BIFURCATION RESULTS FOR THE YAMABE PROBLEM ON RIEMANNIAN MANIFOLDS WITH BOUNDARY 7 1 ELKIND.CARDENAS 0 2 n Abstract. WeconsidertheproductofacompactRiemannianmanifoldwith- a out boundary and null scalar curvature with a compact Riemannian mani- J foldwithboundary,nullscalarcurvatureandconstantmeancurvatureonthe 6 boundary. Weusebifurcationtheorytoprovetheexistenceofainfinitenum- 2 berofconformalclasseswithatleasttwonon-homotheticRiemannianmetrics of nullscalar curvature andconstant meancurvature of the boundaryon the ] productmanifold. G D . h t 1. Introduction a m Let (M,g) be a compact Riemannian manifold without boundary of dimension [ m 3.Itiswellknowthatthemetricsofconstantscalarcurvatureintheconformal cla≥ss [g] of g can be characterized variationally as critical points of the Hilbert- 1 Einsteinfunctionalontheconformalclass[g].Theexistenceofmetricswithconstant v 2 scalar curvature in conformal class [g] was established trough the combined works 5 of Yamabe [18], Trudinger [17], Aubin [3] and Schoen [15]. 5 Once the existence of constantscalarcurvature metrics conformalto g has been 7 established, it is natural do ask about its uniqueness. For instance, if the Yam- 0 abe constant Y[M,[g]] defined as the minimum of the normalized Hilbert-Einstein . 1 functional on [g] is non-positive then there exists only one of these metrics (up to 0 homotheties). Also, if g is an Einstein metric which is not conformal to the round 7 metric on the sphere Sn, there exists one unique metric of unit volume and con- 1 : stant scalar curvature in [g] by a result of Obata [13]. However, Pollack proved v in [14] that if the Yamabe constant Y[M,[g]] is positive then there are conformal i X classes with an arbitrarily large number of metrics of unit volume and constant r scalar curvature sufficiently C0- close to [g]. a Bifurcation techniques have been successfully used to study multiplicity of solutions for the Yamabe problem, both in the compact and the nonimpact case, see for instance [7], [5] and [6]. In particular, Lima et al. in [12] using bifurcation techniquesprovedaresultofmultiplicity inaninfinite numberofconformalclasses on product manifolds. ForcompactRiemannianmanifoldswithboundaryanddimensionm 3similar ≥ problems to those mentioned above have been studied by different authors. For instance, Escobar proved in [?] that almost every compact Riemannian manifold withboundaryisconformallyequivalenttoawithnullscalarcurvatureandconstant mean curvature on the boundary. Uniqueness results were obtained by Escobar in [11]. Date:April2016. 1 2 ELKIND.CARDENAS Motivated by the results in [12] in this paper using the bifurcation theory we studythe multiplicity ofmetrics withnullscalarcurvatureandconstantmeancur- vature on conformal class of a product metric. More precisely, consider (M ,g(1)) 1 a compact Riemannian manifold, without boundary, null scalar curvature and (M ,g(2)) a compact Riemannian manifold with boundary, null scalar curvature 2 and constant mean curvature. Consider now the product manifold