the equality ofCkandCk(g0) follows from this lemma and Proposition 25.

References

[1] Melvin S. Berger, Nonlinearity and Functional Analysis: Lectures on Nonlinear Problemsin Mathematical Analysis, Pure Appl. Math.74, Academic Press, New York, 1977.


[2] Arthur L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb. (3)10, Springer-Verlag, Berlin,1987.

[3] Robert Bryant, Phillip A. Griffiths, and Lucas Hsu, The Poincaré-Cartan form, Institutefor Advanced Study preprint, 1997.

[4] L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math.155 (1985), 261–301.

[5] Élie Cartan, Les espaces à connexion conforme, Ann. Soc. Polon. Math.2 (1923), 171–221.[6] Lars Garding, An inequality for hyperbolic polynomials, J. Math. Mech.8 (1959), 957–965.[7] Robert B. Gardner, A differential geometric generalization of characteristics, Comm. Pure

Appl. Math.22 (1969), 597–626.[8] I. M. Gelfand and S. V. Fomin, Calculus of Variations, rev. ed., Prentice-Hall, Englewood

Cliffs, N.J., 1963.[9] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Math. Lib., Cambridge

Univ. Press, Cambridge, 1988.[10] Shoshichi Kobayashi, Transformation Groups in Differential Geometry, Classics Math.,

Springer-Verlag, Berlin, 1995.[11] John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.)

17 (1987), 37–91.[12] Rafe Mazzeo, Daniel Pollack, and Karen Uhlenbeck, Moduli spaces of singular Yamabe

metrics, J. Amer. Math. Soc.9 (1996), 303–344.[13] Morio Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J.

Math. Soc. Japan14 (1962), 333–340.[14] , The conjectures on conformal transformations of Riemannian manifolds, J. Differential

Geometry6 (1971/72), 247–258.[15] Richard S. Palais, The principle of symmetric criticality, Comm. Math. Phys.69 (1979),

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Math. J.20 (1973), 373–383.[17] Richard M. Schoen, “Variational theory for the total scalar curvature functional for

Riemannian metrics and related topics” inTopics in Calculus of Variations (MontecatiniTerme, 1987)Lecture Notes in Math.1365, Springer-Verlag, Berlin, 1989, 120–154.

[18] Jeff A. Viaclovsky, Conformally invariant Monge-Ampère equations: Global solutions, toappear in Trans. Amer. Math. Soc.

[19] Xu Jia Wang, A class of fully nonlinear elliptic equations and related functionals, IndianaUniv. Math. J.43 (1994), 25–54.

Department of Mathematics, Princeton University, Princeton, New Jersey 08544, USA;[email protected]

Current: Department of Mathematics, The University of Texas at Austin, Austin, Texas78712, USA

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