UNIVERSIDADE FEDERAL DO CEARÁ CENTRO DE CIÊNCIA DEPARTAMENTO DE MATEMÁTICA PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA ISRAEL DE SOUSA EVANGELISTA

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1 UNIVERSIDADE FEDERAL DO CEARÁ CENTRO DE CIÊNCIA DEPARTAENTO DE ATEÁTICA PROGRAA DE PÓS-GRADUAÇÃO E ATEÁTICA ISRAEL DE SOUSA EVANGELISTA COPACT ALOST RICCI SOLITON, CRITICAL ETRICS OF THE TOTAL SCALAR CURVATURE FUNCTIONAL AND P-FUNDAENTAL TONE ESTIATES FORTALEZA 017

2 ISRAEL DE SOUSA EVANGELISTA COPACT ALOST RICCI SOLITON, CRITICAL ETRICS OF THE TOTAL SCALAR CURVATURE FUNCTIONAL AND P-FUNDAENTAL TONE ESTIATES Thesis submitted to the Post-graduate Program of the athematical Department of Universidade Federal do Ceará in partial fulfillment of the necessary requirements for the degree of Ph.D. in athematics. Area of expertise: Differential Geometry Advisor: Prof. Dr. Abdênago Alves de Barros FORTALEZA 017

3 Dados Internacionais de Catalogação na Publicação Universidade Federal do Ceará Biblioteca Universitária Gerada automaticamente pelo módulo Catalog, mediante os dados fornecidos pelo(a) autor(a) E9c Evangelista, Israel de Sousa. Compact almost Ricci soliton, critical metrics of the total scalar curvature functional and p-fundamental tone estimates / Israel de Sousa Evangelista f. Tese (doutorado) Universidade Federal do Ceará, Centro de Ciências, Programa de Pós-Graduação em atemática, Fortaleza, 017. Orientação: Prof. Dr. Abdênago Alves de Barros. 1. Quase Soliton de Ricci.. Funcional Curvatura Escalar Total. 3. P-laplaciano. 4. P-tom Fundamental. 5. étricas de Einstein. I. Título. CDD 510

4 ISRAEL DE SOUSA EVANGELISTA COPACT ALOST RICCI SOLITON, CRITICAL ETRICS OF THE TOTAL SCALAR CURVATURE FUNCTIONAL AND P-FUNDAENTAL TONE ESTIATES Thesis submitted to the Post-graduate Program of the athematical Department of Universidade Federal do Ceará in partial fulfillment of the necessary requirements for the degree of Ph.D. in athematics. Area of expertise: Differential Geometry Aproved in: 04/07/017 EXAINATION BOARD Prof. Dr. Abdênago Alves de Barros (Advisor) Universidade Federal do Ceará (UFC) Prof. Dr. Barnabé Pessoa Lima Universidade Federal do Piauí (UFPI) Prof. Dr. Ernani de Sousa Ribeiro Júnior Universidade Federal do Ceará (UFC) Prof. Dr. Gregório Pacelli Feitosa Bessa Universidade Federal do Ceará (UFC) Prof. Dr. José Nazareno Vieira Gomes Universidade Federal do Amazonas (UFA)

5 To my parents, brothers and wife.

6 AGRADECIENTOS Agradeço primeiramente a Deus, pelo dom da vida e pela família maravilhosa que tenho, família que meu deu todo suporte para realizar meu sonho de fazer um doutorado em atemática. Aos meus pais oisés Araújo Evangelista e aria Augusta de Sousa Evangelista, que sempre me incentivaram a estudar desde de muito cedo e que me deram todo suporte necessário, com todos seus esforços, para que eu pudesse me dedicar exclusivamente aos estudos. Por seu empenho, amor e dedicação, meus mais profundos agradecimentos e respeito. Aos meus irmãos Jonathan, Nathália, oisés e Nathannaelly pelo companheirismo e apoio que foram fundamentais em todas as etapas da minha vida. Que sempre foram entusiasmados com minhas conquistas e sempre me incentivaram. À minha amada esposa Rita de Cássia pela sua compreensão durante todos os anos do doutorado, por ter me dado suporte durante tanto tempo e mesmo a distância sempre se mantendo firme ao meu lado. Vivemos juntos as fases do namoro, noivado e agora do casamento nesse período de cinco anos do doutorado. eus sinceros agradecimentos por seu amor, carinho e atenção durante esses anos. Aos amigos com quem dividi apartamento durante todos esses anos e que foram minha família em Fortaleza Alex Sandro, Franciane, aria Vieira, Antonio Aguiar e Thomaz, pela amizade, paciência e companheirismo durante todos esses anos. Aos amigos Klenio e Ronan com quem dividi apartamento, mas não tive um convívio muito próximo, mas que também fizeram parte dessa jornada. Aos meus amigos de fé e irmãos camaradas Edvalter Sena Filho e Valdir Jr. pela amizade, companheirismo que sempre me deram suporte nessa luta árdua do doutorado, os amigos para todas as horas com quem sempre pude contar. Aos meus amigos do Amazonas Cleiton Cunha e Cris, pessoas muito especiais com quem sempre pude contar no doutorado. Aos meus amigos Kelson, Fabrício, Emanuel, Adam, Halyson, Jocel, Tiago, Diego Eloi, Diego Sousa, arcos Ranieri, Wanderley, Davi Lustosa e Davi Ribeiro. Aos meus amigos e colegas do Curso de atemática em Parnaíba Roberto, Cleyton, Carla e Haroldo pela amizade e apoio. Aos amigos alunos do doutorado que não citei acima, que fizeram parte dessa conquista.

7 Ao amigo Alexandro arinho que foi meu orientador de iniciação científica e mestrado e me incentivou muito a fazer doutorado em Fortaleza. Ao amigo arcondes Clark pelo incentivo a fazer doutorado e pelos valiosos conselhos. Ao professor Abdênago Barros pelos ensinamentos e paciência durante esse processo de construção da tese. Aos professores Ernani Ribeiro Jr., José Nazareno Vieira Gomes, Barnabé Pessoa Lima e Gregório Pacelli Bessa, por terem aceito o convite para compor a banca examinadora da tese, e aos professores Rafael Jorge Pontes Diógenes e Fernanda Ester Camilo Camargo por terem aceito convite para serem suplentes da banca. Aos professores Ernani Ribeiro Jr., Diego oreira, Pacelli Bessa, Luciano ari e Alexandre Fernandes, com quem sempre pude contar para tirar dúvidas durante a elaboração da tese e sempre muito atenciosamente de pronto me atenderam. Ao professor Keomkyo Seo, com quem fiz um artigo que foi crucial nesse processo de crescimento profissional. E também ao professor Pacelli pelas sugestões que muito contribuíram para melhorar o trabalho feito com o professor Seo. Ao professor João Xavier da Cruz Neto que foi o primeiro professor a incentivar-me a estudar matemática visando uma pós-graduação. Às secretárias da pós-graduação Andrea e Jessyca pelo atendimento e atenção nos problemas burocráticos da universidade. Ao colegiado do curso de atemática do Campus inistro Reis Velloso da Universidade Federal do Piauí, que concedeu minha liberação das atividades docentes por um ano para conclusão do doutorado. Ao CNPq e Capes pelas bolsas concedidas, respectivamente, na segunda e primeira metade do doutorado. Auxílio financeiro imprescindível para minha permanência em Fortaleza.