M =M M 1 2 × and the family of Riemannian metrics (g ) with null scalar curvature and con- t t>0 stant mean curvature on M defined by g = g(1) tg(2). The main result of the t ⊕ paper (Theorem11) states that if (M ,g(2)) has positive constant curvature mean 2 onthe boundary,then thereisa sequence(t )strictlydecreasingtending to 0such n that every element of this sequence is a bifurcation instant of the family (g ) . t t>0 For all other value of t the family (g ) is locally rigid. The fact that every t t t>0 n is a bifurcation instant gives an entirely new sequence (g ) of Riemannian metrics i withnullscalarcurvatureandconstantmeancurvatureonboundarysuchthateach g [g ] for some t close to t with g non-isometric to g . This proves multiplicity i t n i t ∈ for an infinite number of conformal classes. The paper is organized as follows: in section 2 we recall some basic fact about variational characterization of the Riemannian metrics with null scalar curvature and constant mean curvature on boundary conformally related to a metric g. In section3 we give an abstractresult of bifurcationand we examine the convergence of a bifurcating branch. Finally, in section 4 we show our main result. 2. Variational setting of the Yamabe problem Let(M,g)beacompactRiemannianmanifoldwithboundaryandm=dimM 3. We will denote by H1(M) the space of functions in L2(M) with first wea≥k derivatives in L2(M). Endowed with the inner product ϕ ,ϕ = g( ϕ , ϕ )+ϕ ϕ υ , ϕ ,ϕ H1(M), 1 2 1 2 1 2 g 1 2 h i ZM(cid:16) ∇ ∇ (cid:17) ∈ H1(M) is a Hilbert space. Let E :H1(M) R be defined by −→ m 2 m 2 E(ϕ)= g( ϕ, ϕ)+ − R ϕ2 υ + − H ϕ2σ . ZM(cid:16) ∇ ∇ 4(m−1) g (cid:17) g 2 Z∂M g g It is well known in literature that ϕ H1(M) is a critical point of E under the constraint C = {ϕ ∈ H1(M)| ∂Mϕ2∈(mm−−21) σg = 1} if and only if the conformal metric g˜ = ϕm4−2g has null scaRlar curvature and constant mean curvature, see for instance [?, Proposition 2,1] . 2.1. The manifold of the normalized harmonic functions. To study the bifurcation of metrics with null scalar curvature and constant mean curvature we will need to characterize these as critical points of E on a special submanifold of H1(M). For this purpos, remember that a function ϕ H1(M) is harmonic if ∈ g( ϕ, φ)υ =0, g Z ∇ ∇ M for all φ H1(M), where H1(M) is the kernel of trace operator, i.e., ∈ 0 0 H1(M)= φ H1(M) φ =0 . 0 { ∈ (cid:12) |∂M } (cid:12) (cid:12) BIFURCATION FOR YAMABE PROBLEM IN MANIFOLDS WITH BOUNDARY 3 Let H1(M) denote the subspace of H1(M) given by all harmonic functions. ∆ We know that H1(M)=H1(M) H1(M). Hence, H1(M) is a closed subspace of HilbertspaceH1(M),andt0herefo⊕rea∆nembeddedsubm∆anifoldofit. BytheSobolev embedding, there is a continuousinclusionH1(M) Lp(∂M)with p= 2(m−1), see ⊂ m 2 for instance [10, Th. 2.21, pg. 19]. Let (M,g) denote the subset of−H1(M) H1 2(m−1) ∆ consisting of those functions ϕ such that ∂Mϕ m−2 σg =1. R Proposition 1. (1) (M,g) is an embedded codimension 1 submanifold of H1(M). In addi- H1 ∆ tion, for ϕ (M,g) the tangent space T (M,g) of (M,g) in ϕ is 1 ϕ 1 1 ∈ H H H given by (2.1) TϕH1(M,g)=nφ∈H1∆(M)(cid:12) Z∂Mϕmm−2φσg =0o. (cid:12) (2) Suppose that the metric g has null s(cid:12)calar curvature. Then ϕ (M,g) 1 ∈ H is a critical point of E on (M,g) if and only if ϕ is a smooth function, 1 H positive and the conformal metric g˜=ϕm4−2g has null scalar curvature and ∂M has constant g˜-mean curvature.1 Assume that σ = 1, and ϕ = 12 is a critical point of E on ∂M g 0 1(M,g) (i.e., ifRg has null scalar curvature and constant mean curvature) H then: (3) The second variation d2E(ϕ ) of E at ϕ is given by 0 0 (2.2) (φ ,φ ) 2 g( φ , φ )υ H φ φ σ , 1 2 1 2 g g 1 2 g −→ nZM ∇ ∇ − Z∂M o where φ H1 (M) is such that φ σ =0, i=1,2. (4) There exii∈sts a∆(unbounded) self-aRd∂Mjoinit ogperator Jg :L2(∂M) L2(∂M) −→ such that (2.3) d2E(ϕ )(φ ,φ )=2 J (φ )φ σ , 0 1 2 g 1 ∂M 2 g Z | ∂M where φ H1 (M) satisfies φ σ =0, i=1,2. i ∈ ∆ ∂M i g R Proof. Consider the function :H1(M) R defined by Vg ∆ −→ 2(m−1) g(ϕ)= ϕ m−2 σg. V Z ∂M In order to prove (1), it is sufficient to show that is a submersion at ϕ, for all g V ϕ (M,g). Note than is smooth and its differential in ϕ (M,g) is 1 g 1 ∈ H V ∈ H given by dVg(ϕ)φ = 2(mm−21) ∂Mϕmm−2φσg, for all φ ∈ H1∆(M). In particular, for φ = ϕ we get dVg(ϕ)φ =− 2(mRm−21) > 0, therefore dVg(ϕ) is surjective. Clearly the kernel of d (ϕ) is compleme−nted in H1 (M), hence is a submersion. Now for Vg ∆ Vg ϕ (M,g), the tangent space T (M,g) of (M,g) in ϕ is the kernel of 1 ϕ 1 1 ∈ H H H d (ϕ) which establishes (2.1). g V 1Withaslightabuseofterminology,wewillsaythatametricgonamanifoldM withboundary has constant mean curvature meaningthattheboundary∂M hasconstant meancurvaturewith respecttog. 2here1denote theconstant function1onM 4 ELKIND.CARDENAS If the metric g has null scalar curvature, it is easy to see that ϕ (M,g) 1 ∈ H is a critical point of E on (M,g) if and only if it is a critical point of E under 1 the constraint C = {ϕ ∈ HH1(M)| ∂Mϕ2(mm−−21) σg = 1}. This proves (2). Assume σ =1andletϕ =1 be acrRiticalpointofE on (M,g). The formula(2.2) ∂M g 0 H1 iRs a straightforwardcomputation based on the method of Lagrange multipliers. Finally, given φ , φ H1(M) we have 1 2 ∈ g( φ , φ )υ = (φ )φ σ , 1 2 g g 1 ∂M 2 g Z ∇ ∇ Z N | M ∂M where denotes the Direchlet-Neumann map, see for instance [2]. Hence from g N (2.2) we have d2E(ϕ )(φ ,φ )=2 ( (φ ) H φ )φ σ , 0 1 2 g 1 ∂M g 1 2 g Z N | − ∂M which proves (3) with J (φ )= (φ ) H φ . g 1 g 1 ∂M g 1 N | − (cid:3) We will call (M,g) the manifold of normalized harmonic functions. 1 H 2.2. Jacobi Operator. TheunboundedlinearoperatorJ :L2(∂M) L2(∂M) g −→ defined by J (φ)= (φ) H φ g g g N − is called the Jacobi operator. From the equation (2.3) follows that the dimension of KerJ and the number (counted with multiplicity) of negative eigenvalues of J g g are the nullity and the Morse index of ϕ as a critical point of d2E in (M,g). 