Le but dans cette thèse est d'expliciter les liens entre les propriétés analytiques, géométriques et topologiques des variétés compactes de dimension n supérieure ou égale à 3 et le comportement des valeurs propres de l'opérateur de Yamabe. On commence par étudier les propriétés de ces valeurs propres : l'un des remarques principales est que leur signe est invariant par un changement conforme de métriques. On s'intéresse plus particulièrement à la deuxième valeur propre de l'opérateur de Yamabe et on fait le lien entre son signe et l'existence des solutions nodales de l'équation de Yamabe. Pour finir, nous donnons une formule de chirurgie pour le second invariant de Yamabe, qui nous permet d'en obtenir une borne inférieure sous certaines hypothèses topologiques / The goal of this thesis is to study the relationships between the analytical, geometrical and topological properties of compact manifolds of dimension n greater or equal to 3 and the behavior of the eigenvalues of the Yamabe operator. We start by studying the properties of these eigenvalues : One of the most important observation is that their sign is a conformal invariant. We are interested particulary by the study of the seconf eigenvalue of the Yamabe operator and we enlight the relations between its sign and the existence of nodal solutions of the Yamabe equation. At last, we etablish a surgery formula for the second Yamabe invariant which allows to obtain a lower bond for the second Yamabe invariant under some topological hypothesis

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. Programa de Pós-Graduação em Matemática e Computação Científica. / Made available in DSpace on 2012-10-23T19:52:15Z (GMT). No. of bitstreams: 1 248382.pdf: 390919 bytes, checksum: 7fcbdb3407b73db7794b8e411f407887 (MD5) / Definimos o Laplaciano e a Curvatura Escalar sobre uma variedade M e os invariantes de Yamabe e de Perelman. Provamos que eles são iguais quando o primeiro é não positivo e que o invariante de Perelman é igual a mais infinito quando o invariante de Yamabe é positivo.

CRUZ, Cícero Tiarlos Nogueira. Hipersuperfícies com bordo livre e rigidez de superfícies mínimas. 2015. 56 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2015. / Submitted by Erivan Almeida (eneiro@bol.com.br) on 2015-05-06T19:44:15Z No. of bitstreams: 1 2015_tese_ctncruz.pdf: 975083 bytes, checksum: 9407fc51d29686a5e14baf7d34105f98 (MD5) / Approved for entry into archive by Rocilda Sales(rocilda@ufc.br) on 2015-05-07T11:35:24Z (GMT) No. of bitstreams: 1 2015_tese_ctncruz.pdf: 975083 bytes, checksum: 9407fc51d29686a5e14baf7d34105f98 (MD5) / Made available in DSpace on 2015-05-07T11:35:24Z (GMT). No. of bitstreams: 1 2015_tese_ctncruz.pdf: 975083 bytes, checksum: 9407fc51d29686a5e14baf7d34105f98 (MD5) Previous issue date: 2015 / In this thesis, we prove estimates for the volume and boundary area of stable hypersurfaces ∑n-1 with nonpositive Yamabe invariant satisfying the free boundary condition in a Riemannian manifold Mn with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that ∑ is locally volume-minimizing in a manifold M with scalar curvature bounded below by a nonpositive constant, we conclude that locally M splits along ∑ as (-Є, Є)x ∑, for some Є > 0. In the case that ∑ locally minimizes a certain functional inspired by the work of Yau (2001), a neighborhood of ∑ in M is isometric to ((-Є, Є) x ∑, dt2 + e2tg), where g is Ricci at. In the second part, we study other scalar curvature rigidity phenomena adapting a technique developed by Máximo e Nunes (2013) to show a local rigidity result for three-dimensional Riemannian manifold M3 whose scalar curvature is bounded from below by a negative constant. We prove the following result: Let ∑2 ⊂ M3 be a stable minimal surface which locally maximizes the Hawking mass on M. Then M near ∑ is a piece of one the Kottler space. / Nesta tese, provamos estimativas para o volume e área do bordo de hipersuperficies estáveis ∑n-1 com invariante de Yamabe não positivo satisfazendo à condição de bordo livre em uma variedade Riemanniana de dimensão n com limitação na curvatura escalar e curvatura média do bordo. Supondo ainda que ∑ é localmente minimizante de volume em uma variedade M com curvatura escalar limitada inferiormente por uma constante não positiva, concluímos que localmente M divide-se ao longo ∑ como (-Є, Є)x ∑, para algum Є > 0. No caso em que ∑ localmente minimiza um funcional adequado inspirado pelo trabalho de Yau (2001), uma vizinhança de ∑ em M é isométrica a ((-Є, Є) x ∑, dt2 +e2tg), onde g é Ricci plana. Na segunda parte, estudamos outro fenômeno de rigidez pela curvatura escalar adaptando a técnica desenvolvida por Máximo e Nunes (2013) para mostrar um resultado local de rigidez para uma variedade Riemanniana tridimensional M3 cuja curvatura escalar é limitada inferiormente por um constante negativa. Provamos o seguinte resultado: Seja ∑2 ⊂ M3 uma superfície mínima estritamente estável que localmente maximiza a massa Hawking em M. Então M perto de ∑ é um pedaço de um dos espaços de Kottler.

