<p> The aim of this thesis is to construct some smooth Einstein manifolds with nonzero Einstein constant, and then to investigate their topological and geometric properties.</p> <p> In the negative case, we are able to construct conformally compact Einstein metrics on 1. products of an arbitrary closed Einstein manifold and a certain even-dimensional ball bundle over products of Hodge Kähler-Einstein manifolds, 2. certain solid torus bundles over a single Fano Kähler-Einstein manifold. We compute the associated conformal invariants, i.e., the renormalized volume in even dimensions and the conformal anomaly in odd dimensions. As by-products, we obtain many Riemannian manifolds with vanishing Q-curvature.</p> <p> In the positive case, we are able to construct complete Einstein metrics on certain 3-sphere bundles over a Fano Kähler-Einstein manifold. We classify the homeomorphism and diffeomorphism types of the total spaces when the base manifold is the complex projective plane.</p> / Thesis / Doctor of Philosophy (PhD)

In this thesis, we explore short time existence and uniqueness of solutions to the Ricci flow on manifolds with boundary, as well as the preservation of natural curvature positivity conditions along the flow. In chapter 2, we establish the existence and uniqueness for linear parabolic systems on vector bundles for Hölder continuous initial data. We introduce appropriate weighted parabolic Hölder spaces to study the existence and uniqueness problem. Having developed the linear theory, we apply it to establish the existence and uniqueness for the Ricci-DeTurck flow, the harmonic map heat flow, and the Ricci flow with Hölder continuous initial data in Chapter 3. In chapter 4, we discuss a general preservation result concerning the preservation of various curvature conditions during boundary deformation. Using a perturbation argument, we construct a family of metrics which interpolate between two metrics that agree on the boundary, and such family of metrics preserves various natural curvature conditions under suitable assumptions on the boundary data. The results from chapters 2 through 4 will be utilized in proving the Main Theorems in chapter 5. In particular, we construct canonical solutions to the Ricci flow on manifolds with boundary from canonical solutions to the Ricci flow on closed manifolds with Hölder continuous initial data via doubling.

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Being able to identify fuel efficient transfers between orbits is critical to planning and executing missions involving spacecraft. With a renewed focus on missions in cislunar space, identifying efficient transfers in the dynamical environment characterized by the Circular Restricted Three-Body Problem (CR3BP) will be especially important, both now and in the immediate future. The focus of this thesis is to develop a methodology that can be used to identify a valid low-cost transfer between a variety of orbits in the CR3BP. The approach consists of two distinct parts. First, tools related to dynamical systems theory and manifolds are used to create an initial set of possible transfers. An optimization scheme is then applied to the initial transfers to obtain an optimized set of transfers. Code was developed in MATLAB to implement and test this approach. The methodology and its implementation were evaluated by using the code to identify a low-cost transfer in three different transfer cases. For each transfer case, the best transfers from each set were compared, and important characteristics of the transfers in the first and final sets were examined. The results from those transfer cases were analyzed to determine the overall efficacy of the approach and effectiveness of the implementation code. In all three cases, in terms of cost and continuity characteristics, the best optimized transfers were noticeably different compared to the best manifold transfers. In terms of the transfer path identified, the best optimized and best manifold transfers were noticeably different for two of the three cases. Suggestions for improvements and other possible applications for the developed methodology were then identified and presented. / Master of Science / Being able to identify fuel efficient transfers between orbits is critical to planning and executing missions involving spacecraft. With a renewed focus on lunar missions, identifying efficient transfers between orbits in the space around the Moon will be especially important, both now and in the immediate future. The focus of this thesis is to develop a methodology that can be used to identify a valid low-cost transfer between a variety of orbits in the space around the Moon. The approach was evaluated by using the code to identify a low-cost transfer in three different transfer cases. The results from those transfer cases were analyzed to determine the overall efficacy of the approach and effectiveness of the implementation code. Suggestions for improvements and other possible applications for the developed methodology were then identified and presented.

