Academic literature on the topic 'Minkowski mean curvature operator'

In this paper, we try to give a classification of spacelike hypersurfaces of the Lorentz-Minkowski space-time E1n+1, whose mean curvature vector field of order (k+ 1) is an eigenvector of the kth linearized operator Lk, for a non-negative integer k less than n. The operator Lk is defined as the linear part of the first variation of the (k + 1)th mean curvature of a hypersurface arising from its normal variations. We show that any spacelike hypersurface of E1n+1 satisfying the condition LkHk+1 = λHk+1 (where 0 ≤ k ≤ n − 1) belongs to the class of Lk-biharmonic, Lk-1-type or Lk-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of spacelike hypersurfaces of Lorentz-Minkowski spaces, named the weakly convex hypersurfaces (i.e. on which all of principle curvatures are nonnegative). We prove that, on any weakly convex spacelike hypersurface satisfying the above condition for an integer k (where, 0 ≤ r ≤ n−1), the (k + 1)th mean curvature will be constant. As an interesting result, any weakly convex spacelike hypersurfaces, having assumed to be Lk-biharmonic, has to be k-maximal.

AbstractWe discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz–Minkowski space$\left\{\begin{aligned} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega.\end{aligned}\right.$The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.

AbstractIn this paper, we use the critical point theory for convex, lower semicontinuous perturbations of{C^{1}}-functionals to obtain the existence of multiple nontrivial solutions for one parameter potential systems involving the operator{u\mapsto\operatorname{div}(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}})}. The solvability of a general non-potential system is also established.

