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The global log canonical threshold, algebraic counterpart to Tian's alpha-invariant, plays an important role when studying the geometry of Fano varieties. In particular, Tian showed that Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In recent years Donaldson drafted a programme to precisely determine when a smooth Fano variety X admits a Kahler-Einstein metric. It was conjectured that the existence of such a metric is equivalent to X being K-stable, an algebraic-geometric property. A crucial step in Donaldson's programme consists on finding a Kahler-Einstein metric with edge singularities of small angle along a smooth anticanonical boundary. Jeffres, Mazzeo and Rubinstein showed that a dynamic version of the alpha-invariant could be used to find such metrics. The global log canonical threshold measures how anticanonical pairs fail to be log canonical. In this thesis we compute the global log canonical threshold of del Pezzo surfaces in various settings. First we extend Cheltsov's computation of the global log canonical threshold of complex del Pezzo surfaces to non-singular del Pezzo surfaces over a ground field which is algebraically closed and has arbitrary characteristic. Then we study which anticanonical pairs fail to be log canonical. In particular, we give a very explicit classifiation of very singular anticanonical pairs for del Pezzo surfaces of degree smaller or equal than 3. We conjecture under which circumstances such a classifcation is plausible for an arbitrary Fano variety and derive several consequences. As an application, we compute the dynamic alpha-invariant on smooth del Pezzo surfaces of small degree, where the boundary is any smooth elliptic curve C. Our main result is a computation of the dynamic alpha-invariant on all smooth del Pezzo surfaces with boundary any smooth elliptic curve C. The values of the alpha-invariant depend on the choice of C. We apply our computation to find Kahler-Einstein metrics with edge singularities of angle β along C.

This thesis is dedicated to the study of the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for the existence of Einstein metrics with totally geodesic fibers in terms of Casimir operators. Some particular cases are studied, for instance, for normal base or fiber, symmetric fiber, Einstein base or fiber, for which the Einstein equations are manageable. We investigate the existence of such Einstein metrics for invariant bisymmetric fibrations of maximal rank, i.e., when both the base and the fiber are symmetric spaces and the base is an isotropy irreducible space of maximal rank. We find this way new Einstein metrics. For such spaces we describe explicitly the isotropy representation in terms subsets of roots and compute the eigenvalues of the Casimir operators of the fiber along the horizontal direction. Results for compact simply connected 4-symmetric spaces of maximal rank follow from this. Also, new invariant Einstein metrics are found on Kowalski n-symmetric spaces.

This Dissertation covers three aspects of General Relativity: inequivalent Einstein metrics on Lie Group Manifolds, proving the Hoop Conjecture for Black Rings, and investigating ergoregions in magnetised black hole spacetimes. A number of analytical and numerical techniques are employed to that end. It is known that every compact simple Lie Group admits a bi-invariant homogeneous Einstein metric. We use two ansatze to probe the existence of additional inequivalent Einstein metrics on the Lie Group SU (n). We provide an explicit construction of 2k + 1 and 2k inequivalent Einstein metrics on SU (2k) and SU (2k + 1) respectively. We prove the Hoop Conjecture for neutral and charged, singly and doubly rotating black rings. This allows one to determine whether a rotating mass distribution has an event horizon, that it is in fact a black ring. We investigate ergoregions in magnetised black hole spacetimes. We show that, in general, rotating charged black holes (Kerr-Newman) immersed in an external magnetic field have ergoregions that extend to infinity near the central axis unless we restrict the charge to q = amB and keep B below a maximal value. Additionally, we show that as B is increased from zero the ergoregion adjacent to the event horizon shrinks, vanishing altogether at a critical value, before reappearing and growing until it is no longer bounded as B becomes greater than the maximal value.