Дисертації з теми "Curvature bounds"

Any Riemannian manifold possesses a minimal solution of the heat equation for the Dirichlet Laplacian, called the heat kernel. During the last decades many authors investigated geometric properties of the manifold such that its heat kernel fulfills a so-called Gaussian upper bound. Especially compact and non-compact manifolds with lower bounded Ricci curvature have been examined and provide such Gaussian estimates. In the compact case it ended even with integral Ricci curvature assumptions. The important techniques to obtain Gaussian bounds are the symmetrization procedure for compact manifolds and relative Faber-Krahn estimates or gradient estimates for the heat equation, where the first two base on isoperimetric properties of certain sets. In this thesis, we generalize the existing results to the following. Locally uniform integral bounds on the negative part of Ricci curvature lead to Gaussian upper bounds for the heat kernel, no matter whether the manifold is compact or not. Therefore, we show local isoperimetric inequalities under this condition and use relative Faber-Krahn estimates to derive explicit Gaussian upper bounds. If the manifold is compact, we can even generalize the integral curvature condition to the case that the negative part of Ricci curvature is in the so-called Kato class. We even obtain uniform Gaussian upper bounds using gradient estimate techniques. Apart from the geometric generalizations for obtaining Gaussian upper bounds we use those estimates to generalize Bochner’s theorem. More precisely, the estimates for the heat kernel obtained above lead to ultracontractive estimates for the heat semigroup and the semigroup generated by the Hodge Laplacian. In turn, we can formulate rigidity results for the triviality of the first cohomology group if the amount of curvature going below a certain positive threshold is small in a suitable sense. If we can only assume such smallness of the negative part of the Ricci curvature, we can bound the Betti number by explicit terms depending on the generalized curvature assumptions in a uniform manner, generalizing certain existing results from the cited literature
Jede Riemannsche Mannigfaltigkeit besitzt eine minimale Lösung für die Wärmeleitungsgleichung des zur Mannigfaltigkeit gehörigen Dirichlet-Laplaceoperators, den Wärmeleitungskern. Während der letzten Jahrzehnte fanden viele Autoren geometrische Eigenschaften der Mannigfaltigkeiten unter welchen der Wärmeleitungskern eine sogenannte Gaußsche obere Abschätzung besitzt. Insbesondere bestizen sowohl kompakte als auch nichtkompakte Mannigfaltigkeiten mit nach unten beschränkter Ricci-Krümmung solche Gaußschen Abschätzungen. Im kompakten Fall reichten bisher sogar Integralbedingungen an die Ricci-Krümmung aus. Die wichtigen Techniken, um Gaußsche Abschätzungen zu erhalten, sind die Symmetrisierung für kompakte Mannigfaltigkeiten und relative Faber-Krahn- und Gradientenabschätzungen für die Wärmeleitungsgleichung, wobei die ersten beiden auf isoperimetrischen Eigenschaften gewisser Mengen beruhen. In dieser Arbeit verallgemeinern wir die bestehenden Resultate im folgenden Sinne. Lokal gleichmäßig beschränkte Integralschranken an den Negativteil der Ricci-Krümmung ergeben Gaußsche obere Abschätzungen sowohl im kompakten als auch nichtkompakten Fall. Dafür zeigen wir lokale isoperimetrische Ungleichungen unter dieser Voraussetzung und nutzen die relativen Faber-Krahn-Abschätzungen für eine explizite Gaußsche Schranke. Für kompakte Mannigfaltigkeiten können wir sogar die Integralschranken an den Negativteil der Ricci-Krümmung durch die sogenannte Kato-Bedingung ersetzen. In diesem Fall erhalten wir gleichmäßige Gaußsche Abschätzungen mit einer Gradientenabschätzung. Neben den geometrischen Verallgemeinerungen für Gaußsche Schranken nutzen wir unsere Ergebnisse, um Bochners Theorem zu verallgemeinern. Wärmeleitungskernabschätzungen ergeben ultrakontraktive Schranken für die Wärmeleitungshalbgruppe und die Halbgruppe, die durch den Hodge-Operator erzeugt wird. Damit können wir Starrheitseigenschaften für die erste Kohomologiegruppe zeigen, wenn der Teil der Ricci-Krümmung, welcher unter einem positiven Level liegt, in einem bestimmten Sinne klein genug ist. Wenn der Negativteil der Ricci-Krümmung nicht zu groß ist, können wir die erste Betti-Zahl noch immer explizit uniform abschätzen

In this dissertation we study the Laplace operator acting on functions on a smooth, compact Riemannian manifold. Our approach is based on the study of the spectrum of the aforementioned operator. The main objects of our interest are the counting function of the Laplacian and its Riesz means. We discuss the asymptotics of aforementioned functions when the argument approaches infinity.

Concrete topological properties of a manifold can be found by examining its geometry. Theorem 17 of his thesis, due to Myers [Mye41], is one such example of this; it gives an upper bound on the length of any minimizing geodesic in a manifold N in terms of a lower positive bound on the Ricci curvature of N, and concludes that N is compact. Our main result, Theorem 40, is of the same flavour as this, but we are instead concerned with stable, minimal surfaces in manifolds of positive scalar curvature. This result is a version of Proposition 1 in the paper of Schoen and Yau [SY83], written in the context of Riemannian geometry. It states: a stable, minimal 2-submanifold of a 3-manifold whose scalar curvature is bounded below by κ > 0 has a inradius bound of ≤√(8/3) π/√κ, and in particular is compact.

