Dissertations / Theses: 'Curvature'

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Fonseca, Aurineide Castro. "Conjectura da curvatura escalar normal." Universidade Federal do CearÃ, 2008. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=2846.

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Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico
O objetivo desta dissertaÃÃo Ã apresentar uma demonstraÃÃo para uma desigualdade pontual, denominada conjectura da curvatura escalar normal, a qual Ã vÃlida para subvariedades n-dimensionais, Mn, imersas isometricamente em formas espaciais Nn+m(c) de curvatura seccional constante c.
In this work we present a proof of the Normal Scalar Curvature Conjecture for submanifolds Mn, isometrically immersed into space forms Nn+m(c) of constant sectional curvature c.

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Choi, Yang Ho. "Curvature arbitrage." Diss., University of Iowa, 2007. http://ir.uiowa.edu/etd/166.

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Pereira, José Ilhano da Silva. "Hipersuperfícies mínimas de R4 com curvatura de Gauss-Kronecker nula." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/27052.

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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.
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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)
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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.

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Eskandari, Sam. "Curvature based Rendering." Thesis, Uppsala University, Department of Information Technology, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-129722.

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In this thesis work, mean curvature and Gaussian curvature are taken into account. Vertex normal calculation is one of the important parts of this thesis work. Vertex normal calculation is done in order to improve the appearance and smoothness of the surface in our model. Vertex normal is also one of the parts of the mean curvature calculation. Curvature based illumination is the main goal of this thesis work. To achieve our goal we have to define locally backscattered light, ambient attenuation and subsurface scattering based on both mean curvature and Gaussian curvature calculations. All of these calculations are also done in this thesis work. We cannot see anything without having the light, thus we need also a light source which acts as sun with different angles to the horizon. The sun and its angle are also simulated in this thesis work. Since this thesis work is based on local curvature based lighting model for rendering of snow, curvature calculations are applied to the lighting model. Then, mean and Gaussian curvature calculations also evaluated for this model and finally mean curvature and Gaussian curvature values are compared to another calculations method which those values are available in an ASCII file. The idea behind these comparisons is to determine whether the mean and Gaussian curvatures of my calculations or from the ASCII file are more suitable in general if it is possible to say and the advantages and disadvantages of these calculations if any.

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Dobbins, Allan C. (Allan Chalmers). "Endstopping and curvature." Thesis, McGill University, 1988. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=61913.

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Umur, Habib. "Flows with curvature." Thesis, Imperial College London, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.283576.

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Lope, Vicente Joe Moises. "Curvatura y fibrados principales sobre el círculo (Curvature and principal S 1 -bundles)." Master's thesis, Pontificia Universidad Católica del Perú, 2018. http://tesis.pucp.edu.pe/repositorio/handle/123456789/12829.

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The aim of this thesis is to study in detail the work of S. Kobayashi on the Riemannian geometry on principal S1-bundles. To be more precise, we explain how to obtain metrics with constant scalar curvature on these bundles. The method that we use is based in [18]. The basic idea behind Kobayashi’s construction is to slightly deform the Hopf fibration S1 ‹→ S2n+1 −→ CPn in a such a way that the corresponding sectional curvatures are not far from the produced by the standard metrics on the sphere and the complex projective space on the Hopf fibration. This deformations can be controlled applying the notions of Riemaniann and Kahlerian pinching (see Chapter 3). Furthermore, thanks to a technique developed by Hatakeyama in [14], it is possible to obtain less generic metrics but with a larger set of symmetries on the total space: Sasaki metrics. Actually, If one chooses as a base space a K¨ahler-Einstein manifold with positive scalar curvature one can obtain a Sasaki-Einstein metric.
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Junior, Ernani de Sousa Ribeiro. "Stability of spacelike hypersurfaces in foliated spacetimes." Universidade Federal do CearÃ, 2009. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=5097.

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CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior
Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico
Dado um espaÃo-tempo M─n+1 = I x Ã Fn Robertson-Walker generalizado onde Ã Ã a funÃÃo warping que verifica uma certa condiÃÃo de convexidade, vamos classificar hipersuperfÃcies tipo-espaÃo fortemente estÃveis com curvatura mÃdia constante. Mais precisamente, vamos mostrar que, considerando x : Mn→ M─n+1 uma hipersuperfÃcie tipo-espaÃo fortemente estÃvel, fechada imersa em M─n+1 com curvatura mÃdia constante H, se a funÃÃo warping Ã satisfaz Ãâ ≥ max {H Ãâ, 0} ao longo de M, entÃo Mn Ã maximal ou uma folha tipo-espaÃo Mto={to} x F, para algum to Є I.
Give a generalized M─n+1 = I xÃ Fn Robertson-Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given x : Mn → M─n+1 a closed, strongly stable spacelike hypersurfaces of M─n+1 with constant mean curvature H, if the warping function Ã satisfying Ã ≥ max {HÃ', 0} along M, is either maximal or a spacelike slice Mto = {to} x F, for some to Є I.

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Lubbe, Felix [Verfasser]. "Curvature estimates for graphical mean curvature flow in higher codimension / Felix Lubbe." Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2015. http://d-nb.info/1075867371/34.

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Robinson, Sebastian Thomas. "Curvature-based surface fairing." Thesis, University of Bath, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.488895.

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In the computer aided engineering environment, exceptionally smooth but irregular surfaces are often required, such as car bonnets. It is often a lengthy process to design these surfaces to the degree of smoothness and aesthetic beauty that is required by the designer. Smoothing these surfaces is known as fairing and a variety of techniques exist to tackle the problem in different ways. A new method of surface fairing is proposed and demonstrated in this thesis. Many conventional fairing methods use an agreeable curvature plot across the surface as proof of fairness, the method documented here takes the more holistic approach of constructing the improved surface from an agreeable curvature plot.

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Groenewald, Ben. "Substrate curvature measurement system." Master's thesis, University of Cape Town, 1999. http://hdl.handle.net/11427/19376.