8 To humans belong the plans of the heart, but from the LORD comes the proper answer of the tongue. (Proverbs 16:1)

9 RESUO A presente tese está dividida em três partes diferentes. O objetivo da primeira parte é provar que um quase soliton de Ricci compacto com tensor de Cotton nulo é isométrico a uma esfera canônica desde que uma das seguintes condições associadas ao tensor de Schouten seja válida: a segunda função simétrica é constante e positiva; duas funções simétricas consecutivas são múltiplas, não nulas, ou alguma função simétrica é constante e o tensor de Schouten é positivo. O objetivo da segunda parte é estudar as métricas críticas do funcional curvatura escalar total em variedades compactas com curvatura escalar constante e volume unitário, por simplicidade, métricas CPE. Foi conjecturado que toda métrica CPE deve ser Einstein. Prova-se que a conjectura é verdadeira para as métricas CPE sob uma condição integral adequada e também se prova que é suficiente que a métrica seja conforme a uma métrica Einstein. Na terceira parte, estima-se o p-tom fundamental de subvariedades em uma variedade tipo Cartan- Hadamard. Primeiramente, obtém-se estimativas por baixo para o p-tom fundamental de bolas geodésicas e em subvariedades com curvatura média limitada. Além disso, obtém-se estimativas do p-tom fundamental de subvariedades mínimas com certas condições sobre a norma da segunda forma fundamental. Por fim, estudam-se folheações de classe C transversalmente orientadas de codimensão 1 de subconjuntos abertos Ω de variedades riemannianas e obtêm-se estimativas por baixo para o ínfimo da curvatura média das folhas em termos do p-tom fundamental de Ω. Palavras-chave: Quase soliton de Ricci. Funcional curvatura escalar total total. P-laplaciano. P-tom fundamental. étricas de Einstein. Curvatura escalar. Tensor de cotton.

10 ABSTRACT The present thesis is divided in three different parts. The aim of the first part is to prove that a compact almost Ricci soliton with null Cotton tensor is isometric to a standard sphere provided one of the following conditions associated to the Schouten tensor holds: the second symmetric function is constant and positive; two consecutive symmetric functions are non null multiple or some symmetric function is constant and the quoted tensor is positive. The aim of the second part is to study the critical metrics of the total scalar curvature funcional on compact manifolds with constant scalar curvature and unit volume, for simplicity, CPE metrics. It has been conjectured that every CPE metric must be Einstein. We prove that the Conjecture is true for CPE metrics under a suitable integral condition and we also prove that it suffices the metric to be conformal to an Einstein metric. In the third part we estimate the p-fundamental tone of submanifolds in a Cartan-Hadamard manifold. First we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. oreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension one C -foliations of open subsets Ω of Riemannian manifolds and obtain lower bounds estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of Ω. Keywords: Almost Ricci soliton. Total scalar curvature functional. P-Laplacian. P-fundamental tone. Einstein metrics. Scalar curvature. Cotton tensor.

11 CONTENTS 1 INTRODUCTION PRELIINARIES Basic notations Newton transformations Algebraic lemmas Algebraic tools CPE lemmas Conformal geometry On the p-fundamental tone Isometric immersions Newton transformations for the shape operator Volume growth of submanifolds Foliations ALOST RICCI SOLITONS Introduction Auxiliaries lemmas Integral formulae Rigidity result CPE CONJECTURE Introduction Integral Formulae Integral condition for the CPE conjecture The conformal case ON THE P-FUNDAENTAL TONE ESTIATES introduction Lower bounds for the p-fundamental tone on geodesic balls Submanifolds with locally bounded mean curvature The p-fundamental tone of minimal submanifolds with controlled extrinsic curvature Bernstein-Heinz-Chern-Flanders type inequalities

12 5.6 Haymann-akai-Osserman type inequality CONCLUSION REFERENCES

13 1 1 INTRODUCTION This thesis deal with three different problems. First we study compact almost Ricci solitons, we investigate which geometric implication has the assumption of the second symmetric function S (A) associated to the Schouten tensor to be constant and positive on a compact almost Ricci soliton. ore precisely, we have the following result. Theorem Let ( n,g,x,λ), n 3, be a non-trivial compact oriented almost Ricci soliton such that the Cotton tensor is identically zero. Then, n is isometric to a standard sphere S n provided that one of the next conditions is satisfied: 1. S (A) is constant and positive.. S k (A) is nowhere zero on and S k+1 (A) = cs k (A), where c R\{0}, for some k = 1,,n Ric R n g, with R > 0, and S k(a) h 0 for some k n S k (A) is constant for some k =,,n 1, and A > 0. We highlight that the symmetric functions associated to the Schouten tensor were used by Hu, Li and Simon (HU et al., 008) to study locally conformally flat manifolds. By assuming that the Weyl tensor vanishes, the conclusion of item 4 in the above theorem follows directly from Theorem 1 obtained in (HU et al., 008). In this direction, we point out that item 1 and item 4 of the above theorem improve Theorem 1 in (HU et al., 008) for compact almost Ricci solitons under the hypothesis of Cotton tensor identically zero. Second, we study the CPE metric which is a 3-tuple ( n, g, f ) where ( n,g), n 3 is a n-dimensional compact oriented Riemannian manifold with constant Ricci scalar curvature and f is a smooth potential function that satisfies the equation Ric R n g = Hess f f ( Ric R n 1 g), There is a famous conjecture proposed in (BESSE, 007) which says that a CPE metric is always Einstein. Recently, by considering the function h = f + R n(n 1) f, Leandro (NETO, 015) was able to show that CPE conjecture is true under the condition that h is a constant. Whereas, Benjamin Filho (FILHO, 015) improved this result requiring that h is constant along of the flow of f. Taking into account that height functions are eigenfunctions of the Laplacian on a sphere S n with standard metric g, we may conclude that (S n, g, h v ) is a CPE metric, where h v is

14 13 a height function for an arbitrary fixed vector field v S n R n+1. Indeed, the existence of a non constant solution is only known in the standard sphere for some height function. Now we define ρ m ( f, f ), which for simplicity we denote by ρ m, according to ρ m = (m 1) f m f 4 (n + )R d f m f d, (1.0.1) n(n 1) where m N. It is easy to check that on S n we have ρ m = 0 for every m = {1,,3, }. We also recall that Benjamim Filho proved in (FILHO, 015) that the CPE conjecture is true provided ρ 1 0 and ρ = 0. In this spirit, inspired by the historical development on the study of CPE conjecture, we shall prove that the assumptions considered in (FILHO, 015) as well as (NETO, 015) can be replaced by a weaker integral condition. We point out that this integral condition is satisfied in the standard sphere, hence it is a natural hypothesis to consider. In this sense, we have established the following result. Theorem 1.0. The CPE conjecture is true provided that the function (1.0.1) satisfies ρ k + ρ m 0, for m > k, where m is even and k is odd. We now deal another approach. In order to do so, we say that a conformal mapping between two Riemannian manifolds (,g) and (N,h) is a smooth mapping F : (,g) (N,h) which satisfies the property F h = α g for a smooth positive function α : R +. Here, we ask what happens if a CPE metric is conformal to an Einstein manifold? The answer is the following result: Theorem Let ( n,g, f ) be a CPE metric. If g is conformal to an Einstein metric g, then is isometric to the standard sphere. It is important to remark that, if a compact 4-dimension manifold is locally conformal to an Einstein manifold then its Bach tensor vanishes, since the Bach tensor is a conformal invariant in dimension four. On the other hand, Qing and Yuan in (QING; YUAN, 013) proved that the CPE conjecture is true provided the metric is Bach flat. Therefore, the previous theorem is an extension of this result for any dimension. Remark We remark that the previous theorem is an extension for any dimension of a particular result which is already true in dimension four. In fact, it is well known that if a