This thesis is concerned with the study of qualitative properties of solutions of the minimal surface equation and of a class of prescribed mean curvature equations on complete Riemannian manifolds. We derive global gradient bounds for non-negative solutions of such equations on manifolds satisfying a uniform Ricci lower bound and we obtain Liouville-type theorems and other rigidity results on Riemannian manifolds with non-negative Ricci curvature. The proof of the aforementioned global gradient bounds for non-negative solutions u is based on the application of the maximum principle to an elliptic differential inequality satisfied by a suitable auxiliary function z=f(u,|Du|), in the spirit of Bernstein’s method of a priori estimates for nonlinear PDEs and of Yau’s proof of global gradient bounds for harmonic functions on complete Riemannian manifolds. The particular choice of the auxiliary function z parallels the one in Korevaar’s proof of a priori gradient estimates for the prescribed mean curvature equation in Euclidean space. The rigidity results obtained in the last part of the thesis include a Liouville theorem for positive solutions of the minimal surface equation on complete Riemannian manifolds with non-negative Ricci curvature, a splitting theorem for complete parabolic manifolds of non-negative sectional curvature supporting non-constant solutions with linear growth of the minimal surface equation, and a splitting theorem for domains of complete parabolic manifolds with non-negative Ricci curvature supporting non-constant solutions of overdetermined problems involving the mean curvature operator.

We consider the problem of finding a bounded-curvature path in the plane from one configuration αs to another configuration αt that avoids the interior of a set of polygonal obstacles Ε. We call any such path from αs to αt a feasible path. In this thesis, we develop algorithms to find feasible paths that have explicit guarantees on when they will return a feasible path. We phrase our guarantees and run time analysis in terms of the complexity of the desired solution (see k and λ below). In a sense, our algorithms are output sensitive, which is particularly desirable because there are no known bounds on the solution complexity amid arbitrary polygonal environments. Our first major result is an algorithm that given Ε, αs, αt, and a positive integer k either (i) verifies that every feasible path has a descriptive complexity greater than k or (ii) outputs a feasible path. The run time of this algorithm is bounded by a polynomial in n (the total number of obstacle vertices in Ε), m (the bit precision of the input), and k. This result complements earlier work by Fortune and Wilfong: their algorithm considers paths of arbitrary descriptive complexity (it has no dependence on k), but it never outputs a path, just whether or not a feasible path exists. Our second major result is an algorithm that given E, αs, αt, a length λ, and an approximation factor Ε, either (i) verifies that every feasible path has length greater than λ or (ii) constructs a feasible path that is at most (1+ Ε) times longer than the shortest feasible path. The run time of this algorithm is bounded by a polynomial in n, m, Ε-1, and λ. This algorithm is the result of applying the techniques developed earlier in our thesis to the previous approximation approaches. A shortcoming of these prior approximation algorithms is that they only search a special class of feasible paths. This restriction implies that the path that they return may be arbitrarily longer than the shortest path. Our algorithm returns a true approximation because we search for arbitrary shortest paths.

This thesis is devoted to the study of structural properties of non-smooth spaces with Ricci curvature bounded from below. The first part concerns with the structure theory of RCD(K,N) spaces: we prove the existence of the so-called essential dimension, along with rectifiability properties of the regular set. This theory is a result of many contributions [43,72,91,95,109,121], in our presentation we closely follow the recent works [41,43]. The second part of this thesis deals with codimension-1 structures on RCD(K,N) spaces. More precisely we study structural properties of boundaries of sets with finite perimeter, generalising the celebrated De Giorgi theory [65, 66] to this framework. This is based on the works [7,40].

This PhD thesis considers the analysis and the interpretation of the complex turbulent flows subjected to curvature and rotation effects. To achieve this goal, largeeddy simulation (LES) is performed for various wall-bounded flow problems. For the validation and verification purposes, the adopted finite-volume code is tested by considering fully developed channel and duct flow problems. The behaviour of the subgrid-scale (SOS) models and the spatial schemes are investigated in detail for the channel and duct flows subjected to orthogonal rotation. Among the tested SOS models and the spatial schemes, the Wall-Adapting Local Eddy Viscosity (WALE) model and the bounded central differencing (BCD) scheme are found to perform the best. During the validation and verification processes, the turbulence mechanism in the channel and duct flows for various rotation rates are reviewed and the laminarization process due to Coriolis force is revealed by considering a wide range of data processing. Using the experience gained from the rotating channel and duct flow cases, more challenging flow cases are considered. The flow in the square-sectioned U-duct and the centrifugal compressor are simulated with LES at high Reynolds numbers. Predictions are extensively validated for both flow problems with the available experimental data. Grid convergence and appropriate near-wall resolutions are provided in order to avoid errors associated with the filter width and the wall functions. For both flow problems, Reynolds-averaged Navier-Stokes (RANS) results are included to determine the impact level of LES. Upon encouraging results obtained via LES, the effects of strong curvature and Coriolis forces are explored on mean, secondary flows and turbulence.

L’objectif de la thèse est de présenter de nouveaux résultats d’analyse sur les espaces métriques mesurés. Nous étendons d’abord à une certaine classe d’espaces avec doublement et Poincaré des inégalités de Sobolev pondérées introduites par V. Minerbe en 2009 dans le cadre des variétés riemanniennes à courbure de Ricci positives. Dans le contexte des espaces RCD(0,N), nous en déduisons une inégalité de Nash pondérée et un contrôle uniforme du noyau de la chaleur pondéré associé. Puis nous démontrons la loi de Weyl sur les espaces RCD(K,N) compactes à l’aide d’un théorème de convergence ponctuelle des noyaux de la chaleur associés à une suite mGH-convergente d’espaces RCD(K,N). Enfin nous abordons dans le contexte RCD(K,N) un théorème de Bérard, Besson et Gallot fournissant, à l’aide du noyau de la chaleur, une famille de plongements asymptotiquement isométriques d’une variété riemannienne fermée dans l’espace de ses fonctions de carré intégrable. Nous introduisons notamment les notions de métrique RCD, de métrique pull-back, et de convergence faible/forte de métriques RCD sur un espace RCD(K,N) compacte, et nous prouvons un résultat de convergence analogue à celui de Bérard, Besson et Gallot
The aim of this thesis is to present new results in the analysis of metric measure spaces. We first extend to a certain class of spaces with doubling and Poincaré some weighted Sobolev inequalities introduced by V. Minerbe in 2009 in the context of Riemannian manifolds with non-negative Ricci curvature. In the context of RCD(0,N) spaces, we deduce a weighted Nash inequality and a uniform control of the associated weighted heat kernel. Then we prove Weyl’s law for compact RCD(K,N) spaces thanks to a pointwise convergence theorem for the heat kernels associated with a mGH-convergent sequence of RCD(K,N) spaces. Finally we address in the RCD(K,N) context a theorem from Bérard, Besson and Gallot which provides, by means of the heat kernel, an asymptotically isometric family of embeddings for a closed Riemannian manifold into its space of square integrable functions. We notably introduce the notions of RCD metrics, pull-back metrics, weak/strong convergence of RCD metrics, and we prove a convergence theorem analog to the one of Bérard, Besson and Gallot