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Industry often requires, in a variety of processes, the measurement of deformation induced in a solid object by mechanical stress. One such process is during the manufacture of very large scale integrated circuits (VLSI). During this process a substrate is coated with a thin film to protect the micro circuitry formed on the substrate. Due to the differences in thermal expansions between film and substrate, mechanical stresses can develop which may lead to deformation of the substrate surface. Any deformation of the substrate surface will result in mechanical stress in the interconnections of the circuitry, which could result in severe damage to the operation of the circuit. Different measurement techniques are available to measure the spherical deformation of substrates, with the latest known technique being a combination of a laser beam deflection and light scattering techniques. Many of the existing techniques reveal shortcomings, one of which is a 2-dimensional scanning capability with a minimum of moving components. Another shortcoming is the incapability of previous techniques to calculate the relative error which the measuring technique induces into the results. The aim of this study has been to develop an electro-optical system embodying the successful principles of these techniques in a system which will eliminate the shortcomings and produce results in excess of those previously recorded. In this work, we have concentrated on discussing the development of a system which will produce in situ real time monitoring of mechanical stresses in a solid. The system includes the minimization of system induced errors through the calculation of error voltage gains, and the introduction of a 2-dimensional scanning capability to determine the true position of the laser beam without prior knowledge of the initial substrate curvature. A four-quadrant position sensitive detector (PSD) with relevant Lab View software and programs were also introduced into the system.

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Colussi, Adeline. "Understanding membrane curvature sensing." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/270350.

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Eukaryotic cells are characterised by membranes with varied and dynamic compositions and shapes. Consequently, membrane-binding proteins are tuned to recognise and modify these membrane states to perform their functions. To study the curvature sensitivity of proteins, I have developed a single-particle assay using NanoSight technology that tracks the Brownian motion of particles to measure their size. I optimised this system to track fluorescently labelled lipid-binding domains bound to liposomes of different sizes moving freely in solution. The comparison of the size distribution of the total liposomes with the fluorescently labelled population allowed me to determine their curvature preferences. To validate the method I tested proteins from the Bin/Amphiphysin/Rvs (BAR) superfamily, which are inherently curved and have known curvature preferences. My method was capable of recapitulating the behaviour of BAR domains with different curvature preferences. I then expanded the range of targets and showed that this assay is also capable of detecting curvature preferences for a variety of other lipid-binding domain families. As such, I identified AKT PH domain as a new curvature-sensing domain. Finally, using the ENTH domain of Epsin1 that causes vesicle budding, I demonstrated that this method can also be used to study membrane remodelling. Trafficking involves generation and sensing of membrane curvature combined with recognition of specific cargo. Endophilin consists of a curvature-sensitive BAR domain followed by an SH3 (Src-homology 3) domain and has recently been identified in a clathrin-independent endocytosis pathway, FEME (fast endophilin-mediated endocytosis), involved in the uptake of cell surface receptors. Endophilin recognises ligands via its SH3 domain, binding G-protein coupled receptors (GPCRs) directly in their intracellular loop 3 and receptor tyrosine kinase (RTKs) via adaptor proteins. However, a specific recognition motif has not been identified yet. Here, using a combination of biophysical approaches and NMR spectroscopy, I characterised the Endophilin binding motif of ALIX (ALG-2-interacting protein X) adaptor protein and of the GPCR $\alpha$2A adrenergic receptor. Comparison of SH3-peptide models resulted in a putative Endophilin recognition site.

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Wang, Jianguo. "Global Finish Curvature Matched Machining." Diss., CLICK HERE for online access, 2005. http://contentdm.lib.byu.edu/ETD/image/etd1102.pdf.

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Karkanis, Tasso. "Curvature dependent implicit surface tiling." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape7/PQDD_0003/MQ45948.pdf.

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Dunn, Corey. "Curvature homogeneous pseudo-Riemannian manifolds /." view abstract or download file of text, 2006. http://proquest.umi.com/pqdweb?did=1188874491&sid=3&Fmt=2&clientId=11238&RQT=309&VName=PQD.

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Thesis (Ph. D.)--University of Oregon, 2006.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 146-147). Also available for download via the World Wide Web; free to University of Oregon users.

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Miller, James. "Shape curve analysis using curvature." Thesis, University of Glasgow, 2009. http://theses.gla.ac.uk/854/.

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Statistical shape analysis is a field for which there is growing demand. One of the major drivers for this growth is the number of practical applications which can use statistical shape analysis to provide useful insight. An example of one of these practical applications is investigating and comparing facial shapes. An ever improving suite of digital imaging technology can capture data on the three-dimensional shape of facial features from standard images. A field for which this offers a large amount of potential analytical benefit is the reconstruction of the facial surface of children born with a cleft lip or a cleft lip and palate. This thesis will present two potential methods for analysing data on the facial shape of children who were born with a cleft lip and/or palate using data from two separate studies. One form of analysis will compare the facial shape of one year old children born with a cleft lip and/or palate with the facial shape of control children. The second form of analysis will look for relationships between facial shape and psychological score for ten year old children born with a cleft lip and/or palate. While many of the techniques in this thesis could be extended to different applications much of the work is carried out with the express intention of producing meaningful analysis of the cleft children studies. Shape data can be defined as the information remaining to describe the shape of an object after removing the effects of location, rotation and scale. There are numerous techniques in the literature to remove the effects of location, rotation and scale and thereby define and compare the shapes of objects. A method which does not require the removal of the effects of location and rotation is to define the shape according to the bending of important shape curves. This method can naturally provide a technique for investigating facial shape. When considering a child's face there are a number of curves which outline the important features of the face. Describing these feature curves gives a large amount of information on the shape of the face. This thesis looks to define the shape of children's faces using functions of bending, called curvature functions, of important feature curves. These curvature functions are not only of use to define an object, they are apt for use in the comparison of two or more objects. Methods to produce curvature functions which provide an accurate description of the bending of face curves will be introduced in this thesis. Furthermore, methods to compare the facial shape of groups of children will be discussed. These methods will be used to compare the facial shape of children with a cleft lip and/or palate with control children. There is much recent literature in the area of functional regression where a scalar response can be related to a functional predictor. A novel approach for relating shape to a scalar response using functional regression, with curvature functions as predictors, is discussed and illustrated by a study into the psychological state of ten year old children who were born with a cleft lip or a cleft lip and palate. The aim of this example is to investigate whether any relationship exists between the bending of facial features and the psychological score of the children, and where relationships exist to describe their nature. The thesis consists of four parts. Chapters 1 and 2 introduce the data and give some background to the statistical techniques. Specifically, Chapter 1 briefly introduces the idea of shape and how the shape of objects can be defined using curvature. Furthermore, the two studies into facial shape are introduced which form the basis of the work in this thesis. Chapter 2 gives a broad overview of some standard shape analysis techniques, including Procrustes methods for alignment of objects, and gives further details of methods based on curvature. Functional data analysis techniques which are of use throughout the thesis are also discussed. Part 2 consists of Chapters 3 to 5 which describe methods to find curvature functions that define the shape of important curves on the face and compare these functions to investigate differences between control children and children born with a cleft lip and/or palate. Chapter 3 considers the issues with finding and further analysing the curvature functions of a plane curve whilst Chapter 4 extends the methods to space curves. A method which projects a space curve onto two perpendicular planes and then uses the techniques of Chapter 3 to calculate curvature is introduced to facilitate anatomical interpretation. Whilst the midline profile of a control child is used to illustrate the methods in Chapters 3 and 4, Chapter 5 uses curvature functions to investigate differences between control children and children born with a cleft lip and/or palate in terms of the bending of their upper lips. Part 3 consists of Chapters 6 and 7 which introduce functional regression techniques and use these to investigate potential relationships between the psychological score and facial shape, defined by curvature functions, of cleft children. Methods to both display graphically and formally analyse the regression procedure are discussed in Chapter 6 whilst Chapter 7 uses these methods to provide a systematic analysis of any relationship between psychological score and facial shape. The final part of the thesis presents conclusions discussing both the effectiveness of the methods and some brief anatomical/psychological findings. There are also suggestions of potential future work in the area.