15 14 compact 4-dimension manifold is locally conformal to an Einstein manifold then its Bach tensor vanishes. In particular, any 4-dimensional CPE metric conformal to an Eienstein metric is Bach flat. Thus, by Theorem 3.10 in (QING; YUAN, 013) the CPE conjecture is true in this case. In the third part we study with the p-fundamental tone. We denote by λ p() the p-fundamental tone of, which is defined by { λp() = inf f } p d f p d : f W 1,p 0 (), f 0. In this part, inspired in (BESSA; ONTENEGRO, 003), we study lower bounds estimates for the p-fundamental tone on geodesic ball and as a consequence we get lower bounds for the p-fundamental tone of submanifolds with locally bounded mean curvature. We obtain the following results. Theorem Let be an n-dimensional complete Riemannian manifold. Denote by B (q,r) a geodesic ball with radius r < inj(q). Let κ(q,r) = sup{k (x) : x B (q,r)}, where K (x) denotes the sectional curvature of at x. Then 1 p p max{ np r p,[(n 1)k coth(kr)] p } if κ(q,r) = k, λp(b (q,r)) n p p p r p if κ(q,r) = 0, ((n 1)k cot(kr)+1) p p p r p if κ(q,r) = k and r < k π, where k is a positive constant. (1.0.) Theorem Let ϕ : n N m be an isometric immersion with locally bounded mean curvature and let Ω be any connected component of ϕ 1 (B N (q,r)) for q N\ϕ() and r > 0. Denote by κ(q,r) the supremum of the sectional curvature of in B (q,r) as in Theorem 5... Then, for a constant k > 0, we have the following: 1. If κ(q,inj(q)) = k < + and cot 1 r < min inj(q), ( h(q,inj(q)) (n 1)k k ), then λ p(ω) [(n 1)k cot(kr) h(q,r)]p p p.. If κ(q,r) > 0 for all r > 0, lim r κ(q,r) =, inj(q) =, and ( ) cot 1 h(q,s) (n 1) κ(q,s) r < r 0 := max, s>0 κ(q,s)

16 15 then λ p(ω) [(n 1) κ(q,r)cot( κ(q,r)r) h(q,r)] p p p. n n 3. If κ(q,inj(q)) = 0 and r < min{inj(q), h(q,inj(q)) }, where h(q,inj(q)) = + when h(q,inj(q)) = 0, then λ p(ω) [ n r h(q,r)]p p p. 4. If κ(q,inj(q)) = k, h(q,inj(q)) < (n 1)k, and r < inj(q), then λ p(ω) [(n 1)k h(q,r)]p p p. 5. If κ(q,inj(q)) = k, h(q,inj(q)) (n 1)k, and coth 1 r < min inj(q), ( h(q,inj(q)) (n 1)k k ), then λ p(ω) [(n 1)k coth(kr) h(q,r)]p p p. Proceeding we study upper bound for the p-fundamental tone which combined with the above lower bound give us the following result. Theorem Let n be an n-dimensional complete properly immersed minimal submanifold in a Cartan-Hadamard manifold N of sectional curvature K N bounded from above by K N κ 0. Suppose that lim Q(R) <. R Then λ p() = (n 1)p κ p p p. As a consequence of the previous theorem, we get the following interesting intrinsic result in the direction of the generalized ckean s Theorem obtained by Lima, ontenegro and Santos in (LIA et al., 010).

17 16 Theorem Let n be a complete simply connected manifold with sectional curvature bounded from above K κ < 0. Furthermore, suppose that there exists a point q such that sup R>0 Vol(B q R ) Vol(B κ < +, (1.0.3) R ) where B q R is the geodesic ball in centered at q of radius R, and Bκ R is the geodesic ball in H n (κ) of the same radius R. Then λ p() = (n 1)p κ p p p. (1.0.4) Finally, we study transversely oriented codimension one C -foliations of open subsets Ω of Riemannian manifolds and obtain lower bounds estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental λ p(ω) tone of Ω, which are called Bernstein-Heinz-Chern-Flanders type inequalities. Following Barbosa, Bessa, and ontenegro s idea (BARBOSA et al., 008), we have the following result. Theorem Let F be a transversely oriented codimension one C -foliation of a connected open set Ω of (n + 1)-dimensional Riemannian manifold. Then p p λ p(ω) n inf F F inf x F HF (x), where H F denotes the mean curvature function of the leaf F. As an interesting consequence the above theorem we obtain a Haymann-akai- Osserman type inequality, which is given in the following result. Theorem Let γ : (α,β) R n be a simple smooth curve and T γ (ρ(t)) be an embedded tubular neighborhood of γ with variable radius ρ(t) and a smooth boundary T γ (ρ(t)). Let ρ 0 = sup t ρ(t) > 0 be its inradius. Then λ p(t γ (ρ(t))) (n 1)p p p ρ p. (1.0.5) 0

18 17 PRELIINARIES In this chapter we introduce the basic notations necessary to a properly comprehension of the results presents in this work. We recommend for the reader as a complementary reading (BESSE, 007), (PETERSEN, 006) and (SCHOEN; YAU, 1994)..1 Basic notations Let ( n,g) be a smooth, n-dimensional Riemannian manifold with metric g. We denote by Rm(X,Y )Z the Riemann curvature operator defined as follows Rm(X,Y )Z = X Y Z Y X Z [X,Y ] Z, and we also denote by Ric(X,Y ) = tr(z Rm(Z,X)Y ) the Ricci tensor, and R = tr(ric) the scalar curvature. We have the well known formula (divrm)(x,y,z) = X Ric(Y,Z) Y Ric(X,Z), (.1.1) where div means the divergence of the tensor. Let A = Ric which is a (0,) symmetric tensor. The Weyl tensor is given by R (n 1) g denote the Schouten tensor, Rm = W + 1 (A g), (.1.) n where means the Kulkarni-Nomizu product defined by the following formula (α β) i jkl = α il β jk + α jk β il α ik β jl α jl β ik, (.1.3) and α,β are (0,) tensors. Finally, we define the Cotton tensor as follows C i jk = i A jk j A ik. (.1.4) It is well known that l W i jkl = n 3 n C i jk. (.1.5) From identities (.1.4) and (.1.5) we see that for n 4 if the Weyl tensor vanishes, then the Cotton tensor also vanishes. It is not difficult to check that when n = 3 the Weyl tensor always vanishes, but the Cotton tensor does not vanish in general. Next, we say that a manifold has harmonic Weyl tensor provided that divw = 0. Since (divw) i jk = l W i jkl, by (.1.5) we also deduce that for n 4, the Cotton tensor is identically zero, if and only if, the Weyl tensor is harmonic.