[from the introduction]: The aim of this thesis is to study metric measure spaces with a synthetic notion of Ricci curvature bounded below. We study them from the point of view of Sobolev/Nash type functional inequalities in the non-compact case, and from the point of view of spectral analysis in the compact case. The heat kernel links the two cases: in the first one, the goal is to get new estimates on the heat kernel of some associated weighted structure; in the second one, the heat kernel is the basic tool to establish our results. The topic of synthetic Ricci curvature bounds has known a constant development over the past few years. In this introduction, we shall give some historical account on this theory, before explaining in few words the content of this work. The letter K will refer to an arbitrary real number and N will refer to any finite number greater or equal than 1.

In dieser Arbeit studieren wir das ekpyrotische Szenario, welches ein kosmologisches Modell des frühen Universums ist. Dieses Modell erklärt mit Hilfe einer kontrahierenden ekpyrotischen Phase die "Anfangsbedingungen" des Universums. Das bedeutet, dass der konventionelle "Urknall" durch einem Rückprall ersetzt wird. In dieser Arbeit versuchen wir Unstimmigkeiten zwischen den Vorhersagen der ekpyrotischen Modelle und den Messungen der Kosmologischen Hintergrundstrahlung des Planck Satelliten zu lösen. Den Planck Messungen zufolge sind die ursprünglichen adiabatischen Fluktuationen fast skaleninvariant und gaußverteilt. Während der ekpyrotischen Phase werden typischer Weise Flutuationen mit nicht-Gaußschen Korrekturen erzeugt. Wir schlagen zwei Ansätze vor, um diese Unstimmigkeit zu beheben. In dem nicht-minimalen entropischen Mechanismus werden fast skaleninvariante entropische Fluktuationen mit Hilfe einer nicht-minimalen kinetischen Kopplung zwischen zwei Skalarfeldern erzeugt. Wir werden zeigen, dass die nicht-Gaußschen Korrekturen während der ekpyrotischen Phase genau Null sind. Dies führt zu insgesamt kleinen nicht-Gaußschen Korrekturen nach der Umwandlung von entropischen zu adiabatischen Fluktuationen. Im Folgendem werden wir eine kinetische Umwandlung untersuchen, die nach einem nicht-singulären Rückprall stattfindet. Das Wachstum der entropischen Fluktuationen während des Rückpralls hat zur Folge, dass die möglichen nicht-Gaußschen Korrekturen, die zur Zeit der ekpyrotischen Phase erzeugt wurden, während des Rückpralls unterdrückt werden. Im letzten Teil der Arbeit gehen wir ein gravierendes Problem des inflationären Paradigmas an, welches "slow-roll eternal inflation" genannt wird. Wir schlagen ein Modell vor, das Ideen von Inflation und Ekpyrosis verbindet. Während der Konflation expandiert das Universum beschleunigt. Die adiabatischen Fluktuationen verhalten sich jedoch wie bei ekpyrotischen Modellen und wird "slow-roll eternal inflation" verhindert.
In this thesis, we study the ekpyrotic scenario, which is a cosmological model of the early universe. In this model the ``initial conditions'' of the universe are determined by a contracting ekpyrotic phase, which means that the conventional ``Big Bang'' is replaced by a bounce. The following thesis addresses the tension between ekpyrotic predictions and the observations of the Cosmic Microwave Background radiation by the Planck team. According to the Planck data, the primordial curvature fluctuations are nearly scale-invariant and Gaussian. However, during ekpyrosis, the fluctuations have typically sizable non-Gaussian signatures. In this thesis, we propose two approaches in order to resolve the tension with observations. In the non-minimal entropic mechanism, nearly scale-invariant entropy perturbations are created due to a non-minimal kinetic coupling between two scalar fields. We will show that the non-Gaussian corrections during ekpyrosis are precisely zero leading to overall small non-Gaussian signatures after the conversion process from entropy perturbations to curvature perturbations. In the following, we will consider a kinetic conversion phase, which takes place after a non-singular bounce. Due to the growth of entropy perturbations during the bounce phase, the possibly large non-Gaussian corrections created during the ekpyrotic phase become suppressed during the bounce. The last part of this thesis addresses a major problem of the inflationary paradigm: Due to large adiabatic fluctuations, slow-roll eternal inflation creates infinitely many physically distinct pocket universes. We propose a model in the framework of scalar-tensor theories, which conflated ideas of both inflation and ekpyrosis. During conflation, the universe undergoes accelerated expansion, but there are no large adiabatic fluctuations like during ekpyrosis resulting in the absence of slow-roll eternal inflation.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
We study problems of existence and uniqueness of constant mean curvature surfaces with prescribed boundary satisfying the bounded slope condition. The surfaces are given as Euclidean graphs in R3 and as parabolic graphs in H3, over bounded domains contained in totally geodesic surfaces in these ambients, or moreover, as radial graphs over bounded domains contained in S2.
Estudamos problemas de existência e unicidade de superfícies de curvatura média constante com bordo prescrito satisfazendo a condição de declividade limitada (CDL). Tais superfícies são dadas como gráficos euclidianos (verticais) em R3 e como gráficos parabólicos em H3, definidos sobre domínios limitados contidos em superfícies totalmente geodésicas destes ambientes, ou ainda como gráficos radiais em R3 sobre domínios limitados contidos em S2.