0 1 H The spectrum of J restricted to L (M), where g 0 L (M)= φ L2(∂M) φ=ϕ , ϕ H1 (M) , 0 { ∈ | |∂M ∈ ∆ } is given by H <ρ ( ) H ρ ( ) H ρ ( ) H , g 1 g g 2 g g 3 g g − N − ≤ N − ≤ N − ≤··· where the ρ ( ) are repeated according to multiplicity. This proves the following j g N Proposition 2. Assume that ϕ 1 is a critical point of E : (M,g) R. 0 1 ≡ H −→ Then: (1) The Morse index i(g) of ϕ is given by 0 i(g)=max j ρ ( )s 1+ m, see for instance →C − p [?, Th.2.30, pag. 50]. (cid:3) Remark 7. It is important to stress that, for all t (0, ), one has ∆ = tg ∈ ∞ 1∆, H = 1 H and ηtg = 1 ηg. This means that the spectrum of the t tg √t g √t operator H is invariant by homethety of the metric. On the other hand, g g N − σ =tm 1/2σ . When needed, we will normalize metrics to have volume 1 on the tg − g boundary, without changing the spectral theory of the operator H . g g N − 4. Bifurcation of solutions for the Yamabe problem on product manifolds Let (M ,g(1)) be a compact Riemannian manifold without boundary and null 1 scalarcurvature,andlet(M ,g(2))beacompactRiemannianmanifoldwithbound- 2 ary, null scalar curvature and constant mean curvature. Consider the product manifold, M = M M , which boundary is given by ∂M = M ∂M . Let 1 2 1 2 × × m and m be the dimensions of M and M , respectively, and assume that 1 2 1 2 m = dimM = m +m 3. For each t (0,+ ), define g = g(1) tg(2) a 1 2 t ≥ ∈ ∞ ⊕ metric on the product manifold M. It is easily see that the Riemannian manifold (M,g ) have null scalar curvature and constant mean curvature t m 1 Hˆ H = 2− H = g(2), for all t>0, gt (m 1)√t g(2) √t − where Hˆg(2) = (mm2−11)Hg(2). Let Ht denote the subspace of L2(∂M) consisting of functions ψ such t−hat ψ =ϕ ,ϕ H1 (M) and ψσ =0 and let denote |∂M ∈ ∆ ∂M gt Et thesubspaceofL2(∂M)ofthosemapsφsuchthat R φσ =0. Let : ∂M gt Jt Ht −→Et be the Jacobi operator defined by R (ψ)= (ψ) H ψ. Jt Ngt − gt Denote by 0=ρ(0) <ρ(1) <ρ(2) <... the sequence of all distinct eigenvalues of ∆ ,withgeometricmultiplicityµ(i), i 0,andbyρ(i)(t)thej ésimoeigenvalue g(1) ≥ j − of the problem ∂ϕ (4.1) ∆ ϕ+tρ(i)ϕ=0 in M , =ρϕ on ∂M , g(2) 2 ∂η2 2 with t > 0. We will denote by µ(i) the geometric multiplicity of ρ(i)(t) and we j j emphasize that ρ(i)(t)’s are not necessarily all distinct. The spectrum Σ( ) of the j Jt Jacobi operator is given by t J ρ(i)(t) Hˆ Σ( )= j − g(2) i+j >0, i,j N , t ∗ J n √t (cid:12) ∈ o (cid:12) where N = N 0 . We are interested(cid:12)in determining all values of t for which ∗ ∪{ } ρ(i)(t)= Hˆ , for some i,j N with i+j > 0. In this case, the Jacobi operator j g(2) ∈ ∗ is degenerate. t J Remark 8. BIFURCATION FOR YAMABE PROBLEM IN MANIFOLDS WITH BOUNDARY 9 (a) For i = 0 and j N , ρ(i)(t) is an eigenvalue for . Hence if Hˆ ∈ ∗ j Ng(2) g(2) is an eigevalue for then the Jacobi operator is degenerate for all Ng(2) Jt t>0. (b) Given ρ(i) >0 and j N , using the variational characterization of ρ(i)(t) ∈ ∗ j it is easily seen that ρ(i)(t )<ρ(i)(t ), j 1 j 2 for all 00, there exists an unique t >0 satisfying ρ(i)(t )=Hˆ . g(2) i 0 i g(2) Proof. Since the map (0, ) t ρ(i)(t) R is strictly increasing, in order to ∞ ∋ −→ 0 ∈ establish (1) we will show that given t > 0 and any two sequences (t(1)) , (t(2)) 0 n n n n such that t(1) t and t(2) t we have n 0 n 0 ց ր (4.2) infρ(i)(t(1))=ρ(i)(t )=supρ(i)(t(2)). 