19 18. Newton transformations to T defined as follows Let T be a symmetric (0,) tensor and σ k (T ) be the symmetric functions associated det(i + st ) = n σ k (T )s k, k=0 where σ 0 = 1 and s R. Since T is symmetric, then ( n k) Sk (T ) = σ k (T ) coincides with the k-th elementary symmetric polynomial of the eigenvalues λ i (T ) of T, i.e., σ k (T ) = σ(λ 1 (T ),...,λ n (T )) = i 1 < <i k λ i1 (T ) λ ik (T ), 1 k n. (..1) For simplicity we do not distinguish between the (0,) tensor T and the operator T : X() X(), that is a (1,1) tensor, such that T (X,Y ) = T X,Y. We introduce the Newton transformations P k (T ) : X() X(), arising from the operator T, by the following inductive law P 0 (T ) = I, P k (T ) = ( ) n S k (T )I T P k 1 (T ), 1 k n (..) k or, equivalently, ( ) ( ) ( ) n n n P k (T ) = S k (T )I S k 1 (T )T + + ( 1) k 1 S 1 (T )T k 1 + ( 1) k T k. k k 1 1 Using the Cayley-Hamilton Theorem we get P n (T ) = 0. Note that P k (T ) is a self-adjoint operator that commutes with T for any k. Furthermore, if {e 1,...,e n } is an orthonormal frame on T p diagonalizing T, then (P k (T )) p (e i ) = µ i,k (T ) p e i, (..3) where µ i,k (T ) = λ i1 (T ) λ ik (T ) = σ k+1 (λ 1 (T ),...,λ n (T )). i 1 < <i k,i j i x i oreover, we have the well known formulae tr(t P k (T )) = c k S k+1 (T ) tr(p k (T )) = c k S k (T ), (..4) where ( ) n c k = (n k) k ( ) n = (k + 1). k + 1

20 19 The divergence of P k (T ) is defined as follows divp k (T ) = tr( P k (T )) = n i=1 ei P k (T )(e i ), where {e 1,...,e n } is a local orthonormal frame on. Our aim is to compute the divergence of P k (T ). The following definition is important in the sequel. Define the tensor D by D i jk = i T jk j T ik. (..5) Note that when T is the Ricci tensor, then by equation (.1.1) D = divrm, and when T is the Schouten tensor, then D is just the Cotton tensor..3 Algebraic lemmas In this section we present some results that are essential for our purpose. We prove some useful algebraic results..3.1 Algebraic tools First of all we show a lemma which concerns to suitable polynomials. Letting I j (x) = x j, let us consider the polynomials p m,q m,r m,s m : R R given by m 1 k 1 k(m + 1 k) 1. p m = ( 1) I m 1 k.. q m = k=1 m 1 j=1 3. r m = mi m + ( 1) j+1 j( j + 1)I j 1. m+1 i= ( 1) i I m+1 i. 4. s m = m(i m + I m 1 ) r m. We also set 1. τ m = r m + m(m+1) (I + 1).. υ m,k = k(k + 1)p m + m(m + 1)p k. 3. µ m = r m + m+1 m 1 s m. 4. λ m,k = k(k + 1)r m + m(m + 1)r k. Lemma.3.1 For m > k, where m is even and k is odd, the above polynomials satisfy: 1. τ m = (I + 1) p m,. µ m = 1 m 1 (I + 1) q m, 3. λ m,k = (I + 1) υ m,k.

21 0 Proof: Since r m ( 1) = 0 we can decompose τ m (x) = (x + 1) τ m (x), where τ m (x) = mx m 1 (m 1)x m + (m )x m 3 + x + = m 1 i=1 ( 1) i 1 (m + 1 i)x m i + (m 1)(m + ). (m 1)(m + ) In the same way, τ m ( 1) = 0, enables us to write ( τ m (x) = (x + 1) mx m (m 1)x m 3 + (3m 3)x m 4 (4m 6)x m 5 + (m )(m + 3) (m 1)(m + ) ) x + = m 1 k 1 k(m + 1 k) (x + 1) ( 1) x m 1 k = (x + 1)p m (x), k=1 which gives τ m (x) = (x + 1) p m (x) that corresponds to the first item. Proceeding analogously we write µ m (x) = (x + 1) µ m (x), where µ m (x) = mx m 1 m 1 + k=1 k+1 (m k) ( 1) m 1 xm 1 k. Arguing as in the first item we obtain ( µ m (x) = (x + 1) mx m (m )x m 3 (m )(m 3) + m 1 (m 3)(m 4) x m m 1 m 1 x + ) m 1 = (x + 1) ( mx m m k (m k)(m k 1) + ( 1) m 1 = = = k=1 1 m 1 (x + 1)( m(m 1)x m + m k=1 m 1 m 1 (x + 1) ( 1) k (m k)(m k 1)x m k k=0 x m 4 x m k) ( 1) k (m k)(m k 1)x m k) m 1 1 m 1 (x + 1) ( 1) k+1 k(k + 1)x k 1 = 1 m 1 (x + 1)q m(x), k=1 and this completes the proof of the second item. Following the same argument used in the first item we write λ m,k (x) = (x + 1) υ m,k (x), where υ m,k (x) = k(k + 1) m i=1 and υ m,k ( 1) = 0 enable us to write ( 1) i+1 (m + 1 i)x m i + m(m + 1) k j=1 ( 1) j+1 (k + 1 j)x k j, υ m,k (x) = (x + 1)(k(k + 1)p m (x) + m(m + 1)p k (x)) = (x + 1)υ m,k (x), which proves the last item.

22 1 Lemma.3. Let p m (x) be the polynomial defined above. If m is even, then p m (x) > 0 for all x R. Proof: First note that p (x) =. Hence it suffices to show that p m+ > p m. Let us set J m+ = p m+ p m. Then we claim Indeed, we can write J m+ (x) = m+1 i=1 J m+ (x) = (m + )x m + (m + 3) i 1 i(m + 5 i) ( 1) = (m + )x m (m + 3)x m 1 + m+1 i=3 i 1 i(m + 5 i) ( 1) = (m + )x m (m + 3)x m 1 + m 1 i=1 i 1 (i + )(m + 3 i) ( 1) = (m + )x m + (m + 3) m i=1 x m+1 i m 1 i=1 x m+1 i m 1 ( 1) i x m i, i=1 m i=1 x m 1 i m 1 ( 1) i x m i. (.3.1) i 1 i(m + 1 i) ( 1) i 1 i(m + 1 i) ( 1) i=1 x m 1 i x m 1 i i 1 i(m + 1 i) ( 1) x m 1 i which gives the claim. Since m is even, by (.3.1) we deduce that J m+ ( x) > 0 if x 0, with J m+ (0) = m + 3. Now, we define u m+ (x) = (x + 1)J m+ (x) and using once more (.3.1) we infer (x + 1)J m+ (x) = (m + )x m (x + 1) + (m + 3) m i=1 = (m + )x m+1 (m + 1)x m + m + 3. Next noticing that u m+ achieves its minimum at x = (( 1) i x m+1 i + ( 1) i x m i) m m+ we conclude that J m+(x) > 0 for any x R, which gives that p m (x) for x R provided that m is even, which finishes the proof. Lemma.3.3 Let q m (x) be the polynomial defined above. If m is even, then q m (x) > 0 for all x R. Proof: We notice that q m ( x) > 0 for every x 0, since m is even. From now on we suppose that x > 0. Under this choice we can write q m (x) = x m L m (x 1 ),