A theoretical explanation of the existence of lipid rafts in cell membranes remains a topic of lively debate. Large, micrometer sized rafts are readily observed in artificial membranes and can be explained using thermodynamic models for phase separation and coarsening. In live cells such domains are not observed and various models are proposed to describe why the systems do not coarsen. We review these attempts critically and show within a phase field approach that membrane bound proteins have the potential to explain the different behaviour observed in vitro and in vivo. Large scale simulations are performed to compute scaling laws and size distribution functions under the influence of membrane bound proteins and to observe a significant slow down of the domain coarsening at longer times and a breakdown of the self-similarity of the size-distribution function
Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior
Este trabalho à baseado no artigo The Mean Curvature Cylindrically Bounded Submanifolds, nele abordaremos uma estimativa para a curvatura mÃdia de subvariedades completas cilindricamente limitadas. Ademais apresentaremos uma relaÃÃo entre uma estimativa da curvatura mÃdia e o fato de M ser estocasticamente incompleta.
This work is based on the article The Mean Curvature Cylindrically Bounded Submanifolds, it will discuss an estimate for the mean curvature of complete cylindrically submanifolds bounded. Furthermore we present a relationship between an estimate of the mean curvature and the fact that M is stochastically incomplete.

A theoretical explanation of the existence of lipid rafts in cell membranes remains a topic of lively debate. Large, micrometer sized rafts are readily observed in artificial membranes and can be explained using thermodynamic models for phase separation and coarsening. In live cells such domains are not observed and various models are proposed to describe why the systems do not coarsen. We review these attempts critically and show within a phase field approach that membrane bound proteins have the potential to explain the different behaviour observed in vitro and in vivo. Large scale simulations are performed to compute scaling laws and size distribution functions under the influence of membrane bound proteins and to observe a significant slow down of the domain coarsening at longer times and a breakdown of the self-similarity of the size-distribution function.
Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.

Autonomous mobile robots - both aerial and terrestrial vehicles - have gained immense importance due to the broad spectrum of their potential military and civilian applications. One of the indispensable requirements for the autonomy of a mobile vehicle is the vehicle's capability of planning and executing its motion, that is, finding appropriate control inputs for the vehicle such that the resulting vehicle motion satisfies the requirements of the vehicular task. The motion planning and control problem is inherently complex because it involves two disparate sub-problems: (1) satisfaction of the vehicular task requirements, which requires tools from combinatorics and/or formal methods, and (2) design of the vehicle control laws, which requires tools from dynamical systems and control theory. Accordingly, this problem is usually decomposed and solved over two levels of hierarchy. The higher level, called the geometric path planning level, finds a geometric path that satisfies the vehicular task requirements, e.g., obstacle avoidance. The lower level, called the trajectory planning level, involves sufficient smoothening of this geometric path followed by a suitable time parametrization to obtain a reference trajectory for the vehicle. Although simple and efficient, such hierarchical separation suffers a serious drawback: the geometric path planner has no information of the kinematic and dynamic constraints of the vehicle. Consequently, the geometric planner may produce paths that the trajectory planner cannot transform into a feasible reference trajectory. Two main ideas appear in the literature to remedy this problem: (a) randomized sampling-based planning, which eliminates altogether the geometric planner by planning in the vehicle state space, and (b) geometric planning supported by feedback control laws. The former class of methods suffer from a lack of optimality of the resultant trajectory, while the latter class of methods makes a restrictive assumption concerning the vehicle kinematic model. In this thesis, we propose a hierarchical motion planning framework based on a novel mode of interaction between these two levels of planning. This interaction rests on the solution of a special shortest-path problem on graphs, namely, one using costs defined on multiple edge transitions in the path instead of the usual single edge transition costs. These costs are provided by a local trajectory generation algorithm, which we implement using model predictive control and the concept of effective target sets for simplifying the non-convex constraints involved in the problem. The proposed motion planner ensures "consistency" between the two levels of planning, i.e., a guarantee that the higher level geometric path is always associated with a kinematically and dynamically feasible trajectory. We show that the proposed motion planning approach offers distinct advantages in comparison with the competing approaches of discretization of the state space, of randomized sampling-based motion planning, and of local feedback-based, decoupled hierarchical motion planning. Finally, we propose a multi-resolution implementation of the proposed motion planner, which requires accurate descriptions of the environment and the vehicle only for short-term, local motion planning in the immediate vicinity of the vehicle.

