Orientador: Caio Jose Colletti Negreiros / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T10:04:33Z (GMT). No. of bitstreams: 1 Grama_LinoAndersondaSilva_M.pdf: 862309 bytes, checksum: ac6a88c1ff96ef74d7a840ce591336f5 (MD5) Previous issue date: 2008 / Resumo: O principal objetivo desta dissertação 'e apresentar a construção e a classificação das aplicações harmônicas de S2 em U(n), baseado nas idéias de K.Uhlenbeck. Apresentamos um exemplo de aplicação harmônica em U(4) e provamos que tal exemplo 'e, de fato, uma aplicação harmônica não-holomorfa na variedade de Grassman G2(C4), de 2-planos em C4.Demonstramos o teorema de Valli sobre o espectro da energia e, por fim, parametrizamos o conjunto Harm(S2, U(n)), de todas aplicações harmônicas de S2 em U(n), fornecendo uma classifica¸c¿ao para tais aplicações, seguindo o trabalho de J.C.Wood / Abstract: This dissertation is concerned with the construction and classification of harmonic maps from S2 on U(n), according to K. Uhlenbeck. We construct an example of harmonic map on U(4) and prove that this example is, in fact, a non-holomorphic harmonic map in the Grassmann manifold G2(C4) of 2-plans on C4. We also prove the theorem of Valli on the spectrum of energy and, finally, describe the arametrization of the space Harm(S2, U(n)), of all harmonics maps from S2 in U(n), provide the classification for such maps, following the work of J.C.Wood / Mestrado / Mestre em Matemática

Orientador: Caio José Colletti Negreiros / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T12:45:02Z (GMT). No. of bitstreams: 1 Grama_LinoAndersondaSilva_D.pdf: 1119551 bytes, checksum: d2dc2c993629f40f7976e91497c5d219 (MD5) Previous issue date: 2011 / Resumo: O principal objetivo deste trabalho é o estudo de aplicações harmônicas em variedades flag generalizadas. Na primeira parte do trabalho, consideramos aplicações cujo domínio é uma superfície de Riemann. Provamos que toda aplicação holomorfa-horizontal na variedade flag é uma aplicação equiharmônica (ie, harmônica com respeito a cada métrica invariante na variedade flag). Obtemos também as fórmulas de Plucker para curvas holomorfa-horizontais na variedade flag maximal. Na segunda parte do trabalho, consideramos aplicações harmônicas cujo domínio possui dimensão 1 ( ie, geodésicas) na variedade flag. Provamos que toda variedade ag generalizada admite curvas que são geodésicas com respeito a cada métrica invariante. Tais curvas são chamadas equigeodésicas. Fornecemos uma descrição algébrica para tais curvas e exibimos famílias de equigeodésicas em diversas famílias de variedades flag / Abstract: The main goal of this work is the study of harmonic maps in generalized flag manifolds. In the first part of the work, we consider maps whose domain is a Riemann surface. We prove that every holomorphic-horizontal map in the flag manifold is an equiharmonic map (i.e. harmonic with respect to each invariant metric in the flag manifold). We also obtain the Plucker formulae for holomorphic-horizontal curves in full flag manifolds. In the second part of the work, we consider harmonic maps whose domain has dimension one (i.e. geodesics) in the ag manifold. We prove that every generalized flag manifold admit curves that are geodesics with respect to each invariant metric. Such curves are called equigeodesics. We provide an algebraic characterization for such curves and exhibit families of equigeodesics in several families of flag manifolds / Doutorado / Doutor em Matemática