0 n 0 0 0 n n n It is clear that ρ(i)(t ) < ρ(i)(t(1)) for all n N, hence ρ(i)(t ) infρ(i)(t(1)). 0 0 0 n ∈ 0 0 ≤ n 0 n On the other hand, by the variational characterizationof ρ(i)(t ), there exists ϕ 0 0 ∈ (M ) such that ∞ 2 C ρ(i)(t )=E (ϕ):= g( ϕ, ϕ)+t ρ(i)ϕ2 υ . 0 0 t0 ZM2(cid:16) ∇ ∇ 0 (cid:17) g(2) Thus of the variational characterizationof ρ(i)(t(1)) follows that 0 n ρ(i)(t(1)) E (ϕ):= g( ϕ, ϕ)+t(1)ρ(i)ϕ2 υ , for all n N. 0 n ≤ t(n1) ZM2(cid:16) ∇ ∇ n (cid:17) g(2) ∈ Therefore infρ(i)(t(1)) infE (ϕ)=E (ϕ)=ρ(i)(t ), n 0 n ≤ n t(n1) t0 0 0 which proves the left equality in (4.2). We will show now the equality of the right handside in(4.2). Giventhe sequence(t(2)) suchthatt(2) t ,itis evidentthat n n n 0 ր supρ(i)(t(2)) ρ(i)(t ). Given n N, let ϕ (M ) be such that 0 n ≤ 0 0 ∈ n ∈C∞ 2 n ρ(i)(t(2))=E (ϕ ), ϕ2 σ =1, and ϕ σ =0, 0 n t(n2) n Z∂M2 n g(2) Z∂M2 n g(2) this ϕ exists because of the variational characterization of ρ(i)(t(2)). Note that n 0 n ϕ is a bounded subset in H1(M ). Since bounded sets are weakly compact in n 2 { } a Hilbert space, there exists a subsequence of ϕ , denoted also by ϕ , that n n converges weakly to a function ψ H1(M ). {The}n ψσ = 0.{Sin}ce the ∈ 2 ∂M2 g(2) R 10 ELKIND.CARDENAS embeddingsH1(M )intoL2(∂M )andintoL2(M )arecompact,thenthesequence 2 2 2 ϕ satisfies n { } ϕ2 σ ψ2σ and ϕ2σ ψ2σ . Z n g(2) −→Z g(2) Z n g(2) −→Z g(2) ∂M2 ∂M2 M2 M2 Hence ψ2σ = 1. From the variational characterization of ρ(i)(t ) we get ∂M2 g(2) 0 0 ρ(i)(t )R E (ψ). Since 0 0 ≤ t0 g( ψ, ψ)σ limsup g( ϕ2, ϕ2)σ , ZM2 ∇ ∇ g(2) ≤ n ZM2 ∇ n ∇ n g(2) then E (ψ) limsupE (ϕ )=limsupρ(i)(t(2))=supρ(i)(t(2)), t0 ≤ n t(n2) n n 0 n n 0 n the inequality above establishes the equality of the right hand side in (4.2), which completesthe proofof(1). Nowwewillprove(2). Tothisendnotethat0 ρ(i)(t) ≤ 0 for all t>0. Thus 0 infρ(i)(t). On the other hand, it is known that there exists ≤t>0 0 C >0 such that ϕ2υ C g( ϕ, ϕ)υ + ϕ2σ , Z g(2) ≤ (cid:26)Z ∇ ∇ g(2) Z g(2)(cid:27) M2 M2 ∂M2 for all ϕ H1(M ). Thus 2 ∈ E (ϕ) g( ϕ, ϕ)υ +tρ(i)C g( ϕ, ϕ)υ + ϕ2σ . t ≤Z ∇ ∇ g(2) (cid:26)Z ∇ ∇ g(2) Z g(2)(cid:27) M2 M2 ∂M2 Hence Et(ϕ) (1+tρiC) M2g(∇ϕ,∇ϕ)υg(2) +tρiC, ϕ2σ ≤ R ϕ2σ ∂M2 g(2) ∂M2 g(2) R R for all ϕ H1(M ) 0 . Using the variational characterizationof ρ(i)(t), it follows ∈ 2 \{ } 0 from the inequality above that 0 ρ(i)(t) tρ(i)C, for all t>0. ≤ 0 ≤ Therefore ρ(i)(t) 0 as t 0. 0 −→ −→ In order to prove (3) suppose the assertion is false. Let ρ R be such that ∈ lim ρ(i)(t)=ρ. Clearly ρ(i)(t) ρ. Let ϕ (M ) be such that t 0 0 ≤ t ∈C∞ 2 −→∞ ρ(i)(t)=E (ϕ ) and ϕ2σ =1. 0 t t Z t g(2) ∂M We claim that ϕ2υ 0 at t . In fact, otherwise M2 t g(2) −→ −→∞ R inf ϕ2υ >0, t>0ZM2 t g(2) hence there exists t>0 such that ρ0ZM2 t g(2) ≤ 0

Bifurcation results for the Yamabe problem on Riemannian manifolds with boundary

Similar Bifurcation results for the Yamabe problem on Riemannian manifolds with boundary