Le flot de Ricci est une équation aux dérivées partielles qui régit l’évolution d’une métrique riemannienne dépendant d’un paramètre de temps sur une variété différentielle. D’abord introduit et étudié par R. Hamilton, il est à l’origine de la solution de la conjecture de géométrisation des variétés compactes de dimension 3 par G. Perelman en 2001. La théorie classique concernant l’existence en temps court des solutions, due à Hamilton et à Shi, garantit (en dimension quelconque) l’existence d’un flot soit sur une variété compacte, soit lorsque la métrique initiale est complète avec une borne sur la norme du tenseur de courbure. En l’absence de cette borne, on conjecture qu’on peut trouver, à partir de la dimension 3, des données initiales pour lesquelles il n’existe pas de solution. Dans cette thèse, on démontre des théorèmes d’existence en temps court du flot sous des hypothèses plus faibles qu’une borne sur la norme du tenseur de courbure. Pour cela, on introduit une construction générale qui, pour une métrique riemannienne g quelconque sur une variété M, pas nécessairement complète, permet de produire une solution de l’équation du flot sur un domaine ouvert D de l’espace-temps M * [0,T] qui contient la tranche de temps initiale, avec g pour donnée initiale. On montre ensuite que sous des hypothèses adaptées sur la métrique g, on contrôle la forme du domaine D. En particulier, lorsque la métrique g est complète, D contient un ensemble de la forme M * [0,t], avec t>0, ce qui revient à dire qu’il existe un flot au sens classique dont la donnée initiale est g. Les « hypothèses adaptées » qui conduisent à des théorèmes d’existence sont de trois types. Dans tout les cas, on suppose une minoration uniforme du volume des boules de rayon au plus 1, à quoi on ajoute : a) en dimension 3, une minoration du tenseur de Ricci, b) en dimension n, une minoration d’une notion de courbure dite « courbure isotrope I » ou bien c) en dimension n, une borne sur la norme du tenseur de Ricci et une hypothèse qui garantit la proximité au sens métrique des boules de rayon au plus 1 avec une boule de même rayon dans un espace métrique obtenu comme le produit cartésien d’un espace de dimension 3 et d’un facteur euclidien de dimension n-3. De plus, avec ces résultats d’existence viennent des estimations sur les propriétés de régularisation du flot quantifiées en fonction des hypothèses sur la donnée initiale. La possibilité ainsi offerte de régulariser, globalement ou localement, pour un temps et avec des estimations quantifiés, une métrique initiale a des conséquence sur les espaces métriques singuliers obtenus comme limites, pour la distance de Gromov-Hausdorff, de suites de variétés satisfaisant uniformément aux conditions a), b) ou c). En effet, des théorèmes de compacité classiques pour le flot de Ricci permettent d’extraire un flot limite, étant donnée une suite de métriques initiales satisfaisant uniformément à ces hypothèses, et possédant donc toutes un flot pour un temps contrôlé. Lorsque les métriques en question approchent, pour la topologie de Gromov-Hausdorff, un espace singulier, cette solution limite s’interprète comme un flot régularisant l’espace singulier en question, et son existence contraint la topologie de cet espace singulier
The Ricci Flow is a partial differential equation governing the evolution of a Riemannian metric depending on a time parameter t on a differential manifold. It was first introduced and studied by R. Hamilton, and eventually led to the solution of the Geometrization conjecture for closed three-dimensional manifolds by G. Perelman in 2001. The classical short-time existence theory for the Ricci Flow, due to Hamilton and Shi, asserts, in any dimension, the existence of a flow starting from any initial metric when the underlying manifold in compact, or for any complete initial metric with a bound on the norm of the curvature tensor otherwise. In the absence of such a bound, though, the conjecture is that starting from dimension 3 one can find such initial data for which there is no solution. In this thesis, we prove short-time existence theorems under hypotheses weaker than a bound on the norm of the curvature tensor. To do this, we introduce a general construction which, for any Riemannian metric g (not necessarily complete) on a manifold M, allows us to produce a solution to the equation of the flow on an open domain D of the space-time M * [0,T] which contains the initial time slice, with g as an initial datum. We proceed to show that under suitable hypotheses on g, one can control the shape of the domain D, so that in particular, D contains a subset of the form M * [0,t] with t>0 if g is complete. By « suitable hypothesis », we mean one of the following. In any case, we assume a lower bound on the volume of balls of radius at most 1, plus a) in dimension 3, a lower bound on the Ricci tensor, b) in dimension n, a lower bound on the so-called « isotropic curvature I » or c) in dimension n, a bound on the norm of the Ricci tensor, as well as a hypothesis which garanties the metric proximity of every ball of radius at most $1$ with a ball of the same radius in a metric product between a three-dimensional metric space and a $n-3$ dimensional Euclidian factor. Moreover, with these existence results come estimates on the existence time and regularization properties of the flow, quantified in term of the hypotheses on the initial data. The possibility to regularize metrics, locally or globally, with such estimates has consequences in terms of the metric spaces obtained as limits, in the Gromov-Hausdorff topology, of sequences of manifolds uniformly satisfying a), b) or c). Indeed, the classical compactness theorems for the Ricci Flow allow for the extraction of a limit flow for any sequence of initial metrics uniformly satisfying the hypotheses and thus possessing a flow for a controlled amount of time. In the case when these metrics approach a singular space in the Gromov-Hausdorff topology, such a limit solution can be interpreted as a flow regularizing the singular limit space, the existence of which puts constraints on the topology of this space

In this dissertation we study free boundary problems that model the evolution of interfaces in the presence of elasticity, such as thin film profiles and material void boundaries. These problems are characterized by the competition between the elastic bulk energy and the anisotropic surface energy. First, we consider the evolution equation with curvature regularization that models the motion of a two-dimensional thin film by evaporation-condensation on a rigid substrate. The film is strained due to the mismatch between the crystalline lattices of the two materials and anisotropy is taken into account. We present the results contained in [62] where the author establishes short time existence, uniqueness and regularity of the solution using De Giorgi’s minimizing movements to exploit the L2 -gradient flow structure of the equation. This seems to be the first analytical result for the evaporation-condensation case in the presence of elasticity. Second, we consider the relaxed energy introduced in [20] that depends on admissible pairs (E, u) of sets E and functions u defined only outside of E. For dimension three this energy appears in the study of the material voids in solids, where the pairs (E, u) are interpreted as the admissible configurations that consist of void regions E in the space and of displacements u of the atoms of the crystal. We provide the precise mathematical framework that guarantees the existence of minimal energy pairs (E, u). Then, we establish that for every minimal configuration (E, u), the function u is C 1,γ loc -regular outside an essentially closed subset of E. No hypothesis of starshapedness is assumed on the voids and all the results that are contained in [18] hold true for every dimension d ≥ 2.