This thesis investigates the gradient flow of Dirac-harmonic maps. Dirac-harmonic maps are critical points of an energy functional that is motivated from supersymmetric field theories. The critical points of this energy functional couple the equation for harmonic maps with spinor fields. At present, many analytical properties of Dirac-harmonic maps are known, but a general existence result is still missing. In this thesis the existence question is studied using the evolution equations for a regularized version of Dirac-harmonic maps. Since the energy functional for Dirac-harmonic maps is unbounded from below the method of the gradient flow cannot be applied directly. Thus, we first of all consider a regularization prescription for Dirac-harmonic maps and then study the gradient flow. Chapter 1 gives some background material on harmonic maps/harmonic spinors and summarizes the current known results about Dirac-harmonic maps. Chapter 2 introduces the notion of Dirac-harmonic maps in detail and presents a regularization prescription for Dirac-harmonic maps. In Chapter 3 the evolution equations for regularized Dirac-harmonic maps are introduced. In addition, the evolution of certain energies is discussed. Moreover, the existence of a short-time solution to the evolution equations is established. Chapter 4 analyzes the evolution equations in the case that the domain manifold is a closed curve. Here, the existence of a smooth long-time solution is proven. Moreover, for the regularization being large enough, it is shown that the evolution equations converge to a regularized Dirac-harmonic map. Finally, it is discussed in which sense the regularization can be removed. In Chapter 5 the evolution equations are studied when the domain manifold is a closed Riemmannian spin surface. For the regularization being large enough, the existence of a global weak solution, which is smooth away from finitely many singularities is proven. It is shown that the evolution equations converge weakly to a regularized Dirac-harmonic map. In addition, it is discussed if the regularization can be removed in this case. / Die vorliegende Dissertation untersucht den Gradientenfluss von Dirac-harmonischen Abbildungen. Dirac-harmonische Abbildungen sind kritische Punkte eines Energiefunktionals, welches aus supersymmetrischen Feldtheorien motiviert ist. Die kritischen Punkte dieses Energiefunktionals koppeln die Gleichung für harmonische Abbildungen mit Spinorfeldern. Viele analytische Eigenschaften von Dirac-harmonischen Abbildungen sind bereits bekannt, ein allgemeines Existenzresultat wurde aber noch nicht erzielt. Diese Dissertation untersucht das Existenzproblem, indem der Gradientenfluss von einer regularisierten Version Dirac-harmonischer Abbildungen untersucht wird. Die Methode des Gradientenflusses kann nicht direkt angewendet werden, da das Energiefunktional für Dirac-harmonische Abbildungen nach unten unbeschränkt ist. Daher wird zunächst eine Regularisierungsvorschrift für Dirac-harmonische Abbildungen eingeführt und dann der Gradientenfluss betrachtet. Kapitel 1 stellt für die Arbeit wichtige Resultate über harmonische Abbildungen/harmonische Spinoren zusammen. Außerdem werden die zur Zeit bekannten Resultate über Dirac-harmonische Abbildungen zusammengefasst. In Kapitel 2 werden Dirac-harmonische Abbildungen im Detail eingeführt, außerdem wird eine Regularisierungsvorschrift präsentiert. Kapitel 3 führt die Evolutionsgleichungen für regularisierte Dirac-harmonische Abbildungen ein. Zusätzlich wird die Evolution von verschiedenen Energien diskutiert. Schließlich wird die Existenz einer Kurzzeitlösung bewiesen. In Kapitel 4 werden die Evolutionsgleichungen für den Fall analysiert, dass die Ursprungsmannigfaltigkeit eine geschlossene Kurve ist. Die Existenz einer Langzeitlösung der Evolutionsgleichungen wird bewiesen. Es wird außerdem gezeigt, dass die Evolutionsgleichungen konvergieren, falls die Regularisierung groß genug gewählt wurde. Schließlich wird diskutiert, ob die Regularisierung wieder entfernt werden kann. Kapitel 5 schlussendlich untersucht die Evolutionsgleichungen für den Fall, dass die Ursprungsmannigfaltigkeit eine geschlossene Riemannsche Spin Fläche ist. Es wird die Existenz einer global schwachen Lösung bewiesen, welche bis auf endlich viele Singularitäten glatt ist. Die Lösung konvergiert im schwachen Sinne gegen eine regularisierte Dirac-harmonische Abbildung. Auch hier wird schließlich untersucht, ob die Regularisierung wieder entfernt werden kann.