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Skea, T. "Anisotropic cosmology and curvature invariants." Thesis, University of Sussex, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.381633.

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Yu, Fei. "Hollow core negative curvature fibres." Thesis, University of Bath, 2013. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.648936.

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Hollow core fibre (HCF) is a type of novel optical fibre which has lower refractive index in the hollow core than in the cladding. Total internal reflection (TIR) cannot explain the confinement of light to the core of a HCF. According to the confinement mechanism, the HCFs can be generally divided into hollow core photonic bandgap fibres and hollow core leaky mode fibres depending on their optical properties of the cladding structure. Hollow core negative curvature fibre (HC-NCF) is a kind of hollow core leaky mode fibre, which is defined by the negative curvature of the core boundary. This thesis presents my study of HC-NCFs over the last two years. My research has focused on developing low loss silica HC-NCFs and exploring the attenuation limit of HC-NCFs. Fifty different HC-NCFs were fabricated, which covered the spectral range from 800 nm to 4.5 μm. Minimum attenuations of 24.4 dB/km and 85 dB/km were measured at around 2400 nm wavelength and 4000 nm respectively, which are the best achieved in HCFs at these wavelengths to the best of my knowledge. The limits of HC-NCF attenuation were revealed by analysing the data from HC-NCFs scaled for minimum attenuation in different spectral regions. Other properties of HC-NCFs, including bending loss and dispersion, were also studied experimentally. By using white light interferometery, a low group velocity dispersion (GVD) was found in HC-NCFs, which agrees well with simulations. The bending loss of HC-NCFs was preliminarily studied by measuring the transmission spectra under different bending conditions. Significant bending loss was found when the bending radius was less than 15 cm. Numerical simulations were performed using COMSOL software to study the properties of HC-NCFs. In the simulations, it was found that the capillary thickness is the most important factor determining the attenuation of HC-NCFs. These results were used to explain the experimental results. This thesis is comprised of seven chapters. Chapter 1 and 2 supply background material that helps to understand the light guidance mechanism of HC-NCFs. My original work is presented in Chapter 3, 4, 5, and 6. Chapter 7 includes a summary and suggestions for future work.

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Hinde, Colin Douglas. "The essence of Ricci curvature." Diss., Restricted to subscribing institutions, 2008. http://proquest.umi.com/pqdweb?did=1619436071&sid=1&Fmt=2&clientId=1564&RQT=309&VName=PQD.

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Wang, Yuhao. "Advanced Theory of Field Curvature." Diss., The University of Arizona, 2014. http://hdl.handle.net/10150/325494.

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Classical field curvature theory emphasizes the Petzval theorem, which models field curvature aberration to the 4th order. However, modern lens designs use aspheric surfaces. These surfaces strongly induce higher order field curvature aberration which is not accounted for Petzval field curvature. This dissertation focuses on developing higher order field curvature theories that are applied to highly aspheric designs. Three new theories to control field curvature aberration are discussed. Theory 1: an aspheric surface that is close to the image and has two aspheric terms sharply reduces field curvature by 85%. Theory 2: an aspheric surface that is farther from the image plane induces astigmatism to balance Petzval field curvature. Theory 3: oblique spherical aberration can be induced to balance Petzval field curvature. All three theories are applied to real design examples including the following lenses: cellular phone, wide angle, fast photographic, and zoom lenses. All of the analyses results are consistent with the theories. Moreover, two types of novel aspheric surfaces are proposed to control field curvature. Neither of the surfaces are polynomial-type surfaces. Examples show that the novel aspheric surfaces are equivalent to even aspheric surfaces with two aspheric coefficients in terms of field curvature correction. The study on field curvature correction using aspheric surfaces provides an alternative method to use when aspheres are accessible. Overall, this dissertation advances the theory of field curvature aberration, and it is particularly valuable to evaluate highly aspheric designs when Petzval theory is inapplicable.

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Ligo, Richard G. "Conformal transformations, curvature, and energy." Diss., University of Iowa, 2017. https://ir.uiowa.edu/etd/5550.