La presente tesi tratta due argomenti distinti. Il Capitolo 1 e il Capitolo 2 riguardano problemi di evoluzione di interfacce nel piano. Nel Capitolo 1 viene considerata l’evoluzione di un materiale policristallino con tre (o più) fasi, in presenza di un’anisotropia cristallina (pari) ϕo la cui linea di livello 1, Fϕ :={ϕo ≤1} (Frank diagram), è un poligono regolare di n lati. La funzione duale ϕ : R2 →R definita da ϕ(ξ) := sup{ξ·η : ϕo(η)≤1}´e anch’essa un’anisotropia cristallina e Wϕ := {ϕ ≤ 1} è detta Wulff shape. In particolare, viene studiato il moto per curvatura cristallina di triodi elementari, ossia speciali reti piane di curve che sono frontiere regolari di insiemi rappresentanti tre fasi distinte di un materiale. Un triodo elementare è formato dall’unione di tre curve Lipschitziane, le interfacce, che si intersecano in un unico punto detto giunzione tripla. Ogni interfaccia è l’unione di un segmento di lunghezza finita e di una semiretta che riproduce due lati consecutivi della Wulff shape Wϕ. Viene analizzata l’esitenza locale e globale e la stabilità del flusso. Si dimostra l’esistenza locale di un unico flusso regolare stabile a partire da un dato iniziale regolare stabile: se n, il numero dei lati della Wulff shapeWϕ, è un multiplo di 6 allora il flusso è globale e converge a un flusso omotetico per t →+∞. L’analisi del comportamento del flusso per tempi grandi richiede lo studio della stabilità. La stabilità è l’ingrediente che assicura che nessun segmento si sviluppa dalla giunzione tripla durante il flusso. In generale, il flusso può diventare instabile in un tempo finito: se ciò accade e tutte le lunghezze dei segmenti finiti sono strettamente positive per tale tempo,è possibile costruire un flusso regolare per tempi successivi aggiungendo in corrispondenza della giunzione tripla in una delle tre interfacce un segmento infinitesimo opportuno (o addirittura un arco di curva a curvatura cristallina nulla). ´E anche possibile che durante il flusso uno dei tre segmenti scompaia in un tempo finito. In tal caso, in tale tempo il campo vettoriale di Cahn-Hoffman ha un salto di discontinuità e ai tempi successivi la giunzione tripla si muove traslando lungo la semiretta adiacente. Ognuno di questi flussi ha la proprietà che tutte le curvature cristalline rimangono limitate (persino se un segmento appare o scompare). ´E importante sottolineare che Taylor aveva già predetto la nascita di nuovi segmenti dalla giunzione tripla (senza però dimostrarlo). Viene inoltre considerato il flusso per curvatura cristalina di una partizione regolare stabile formata da due triodi elementari adiacenti. Vengono discussi alcuni esempi di situazioni di colasso che portano a cambi di topologia, come ad esempio la collisione di due giunzioni triple. Questi esempi (come anche il risultato di esistenza per tempi piccoli) mostrano uno dei vantaggi del flusso per curvatura cristallino rispetto, ad esempio, all’usuale moto per curvatura: calcoli espliciti possono essere fatti, e nel caso di non unicità, è possibile confrontare le energie delle diverse evoluzioni (difficile nel caso euclideo). Nel Capitolo 2 viene introdotta, usando la teoria delle funzioni a variazione limitata a valori in S1, la sfera diR2, una nuova classe di funzionali energia definiti su partizioni. Attraverso la variazione prima del funzionale energia, viene fornito un nuovo modello per l’evoluzione di interfacce che parzialmente estende quello introdotto nel Capitolo 1 e che consiste in un problema di frontiera libera definito sulle funzioni a variazione limitata a valori in S1. Questo modello è legato all’evoluzione di materiali policristallini dove è consentito alla Wulff shape di ruotare. Assumendo l’esitenza locale del flusso, si dimostra che durante il flusso curve chiuse convesse rimangono convesse e curve chiuse embedded rimangono embedded. Il secondo argomento della tesi è trattato nel Capitolo 3: l’obiettivo è quello di estendere il metodo delle linee di livello a sistemi di equazioni differenziali alle derivate parziali. Il metodo che viene proposto è consistente con la precedente ricerca portata avanti da Evans per l’equazione del calore e da Giga e Sato per equazioni di Hamilton-Jacobi. Il nostro approccio segue una costruzione geometrica che è legate alla nozione di barriera introdotta da De Giorgi. L’idea principale è quella di forzare un principio di confronto tra varietà di diversa codimensione e richiedere che ogni sottolivello di una soluzione dell’equazione per le linee di livello, detta level set equation, sia una barriera per i grafici di soluzioni del corrispondente sistema. Tale metodo ben si applica a una classe di sistemi di equazioni quasi-lineari del primo ordine. Viene fornita la level set equation associata ad opportuni sitemi di leggi di conservazione del primo ordine, al flusso per curvatura media di una varietà di codimensione arbitraria e a sitemi di equazioni di reazione-diffusione. Infine, viene calcolata la level set equation associata al sistema soddisfatto dalle parametrizzazioni di curve piane che si muovono per curvatura.
The present thesis deals with two different subjects. Chapter 1 and Chapter 2 concern interfaces evolution problems in the plane. In Chapter 1 I consider the evolution of a polycrystalline material with three (or more) phases, in presence of for an even crystalline anisotropy ϕo whose one-sublevel set Fϕ := {ϕo ≤ 1} (the Frank diagram) is a regular polygon of n sides. The dual function ϕ : R2 → R defined by ϕ(ξ) := sup{ξ ·η : ϕo(η) ≤ 1} is crystalline too and Wϕ := {ϕ ≤ 1} is called the Wulff shape. I am particularly interested in the motion by crystalline curvature of special planar networks called elementary triods, namely a regular three-phase boundary given by the union of three Lipschitz curves, the interfaces, intersecting at a point called triple junction. Each interface is the union of a segment of finite length and a half-line, reproducing two consecutive sides of Wϕ. I analyze local and global existence and stability of the flow. I prove that there exists, locally in time, a unique stable regular flow starting from a stable regular initial datum. I show that if n, the number of sides of Wϕ, is a multiple of 6 then the flow is global and converge to a homothetic flow as t → +∞. The analysis of the long time behavior requires the study of the stability. Stability is the ingredient that ensures that no additional segments develop at the triple junction during the flow. In general, the flow may become unstable at a finite time: if this occurs and none of the segments desappears, it is possible to construct a regular flow at subsequent times by adding an infinitesimal segment (or even an arc with zero crystalline curvature) at the triple junction. I also show that a segment may desappear. In such a case, the Cahn-Hoffman vector field Nmin has a jump discontinuity and the triple junction translates along the remaining adjacent half-line at subsequent times. Each of these flows has the property that all crystalline curvatures remain bounded (even if a segment appears or disappears). I want to stress that Taylor already predicted the appearance of new edges from a triple junction. I also consider the crystalline curvature flow starting from a stable ϕ-regular partition formed by two adjacent elementary triods. I discuss some examples of collapsing situations that lead to changes of topology, such as for instance the collision of two triple junctions. These examples (as well as the local in time existence result) show one of the advantages of crystalline flows with respect, for instance, to the usual mean curvature flow: explicit computations can be performed to some extent, and in case of nonuniqueness, a comparison between the energies of different evolutions (difficult in the euclidean case) can be made. In Chapter 2 we introduce, using the theory of S1-valued functions of bounded variations, a class of energy functionals defined on partitions and we produce, through the first variation, a new model for the evolution of interfaces which partially extends the one in Chapter 1 and which consists of a free boundary problem defined on S1-valued functions of bounded variation. This model is related to the evolution of polycrystals where the Wulff shape is allowed to rotate. Assuming the local existence of the flow, we show convexity preserving and embeddedness preserving properties. The second subject of the thesis is considered in Chapter 3 where we aim to extend the level set method to systems of PDEs. The method we propose is consistent with the previous research pursued by Evans for the heat equation and by Giga and Sato for Hamilton-Jacobi equations. Our approach follows a geometric construction related to the notion of barriers introduced by De Giorgi. The main idea is to force a comparison principle between manifolds of different codimension and require each sub-level of a solution of the level set equation to be a barrier for the graph of a solution of the corresponding system. We apply the method for a class of systems of first order quasi-linear equations. We compute the level set equation associated with suitable first order systems of conservation laws, with the mean curvature flow of a manifold of arbitrary codimension and with systems of reaction-diffusion equations. Finally, we provide a level set equation associated with the parametric curvature flow of planar curves.