This dissertation considers several problems related to Ricci flow, including the existence and behavior of solutions. The first goal is to obtain explicit, coordinate-based descriptions of Ricci flow solutions--especially those corresponding to Ricci solitons--on two classes of nilpotent Lie groups. On the odd-dimensional classical Heisenberg groups, we determine the asymptotics of Ricci flow starting at any metric, and use Lott's blowdown method to demonstrate convergence to soliton metrics. On the groups of real unitriangular matrices, which are more complicated, we describe the solitons and corresponding solutions using a suitable ansatz. Next, we consider solsolitons involving the nilsolitons in the Heisenberg case above. This uses work of Lauret, which characterizes solsolitons as certain extensions of nilsolitons, and work of Will, which demonstrates that the space of solsolitons extensions of a given nilsoliton is parametrized by the quotient of a Grassmannian by a finite group. We determine these spaces of solsoliton extensions of Heisenberg nilsolitons, and we also explicitly describe many-parameter families of these solsolitons in dimensions greater than three. Finally, we explore Ricci flow coupled with harmonic map flow, both as it arises naturally in certain bundle constructions related to Ricci flow and as a geometric flow in its own right. In the first case, we generalize a theorem of Knopf that demonstrates convergence and stability of certain locally R[superscript N]-invariant Ricci flow solutions. In the second case, we prove a version of Hamilton's compactness theorem for the coupled flow, and then generalize it to the category of etale Riemannian groupoids. We also provide a detailed example of solutions to the flow on the three-dimensional Heisenberg group. / text

In the first chapter of this report, our aim is to introduce harmonic maps between Riemann surfaces using the Energy integral of a map. Once we have the desired prerequisites, we move on to show how to continuously deform a given map to a harmonic map (i.e., find a harmonic map in its homotopy class). We follow J¨urgen Jost’s approach using classical potential theory techniques. Subsequently, we analyze the additional conditions needed to ensure a certain uniqueness property of harmonic maps within a given homotopy class. In conclusion, we look at a couple of applications of what we have shown thus far and we find a neat proof of a slightly weaker version of Hurwitz’s Automorphism Theorem. In the second chapter, we introduce the concept of minimal surfaces. After exploring a few examples, we mathematically formulate Plateau’s problem regarding the existence of a soap film spanning each closed, simple wire frame and discuss a solution. In conclusion, a partial result (due to Rad´o) regarding the uniqueness of such a soap film is discussed.

In this Thesis, we analyze three variational and geometric problems, that extend classical Euclidean issues of the calculus of variations to more general classes of spaces. The results we outline are based on the articles [Ved21; MV21] and on a forthcoming joint work with Nicolussi Golo and Serra Cassano. In the first place, in Chapter 1 we provide a general introduction to metric measure spaces and some of their properties. In Chapter 2 we extend the classical Talenti’s comparison theorem for elliptic equations to the setting of RCD(K,N) spaces: in addition the the generalization of Talenti’s inequality, we will prove that the result is rigid, in the sense that equality forces the space to have a symmetric structure, and stable. Chapter 3 is devoted to the study of the Bernstein problem for intrinsic graphs in the first Heisenberg group H^1: we will show that under mild assumptions on the regularity any stationary and stable solution to the minimal surface equation needs to be intrinsically affine. Finally, in Chapter 4 we study the dimension and structure of the singular set for p-harmonic maps taking values in a Riemannian manifold.

On étudie les déformations des applications harmoniques $f$ tordues par rapport à une représentation. Après avoir construit une application harmonique tordue "universelle", on donne une construction de toute déformations du premier ordre de $f$ en termes de la théorie de Hodge ; on applique ce résultat à l'espace de modules des représentations réductives d'un groupe de Kähler, pour démontrer que les points critiques de la fonctionnelle de l'énergie $E$ coïncident avec les représentations de monodromie des variations complexes de structures de Hodge. Ensuite, on procède aux déformations du second ordre, où des obstructions surviennent ; on enquête sur l'existence de ces déformations et on donne une méthode pour le construire. En appliquant ce résultat à la fonctionnelle de l'énergie comme ci-dessus, on démontre (pour n'importe quel groupe de présentation finie) que la fonctionnelle de l'énergie est strictement pluri sous-harmonique sur l'espace des modules des représentations. En assumant de plus que le groupe soit de Kähler, on étudie les valeurs propres de la matrice hessienne de $E$ dans les points critiques. / We study the deformations of twisted harmonic maps $f$ with respect to a representation. After constructing a continuous ``universal'' twisted harmonic map, we give a construction of every first order deformation of $f$ in terms of Hodge theory; we apply this result to the moduli space of reductive representations of a K\"ahler group, to show that the critical points of the energy functional $E$ coincide with the monodromy representations of polarized complex variations of Hodge structure. We then proceed to second order deformations, where obstructions arise; we investigate the existence of such deformations, and give a method for constructing them, as well. Applying this to the energy functional as above, we prove (for every finitely presented group) that the energy functional is strictly pluri sub-harmonic on the moduli space of representations; assuming furthermore that the group is Kähler, we study the eigenvalues of the Hessian of $E$ at critical points.