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Space curves have a variety of uses within mathematics, and much attention has been paid to calculating quantities related to such objects. The quantities of curvature and energy are of particular interest to us. While the notion of curvature is well-known, the Mobius energy is a much newer concept, having been first defined by Jun O'Hara in the early 1990s. Foundational work on this energy was completed by Freedman, He, and Wang in 1994, with their most important result being the proof of the energy's conformal invariance. While a variety of results have built those of Freedman, He, and Wang, two topics remain largely unexplored: the interaction of curvature and Mobius energy and the generalization of the Mobius energy to curves with a varying thickness. In this thesis, we investigate both of these subjects. We show two fundamental results related to curvature and energy. First, we show that any simple, closed, twice-differentiable curve can be transformed in an energy-preserving and length-preserving way that allows us to make the pointwise curvature arbitrarily large at a point. Next, we prove that the total absolute curvature of a twice-differentiable curve is uniformly bounded with respect to conformal transformations. This is accomplished mainly via an analytic investigation of the effect of inversions on total absolute curvature. In the second half of the thesis, we define a generalization of the Mobius energy for simple curves of varying thickness that we call the "nonuniform energy." We call such curves "weighted knots," and they are defined as the pairing of a curve parametrization and positive, continuous weight function on the same domain. We then calculate the first variation formulas for several different variations of the nonuniform energy. Variations preserving the curve shape and total weight are shown to have no minimizers. Variations that "slide" the weight along the curve are shown to preserve energy is special cases.

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Feitosa, Samuel Barbosa. "FolheaÃÃes por hipersuperfÃcies de curvatura mÃdia constante." Universidade Federal do CearÃ, 2009. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=3552.

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O presente trabalho apresenta resultados objetivando classificar folheaÃÃes de codimensÃo 1 em variedades Riemannianas cujas folhas tem curvatura mÃdia constante. O principal resultado Ã o teorema de Barbosa-Kenmotsu-Oshikiri([3]), Teorema: Seja M uma variedade Riemanniana compacta com curvatura de Ricci nÃo negativa e F um folheaÃÃo de codimensÃo 1 e classe C3 de M, transversalmente orientÃvel, cujas folhas tem curvatura mÃdia constante. EntÃo, qualquer folha de F Ã uma subvariedade totalmente geodÃsica de M. AlÃm disso, M Ã localmente um produto Riemanniano de uma folha de F e uma curva normal e a curvatura de Ricci na direÃÃo normal Ãs folhas Ã zero. O resultado anterior nÃo pode ser estendido para o caso onde M Ã nÃo compacta. Uma folheaÃÃo contra-exemplo pode ser construÃda a partir de uma funÃÃo f que nÃo satisfaz a conjectura de Bernstein. No final, sÃo apresentados resultados recentes sobre os problemas abordados e uma prova da desigualdade de Heinz-Chern
In this paper, we work showing results aiming classify foliations of codimension-one in Riemannian manifolds whose leaves have constant mean curvature. The main result is the theorem by Barbosa-Kenmotsu-Oshikiri([3]). Theorem: LetM be a compact Riemannian manifold with nonnegative Ricci curvature e F, a codimensiononeC3-foliation of M whose leaves have constant mean curvature. The any leaf of F is totally geodesic submanifold of M. Futhermore M is locally a Riemannian product of a leaf of F and a normal curve,and the Ricci curvature in the direction normal to the leaves is zero. The previous result can not be extended for the case where M is not compact. A foliation counterexample can be built from a function f that does not satisfy the Bernsteinâs conjecture. At the end, they are present recent results about the boarded problems and a proof of the Heinz-Chern inequality.

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Lima, Filipe MendonÃa de. "Estimativas extrÃnsecas de autovalores de operadores elÃpticos em hipersuperfÃcies." Universidade Federal do CearÃ, 2010. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=5098.

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CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior
O objetivo desse trabalho Ã mostrar estimativas superiores para o menor autovalor nÃo-nulo lambda1 do operador de Laplace-Beltrami delta. Os resultados que se seguem foram encontrados por R. Reilly [1] e a dupla A. El Soufi e S. Ilias [2]. A estimativa de Reilly Ã feita para variedades imersas no espaÃo euclidiano Rn, e a de Soufi-Ilias para variedades conformemente imersas na esfera Sn. A partir daÃ concluiremos o resultado, tambÃm de Soufi-Ilias [2], para subvariedades do espaÃo hiperbÃlico Hn.
The aim of this works is to show superior estimatives to the least non-zero eingenvalue lambda1 of the Laplace-Beltrami operator delta. The forthcoming results were discovered by Reilly [1] and the duo A. El Soufi and S. Ilias [2]. Reillyâs Estimative was calculated for immersed manifolds in the Euclidian Space Rn, and Soufi-Ilias for conformally immersed manifolds in the sphere Sn.Then, we conclude the result, again by Soufi-Ilias [2], for submanifolds of the hyperbolic space Hn.

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Oliveira, Antonio Edinardo de. "Uma caracterizaÃÃo do produto Sk (cos θ) x Sn-k (sen θ) na esfera euclidiana S^ (n+1)." Universidade Federal do CearÃ, 2009. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=11689.

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Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico
Neste trabalho consideraremos hipersuperfÃcies n-dimensionais com curvaturas escalar constante na esfera unitÃria S^(n+1). Caracterizaremos as hipersuperfÃcies dadas por produtos de esferas, cuja dimensÃo Ã n, na esfera unitÃria S^(n+1) e mostraremos que existe vÃrias hipersuperfÃcies compactas com curvaturas escalar constante na esfera unitÃria S^(n+1) que nÃo sÃo congruentes entre si. Em particular, provaremos que se M Ã uma hipersuperfÃcie n-dimensional (n>3) completa, localmente conformemente plana com curvatura escalar constante n(n-1)r na esfera unitÃria S^(n+1), entÃo r Ã maior do que um valor prÃ-estabelecido e sÃo provados dois resultados, um envolvendo isometrias e o outro de existÃncia, quando r e S satisfazem determinadas condiÃÃes, onde S Ã o quadrado da norma se segunda forma fundamental de M.
In this paper we consider n-dimensional hypersurfaces with constant scalar curvature in the unit sphere S ^ (n +1). Characterize the hypersurfaces given by products of spheres whose size is n, the unit sphere S ^ (n +1) and show that there is more compact hypersurfaces with constant scalar curvature in the unit sphere S ^ (n +1) that are not congruent itself. In particular, prove that M is an n-dimensional hypersurface (n> 3) complete with buckle locally flat accordingly constant scalar n (n-1) on the unit sphere S r ^ (n +1) is greater than r a pre-established and two results are proven value, and the other one involving isometries of existence, when are S satisfy certain conditions, where S is the square of the standard is second fundamental form of M.