La géométrie de Cauchy-Riemann, CR en abrégé, est la géométrie naturelle des hypersurfaces réelles pseudoconvexes de $C^{n+1}$, lorsque $ngeq 1$. Nous considérons le cas générique où les variétés CR considérées sont de contact. La géométrie CR présente de nombreuses similarités avec la géométrie conforme ; les invariants mis au jour et les techniques éprouvées en géométrie conforme peuvent donc être adaptées dans ce contexte. Nous nous intéressons dans cette thèse à deux invariants de ce type. Dans une première partie, en utilisant la géométrie asymptotiquement hyperbolique complexe, nous introduisons un opérateur différentiel CR covariant agissant sur les applications allant d'une variété CR vers une variété riemannienne, égal pour les fonctions à l'opérateur de Paneitz CR. Dans une seconde partie, nous proposons un invariant de Yamabe pour les variétés de contact admettant une structure CR, et nous étudions son comportement sous somme connexe. / Cauchy-Riemann geometry, CR for short, is the natural geometry of real pseudoconvex hypersurfaces of $C^{n+1}$ for $ngeq 1$. We consider the generic case when CR manifolds are contact manifolds. CR geometry presents strong analogies with conformal geometry; hence, known invariants and techniques of conformal geometry can be transported to that context. We focus in this thesis on two such invariants. In a first part, using asymptotically complex hyperbolic geometry, we introduce a CR covariant differential operator on maps from a CR manifold to a Riemannian manifold, which coincides on functions with the CR Paneitz operator. In a second part, we propose a Yamabe invariant for contact manifolds which admit a CR structure, and we study its behaviour under connected sum.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Apresentamos vÃrios resultados de existÃncia, unicidade, rigidez e bifurcaÃÃo para o problema da prescriÃÃo de diversas estruturas geomÃtricas em variedades Riemannianas, entre os quais incluem-se: i) deformaÃÃo e rigidez para estruturas 2k-Einstein em variedades com (2k − 2)-curvatura seccional constante; ii) deformaÃÃo conforme de mÃtricas no contexto do problema de Yamabe para curvaturas de Gauss-Bonnet; iii) unicidade, bifurcaÃÃo e rigidez local no Ãmbito do problema de Yamabe para as funÃÃes simÃtricas dos autovalores do tensor de Schouten. / We present several results of existence, uniqueness, rigidity and bifurcation for the problem of prescribing various geometric structures on Riemannian manifolds, among which include: i) deformation and rigidity for 2k-Einstein structures on manifolds with constant (2k − 2)-sectional curvature; ii) conformal deformation of metrics in the context of the Yamabe Problem for Gauss-Bonnet curvatures; iii) uniqueness, bifurcation and local rigidity in scope of the Yamabe Problem for symmetric functions of eigenvalues of the Schouten tensor.