[pt] Primeiro examinamos soluções de viscosidade Lp para equações elípticas totalmente não lineares com ingredientes de fronteira mensuráveis. Ao considerar p0 < p < d, focamos nas estimativas da regularidade dos gradientes derivadas de potenciais não lineares. Encontramos condições para Lipschitz-continuidade local das soluções e continuidade do gradiente. Examinamos avanços recentes na teoria da regularidade decorrentes de estimativas potenciais (não lineares). Nossas descobertas decorrem de – e são inspiradas por – fatos fundamentais na teoria de soluções de Lp-viscosidade, e resultados do trabalho de Panagiota Daskalopoulos, Tuomo Kuusi e Giuseppe Mingione (DKM2014). Na segunda parte provamos a regularidade parcial de mapas harmônicos com peso fracamente estacionários com dados de fronteira livre em um cone. Como ponto de partida, damos uma olhada na teoria da regularidade parcial interior para mapas harmônicos fracionários de minimização de energia intrínseca do espaço euclidiano em variedades Riemannianas compactas e suaves para potências fracionárias estritamente entre zero e um. Mapas harmônicos fracionários intrínsecos podem ser estendidos para mapas harmônicos com peso, então provamos regularidade parcial para mapas harmônicos minimizantes locais com dados de fronteira (parcialmente) livres em meios-espaços, mapas harmônicos fracionários então herdam essa regularidade. / [en] We first examine Lp-viscosity solutions to fully nonlinear elliptic equations with bounded measurable ingredients. By considering p0 < p < d, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for local Lipschitz-continuity of the solutions and continuity of the gradient. We survey recent breakthroughs in regularity theory arising from (nonlinear) potential estimates. Our findings follow from – and are inspired by – fundamental facts in the theory of Lp-viscosity solutions, and results in the work of Panagiota Daskalopoulos, Tuomo Kuusi and Giuseppe Mingione (DKM2014). In the second part we prove partial regularity of weakly stationary weighted harmonic maps with free boundary data on a cone. As a starting point we take a look at the interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps can be extended to weighted harmonic maps, so we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces, fractional harmonic maps then inherit this regularity.

<p>This thesis consists of the structure-preserving numerical methods for PNP-NS equation and dynamic liquid crystal systems in Oseen-Frank energy. </p> <p>In Chapter 1, we give a brief introduction of the Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system, and the dynamical liquid system in Oseen-Frank energy in one-constant approximation case and a special non-one-constant case. Each of those systems has a special structure and properties we want to keep at the discrete level when designing numerical methods.</p> <p>In Chapter 2, we introduce a first-order numerical scheme for the PNP-NS system that is decoupled, positivity preserving, mass conserving, and unconditionally energy stable. The numerical scheme is designed in the context of Wasserstein gradient flow based on the form ∇ · (c∇ ln c). The mobility terms are treated explicitly, and the chemical potential terms are treated implicitly so that the solution of the numerical scheme is the minimizer of a convex functional, which is the key to the unique solvability and positivity preserving of the numer-<br> ical scheme. Proper boundary conditions for chemical potentials are chosen to guarantee the mass-conservation property. The convection term in Poisson-Nernst-Planck(PNP) equation part is treated explicitly with an O(∆t) term introduced so that the numerical scheme is decoupled and unconditionally energy stable. Pressure correction methods are used for the Navier-Stokes(NS) equation part. And we proved the optimal convergence rate with an irregular high-order asymptotic expansion technique.</p> <p>In Chapter 3, we propose a first-order implicit numerical method for a dynamic liquid crystal system in a one-constant-approximation case(which is also known as heat flow of harmonic maps to S2). The solution is the minimizer of a convex functional under the unit length constraint, and from this point, the weak convergence of the numerical scheme could be proved. The numerical scheme is solved in an iterative procedure. This procedure could be proved to be energy decreasing and this implies the convergence of the algorithm.</p> <p>In Chapter 4, we study the dynamic liquid crystal system in a more generalized Oseen- Frank energy compared to Chapter 3. We are assuming K2 = K3 = −K4, the domain Ω is a rectangular region in R3, and d satisfies the periodic boundary condition on ∂Ω. And we propose a class of numerical schemes for this system that preserve the unit length constraint. The convergence of the numerical scheme has been proved under necessary assumptions. And numerical experiments are presented to validate the accuracy and demonstrate the performance of the proposed numerical scheme.</p>