23 where L m (x) = m 1 k=1 ( 1) k+1 k(k + 1)x m k 1. Hence, it suffices to show that L m (x) is strictly positive for every x > 0. Proceeding, it is easy to verify that L (x) =, then it is enough to prove that L m+ (x) > L m (x) for every x R and m even. Letting T m+ = L m+ L m we have Whence we get T m+ (x) = m+1 k=1 ( 1) k+1 k(k + 1)x m k+1 = x m 6x m 1 + m 1 k=1 m 1 k=1 ( 1) k+1 (k + 3)x m k 1. ( 1) k+1 k(k + 1)x m k 1 1 T m+(x)(x + 1) = x m+1 ( x m x m x x + 1 ) + m + 3. (.3.) Since m is even we have x m x m x x + 1 = 1 x+1( x m ). Hence, we deduce 1 T m+(x)(x + 1) = x m+ x m+1 + (m + 3)x + m + 1. (.3.3) Since the right hand side of (.3.3) is strictly positive for x > 0 we have the same conclusion for T m+ and we complete the proof of the lemma. Lemma.3.4 Let υ m,k (x) be the polynomial defined above. If m is even, k is odd and m > k, then, υ m,k (x) > 0 for all x R. Proof: Note that it suffices to prove that υ m+,k (x) υ m,k (x) > 0, since υ k+1,k = (k + 1)q k+1 and in the proof of Lemma.3.3 we showed that q k+1 > 0. After a straightforward computation we obtain which implies that υ m,k (x) = k(k + 1) + k j=0 ( 1) m j (m 1 j)(m + + j) ( 1) x j j=k 1 j ( j + 1)( j + )(m k)(m + k + 1) x j, υ m+,k (x) υ m,k (x) = k(k + 1)(m + )x m + k(k + 1)(m + 3) k + (m + 3) j=0 ( 1) j ( j + 1)( j + )x j. m 1 ( 1) j x j j=k 1 By the above expression υ m+,k (x) υ m,k (x) > 0 for every x 0. Now it remains to prove that υ m+,k (x) υ m,k (x) > 0 for every x > 0. Defining Q m,k = (x + 1)(υ m+,k υ m,k ) a straightforward computation gives Q m,k (x) = k(k + 1)(m + )x m+1 k(k + 1)(m + 1)x m k 1 + (m + 3) j=0 ( 1) j ( j + 1)x j.

24 3 Thus, for every x 1 we have υ m+,k (x) υ m,k (x) > 0. Hence, we need to treat only the case 0 < x < 1. If we define η m,k (x) = x m υ m,k (x 1 ), we get m k 1 η m,k (x) = k(k + 1) i=0 + (m k)(m + k + 1) i (i + 1)(m i) ( 1) m i=m k x i i (m i 1)(m i) ( 1) x i. Defining V m,k (x) = η m+,k (x) η m,k (x), we obtain after a direct computation that m k 1 V m,k (x) = k(k + 1) ( 1) i (i + 1)x i + (m + 3) i=0 m i=m k + (m k + )(m + k + 3) ( 1) i (m i 1)(m i)x i m i=m k ( 1) i (m i + 1)x i + ((k + 1)(m + )(m + 3) + k(k + 1) )x m k. We aim to prove that η m+,k (x) η m,k (x) > 0 provided that x > 1. To do so, we consider P m,k (x) = (x + 1) 3 (η m+,k (x) η m,k (x)) = (x + 1) 3 V m,k (x). Whence we get P m,k (x) = k(k + 1)(x + 1) 3 m k 1 ( 1) i (i + 1)x i i=0 + (m + 3)(x + 1) 3 m ( 1) i (m i 1)(m i)x i i=m k + (m k + )(m + k + 3)(x + 1) 3 m ( 1) i (m i + 1)x i i=m k + ((k + 1)(m + )(m + 3) + k(k + 1) )x m k (x + 1) 3. = Z 1 + Z + Z 3 + Z 4. Now, calculating separately Z 1, Z, andz 3 we get Z 1 = (x + 1) 3 m k 1 ( 1) i (i + 1)x i i=0 = (m k)x m k+ + (m k + 1)x m k+1 + (m k + 1)x m k + x + 1, Z = (x + 1) 3 m ( 1) i (m i 1)(m i)x i i=m k = x m+1 k(k + 1)x m k+ (k + 1)(k 1)x m k+1 k(k 1)x m k

25 4 and Z 3 = (x + 1) 3 m ( 1) i (m i + 1)x i i=m k = x m+3 x m+1 (k + 3)x m k+ 4(k + 1)x m k+1 (k + 1)x m k. So, it is not difficult to check that P m,k (x) = k(k + 1)(x + 1) + (m + 3)x m+1 + (m k + )(m + k + 3)(x m+3 x m+1 ) + (k + 1)(m + 5m + k + k + 6)x m k+3 + k(m + 5m + 3k + 6k + 9)x m k+ + (k + 1)( m m + 3k + 3k)x m k+1 + k( m m + k + k + 1)x m k. Then, by the above expression P m,k (x) > 0 for every x > 1, which implies η m+,k (x) η m,k (x) for all x > 1. Now we note that υ k+1,k (x) = (k + 1)q k+1 (x) > 0 for every x. Since η k+1,k (x) = x k 1 υ k+1,k (x 1 ) we obtain η k+1,k (x) > 0 for all x. Therefore, η m,k (x) > 0 for every x > 1. Finally, since υ m,k (x) = x m η m,k (x 1 ) > 0 for all x 1 > 1, we have υ m,k (x) > 0 for 0 < x < 1, which finishes the proof..4 CPE lemmas Let S,T : H H be operators defined over a finite dimensional Hilbert space H. The Hilbert-Schmidt inner product is defined by S,T = tr ( ST ), where tr and denote, respectively, the trace and the adjoint operation. oreover, if I denotes the identity operator on H of dimension n the traceless of an operator T is given by T = T trt n I. Using this notation we have the following lemmas. Lemma.4.1 (FILHO, 015) Let ( n, g, f ) be a CPE metric. Then we have: 1. ( f + 1) Ric = f. In particular, ( n, g, f ) is Einstein if and only if f is a conformal vector field.. f m Ric, f d = m f m 1 Ric( f, f )d. 3. ( f + 1 ) f d = f ( f, f )d. 4. f m Ric, f = m i=1 ( 1)i+1 f m i f.