Dans tout ce qui suit, nous considérons une variété riemannienne compacte de dimension au moins égale à 2. A tout domaine (suffisamment régulier) , on peut associer la première valeur propre ?Ù de l’opérateur de Laplace-Beltrami avec condition de Dirichlet au bord. Nous dirons qu’un domaine est extrémal (sous entendu, pour la première valeur propre de l’opérateur de Laplace-Beltrami) si est un point critique de la fonctionnelle Ù? ?O sous une contrainte de volume V ol(Ù) = c0. Autrement dit, est extrémal si, pour toute famille régulière {Ot}te (-t0,t0) de domaines de volume constant, telle que Ù 0 = Ù, la dérivée de la fonction t ? ?Ot en 0 est nulle. Rappelons que les domaines extrémaux sont caractérisés par le fait que la fonction propre, associée à la première valeur propre sur le domaine avec condition de Dirichlet au bord, a une donnée de Neumann constante au bord. Ce résultat a été démontré par A. El Soufi et S. Ilias en 2007. Les domaines extrémaux sont donc des domaines sur lesquels peut être résolu un problème elliptique surdéterminé. L’objectif principal de cette thèse est la construction de domaines extrémaux pour la première valeur propre de l’opérateur de Laplace-Beltrami avec condition de Dirichlet au bord. Nous donnons des résultats d’existence de domaines extrémaux dans le cas de petits volumes ou bien dans le cas de volumes proches du volume de la variété. Nos résultats permettent ainsi de donner de nouveaux exemples non triviaux de domaines extrémaux. Le premier résultat que nous avons obtenu affirme que si une variété admet un point critique non dégénéré de la courbure scalaire, alors pour tout volume petit il existe un domaine extrémal qui peut être construit en perturbant une boule géodésique centrée en ce point critique non dégénéré de la courbure scalaire. La méthode que nous utilisons pour construire ces domaines extrémaux revient à étudier l’opérateur (non linéaire) qui à un domaine associe la donnée de Neumann de la première fonction propre de l’opérateur de Laplace-Beltrami sur le domaine. Il s’agit d’un opérateur (hautement non linéaire), nonlocal, elliptique d’ordre 1. Dans Rn × R/Z, le domaine cylindrique Br × R/Z, o`u Br est la boule de rayon r > 0 dans Rn, est un domaine extrémal. En étudiant le linéarisé de l’opérateur elliptique du premier ordre défini par le problème précédent et en utilisant un résultat de bifurcation, nous avons démontré l’existence de domaines extrémaux nontriviaux dans Rn × R/Z. Ces nouveaux domaines extrémaux sont proches de domaines cylindriques Br × R/Z. S’ils sont invariants par rotation autour de l’axe vertical, ces domaines ne sont plus invariants par translations verticales. Ce deuxi`eme r´esultat donne un contre-exemple à une conjecture de Berestycki, Caffarelli et Nirenberg énoncée en 1997. Pour de grands volumes la construction de domaines extrémaux est techniquement plus difficile et fait apparaître des phénomènes nouveaux. Dans ce cadre, nous avons dû distinguer deux cas selon que la première fonction propre Ø0 de l’opérateur de Laplace-Beltrami sur la variété est constante ou non. Les résultats que nous avons obtenus sont les suivants : 1. Si Ø0 a des points critiques non dégénérés (donc en particulier n’est pas constante), alors pour tout volume assez proche du volume de la variété, il existe un domaine extrémal obtenu en perturbant le complément d’une boule géodésique centrée en un des points critiques non dégénérés de Ø0. 2. Si Ø0 est constante et la variété admet des points critiques non dégénérés de la courbure scalaire, alors pour tout volume assez proche du volume de la variété il existe un domaine extrémal obtenu en perturbant le complément d’une boule géodésique centrée en un des points critiques non dégénérés de la courbure scalaire
In what follows, we will consider a compact Riemannian manifold whose dimension is at least 2. Let Ù be a (smooth enough) domain and ?O the first eigenvalue of the Laplace-Beltrami operator on Ù with 0 Dirichlet boundary condition. We say that Ù is extremal (for the first eigenvalue of the Laplace-Beltrami operator) if is a critical point for the functional Ù? ?O with respect to variations of the domain which preserve its volume. In other words, Ù is extremal if, for all smooth family of domains { Ù t}te(-t0,t0) whose volume is equal to a constant c0, and Ù 0 = Ù, the derivative of the function t ? ?Ot computed at t = 0 is equal to 0. We recall that an extremal domain is characterized by the fact that the eigenfunction associated to the first eigenvalue of the Laplace-Beltrami operator over the domain with 0 Dirichlet boundary condition, has constant Neumann data at the boundary. This result has been proved by A. El Soufi and S. Ilias in 2007. Extremal domains are then domains over which can be solved an elliptic overdeterminated problem. The main aim of this thesis is the construction of extremal domains for the first eigenvalue of the Laplace-Beltrami operator with 0 Dirichlet boundary condition. We give some existence results of extremal domains in the cases of small volume or volume closed to the volume of the manifold. Our results allow also to construct some new nontrivial exemples of extremal domains. The first result we obtained states that if the manifold has a nondegenerate critical point of the scalar curvature, then, given a fixed volume small enough, there exists an extremal domain that can be constructed by perturbation of a geodesic ball centered in that nondegenerated critical point of the scalar curvature. The methode used is based on the study of the operator that to a given domain associes the Neumann data of the first eigenfunction of the Laplace-Beltrami operator over the domain. It is a highly nonlinear, non local, elliptic first order operator. In Rn × R/Z, the circular-cylinder-type domain Br × R/Z, where Br is the ball of radius r > 0 in Rn, is an extremal domain. By studying the linearized of the elliptic first order operator defined in the previous problem, and using some bifurcation results, we prove the existence of nontrivial extremal domains in Rn × R/Z. Such extremal domains are closed to the circular-cylinder-type domains Br × R/Z. If they are invariant by rotation with respect to the vertical axe, they are not invariant by vertical translations. This second result gives a counterexemple to a conjecture of Berestycki, Caffarelli and Nirenberg stated in 1997. For big volumes the construction of extremal domains is technically more difficult and shows some new phenomena. In this context, we had to distinguish two cases, according to the fact that the first eigenfunction Ø0 of the Laplace-Beltrami operator over the manifold is constant or not. The results obtained are the following : 1. If Ø0 has a nondegenerated critical point (in particular it is not constant), then, given a fixed volume closed to the volume of the manifold, there exists an extremal domain obtained by perturbation of the complement of a geodesic ball centered in a nondegenerated critical point of Ø0. 2. If Ø0 is constant and the manifold has some nondegenerate critical points of the scalar curvature, then, for a given fixed volume closed to the volume of the manifold, there exists an extremal domain obtained by perturbation of the complement of a geodesic ball centered in a nondegenerate critical point of the scalar curvature