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Kevin, Tom. "Sub-1V Curvature Compensated Bandgap Reference." Thesis, Linköping University, Department of Electrical Engineering, 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-2585.

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This thesis investigates the possibility of realizing bandgap reference crcuits for processes having sub-1V supply voltage. With the scaling of gate oxide thickness supply voltage is getting reduced. But the threshold voltage of transistors is not getting scaled at the same rate as that of the supply voltage. This makes it difficult to incorporate conventional designs of bandgap reference circuits to processeshaving near to 1V supply voltage. In the first part of the thesis a comprehensive study on existing low voltage bandgap reference circuits is done. Using these ideas a low-power, low-voltage bandgap reference circuit is designed in the second part of the thesis work.

The proposed bandgap reference circuit is capable of generating a reference voltage of 0.730V. The circuit is implemented in 0.18µm standard CMOS technology and operates with 0.9V supply voltage, consuming 5µA current. The circuit achieves 7 ppm/K of temperature coefficient with supply voltage range from 0.9 to 1.5V and temperature range from 0 to 60C.

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Johansson, Björn. "Multiscale Curvature Detection in Computer Vision." Licentiate thesis, Linköping University, Linköping University, Computer Vision, 2001. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-54966.

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This thesis presents a new method for detection of complex curvatures such as corners, circles, and star patterns. The method is based on a second degree local polynomial model applied to a local orientation description in double angle representation. The theory of rotational symmetries is used to compute curvature responses from the parameters of the polynomial model. The responses are made more selective using a scheme of inhibition between different symmetry models. These symmetries can serve as feature points at a high abstraction level for use in hierarchical matching structures for 3D estimation, object recognition, image database search, etc.

A very efficient approximative algorithm for single and multiscale polynomial expansion is developed, which is used for detection of the complex curvatures in one or several scales. The algorithm is based on the simple observation that polynomial functions multiplied with a Gaussian function can be described in terms of partial derivatives of the Gaussian. The approximative polynomial expansion algorithm is evaluated in an experiment to estimate local orientation on 3D data, and the performance is comparable to previously tested algorithms which are more computationally expensive.

The curvature algorithm is demonstrated on natural images and in an object recognition experiment. Phase histograms based on the curvature features are developed and shown to be useful as an alternative compact image representation.

The importance of curvature is furthermore motivated by reviewing examples from biological and perceptual studies. The usefulness of local orientation information to detect curvature is also motivated by an experiment about learning a corner detector.

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Bårman, Håkan. "Hierarchical curvature estimation in computer vision." Doctoral thesis, Linköpings universitet, Bildbehandling, 1991. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-54887.

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This thesis concerns the estimation and description of curvature for computer vision applications. Different types of multi-dimensional data are considered: images (2D); volumes (3D); time sequences of images (3D); and time sequences of volumes (4D). The methods are based on local Fourier domain models and use local operations such as filtering. A hierarchical approach is used. Firstly, the local orientation is estimated and represented with a vector field equivalent description. Secondly, the local curvature is estimated from the orientation description. The curvature algorithms are closely related to the orientation estimation algorithms and the methods as a whole give a unified approach to the estimation and description of orientation and curvature. In addition, the methodology avoids thresholding and premature decision making. Results on both synthetic and real world data are presented to illustrate the algorithms performance with respect to accuracy and noise insensitivity. Examples illustrating the use of the curvature estimates for tasks such as image enhancement are also included.

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Raza, Aleksis. "Scalar curvature and multiplicity-free actions." Thesis, Imperial College London, 2006. http://hdl.handle.net/10044/1/1264.

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This thesis comprises of three main results. First, we use Guillemin-Abreu theory from KÃƒ?Ã‚Â¤hler toric geometry to derive a formula for the scalar curvature of SU(n)-invariant KÃƒ?Ã‚Â¤hler metrics on Cn n f0g or, equivalently, spaces of the form S2nÃƒ?Ã‚Â¡1 Ãƒ?Ã‚Â£ (0,Ãƒ?Ã‚Â¥). Second, we use the aforementioned formula to describe a U(n)-invariant, scalar-flat, KÃƒ?Ã‚Â¤hler metric on the blow-up bCn of Cn at the origin in symplectic coordinates. This metric is the generalization of the well-known Burns metric on bC2. Third, we use an equivariant version of an Ãƒ?Ã‚?infinite dimensional moment map frameworkÃƒ?Ã‚? to derive a formula for the scalar curvature of SU(n)-invariant, KÃƒ?Ã‚Â¤hler metrics on the multiplicity-free SU(n)-space Xn of the form SU(n) Ãƒ?Ã‚Â£ U where U is a certain open connected subset of the Lie algebra of a Cartan torus in SU(n) i.e. RnÃƒ?Ã‚Â¡1.

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Suen, Tsz-yin Simon. "Curvature domain stitching of digital photographs." Click to view the E-thesis via HKUTO, 2007. http://sunzi.lib.hku.hk/hkuto/record/B38800901.

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Ford, Robert Kuperberg Krystyna. "Path curvature on a convex roof." Auburn, Ala., 2007. http://repo.lib.auburn.edu/2007%20Fall%20Dissertations/Ford_Robert_45.pdf.

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Murali, Shobhana. "Curvature, isoperimetry, and discrete spin systems." Diss., Georgia Institute of Technology, 2001. http://hdl.handle.net/1853/28843.

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Dammerman, Brandon. "Metrics of special curvature with symmetry." Thesis, University of Oxford, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.408688.

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Foltin, Georg [Verfasser]. "Orientational Order and Curvature / Georg Foltin." Aachen : Shaker, 2005. http://d-nb.info/1186582022/34.