26 5 Lemma.4. (FILHO, 015) Let ( n,g) be a Riemannian manifold and f,ϕ smooth functions on such that f + n 1 R f = 0 and let h = f + R n(n 1) f. Then we have: 1 1. h = f + Ric( f, f ),. 1 ϕ, f = f ( ϕ, f )..5 Conformal geometry A conformal mapping between two Riemannian manifolds (,g) and (N,h) is a smooth mapping F : (,g) (N,h) which satisfies the property F h = α g for a smooth positive function α : R +. We look to conformal variations of, that is, those variations of the form φ g, for a smooth positive function φ : R. In the approach of conformal geometry we have the following lemma, which is a well known result of the conformal geometry theory whose proof is standard. Lemma.5.1 Let ( n,g) be a Riemannian manifold and g = φ g a metric conformal to g. Then, the following geometric data associated to the metric g are given in terms of the metric g by the following expressions: Proof: (i) X Y = X Y (Xφ)Y (Y φ)x + X,Y φ (ii) Rm(X,Y )Z = Rm(X,Y )Z X,Z φ(y ) + Y,Z φ(x) + ( φ(y,z) + (Y φ)(zφ) Y,Z φ )X ( φ(x,z) + (Xφ)(Zφ) X,Z φ )Y + (Xφ Y,Z Y φ X,Z ) φ (iii) Ric = Ric + φ 1( (n ) φ (n 1) φ ) g + φg, φ (iv) R = φ ( R + φ 1( (n 1) φ (n 1)n φ )), φ (v) Ric = Ric + (n )φ 1 φ The proof of (i) and (ii) are straightforward, see (KÜHNEL, 1988). To obtain (iii) we take trace in (ii) to get Taking the trace again we obtain R jk = R jk + φ 1 ((n )φ jk (n 1) φ iφ i φ g jk + φg jk ). R = φ (R + φ 1 ((n 1) φ (n 1)n φ iφ i φ )).

27 6 Finally we deduce R jk R n g jk = R jk + φ 1 ((n )φ jk (n 1) φ iφ i φ g jk + φg jk ) φ n (R + φ 1 ((n 1) φ (n 1)n φ iφ i = R jk + (n )φ 1 (φ jk 1 n φg jk). φ ))φ g jk.6 On the p-fundamental tone Let (,g) be a Riemannian manifold. For any p (1, ) and any function u W 1,p loc (), the p-laplacian is the differential operator defined by p u = div( u p u). The p-laplacian appears naturally on the variational problems associated to the energy functional E p : W 1,p 0 () R E p (u) = u p d. In particular, if p =, the p-laplacian p is the usual Laplace operator. We denote by λ p() the p-fundamental tone of, which is defined by { λp() = inf f } p d f p d : f W 1,p 0 (), f 0. (.6.1) Let be an n-dimensional complete noncompact manifold. Let {Ω i } be an exhaustion of by compact domains, i.e., {Ω i } are compact domains such that i=1ω i = and Ω i Ω i+1 for all i N. Consider the first eigenvalue λ 1,p (Ω i ) of the following Dirichlet boundary value problem: p u = λ u p u in Ω i, u = 0 on Ω i. In (VÉRON, 1991), Veron showed the existence of the above eigenvalue problem and the variational characterization as in (.6.1). Lindqvist (LINDQVIST, 1990) proved that λ 1,p (Ω i ) is simple for each compact domain Ω i, i N (see also (BELLONI; KAWOHL, 00)). By definition, we see that λ p(ω i ) = λ 1,p (Ω i ) for each compact domain Ω i, i N. Using the

28 7 domain monotonicity of λ 1,p (Ω i ), we deduce that λ 1,p (Ω i ) is non-increasing in i N and has a limit which is independent of the choice of the exhaustion of. Therefore λ p() = lim i λ p(ω i ). (.6.).7 Isometric immersions Consider an isometric immersion ϕ : N, where n and N m are complete Riemannian manifolds. Denote by, and the Riemannian connection, the Hessian and the Laplacian on N, respectively, while by, and the Riemannian connection, the Hessian and the Laplacian on, respectively. When is an orientable hypersurface, consider η a unit normal vector field along. Thereby, the Gauss and Weingarten formulae for the hypersurfaces in N are given, respectively, by X Y = X Y αx,y η (.7.1) and α(x) = X η, (.7.) for all tangent vector fields X,Y X(). Here α : X() X() defines the shape operator of with respect to η. For simplicty we identify the operator α with the symetric (0,) tensor α associated to α defined by α(x,y ) = α(x),y. We also define the mean curvature H by H = trα. Thereby, using this notation we have the following well known theorem (for a proof see for instance (ANFREDO, 199)). Theorem.7.1 (Gauss) Let p and X,Y orthonormal vectors on the tangent space T p. Then, K(X,Y ) K(X,Y ) = α(x,x),α(y,y ) α(x,y ). (.7.3) Definition.7.1 Let ϕ : N be a isometric immersion. We say that ϕ is a proper immersion if ϕ is a proper map, i.e., for every compact set K N, the inverse image ϕ 1 (K) is compact.

29 8.7.1 Newton transformations for the shape operator Let α be the shape operator, since α is a symmetric (0,) tensor we may consider the symmetric functions associated to α det(i + sα) = m σ k (α)s k, k=0 where σ 0 = 1. Now we define H k called the k-th mean curvature by ( ) m 1 H k = σ k (α), k which are precisely the average of the k-th elementary symmetric polynomial of the eigenvalues λ i (λ). oreover, the Newton transformations associated to the shape operator are given by P k (T ) = ( ) ( ) ( ) m m m H k I H k 1 α + + ( 1) k 1 H 1 α k 1 + ( 1) k α k. k k 1 1 Note that our choice of the mean curvature H of N m satisfies H = mh Volume growth of submanifolds Denote by n (κ) the n-dimensional simply connected real space form of constant sectional curvature κ 0. Recall that the volume of the geodesic sphere SR κ and the geodesic ball B κ R of radius R in n (κ) are given by Vol(S κ R) = ω n 1 S κ (R) n 1 and Vol(B κ R) = where ω n 1 stands for the volume of the unit sphere in R n and S κ (t) is t if κ = 0, S κ (t) = sinh( κt) κ if κ < 0. R 0 Vol(S κ t )dt, The mean curvature H(t) of the geodesic spheres of radius t in n (κ) is H(t) = (n 1)H κ (t), where H κ (t) = S κ(t) S κ (t). (.7.4) Let ϕ : N be an immersion from a manifold to a Cartan-Hadamard manifold of sectional curvature K N bounded from above by K N κ 0. Given a point q, the extrinsic distance function r q : R + is defined by r q (x) = dist N (ϕ(q),ϕ(x)),

30 9 where dist N denotes the distance function in N. The extrinsic ball R q centered at q of radius R is given by R q := {x : r q (x) < R}. The volume growth function Q : R + R + is defined by Q(R) := Vol(R q ) Vol(B κ R ), where B κ R is the geodesic ball of radius R in n (κ). It is well-known that the volume growth function Q(R) of a minimal submanifold in a Cartan-Hadamard manifold is non-decreasing for 0 < R < dist N (q, ) (see (ALLARD, 197; ANDERSON, 1984; PALER, 1999; SION et al., 1983)). Using this monotonicity property of minimal submanifolds, Gimeno (GIENO, 014) proved the following useful fact: Lemma.7.1 (GIENO, 014) Let ϕ : N be an isometric minimal immersion from a manifold to a Cartan-Hadamard manifold of sectional curvature K N bounded from above by K N κ 0. Then Q(t)Vol(St κ ) Vol(q) t = (lnq(t)) Vol(Bt κ )Q(t) + Q(t)Vol(St κ ). (.7.5) Now we present a couple of definitions which will be necessary for a better understand of the next three theorems presented in this section. Definition.7. A noncompact manifold is of finite topological type if there is a compact domain Ω such that \Ω is homeomorphic to Ω [1, ). Definition.7.3 Let be a complete non-compact Riemannian manifold. Let K be a compact set with non-empty interior and smooth boundary. We denote by E K () the number of connected components U 1,...,U EK () of \K with non-compact closure. Then has E K () ends with respect to K, and the global number of ends E () is given by E () = sup E K (), K where K ranges on the compact sets of with non-empty interior and smooth boundary. In the sequel, we have the following results which relate some properties of the extrinsic geometry of the submanifold with its volume growth.