The subject of this thesis is the study of some geometric problems arising in the context of locally and globally homogeneous Riemannian spaces. In particular, we are mainly interested in investigate the interplay between curvature conditions and the compactness of some classes of locally homogeneous spaces, with respect to appropriate topologies.

碩士
國立中正大學
應用數學研究所
81
Kahler manifolds with negative sectional curvature play an important role in the complex geometry theory, but we have very few examples.We desire to get more examples very much. So in this paper, we will introduce some examples which are known. And we will consider two Riemannian metrics with negative curvature on a domain of .R^2. which are not necessarily conformal, and discuss under what condition, the sum of these metrics still has negative curvature.

Let compact n-dimensional Riemannian manifolds $(M,g),\ (\widehat M,\ g)$ a diffeomorphism $u\sb0: M\to \widehat M,$ and a constant $p > n$ be given. Then sufficiently small $L\sp{p}$ bounds on the curvature of $\widehat M$ and on the difference of $g$ and $u\sbsp{0}{\*}\ g$ guarantee that $u\sb0$ can be continuously deformed to a harmonic diffeomorphism. A vector field $v$ is constructed on the space of mappings $u$ which are $L\sp{2,p}$ close to $u\sb0$ by solving the nonlinear elliptic equation $\Delta v + \widehat{Rc}\ v = -\Delta u.$ It is shown that under sufficient conditions on $u\sb0$ and on the curvature $\widehat{Rm}$ of the target, the integral curve $u\sb t$ of this vector field converges to a harmonic diffeomorphism. Since the objects we work with, such as $v$ and its derivatives, live in bundles over $M$, to prove regularity results we must first adapt standard techniques and results of elliptic theory to the bundle case. Among the generalizations we prove are Moser iteration, a Sobolev embedding theorem, and a Calderon-Zygmund inequality.

General metric spaces satisfying weak and synthetic notions of upper and lower curvature bounds will be studied. The relations between upper and lower bounds will be pointed out, especially the interactions between a packing condition and different forms of convexity of the metric. The main tools will be a new and flexible definition of entropy on metric spaces and a version of the Tits Alternative for groups of isometries of the metric spaces under consideration. The applications can be divided into classical and new results: the former consist in generalizations to a wider context of the theory of negatively curved Riemannian manifolds, while the latter include several compactness and continuity results.