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Suen, Tsz-yin Simon, and 孫子彥. "Curvature domain stitching of digital photographs." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2007. http://hub.hku.hk/bib/B38800901.

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Kell, Martin. "On curvature conditions using Wasserstein spaces." Doctoral thesis, Universitätsbibliothek Leipzig, 2014. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-149614.

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This thesis is twofold. In the first part, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given and a new curvature condition on abstract metric measure spaces is defined. In the second part of the thesis a proof of the identification of the q-heat equation with the gradient flow of the Renyi (3-p)-Renyi entropy functional in the p-Wasserstein space is given. For that, a further study of the q-heat flow is presented including a condition for its mass preservation.

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Hickin, D. G. "Bogomol’nyi equations on constant curvature spaces." Thesis, Durham University, 2004. http://etheses.dur.ac.uk/3037/.

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This thesis is concerned with the anti-self-dual Yang-Mills equations and their reductions to Bogomol’nyi equations on constant curvature spaces. Chapters 1 and 2 contain introductory material. Chapter 1 discusses the origin of the equations in particle physics and their role in integrable systems. Chapter 2 describes the equations and the reduction process and outlines the construction of solutions via the twistor transform. In Chapter 3 we consider Bogomol’nyi equations on (2 + 1)-dimensional manifolds and show that for constant curvature space-times the equations are integrable and consider solutions in the negative scalar curvature case. In Chapter 4 we cover the negative scalar curvature case in more detail, constructing a number of soliton solutions including non-trivial scattering and consider the zero-curvature limit. In Chapter 5 we consider Bogomornyi equations in 3- diniensional hyperbolic space, derive an ansatz for solutions of the equation and use it to construct a number of new solutions. Chapter 6 contains concluding remarks.

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Petursson, Ingi Orn. "Almost positive isotropic curvature and surgery." Thesis, University of Warwick, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.398718.

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JUNIOR, ROGERIO VAZ DE ALMEIDA. "CURVATURE ESTIMATORS FOR CURVES IN R4." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2011. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=21598@1.

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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO
Vamos apresentar neste trabalho dois métodos para calcular as propriedades diferenciais geométricas de uma curva discreta no R4. O primeiro é baseado em aproximações por comprimento de arco. O segundo é baseado na metodologia de derivação discreta. Esses métodos estimam numericamente as curvaturas k1, k2 e k3 e os vetores tangente, normal, binormal e trinormal para cada ponto da curva. São apresentados também cálculos dessas propriedades geométricas para curvas tanto na forma paramétrica como na forma implícita, com o objetivo final de testar a consistência dos métodos propostos comparando-os aos resultados teóricos.
We present new algorithms for computing the diferential geometry properties of a discrete curve in R4 based on two different methods: arc-lenght aproximation and discrete derivatives.

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39

Martin, Kimberly Cordwint. "Direction of cellular protrusions by curvature." Thesis, University of Edinburgh, 2017. http://hdl.handle.net/1842/23553.

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Developmental processes involving symmetry-breaking of homogeneous cell populations into leaders and followers are found in many important contexts. Cells constrained by culture on two-dimensional scaffolds, as well as in three-dimensional shapes, appear to respond to convex curves with an increasing propensity to protrude, while concave curves in contrast appear to inhibit protrusion. This has interesting implications in terms of a potential positive feedback loop. This feedback may act in symmetry-breaking, through amplification of initial stochastic differences in cell shape, and also in collective migration, through reinforcing and directing the coherent movement of collectives. In this study, epithelial cells were cultured on two-dimensional micropatterns with variable curvatures to examine the effect of edge geometry and other variables on the likelihood of protrusions forming. This platform allowed the quantification of F-actin-based protrusions at the periphery of multicellular epithelial clusters, in segments defined by cluster edge curvature. The initial observations confirmed reports in the literature of preferential localisation of protrusions at more convex regions, and relative inhibition at more concave regions. A previously-published work has postulated a role for secreted modulators of motility, with the shape of a group of cells determining the concentration of diffusing morphogen each individual cell is exposed to. To test this hypothesis, a low-shear flow culture chamber was used to disrupt the putative gradients. Despite theoretical and empirical support for the sufficiency of the flow condition to disrupt autocrine signalling, micropatterned cells cultured under flow showed no significant differences from the control condition. These findings form the basis of a manuscript which has been accepted for publication by the Journal of Anatomy. The results of an Atomic Force Microscopy (AFM) study carried out by collaborators were suggestive of a role for cellular mechanotransduction in sensing and responding to micropattern curvature. Differential calcium channel mechanoactivation was hypothesised as being one potential mechanism underlying the response to curvature, given the known involvement of mechanosensitive ion channels in cellular responses to force and substrate stiffness, and the multiple roles of calcium in cellular motility. Artificially increasing cytosolic calcium levels with Ionomycin reduced protrusion rates at convex curves. However, treatment with BAPTA-AM to sequester intracellular calcium had no effect on protrusion rates. ROCK inhibitor, in contrast, increased protrusion rates at concave curves, and Blebbistatin increased protrusion rates globally. These results together are suggestive of differential control of myosin depending on local curvature: cyclic and driven by calcium-activation of MLCK in the convex regions (with lamellipodia undergoing protrusion-retraction cycles), versus sustained and controlled by ROCK in the concave regions (where lamellipodia are inhibited). The unexpected finding that protrusions at convex regions were resistant to the actin cytoskeleton-disrupting drug Cytochalasin D may point to a role for a tropomyosin isoform in defining the differing mechanical characteristics of the actin cytoskeleton in response to local curvature. In addition, the previously-noted lack of effect of BAPTA-AM treatment (which has been shown to interfere with dynamic microtubules) is suggestive of a role for stabilised microtubules in protrusions at convex regions. These indications of unique characteristics to the protrusions promoted by convex curvature give added support to the curvature-protrusion feedback model, and its relevance to tissue morphogenesis. In summary, this work provides evidence against a previously-published suggested mechanism for the curvature-protrusion feedback loop that is proposed to act during epithelial morphogenesis, and evidence in support of a role for a calcium-based mechanism in driving the initiation and maintenance of leader cells in migrating epithelial sheets. Further work is called for in characterising the protrusions promoted by convex curvature, and the mechanisms controlling them. This area is of significance in gaining greater understanding of tissue morphogenesis in pathogenesis and development, and of potential value in tissue engineering applications.

