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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The computation of curvature quantities over discrete geometry is often required when processing geometry composed of meshes. Curvature information is often important for the purpose of shape analysis, feature recognition and geometry segmentation. In this paper we present a method for accurate estimation of curvature on discrete geometry especially those composed of meshes. We utilise a method based on fitting a continuous surface arising from the solution of the Biharmonic equation subject to suitable boundary conditions over a 1-ring neighbourhood of the mesh geometry model. This enables us to accurately determine the curvature distribution of the local area. We show how the curvature can be computed efficiently by means of utilising an analytic solution representation of the chosen Biharmonic equation. In order to demonstrate the method we present a series of examples whereby we show how the curvature can be efficiently computed over complex geometry which are represented discretely by means of mesh models.

Phafin2, an adaptor protein, is involved in various cellular processes, such as apoptosis, autophagy, endosomal cargo transportation, and macropinocytosis. Two domains, namely, PH and FYVE, contribute to Phafin2's cell membrane binding. Phafin2 also contains a poly aspartic acid (polyD) motif in its C-terminal region that can specifically autoinhibit the PH domain binding to membrane phosphatidylinositol 3-phosphate (PtdIns3P). Firstly, the study investigated the domain-based evolutionary pattern of PH, FYVE, and polyD motif of Phafin2 among its orthologs and Phafin2- like proteins. Using different bioinformatics tools and resources, it was concluded that the polyD motif only evolved in Phafin2 and PH- or both PH-FYVE-containing proteins of animals, highlighting the association in cellular functions that might have evolved uniquely in animals. Moreover, PH domain-free FYVE-containing proteins lack polyD motifs. Secondly, intramolecular autoregulatory and membrane binding properties of Phafin2 were studied by employing liposome co-sedimentation assay, isothermal titration calorimetry, and nuclear magnetic resonance spectroscopy. The residues Gly38, Lys45, Leu45, Lys51, Ala52, and Arg53 of the PH domain form a positively charged binding pocket that can bind the negatively charged polyD motif. The mutated Phafin2 PH domain (K51A/R53C and R53C) was unable to bind to synthetic polyD peptides, establishing the significance of those residues for the interaction between the PH domain and polyD motif. Moreover, the study also concluded that Phafin2-mediated membrane binding is not curvature-dependent. / Master of Science / Phafin2 is a protein that plays a crucial role in several important cellular functions, including cell death, recycling of cellular components, and transporting materials within cells. The protein's ability to attach to cell membranes is mainly due to two of its specific regions, the PH and FYVE domains. Additionally, Phafin2 has a section called the polyD motif that can block the PH domain from binding to specific cell membrane molecules. This study explored how these regions of Phafin2 have evolved across different species, focusing on the PH, FYVE, and polyD motifs. The findings suggest that the polyD motif is unique to Phafin2 and similar animal proteins, potentially indicating a unique role in animal cell functions. Further experiments examined how Phafin2 regulates itself and binds to cell membranes. The study identified specific amino acids in the PH domain crucial for interacting with the polyD motif. When these amino acids were altered, Phafin2 could no longer bind to synthetic polyD peptides, highlighting their importance. Finally, the research determined that Phafin2's ability to bind to membranes does not depend on the shape or curvature of the membrane.

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PEREIRA, José Ilhano da Silva. Hipersuperfícies mínimas de R4 com curvatura de Gauss-Kronecker nula. 2017. 44 f. Dissertação (Mestrado em Matemática) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-10-02T15:01:31Z No. of bitstreams: 1 2017_dis_jispereira.pdf: 596580 bytes, checksum: 3c2c1a16d4ce273bfb7c246f7926c01a (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, Estou devolvendo a Dissertação de JOSÉ ILHANO DA SILVA PEREIRA, pois há alguns erros a serem corrigidos. Os mesmos seguem listados a seguir. 1- FOLHA DE APROVAÇÃO (substitua a folha de aprovação, por outra que não contenha as assinaturas dos membros da banca examinadora) 2- NUMERAÇÃO INDEVIDA (a numeração indevida de página que aparece na folha de aprovação deve ser retirada) 3- RESUMO (retire o recuo de parágrafo presente no resumo e no abstract) 4- PALAVRAS-CHAVE (apenas o primeiro elemento de cada palavra-chave deve começar com letra maiúscula, assim reescreva as palavras-chave como no exemplo a seguir: Hipersuperfícies mínimas) 5- SUMÁRIO (Os títulos dos capítulos principais, que aparecem no sumário e no interior do trabalho, devem estar em caixa alta (letra maiúscula). Ex.: 2 PRELIMINARES 2.1 Tensores 6 – REFERÊNCIAS (retire o conjunto de “citações” à autores que aparece no final das referências bibliográficas, pois elas fogem ao padrão ABNT para a página das referências) Atenciosamente, on 2017-10-04T17:50:58Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-10-23T19:57:28Z No. of bitstreams: 1 2017_dis_jispereira.pdf: 333124 bytes, checksum: 37989a2f3787d5914a0c0553afd4e89f (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-11-01T12:35:13Z (GMT) No. of bitstreams: 1 2017_dis_jispereira.pdf: 333124 bytes, checksum: 37989a2f3787d5914a0c0553afd4e89f (MD5) / Made available in DSpace on 2017-11-01T12:35:13Z (GMT). No. of bitstreams: 1 2017_dis_jispereira.pdf: 333124 bytes, checksum: 37989a2f3787d5914a0c0553afd4e89f (MD5) Previous issue date: 2017-08-25 / This work does study the complete minimal hypersurfaces in the Euclidean space R4 , with Gauss-Kronecker curvature identically zero. Our main result is to prove that if f: M3 → R4 is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature boun-ded from below, then f(M3) splits as a Euclidean product L2 × R , where L2 is a complete minimal surface in R3 with Gaussian curvature bounded from below. Moreover, we show a result about the Gauss-Kronecker curvature of f, without any assumption on the scalar curvature. / Este trabalho tem como objetivo estudar as hipersuperfícies mínimas em R4, com curvatura de Gauss-Kronecker identicamente zero. Como resultado principal provamos que se f : M3 → R4 é uma hipersuperfície mínima com curvatura de Gauss-Kronecker identicamente zero, segunda forma fundamental não se anulando em nenhum ponto e curvatura escalar limitada inferiormente, então f(M3) se decompõe como um produto euclidiano do tipo L2 × R , onde L2 é uma superfície mínima de R3 com curvatura Gaussiana limitada inferiormente. Finalmente, apresentamos um resultado sobre a curvatura de Gauss-Kronecker de f sem nenhuma hipótese sobre a curvatura escalar.