31 30 Theorem.7. (ANDERSON, 1984), (CHEN, 1995) and (GIENO; PALER, 013) Let n be a minimal submanifold immersed in the Euclidean space R m. If n has finite total scalar curvature α n d <. Then supq(r) <. R>0 Theorem.7.3 (QING; YI, 000) Let be a minimal surface immersed in the hyperbolic space H n of has constant sectional curvature κ < 0 or in the Euclidean space R n. If has finite total extrinsic curvature, namely α d <, then has finite topological type, and being χ() the Euler characteristic of. sup R>0Q(R) 1 α d + χ(), 4 Theorem.7.4 (GIENO; PALER, 014) Let n be a minimal n dimensional submanifold properly immersed in the hyperbolic space H m of constant sectional curvature κ < 0. If n > and the submanifold is of faster than exponential decay of its extrinsic curvature, namely, there exists a point p such that α (x) δ(r p(x)) e κr p (x), where δ(r) is a function such that δ(r) 0 when r. Then the submanifold has finite topological type, and being E () the (finite) number of ends of. supq(r) E (), R>0.8 Foliations In this section we define and give a few important definition concerning to foliations. Definition.8.1 A family F = {F γ } γ A of connected subsets of a manifold n is said to be an m-dimensional C r foliation, if 1. F γ =, γ A

32 31. γ λ F γ F λ = /0, 3. For any point q there is a C r chart (local coordinate system) ϕ q : U q R n such that q U q, ϕ q (q) = 0, and if U q F γ /0 the connected components of the sets ϕ q (U q F γ ) are given by equations x m+1 = c m+1,...,x n = c n, where c j s are constants. The sets F γ are immersed submanifolds of called the leaves of F. The family of all the vectors tangent to the leaves is the integrable subbundle of T denoted by T F. If carries a Riemannian structure, T F denotes the subbundle of all the vectors orthogonal to the leaves. A foliation F is said to be orientable (respectively, transversely orientable) if the bundle T F (respectively, T F ) is orientable.

33 3 3 ALOST RICCI SOLITONS The results in this chapter can be found in (BARROS; EVANGELISTA, 016), which is a joint work with Professor Abdênago Barros. Here we study almost Ricci solitons with null Cotton tensor. 3.1 Introduction The concept of almost Ricci soliton was introduced by Pigola et al. in (PIGOLA et al., 010), where essentially they modified the definition of a Ricci soliton by permitting to the parameter λ to be a variable function. ore precisely, Definition We say that a Riemannian manifold ( n, g) is an almost Ricci soliton if there exist a complete vector field X and a smooth soliton function λ : n R satisfying Ric + 1 L Xg = λg, (3.1.1) where Ric and L stand for the Ricci curvature tensor and the Lie derivative, respectively. We shall refer to this equation as the fundamental equation of an almost Ricci soliton ( n,g,x,λ). We say that an almost Ricci soliton is shrinking, steady or expanding provided λ > 0, λ = 0 or λ < 0, respectively, otherwise we say that it is indefinite. When X = f for some smooth function f on n, we say that it is a gradient almost Ricci soliton. In this case identity (3.1.1) becomes Ric + f = λg, (3.1.) where f stands for the Hessian of f. Further, an almost Ricci soliton is trivial provided X is a Killing vector field, otherwise it will be called a non-trivial almost Ricci soliton. We point out that when X is a Killing vector field and n 3, we have that is an Einstein manifold since Schur s lemma ensures that λ is constant. We highlight that Ricci solitons also correspond to self-similar solutions of Hamilton s Ricci flow, for more details about Ricci soliton see e.g. (CAO, 009). In this perspective Brozos-Vázquez, García-Río and Valle-Regueiro (BROZOS-VÁZQUEZ et al., 016) observed that some proper gradient almost Ricci solitons correspond to self-similar solutions of the Ricci-Bourguignon flow, which is a geometric flow given by g(t) = (Ric(t) kr(t)g(t)), t

34 33 where k R and R stands for the scalar curvature. This flow can be seen as an interpolation between the flows of Ricci and Yamabe. For more details on Ricci-Bourguignon flow we recommend (GIOVANNI et al., 015). It is important to emphasize that the round sphere does not admit a (nontrivial) Ricci soliton structure. However, Barros and Ribeiro Jr (BARROS; JR, 01) showed an explicit example of an almost Ricci soliton on the standard sphere. From this, it is interesting to know if, in compact case, this example is the unique with soliton function λ non constant. In this sense, Barros and Ribeiro Jr (BARROS; JR, 01) proved that a compact gradient almost Ricci soliton with constant scalar curvature must be isometric to a standard sphere. Afterward, Barros, Batista and Ribeiro Jr (BARROS et al., 014) proved that every compact almost Ricci soliton with constant scalar curvature is gradient. In (BRASIL et al., 014), Costa, Brasil and Ribeiro Jr showed that under a suitable integral condition, a 4-dimensional compact almost Ricci soliton is isometric to standard sphere S 4. While Ghosh (GHOSH, 014) was able to prove that if a compact K-contact metric is a gradient almost Ricci soliton, then it is isometric to a unit sphere. We also remark that Barros, Batista and Ribeiro Jr (BARROS et al., 01) proved that under a suitable integral condition a locally conformally flat compact almost Ricci soliton is isometric to a standard sphere S n. For more details see, for instance, (BARROS et al., 014), (BARROS et al., 01), (GHOSH, 014), (ASCHLER, 015) and (SHARA, 014). When is a compact manifold the Hodge-de Rham decomposition theorem (see for instance (WARNER, 013)) asserts that X can be decomposed as a sum of a gradient of a function h and a divergence-free vector field Y, i.e. X = h +Y, where divy = 0. From now on we consider h the function given by this decomposition. Henceforth, in this chapter, we denote by n, n 3, a compact connected oriented manifold without boundary. 3. Auxiliaries lemmas In this section we calculate the divergence of the Newton transformations of a general symmetric (0,) tensor and obtain an explicit relation with the tensor D defined by D i jk = i T jk j T ik. This result will be useful when T is the Schouten tensor A. In this case the divergence of the Newton transformations will be related to the Cotton tensor.