Το κύριο αντικείμενο της διατριβής συνίσταται στη μελέτη της γεωμετρίας των τρισδιάστατων H-μετρικών πολλαπλοτήτων επαφής, ή, ισοδύναμα, των μετρικών πολλαπλοτήτων επαφής για τις οποίες το διανυσματικό πεδίο ξ είναι πεδίο ιδιοδιανυσμάτων του τελεστή Ricci Q. Συγκεκριμένα, αποδεικνύεται ότι μια τρισδιάστατη H-μετρική πολλαπλότητα επαφής [Μ, (η, ξ, φ, g)] χαρακτηρίζεται γεωμετρικά από μια συνθήκη που εμπλέκει τον τανυστή καμπυλότητας της Μ και τρεις διαφορίσιμες συναρτήσεις κ, μ και ν της Μ. Η συνθήκη αυτή οδηγεί στην εισαγωγή μιας νέας κλάσης μετρικών πολλαπλοτήτων επαφής: τις (κ, μ, ν)-πολλαπλότητες επαφής. Το ενδιαφέρον με τις (κ, μ, ν)-πολλαπλότητες επαφής είναι ότι για διάσταση μεγαλύτερη του τρία εκφυλίζονται στις (κ, μ)-πολλαπλότητες επαφής, δηλαδή, οι συναρτήσεις κ, μ είναι σταθερές και η συνάρτηση ν είναι η μηδενική συνάρτηση. Αντιθέτως, αποδεικνύεται ότι τέτοιες μετρικές πολλαπλότητες επαφής υπάρχουν στη διάσταση τρία. Ένα άλλο από τα προβλήματα που εξετάζονται σ' αυτή τη διατριβή είναι ο χαρακτηρισμός των διαρμονικών καμπυλών του Legendre και των αντι-αναλλοίωτων επιφανειών εμβυθισμένων σε τρισδιάστατες (κ, μ, ν)-πολλαπλότητες επαφής. Συγκεκριμένα, αποδεικνύεται ότι οι διαρμονικές καμπύλες του Legendre είναι οι γεωδαισιακές αυτών των χώρων. Επιπλέον, αποδεικνύεται ότι οι διαρμονικές και χωρίς ελαχιστικά σημεία αντι-αναλλοίωτες επιφάνειες που είναι εμβυθισμένες σε τρισδιάστατες γενικευμένες (κ, μ)-πολλαπλότητες επαφής και των οποίων το μέτρο του διανυσματικού πεδίου της μέσης καμπυλότητας είναι σταθερό, είναι τοπικά Ευκλείδειες. / The main object of this Doctoral Thesis is the study of the geometry of 3-dimensional H-contact metric manifolds, or, equivalently, the contact metric manifolds whose the vector field ξ is an eigenvector of the Ricci operator Q. More precisely, it is proved that 3-dimensional H-contact metric manifolds [M, (η, ξ, φ, g)] are geometrically characterized by a specific curvature condition and three differentiable functions κ, μ and ν of M. This condition leads to the introduction of a new class of contact metric manifolds: the (κ, μ, ν)-contact metric manifolds. It is remarkable that for dimension greater than three, such manifolds are reduced to (κ, μ)-contact metric manifolds, i.e. the functions κ, μ are constants and the function ν is the zero function. On the contrary, in three dimension (κ,μ,ν)-contact metric manifolds exist. Another problem which is studied is the classification of biharmonic Legendre curves and anti-invariant surfaces immersed in 3-dimensional (κ, μ, ν)-contact metric manifolds. It is proved that biharmonic Legendre curves in 3-dimensional (κ, μ, ν)-contact metric manifolds are necessarily geodesics. Furthermore, it is proved that biharmonic and without minimal points anti-invariant surfaces immersed in 3-dimensional generalized (κ, μ)-contact metric manifolds with constant norm of the mean curvature vector field, are locally flat.