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / O objetivo deste trabalho à estudar a geometria dos sÃlitons de Ricci compactos, os quais correspondem as soluÃÃes auto-similires do fluxo de Ricci. AlÃm disso, essas variedades podem ser vistas como uma generalizaÃÃo das mÃtricas de Einstein. Neste trabalho, mostraremos que todo sÃliton de Ricci compacto tem curvatura escalar positiva. Alem disso, mostraremos que o seu grupo fundamental à sempre finito. Em particular, apresentaremos uma prova feita por Perelman [19] que todo sÃliton de Ricci compacto à do tipo gradiente / The aim of this work is to study the geometry of the compact Ricci soliton, which correspond to self-similar solution of the Ricci flow. These manifolds are natural generalization to Einstein metrics. Here we shall prove that every compact Ricci soliton has positive scalar curvature. Moreover, we show that its fundamental group is finite. Finally, we prove that every compact Ricci soliton must be gradient.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Our aim in this work is threefold. First, we get an extension, to the spherical case, of a theorem due to J. Simons, which concerns unstability of minimal cones constructed over a certain class of minimal submanifolds of the Euclidean sphere. Second, we classify the quasi-Einstein structures of the Riemannian product Hn x R. Third, we get a rigidity theorem for complete hypersurfaces into the De Sitter space, under certain conditions on the mean and scalar curvatures. / Este trabalho aborda trÃs problemas em Geometria Diferencial. Primeiro, obtemos uma extensÃo, para o caso esfÃrico, de um teorema devido a J. Simons sobre instabilidade de cones mÃnimos construÃdos sobre uma certa classe de subvariedades mÃnimas da esfera Euclidiana. Depois, classificamos as estruturas quasi-Einstein existentes sobre o produto Riemanniano Hn X R. Por fim, obtemos um teorema de rigidez para hipersuperfÃcies tipo-espaÃo completas do espaÃo de De Sitter, sob certas condiÃÃes sobre as curvaturas mÃdia e escalar, alÃm de uma condiÃÃo de integrabilidade.

<p>The curvatures in Finsler geometry can be defined in similar ways as in Riemannian geometry. However, since there are fewer restrictions on the metrics, many geometric quantities arise in Finsler geometry which vanish in the Riemannian case. These quantities are generally known as non-Riemannian quantities and interact with the curvatures in controlling the global geometrical and topological properties of Finsler manifolds. In the present work, we study general weighted Ricci curvatures which combine the Ricci curvature and the S-curvature, and define a weighted flag curvature which combines the flag curvature and the T -curvature. We characterize Randers metrics of almost isotropic weighted Ricci curvatures and show the general weighted Ricci curvatures can be divided into three types. On the other hand, we show that a proper open forward complete Finsler manifold with positive weighted flag curvature is necessarily diffeomorphic to the Euclidean space, generalizing the Gromoll-Meyer theorem in Riemannian geometry.</p>

SILVA, A. K. V. Estimativas gradiente para autofunções do V-Laplaciano e métricas m-quasi-Einstein generalizadas compactas com bordo. 2017. 40 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-04-18T14:49:03Z No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322426 bytes, checksum: d6abc5541598409191f635ad25e1f501 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Bom dia Andrea, Por favor repasse esse e-mail para o aluno. Estou devolvendo o trabalho pois tem alguns itens que não estão normalizados. O ano que deve constar na capa, folha de rosto e ficha catalográfica é o da entrega. E na ficha catalográfica o nome do autor e o título do trabalho não é em caixa alta. Somente as iniciais e quando necessário. Atenciosamente, Rocilda Sales on 2017-04-19T11:01:58Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-04-19T13:11:59Z No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322367 bytes, checksum: c9bd22b4cb5917adacde8be99093e8d8 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Alterar o ano. on 2017-04-19T14:52:13Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-04-19T16:22:45Z No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322367 bytes, checksum: 1523e0c8bd0590358cda3a542819aa62 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Correção na folha de aprovação. on 2017-04-19T16:39:17Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-04-19T17:04:43Z No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322112 bytes, checksum: e3087741f0e7bb8b966418fb10253c7b (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-04-24T11:14:14Z (GMT) No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322112 bytes, checksum: e3087741f0e7bb8b966418fb10253c7b (MD5) / Made available in DSpace on 2017-04-24T11:14:14Z (GMT). No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322112 bytes, checksum: e3087741f0e7bb8b966418fb10253c7b (MD5) Previous issue date: 2015-08-17 / The main of this work was to study properties of Riemannian when subjected to conditions on Bakry-Émery-Ricci tensor. Essentially we study two cases. In the first case, motivated by the work of Barros and Ribeiro Jr. (2014), He, Petersen and Wylie (2012) and Miao and Tam (2011), was introduced generalized m-quasi-Einstein metrics compact with boundary, where we get a result that classify these metrics; more specifically, assuming that gradient field of the exponential of potential function is a conformal vector field, we obtain that this must be a geodesic ball in a simply connected space form. That we get some results that implies when these are trivial metrics. In the second case, we work the Bakry-Émery-Ricci tensor bounded bellow, initially in a compact Riemannian, with or without boundary, and later on balls in complete Riemannian. With this study, we obtain gradient estimates for eigenfunctions of V-Laplacian operator, that generalize results of (Li, 2005) and (Li, 2015). Finally, as consequence theses results, we show an Harnack’s inequality. / Este trabalho tem como principal objetivo estudar propriedades de variedades Riemannianas quando submetidas a condições sobre tensores de Ricci-Bakry-Émery. Essencialmente estudamos dois casos. No primeiro caso, motivados pelos trabalhos de Barros e Ribeiro Jr (2014), He, Petersen e Wylie (2012) e por Miao e Tam (2011), introduzimos métricas m-quasi-Einstein generalizadas compactas com bordo, donde obtemos um resultado que garante uma classificação para estas métricas; mais precisamente, assumindo que o gradiente da exponencial da função potencial é um campo conforme, obtemos que aquela deve ser uma bola geodésica de uma forma espacial simplesmente conexa. Disso, obtemos alguns resultados em que garantimos quando estas métricas são triviais. No segundo caso, trabalhos o tensor de Ricci-Bakry-Émery limitado por baixo, inicialmente, em variedades Riemannianas compactas, com bordo ou sem bordo, e posteriormente, sobre bolas em variedades Riemannianas completas. Com esse estudo, obtivemos estimativas do gradiente para autofunções do operador V-Laplaciano, generalizando resultados de (Li, 2005) e (Li, 2015). Finalmente, como consequências desses resultados, exibimos uma desigualdade de Harnack.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2017-03-17T21:10:56Z No. of bitstreams: 2 Dissertação - Laena Furtado Borges - 2017.pdf: 2090414 bytes, checksum: afc3416e502ab5aedc5390b7986a9fcf (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-03-20T13:53:29Z (GMT) No. of bitstreams: 2 Dissertação - Laena Furtado Borges - 2017.pdf: 2090414 bytes, checksum: afc3416e502ab5aedc5390b7986a9fcf (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-03-20T13:53:29Z (GMT). No. of bitstreams: 2 Dissertação - Laena Furtado Borges - 2017.pdf: 2090414 bytes, checksum: afc3416e502ab5aedc5390b7986a9fcf (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-03-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we will present some concepts of quasi-Einstein metrics. From this, we will enunciate and demonstrate rigidity results for quasi-Einstein metrics until we have enough material to demonstrate a stiffness result for quasi-Einstein metrics of dimension two. Finally, we will give some concepts of Kähler metrics, prove a theorem and finally demonstrate a corollary that connects the main theorem of our work with Kähler metrics. / Nesse trabalho, apresentaremos alguns conceitos de métricas quasi-Einstein. A partir disso, enunciaremos e demonstraremos resultados de rigidez para métricas quasi-Einstein, até que tenhamos material suficiente para a demonstração de um resultado de rigidez para métricas quasi-Einstein em dimensão dois. Por fim, daremos alguns conceitos de métricas kähler, provaremos um teorema e por fim demonstraremos um corolário que conecta o teorema principal do nosso trabalho com as métricas Kähler.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / O objetivo deste trabalho à estudar a geometria das mÃtricas tipo-Einstein (solitons de Ricci, quase solitons de Ricci e mÃtricas quasi-Einstein). Mais especificamente, vamos obter equaÃÃes de estrutura, exemplos, fÃrmulas integrais e estimativas que permitirÃo caracterizar estas classes de mÃtricas. / The purpose of this work is study the geometric of the like-Einstein metrics (Ricci soliton, almost Ricci solitons and quasi-Einstein metrics). More specifically, we obtain structure equations, examples, integral formulae and estimates that will enable characterize these classes of metrics.