35 34 Lemma 3..1 Let P k (T ) be the Newton transformations associated with T defined Section. and let {e 1,...,e n } be a local orthonormal frame on. Then, for all Z X(), the divergence of P k (T ) are given recursively as divp 0 (T ) = 0 divp k (T ),Z = T (divp k 1 (T )),Z n i=1 D(e i,z,p k 1 (T )e i ), (3..1) or equivalently divp k (T ),Z = k n j=1 i=1 ( 1) j D(e i,t j 1 Z,P k j (T )e i ). (3..) Proof: Since P 0 (T ) = I, then divp 0 (T ) = 0. By the inductive definition of P k (T ) we have Z P k (T )Y = σ k (T ),Z Y Z (T P k 1 (T ))Y = σ k (T ),Z Y ( Z T P k 1 (T ))Y (T Z P k 1 (T ))Y, so that divp k (T ) = n i=1 Now, by using (..5) we get ( ei P k (T ))e i = σ k (T ) n i=1 ( ei T )(P k 1 (T )e i ),Z = ( ei T )Z,P k 1 (T )e i Therefore, letting ρ = n i=1 = ( ei T )(Z,P k 1 (T )e i ) ( ei T )(P k 1 (T )e i ) (T divp k 1 (T )). = D(e i,z,p k 1 (T )e i ) + Z T (e i,p k 1 (T )e i ) = D(e i,z,p k 1 (T )e i ) + ( Z T )e i,p k 1 (T )e i = D(e i,z,p k 1 (T )e i ) + (P k 1 (T ) Z T )(e i ),e i. D(e i,z,p k 1 (T )e i ), we deduce divp k (T ),Z = σ k (T ),Z tr(p k 1 (T ) Z T ) (T divp k 1 (T )) ρ. (3..3) Now we just need to prove that tr(p k 1 (T ) Z T ) = σ k (T ),Z. (3..4) We prove the above equation using a local orthonormal frame that diagonalizes T. We point out that such a frame does not always exist, since the multiplicity of the eigenvalues may changes. Therefore, we will work in a subset T consisting of points at which the

36 35 multiplicity of the eigenvalues is locally constant. We recall that such subset is open and dense in, and in every connected component of T the eigenvalues form mutually smooth distinct eigenfunctions and, for such a function λ, the assignment p V λ(p) (p) T p defines a smooth eigenspace distribution V λ of T (consult (BESSE, 007), Paragraph 16.10). Therefore, for every p T there exists a local orthonormal frame defined on a neighborhood of p that diagonalizes T, i.e., ( Z T )e i = Z(λ i )e i + (λ i λ j )ω j i (Z)e j, j i where we use the standard notation ω j i (Z) = Ze i,e j. Using (..3) we get tr(p k 1 (T ) Z T ) = = n i=1 n i=1 µ i,k 1 Z(λ i ) Z(λ i ) i 1 < <i k,i j i λ i1 λ ik 1 = Z( i 1 < <i k λ i1 (T ) λ ik (T ) ) = σ k (T ),Z. This proves the statement on T, and by continuity on. Substituting (3..4) into (3..3), we get (3..1). In order to arrive at (3..) it suffices to use an inductive argument. In particular, when D 0 all the Newton transformations are divergence free. In general we have the following. Corollary 3..1 If D = 0, then the Newton transformations are divergence free: divp k (T ) = 0 for each k. 3.3 Integral formulae In this section we take T to be the Schouten tensor, i.e., T = A = Ric R (n 1) g. Now we obtain some integral formulae for the symmetric functions associated to the Schouten tensor A, which will be used to prove our main result. Thereby, we have the following lemma. Lemma Let (,g,x,λ) be a compact oriented almost Ricci soliton. For each k, the following integral formula holds: divp k (A),X d + c k Proof: shows that ( (S1 (A) + 1 n h) S k (A) S k+1 (A)) d = 0. (3.3.1) Note that Y P k (A) is self-adjoint for all Y X(). A straightforward computation div(p k (A)X) = divp k (A),X + n i=1 ei X,P k (A)e i, (3.3.)

37 36 where {e 1,...,e n } is a local orthonormal frame. If we take such a local orthonormal frame that diagonalizes A, then by (..3) we have ei X,P k (A)e i = µ i,k ei X,e i = Pk (A)e i X,e i. By the almost Ricci soliton equation (3.1.1) we get Pk (A)e i X,e i = λ P k (A)e i,e i Ric(P k (A)e i,e i ) ( = λ R (n 1) ) P k (A)e i,e i AP k (A)e i,e i, (3.3.3) hence, using equations (3.3.3) and (..4), equation (3.3.) becomes ( ) div(p k (A)X) = divp k (A),X + λ R (n 1) trp k (A) tr(ap k (A)) ( ) = divp k (A),X + λ c k S k (A) c k S k+1 (A), R (n 1) Taking trace in (3.1.1) we get R + divx = nλ. Since S 1 (A) = (n )R n(n 1) we obtain div(p k (A)X) = divp k (A),X + (S 1 (A) + 1 ) n divx c k S k (A) c k S k+1 (A). (3.3.4) When is a compact manifold and h is the function given by the Hodge-de Rham decomposition theorem it is easy to see that identity (3.3.4) becomes div(p k (A)X) = divp k (A),X + (S 1 (A) + 1 ) n h c k S k (A) c k S k+1 (A). (3.3.5) Integrating (3.3.5) we get the desired result. Note that when the Cotton tensor vanishes Corollary 3..1 implies that Therefore, we obtain the next corollary. divp k (A),X d = 0. (3.3.6) Corollary Let (,g,x,λ) be a compact oriented almost Ricci soliton such that the Cotton tensor vanishes. Then, 3.4 Rigidity result ( (S1 (A) + 1 n h) S k (A) S k+1 (A)) d = 0. (3.3.7) In this section we present our main result which concern to hypothesis on the symmetric functions S k (A) associated to the Schouten tensor A. Thereby, since S 1 (A) = (n )R n(n 1),

38 37 we may conclude that S 1 (A) is constant if and only if R is constant, in this case Barros and others in (BARROS et al., 014) proved that the almost Ricci soliton is isometric to a round sphere. In the next theorem we consider the other symmetric functions S k (A), k =,...,n 1 and under the additional hypothesis on the nullity of the cotton tensor we obtain a rigidity result for the almost Ricci soliton. We also note that n (S 1 (A)) = n(n 1)S (A) + A, hence S (A) constant, in general, does not implies that S 1 (A) constant. We remark that Hu and others in (HU et al., 008) used the symmetric functions of the Schouten tensor to study locally conformally flat manifolds. They proved that under conditions on item 1 and item 4 of the next theorem, a compact locally conformally flat manifold with semi-positive definite Schouten tensor is isometric to a space form of constant sectional curvature. In this direction, the next theorem improves the results obtained in (HU et al., 008) for compact almost Ricci solions with null cotton tensor, since null cotton tensor does not imply locally conformally flat in general. For example, the complex projective space CP n with Fubini Study metric is Einstein, hence has null cotton tensor, however CP n is not locally conformally flat. Remark Before presenting the proofs of the results, we recall that the symmetric functions satisfy Newton s inequalities: S k (A)S k+ (A) Sk+1 (A) for 0 k < n 1, (3.4.1) which is a generalized Cauchy-Schwarz inequality. oreover, if equality occurs for k = 0 or 1 k < n with S k+ (A) 0, then λ 1 (A) = λ (A) =... = λ n (A). As an application, provided that λ k (A) > 0 for 1 k n, we obtain Gårding s inequalities S 1 S 1 S S 1 n n. (3.4.) Here equality holds, for some 1 k < n, if and only if, λ 1 (A) = λ (A) =... = λ n (A). Note that (3.4.) implies that S k+1 k k S k+1 for 1 k < n. For a proof see for instance (HARDY et al., 195) Theorem 51, p. 5 or Proposition 1 in (CAINHA, 006). Theorem Let ( n,g,x,λ), n 3, be a non-trivial compact oriented almost Ricci soliton such that the Cotton tensor is identically zero. Then, n is isometric to a standard sphere S n provided that one of the next condition is satisfied: 1. S (A) is constant and positive.