Cette thèse est consacrée à l'étude d'une famille des flots géométriques associés au problème de la courbure scalaire prescrite sur une variété riemannienne compacte. Plus précisément, si on désigne par (M,g0) une variété riemannienne compacte de dimension n≥3, et si F∈C∞ (M) est une fonction donnée, le problème de la courbure scalaire prescrite consiste à trouver une métrique g conforme à g0 telle que F soit sa courbure scalaire. Ce problème est équivalent à la résolution de l'EDP suivante :-4 (n-1)/(n-2) ∆u+R0 u=Fu((n+2)/(n-2 )) , u>0 , (E), Où R0 est la courbure scalaire de la métrique initiale g0 et ∆ est le laplacien associé à g0. Il s'agit d'une équation elliptique non-linéaire dont la difficulté principale provient du terme u((n+2)/(n-2 )). Hormis le cas de la sphère standard Sn , tous les travaux consacrés à l'étude de l'équation (E) sont basés sur la méthode variationnelle. Dans cette thèse, on développe une autre approche basée sur l'étude d'une famille de flots géométriques qui permet, entre autres, de résoudre l'équation (E). La question dépend bien entendu de la métrique initiale g0 et en particulier du signe de sa courbure scalaire R0. Les flots introduits sont des flots de gradient associés à deux fonctionnelles distinctes dépendant du signe de R0. La première partie de cette thèse est consacrée au cas R0<0 et dans la deuxième partie on traite le cas R0>0. Dans les deux cas, on démontre l'existence globale du flot et on étudie son comportement asymptotique à l'infini. / This thesis is devoted to the study of a family of geometric flows associated with the prescribed scalar curvature problem. More precisely, if we denote by (M,g0) a compact riemannian manifold with dimension n≥3, and if F∈C∞ (M) is a given function, the prescribed scalar curvature problem consists of finding a conformal metric g to g0 such that F is its scalar curvature. This problem is equivalent to the resolution of the following PDE : -4 (n-1)/(n-2) ∆u+R0 u=Fu((n+2)/(n-2 )) , u>0 , (E), Where R0 is the scalar curvature of the initial metric g0 and ∆ is the laplacian associated with g0.It is a nonlinear elliptic equation, whose the main difficulty comes from the term u((n+2)/(n-2 )). Apart from the case of the standard sphere Sn all the works that study the equation (E) are based on the variational method. In this thesis, we develop another approach based on the study of a family of geometric flows which allows to solve equation (E).The flows introduced are gradient flows associated with two distinct functional functions depending on the sign of R0.The first part of this thesis is devoted to the case R0<0 and in the second part we treat the case R0>0. In both cases, our aim is to proof the global existence of the flow and study its asymptotic behavior at infinity.

An intrinsic geometric flow is an evolution of a Riemannian metric by a two-tensor. An extrinsic geometric flow is an evolution of an immersion of a manifold into Euclidean space. An extrinsic flow induces an evolution of a metric because any immersed manifold inherits a Riemannian metric from Euclidean space. In this paper we discuss the inverse problem of specifying an evolution of a metric and then seeking an extrinsic geometric flow which induces the given metric evolution. We limit our discussion to the case of manifolds that are rotationally symmetric and embeddable with codimension one. In this case, we reduce an intrinsic geometric flow to a plane curve evolution. In the specific cases we study, we are able to further simplify the evolution to an evolution of a function of one variable. We provide soliton equations and give proofs that some soliton metrics exist.

No presente trabalho consideramos o produto de uma variedade Riemanniana compacta sem bordo de curvatura escalar zero e uma variedade Riemanniana compacta com bordo, curvatura escalar zero e curvatura media constante no bordo, e fazemos uso da teoria de bifurcação para provar a existência de um numero infinito de classes conforme com, pelo menos, duas métricas Riemannianas não homotéticas de curvatura escalar zero e curvatura média constante no bordo, sobre a variedade produto. / In this work, we consider the product of a compact Riemannian manifold without boundary, null scalar curvature and a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary and we use the bifurcation theory to prove the existence of a infinite number of conformal classes with at least two non homothetic Riemannian metrics of null scalar curvature and constant mean curvature of the boundary on the product manifold.

Apresentaremos alguns resultados de rigidez e de bifurcação para soluções do problema de Yamabe em variedades produto com bordo. / We will discuss some rigidity and bifurcation results for solutions of the Yamabe problem in product manifolds with boundary.

Foi provado por Byde que é possível adicionar um fim do tipo Delaunay a uma variedade compacta não degenerada de curvatura escalar constante positiva; desde que ela seja localmente conformemente plana em alguma vizinhança do ponto de colagem. A variedade resultante é não-compacta e possui a mesma curvatura escalar constante. O principal objetivo desta tese é generalizar este resultado. Construiremos uma família a um parâmetro de soluções para o problema de Yamabe singular positivo em qualquer variedade compacta não degenerada cujo tensor de Weyl anula-se até uma ordem suficientemente grande no ponto singular. Se a dimensão da variedade é no máximo 5; nenhuma condição sobre o tensor de Weyl é necessária. Usaremos técnicas de pertubação e o método de colagem. _________________________________________________________________________________________ ABSTRACT: It has been showed by Byde [5] that it is possible to attach a Delaunay type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main goal of this thesis is to generalize this result. We will construct a one-parameter family of solutions to the positive singular Yamabe problem for any compact non-degenerate manifold with Weyl tensor vanishing to suciently high order at the singular point. If the dimension is at most 5, no condition on the Weyl tensor is needed. We will use perturbation techniques and gluing methods.