[pt] A geometria diferencial provê um conjunto de medidas invariantes sob a ação de um grupo de transformações, em particular rígidas, afins e projetivas. Os invariantes por transformações rígidas são usados em quase todas as aplicações de computação gráfica e modelagem geométrica. O caso afim, por ser mais geral, permite estender essas ferramentas. Neste trabalho, propriedades geométricas são apresentadas no caso de superfícies paramétricas ou implícitas, em particular, a métrica afim, os vetores co-normal e normal afins e as curvaturas Gaussiana e média afins. Alguns resultados usuais de geometria Euclidiana, como a fórmula de Minkowski, são estendidos para o caso afim. Esse estudo permite definir estimadores das estruturas afins no caso de isossuperfícies. Porém, um cálculo direto dessas estruturas resulta em um grande número de operações e instabilidade numérica. Uma redução geométrica é proposta, obtendo fórmulas mais simples e mais estáveis numericamente. As propriedades geométricas incorporadas no Marching Cubes são analisadas e discutidas. / [en] Differential Geometry provides a set of measures invariant under a set of transformations, in particular rigid, affine, and projective. The invariants by rigid motions are using almost all applications of computer graphics and geometric modeling. The affine case, since it is more general, allows to extend these tools. In this work, geometric properties are presented in the case of parametric or implicit surfaces, in particular the affine metric, the conormal and normal vectors, and the affine Gaussian and mean curvatures. Some usual results of Euclidean geometry, as the Minkowski formula, are extended for the affine case. This study allows to define estimators of affines structure in the case of isosurfaces. Although, the direct calculation of these structures greatly increases the number of operations and numerical instabilities. A geometrical reduction is proposed obtaining a much simpler and numerical stabler formulae. The geometrical properties are incorporated in the Marching Cubes algorithms, then they are analyzed and discussed.

<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>

This thesis presents an experimental investigation of the effects of curvature on turbulent boundary layers using the Benchmark Validation Experiment for RANS and LES Investigations (BEVERLI) Hill setup. The study focuses on analyzing the flow behavior over a three-dimensional bump geometry that incorporates both concave and convex surfaces, with the aim of improving the understanding of the complex interactions among curvature, pressure gradients, and turbulence characteristics. The study examines the mean velocity, Reynolds shear stresses, pressure gradient, turbulence intensity, and pressure coefficient variations in relation to the bump curvature. The results are consistent with prior studies on the destabilizing influence of concave curvature with observations such as increased turbulence intensity, a decrease in mean velocity relative to the free-stream velocity U<sub>∞, and higher Reynolds stresses normalized by U<sup>2<sub>∞ throughout entire turbulent boundary layer, particularly in the near-wall region. Convex curvature results are consistent with prior study as well, which exhibits a stabilizing effect, shown to reduce turbulence intensity, an increase mean velocity relative to the free-stream velocity U<sub>∞, and lower Reynolds stresses normalized by U<sup>2<sub>∞ throughout entire boundary layer. This study also highlights the influence of pressure gradient effect, which acts with the curvature effect, impacts the boundary layer stability. This interaction is observed in amplification of turbulence with increasing of turbulence intensity and boundary layer growth. This stability particularly reflects on the embedded shear layers with inflection points which can create conditions for linear instabilities to grow, thus enhancing coherent turbulent motions. Furthermore, the thesis discusses the challenges in separating the influence of curvature from pressure gradient effects in current model, and proposes future research directions to address this issue. By conducting experiments under controlled pressure gradient flow conditions over concave and convex curvature, researchers can analyze the contributions of curvature effect separately from pressure gradient effect. Alternatively, using a hybrid RANS-LES model, will lead to a more precise understanding of flow dynamics over curved surfaces. / Master of Science / This thesis explores the influence of curvature on turbulent boundary layers, as it passes over a three-dimensional bump structure with both concave (curved inward) and convex (curved outward) surfaces. By conducting experimental data computed from prior studies of the BeVERLI project, the study investigates how the curvature of the surface affects the flow's mean velocity, Reynolds shear stresses, pressure gradients, and turbulence intensity. The findings highlighted the complex interactions between the curvature effect and pressure gradient effect on turbulent boundary layers, and provided valuable insights for future work references on the study of curvature effects on turbulent boundary layers.