We prove several theorems about the pseudofree, locally linear and homologically trivial action of finite groups 𝐺 on closed, connected, oriented 4-manifolds 𝑀 with non-zero Euler characteristic. In this setting, the rank𝑝 (𝐺) ≤ 1, for 𝑝 ≥ 5 prime and rank(𝐺) ≤ 2, for 𝑝 = 2, 3. We combine these results into two main theorems: Theorem A and Theorem B in Chapter 1. These results strengthen the work done by Edmonds, and Hambleton and Pamuk. We remark that for low second betti-numbers ( <= 2) there are other examples of finite groups which can act in the above way. / Thesis / Doctor of Philosophy (PhD)

O Teorema de Representação de Weierstrass clássico, que faz uso da análise complexa para descrever uma superfície mínima imersa no espaço Euclidiano em termos de dados holomorfos, tem sido extremamente útil seja para construir novos exemplos de superfícies mínimas, seja para o estudo das propriedades destas superfícies. Em [24], usando a equação harmônica, os autores determinam uma fórmula de representação para superfícies mínimas, simplesmente conexas, imersas em uma variedade Riemanniana qualquer. Neste caso, a condição de holomorficidade dos dados de Weierstrass consiste em um sistema de equações diferenciais parciais com coeficientes não constantes. Logo, em geral, é complicado determinar soluções explícitas. No entanto, escolhendo adequadamente o espaço ambiente, tais equações se simplificam e a fórmula pode ser usada para produzir novos exemplos de imersões mínimas conformes. No espaço de Lorentz-Minkowski tridimensional uma fórmula de representação tipo-Weierstrass foi provada por Kobayashi, para o caso das imersões mínimas de tipo espaço (ver [18]), e por Konderak no caso das imersões mínimas de tipo tempo (ver [20]). Na demonstração destas fórmulas se utilizam as ferramentas da análise complexa e paracomplexa, respectivamente. Recentemente, em [22] os resultados de Kobayashi e Konderak foram generalizados para o caso de superfícies mínimas (de tipo espaço e de tipo tempo) imersas em 3-variedades Lorentzianas. Nesta dissertação estudaremos as fórmulas de representação de Weierstrass para superfícies mínimas imersas em variedades Riemannianas e Lorentzianas, que foram obtidas nos artigos [18], [20], [22] e [24]. / The classic Weierstrass Representation Theorem, which makes use of complex analysis to describe a minimal surface immersed in the Euclidean space in terms of holomorphic data, has been extremely useful either to construct new examples of minimal surfaces, rather than to study structural properties of these surfaces. In [24], using the standard harmonic equation, the authors determine a representation formula for simply connected immersed minimal surfaces in a Riemannian manifold. In this case, the holomorphicity condition of the Weierstrass data is a system of partial differential equations with nonconstant coefficients. Therefore, in geral, it is very difficult to determine explicit solutions. However, for particular ambient spaces, these equations become simpler and the formula can be used to produce new examples of conformal minimal immersions. In the three-dimensional Lorentz-Minkowski space a Weierstrass-type representation formula was proved by Kobayashi for spacelike minimal immersions (see [18]), and by Konderak for the case of timelike minimal immersions (see [20]). In the demonstration of these formulas are used the tools of complex and paracomplex analysis, respectively. Recently, in [22] the results of Kobayashi and Konderak were generalized to the case of (spacelike and timelike) minimal surfaces immersed in 3-Lorentzian manifolds. In this dissertation, we will study the Weierstrass representation formula for immersed minimal surfaces in Riemannian and Lorentzian manifolds, that was obtained in the articles [18], [20], [22] and [24].