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Sakellaris, Zisis. "Minimization of curvature in conformal geometry." Thesis, University of Bath, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.687359.

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Chu-Ketterer, Lucinda-Joi. "Exploration of Curvature Through Physical Materials." Scholarship @ Claremont, 2016. http://scholarship.claremont.edu/pitzer_theses/68.

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Parametric equations are commonly used to describe surfaces. Looking at parametric equations does not provide tangible information about an object. Thus through the use of physical materials, an understanding of the limitations of the materials allows someone to gain a broader understanding of the surface. A M$\ddot{o}$bius strip and Figure 8 Klein bottle were created through knitting due to the precision and steady increase in curvature allowed through knitting. A more standard Klein bottle was created through crochet due to the ease in creating quick increases in curvature. Both methods demonstrate the change in curvature for both surfaces where the M$\ddot{o}$bius strip and Figure 8 Klein bottle have slower changes in curvature, but the classic Klein bottle has a quicker change in curvature.

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Merlin, Gonzalez Cesar Antonio. "Gravitational self-force from curvature scalars." Thesis, University of Southampton, 2015. https://eprints.soton.ac.uk/384183/.

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Betteridge, Benjamin Grant. "LCM Permeability Characterization Over Mold Curvature." BYU ScholarsArchive, 2020. https://scholarsarchive.byu.edu/etd/8508.

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Composite flow simulation tools for LCM processing can be expensive and time-consuming but necessary to design a mold system with proper placement of resin inlets and vacuum outlets. Composites manufacturing engineers would benefit from data regarding the impact of mold curvature radius on resin flow. This could help determine whether or not a particular part and mold would require expensive simulation software designed to handle complex flow paths through curved fabric architectures exhibiting variable permeability over the curvature, or if simple flow modeling would provide accurate enough simulations for sound tooling setup decision making. Four molds, with double curvature having equal radii, were fabricated with radii ranging from 3.2 to 25.4 mm to characterize the permeability of two different fiber reinforcements 1) a carbon biaxial NCF and 2) a fiberglass CSM over the mold curvatures. Three infusions of each material type were conducted on each of the 4 molds for a total of 24 test infusions. Flow front position vs. time data was captured during each experimental infusion. The permeability in the bend regions, KB, was first estimated by the integrated form of Darcy's Law to evaluate the permeability for average flow across the entire bend region. This was done for both the convex and concave regions using a geometric estimate for the increased compaction in the bend regions. The permeability increases as the tool radius increases, and the rate of increase diminishes as the tool radius increases and the permeability approaches the flat region permeability. An estimate of KB for VI was then made by applying a ratio calculated from the resulting permeability from the rigid- and VI-based models in the flat regions. Generic power law fits are reported that could be used in LCM process simulation, to give a model to estimate the permeability for any bend in the reinforcement part geometry. The results suggest that any curve with a radius higher than 25 mm requires no adjustment to the flat permeability.

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Liu, Yang. "Modular curvature for toric noncommutative manifolds." The Ohio State University, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=osu1440121460.

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Lovri?, Miroslav. "L(n+1/2)-curvature pinching /." The Ohio State University, 1990. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487681148541457.

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46

Smit, Philip C. "An electrogoniometer to measure spinal curvature." Thesis, University of Strathclyde, 2014. http://digitool.lib.strath.ac.uk:80/R/?func=dbin-jump-full&object_id=25724.

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Biomechanical motion capture is the process of recording the movements of people or animals. As an analysis tool it offers valuable insight into human motion and is useful to monitor treatment during rehabilitation. The spine in particular receives a significant amount of attention by biomechanical researchers, as spinal health is directly related to the quality-of-life of an individual. Spinal motion capture improves the understanding of the function and vulnerabilities of the spine as a mechanical structure and the analysis of spinal kinematics, in conjunction with spinal loading, offers a method for analysing therapeutic interventions. Numerous motion capture systems and devices are currently available, each with its own strengths and weaknesses. The systems or devices selected by researchers are usually determined by study objectives. For example, a video motion capture system would clearly not be appropriate in a study designed to monitor lower back movements of factory workers. Instead, unobtrusive accelerometry based devices would be more suitable to measure kinematics in a free-living environment. Accelerometry has its drawbacks however. It is limited to only global pitch and roll measurements and requires a subject to make relatively slow movements (i.e. the acceleration component of movement measured by the accelerometer must be significantly smaller than one g). In general, trade-offs exist between accuracy, obtrusiveness, ease-of-use, cost, mobility (degrees-of-freedom) and clinical versus free-living measurements. This thesis proposes an electrogoniometer, which meets many of the above mentioned criteria. The electrogoniometer aims to be accurate yet unobtrusive, low cost (perhaps less than £5 000, compared to the £100 000 price tag of high-end marker-based video motion capture system) and measures high mobility movement (typically the rotation components of a spinal motion segment) and do so within a free-living setting. The electrogoniometer is composed of discrete goniometers, referred to as goniometer-nodes. The goniometer-nodes are chained serially together to construct a multi degree-of-freedom electrogoniometer. The goniometer-nodes consist of mechanical structures embedded with optical sensors, each capable of measuring four degrees-of-freedom (three rotations and one translation). The mechanical structure's articulation is determined by processing the optical sensor data using the principles of triangulation and trilateration. The articulation is measured relative to a local reference frame (i.e. relative to the proximal-end of the node). Since local reference frame measurements are involved, accuracy and precision are important. Poor accuracy and precision will result in measurement errors propagating through the chain. The rotation accuracy is estimated to be better than 2° per axis (which is much less than the typical 5° accuracy of commercial goniometers) and a displacement (translation) accuracy of less than 0.2 mm. Precision is estimated better than 0.5° degrees per axis and 0.1 mm. The device is particularly suitable to measure spinal movement. It is attached to the backof a subject, similar to commercial electrogoniometers. It monitors the spinal kinematics on a continuous basis and transmits the data to a computer via a wireless adaptor. The kinematic data is then available for further analysis. This thesis initially investigates the mechanical and sensor design of the goniometer-nodes. A mechanical composite structure consisting of an universal (two rotations) and cylindrical (one rotation and translation) joint was utilised. Optical emitter-detector pairs were embedded within the structure, and a mathematical model was derived to predict the response of the detectors based upon the kinematic input. A custom instrument was developed to calibrate the nodes. Five nodes were assembled and calibrated, and then chained together to produce the electrogoniometer. The second part of the thesis evaluates the device. Reflective triads were attached to the base of each node within the chain. The device was then manipulated manually and compared against a video motion capture system for accuracy and precision. Analysis of the results showed a local reference frame accuracy and precision of 1.9 ± 1.0° per axis for rotation and 3.5 ± 1.8 mm for translation. The video captured measurements were also compared to the calibration results and proved to be worse than predicted. The cause was traced to the calibration instrument and the measurement method. Although the accuracy and precision specification were not met, it was concluded that the proof-of-concept electrogoniometer demonstrated here has merit as a low-cost motion capture device. The optical measurement method from which the electrogoniometer kinematics are determined, shows promise as a novel kinematic sensing method. It was concluded that with further refinement and improvements of the custom-build calibration instrument, the accuracy and precision requirements can be met. Nonetheless, the concepts and principles have been shown to be valid, and with additional resources, this electrogoniometer can be a viable biomechanical research device.