O Teorema de Representação de Weierstrass clássico, que faz uso da análise complexa para descrever uma superfície mínima imersa no espaço Euclidiano em termos de dados holomorfos, tem sido extremamente útil seja para construir novos exemplos de superfícies mínimas, seja para o estudo das propriedades destas superfícies. Em [24], usando a equação harmônica, os autores determinam uma fórmula de representação para superfícies mínimas, simplesmente conexas, imersas em uma variedade Riemanniana qualquer. Neste caso, a condição de holomorficidade dos dados de Weierstrass consiste em um sistema de equações diferenciais parciais com coeficientes não constantes. Logo, em geral, é complicado determinar soluções explícitas. No entanto, escolhendo adequadamente o espaço ambiente, tais equações se simplificam e a fórmula pode ser usada para produzir novos exemplos de imersões mínimas conformes. No espaço de Lorentz-Minkowski tridimensional uma fórmula de representação tipo-Weierstrass foi provada por Kobayashi, para o caso das imersões mínimas de tipo espaço (ver [18]), e por Konderak no caso das imersões mínimas de tipo tempo (ver [20]). Na demonstração destas fórmulas se utilizam as ferramentas da análise complexa e paracomplexa, respectivamente. Recentemente, em [22] os resultados de Kobayashi e Konderak foram generalizados para o caso de superfícies mínimas (de tipo espaço e de tipo tempo) imersas em 3-variedades Lorentzianas. Nesta dissertação estudaremos as fórmulas de representação de Weierstrass para superfícies mínimas imersas em variedades Riemannianas e Lorentzianas, que foram obtidas nos artigos [18], [20], [22] e [24]. / The classic Weierstrass Representation Theorem, which makes use of complex analysis to describe a minimal surface immersed in the Euclidean space in terms of holomorphic data, has been extremely useful either to construct new examples of minimal surfaces, rather than to study structural properties of these surfaces. In [24], using the standard harmonic equation, the authors determine a representation formula for simply connected immersed minimal surfaces in a Riemannian manifold. In this case, the holomorphicity condition of the Weierstrass data is a system of partial differential equations with nonconstant coefficients. Therefore, in geral, it is very difficult to determine explicit solutions. However, for particular ambient spaces, these equations become simpler and the formula can be used to produce new examples of conformal minimal immersions. In the three-dimensional Lorentz-Minkowski space a Weierstrass-type representation formula was proved by Kobayashi for spacelike minimal immersions (see [18]), and by Konderak for the case of timelike minimal immersions (see [20]). In the demonstration of these formulas are used the tools of complex and paracomplex analysis, respectively. Recently, in [22] the results of Kobayashi and Konderak were generalized to the case of (spacelike and timelike) minimal surfaces immersed in 3-Lorentzian manifolds. In this dissertation, we will study the Weierstrass representation formula for immersed minimal surfaces in Riemannian and Lorentzian manifolds, that was obtained in the articles [18], [20], [22] and [24].

This thesis is concerned with numerical techniques for resolving and continuing closed, compact invariant manifolds in parameter-dependent dynamical systems with specific emphasis on invariant tori under flows. In the first part, we review several numerical methods of continuing invariant tori and concentrate on one choice called the ``orthogonality condition'. We show that the orthogonality condition is equivalent to another condition on the smooth level and show that they both descend from the same geometrical relationship. Then we show that for hyperbolic, periodic orbits in the plane, the linearization of the orthogonality condition yields a scalar system whose characteristic multiplier is the same as the non-unity multiplier of the orbit. In the second part, we demonstrate that one class of discretizations of the orthogonality condition for periodic orbits represents a natural extension of collocation. Using this viewpoint, we give sufficient conditions for convergence of a periodic orbit. The stability argument does not extend to higher-dimensional tori, however, and we prove that the method is unconditionally unstable for some common types of two-tori embedded in R^3 with even numbers of points in both angular directions. In the third part, we develop several numerical examples and demonstrate that the convergence properties of the method and discretization can be quite complicated. In the fourth and final part, we extend the method to the general case of p-tori in R^n in a different way from previous implementations and solve the continuation problem for a three-torus embedded in R^8.