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47

Prasad, M. N. Nagendra. "Moment-curvature relationships in reinforced concrete." Thesis, This resource online, 1993. http://scholar.lib.vt.edu/theses/available/etd-07112009-040255/.

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48

Yokoyama, Shuichiro. "Primordial curvature perturbations in inflationary universe." 京都大学 (Kyoto University), 2008. http://hdl.handle.net/2433/136882.

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ALESSANDRONI, ROBERTA. "Evolution of hypersurfaces by curvature functions." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2008. http://hdl.handle.net/2108/661.

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Consideriamo un'ipersuperficie liscia di ℝⁿ⁺¹, con n≥2, e la sua evoluzione secondo una classe di flussi geometrici. La velocità di questi flussi ha direzione normale alla superficie e il modulo è una funzione simmetrica delle curvature principali. Inizialmente mostriamo alcune proprietà generali di questi flussi e calcoliamo l'equazione di evoluzione per una generica funzione omogenea delle curvature principali. In particolare applichiamo il flusso con velocità S=(H/(logH)), dove H è la curvatura media a meno di una costante, ad una superficie con curvatura media positiva per ottenere delle stime di convessità. Usando solamente il principio del massimo dimostriamo che, su un limite di riscalamenti delle superfici che si evolvono vicino alla singolarità, la parte negativa della curvatura scalare tende a zero. La parte successiva è dedicata allo studio di un'ipersuperficie convessa che si evolve secondo potenze della curvatura scalare: S=R^{p}, con p>1/2. Si dimostra che se la superficie iniziale soddisfa delle stime di "pinching" sulle curvature principali allora si contrae ad un punto in tempo finito e la forma delle superfici che si evolvono approssima sempre più quella di una sfera. In questo caso il grado di omogeneità, strettamente maggiore di uno, permette di concludere la dimostrazione della convergenza ad un "punto rotondo" tramite il solo principio del massimo, evitando l'uso di stime integrali. Viene anche costruito un esempio di superficie convessa che forma una singolarità di tipo "neck pinching". Infine studiamo il caso di un grafico intero su ℝⁿ con crescita al più lineare all'infinito e mostriamo che un grafico che si evolve secondo un qualsiasi flusso nella classe considerata rimane un grafico. Inoltre dimostriamo un risultato di esistenza per tempi lunghi per i flussi con velocità S=R^{p} con p≥1/2 e descriviamo delle soluzioni esplicite per grafici a simmetria di rotazione.
We consider a smooth n-dimensional hypersurface of ℝⁿ⁺¹, with n≥2, and its evolution by a class of geometric flows. The speed of these flows has normal direction with respect to the surface and its modulus S is a symmetric function of the principal curvatures. We show some general properties of these flows and compute the evolution equation for any homogeneous function of principal curvatures. Then we apply the flow with speed S=(H/(logH)), where H is the mean curvature plus a constant, to a mean convex surface to prove some convexity estimates. Using only the maximum principle we prove that the negative part of the scalar curvature tends to zero on a limit of rescalings of the evolving surfaces near a singularity. The following part is dedicated to the study of a convex initial manifold moving by powers of scalar curvature: S=R^{p}, with p>1/2. We show that if the initial surface satisfies a pinching estimate on the principal curvatures then it shrinks to a point in finite time and the shape of the evolving surfaces approaches the one of a sphere. Since the homogeneity degree of this speed is strictly greater than one, the convergence to a "round point" can be proved using just the maximum principle, avoiding the integral estimates. Then we also construct an example of a non convex surface forming a neck pinching singularity. Finally we study the case of an entire graph over ℝⁿ with at most linear growth at infinity. We show that a graph evolving by any flow in the considered class remains a graph. Moreover we prove a long time existence result for flows where the speed is S=R^{p} with p≥1/2 and describe some explicit solutions in the rotationally symmetric case.

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50

Frontini, Matteo. "Pseudoconvessità e curvatura." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2012. http://amslaurea.unibo.it/3853/.

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Questo lavoro parte da un’estensione analitica del concetto di convessità geometrica, dimostrandone l’equivalenza e introducendo la forma di Levi, per arrivare alla definizione di pseudoconvessità. Nel secondo capitolo si introduce il concetto di curvatura di Levi, dandone alcune caratterizzazioni ed esempi, fino a dimostrare la stima isoperimetrica che lega curvatura di Levi e misura di un insieme. Nell’ultimo capitolo si definiscono una serie di operatori di curvatura, in relazione con la forma di Levi, che permettono di dimostrare alcuni teoremi di confronto.

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Dissertations / Theses on the topic 'Curvature'

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