2013/2014
Questa tesi è dedicata allo studio di alcuni modelli differenziali che nascono nell'ambito della fluidodinamica o della relatività generale e che coinvolgono gli operatori di curvatura media nello spazio $N$-dimensionale euclideo o di Minkowski. Entrambi sono operatori ellittici quasi-lineari che non soddisfano la proprietà di uniforme ellitticità, essendo l'operatore di curvatura media euclidea degenere, mentre quello di curvatura media nello spazio di Minkowski singolare. Il lavoro è suddiviso in tre parti. La prima riguarda lo studio delle soluzioni periodiche dell'equazione di curvatura prescritta unidimensionale nello spazio euclideo, equazione che modellizza fenomeni di tipo capillarità. In accordo con la struttura dell'operatore di curvatura e imponendo un opportuno comportamento in 0, o all'infinito, della curvatura prescritta, si dimostra l'esistenza di infinite soluzioni subarmoniche classiche arbitrariamente piccole aventi opportune proprietà nodali, oppure di infinite soluzioni subarmoniche a variazione limitata con oscillazioni arbitrariamente grandi. La tecnica per la ricerca delle soluzioni classiche è topologica e si basa sull'uso del numero di rotazione e su una generalizzazione del teorema di Poincaré-Birkhoff; d'altro lato l'approccio per lo studio delle soluzioni non classiche poggia sulla teoria dei punti critici per funzionali non lisci, in particolare su un lemma di passo di montagna nello spazio delle funzioni a variazione limitata. La seconda parte della tesi è dedicata allo studio del problema di Dirichlet omogeneo associato a un'equazione della curvatura media prescritta anisotropa nello spazio euclideo, il quale fornisce un modello di descrizione della geometria della cornea umana. Il problema è ambientato in un dominio regolare in $\mathbb{R}^N$ con frontiera lipschitziana. Il capitolo è suddiviso a sua volta in tre sezioni, che sono rispettivamente focalizzate sui casi unidimensionale, radiale e $N$-dimensionale. Nel caso unidimensionale e nel caso radiale in una palla, si dimostrano l'esistenza e l'unicità di una soluzione classica, che presenta alcune proprietà qualitative aggiuntive. Le tecniche usate in questo contesto sono di natura topologica. Infine, nel caso $N$-dimensionale in un dominio generale, si provano l'esistenza, l'unicità e la regolarità di una soluzione di tipo forte del problema. In relazione ai possibili fenomeni di scoppio del gradiente, l'approccio è variazionale nello spazio delle funzioni a variazione limitata. Si enunciano e si dimostrano prima di tutto alcuni risultati preliminari riguardo al comportamento del funzionale associato al problema; tra questi, si sottolinea l'importanza di una proprietà di approssimazione. Successivamente si provano l'esistenza e l'unicità del minimizzante globale del funzionale, che è regolare all'interno ma non necessariamente sulla frontiera, e soddisfa il problema secondo un'opportuna definizione. Infine si mostra l'unicità della soluzione del problema. Sotto alcune ipotesi rafforzate sulla geometria del dominio, la soluzione ottenuta è classica. La terza parte della tesi riguarda il problema di Dirichlet associato a un'equazione della curvatura media prescritta nello spazio di Minkowski, che è di interesse in relatività generale. Il problema è ambientato in un dominio limitato regolare in $\mathbb{R}^N$ e un modello di curvatura media prescritta è dato da una funzione $f(x,s)$ che può avere comportamento sublineare, lineare, superlineare o sub-superlineare in $s=0$. L'attenzione è rivolta all'esistenza e alla molteplicità di soluzioni positive del problema. Come il precedente, anche questo capitolo è suddiviso in tre sezioni, che trattano rispettivamente i casi unidimensionale, radiale e $N$-dimensionale in un dominio generale. Nel caso unidimensionale, viene impiegato un approccio di tipo mappa-tempo per studiare una semplice situazione autonoma. Nel caso radiale in una palla, la tecnica è variazionale e lo studio del funzionale associato al problema evidenzia l'esistenza di un punto critico (casi sublineare o lineare), o di due (caso superlineare), o di tre punti critici (caso sub-superlineare): ciascuno di questi è una soluzione positiva del problema. Infine, nel caso generale in dimensione $N$, si adotta un approccio topologico che permette di studiare il problema non variazionale, in cui la funzione $f$ può dipendere dal gradiente della soluzione. Più nel dettaglio, con un metodo di sotto- e sopra-soluzioni specificamente sviluppato per questo problema, proviamo vari risultati di esistenza, molteplicità e localizzazione, in relazione alla presenza di una singola sotto-soluzione, o di una singola sopra-soluzione, o di una coppia di sotto- e sopra-soluzione ordinate o non ordinate. L'Appendice chiude la tesi: qui sono raccolti vari strumenti matematici utilizzati nel corso del lavoro.
This thesis is devoted to the study of some differential models arising in fluid mechanics or general relativity and involving the mean curvature operators in the $N$-dimensional Euclidean or Minkowski spaces. In both cases the operators are quasilinear elliptic operators which do not satisfy the property of uniform ellipticity, the Euclidean mean curvature operator being degenerate, whereas the Minkowski mean curvature operator being singular. This work is subdivided into three parts. The first one concerns the study of the periodic solutions of the one-dimensional prescribed curvature equation in the Euclidean space, which models capillarity-type phenomena. According to the structure of the curvature operator and imposing a suitable behaviour at zero, or at infinity, of the prescribed curvature, we prove the existence of infinitely many arbitrarily small classical subharmonic solutions with suitable nodal properties, or bounded variation subharmonic solutions with arbitrarily large oscillations. The technique for the search of classical solutions is topological and relies on the use of the rotation number and on a generalization of the Poincaré-Birkhoff theorem; whereas the approach for the study of non-classical solutions is based on non-smooth critical point theory, namely on a mountain pass lemma set in the space of bounded variation functions. The second part of the thesis is devoted to the study of the homogeneous Dirichlet problem associated with an anisotropic prescribed mean curvature equation in the Euclidean space, which provides a model for describing the geometry of the human cornea. The problem is set in a bounded domain in $\mathbb{R}^N$ with Lipschitz boundary. This chapter is subdivided into three sections, which are focused on the one-dimensional, the radial and the general $N$-dimensional case, respectively. In the one-dimensional and in the radial case in a ball, we prove an existence and uniqueness result of classical solution, which also displays some additional qualitative properties. Here the techniques used are topological in nature. Finally, in the $N$-dimensional case, we prove the existence, the uniqueness and the regularity of a strong-type solution of the problem. In order to tackle the possible gradient blow-up phenomena, the approach is variational and the framework is the space of bounded variation functions. We first collect some preliminary results about the behaviour of the action functional associated with the problem; among them, we remark the importance of an approximation property. We then prove the existence and uniqueness of the global minimizer of the action functional, which is smooth in the interior but non necessarily on the boundary, and satisfies the problem in a suitable sense. We finally prove the uniqueness of solution. Under some strengthened assumptions on the geometry of the domain, the solution obtained is classical. The third part of the thesis deals with the Dirichlet problem associated with a prescribed mean curvature equation in the Minkowski space, which is of interest in general relativity. The problem is set in a bounded regular domain in $\mathbb{R}^N$ and a model prescribed curvature is given by a function $f(x,s)$ whose behaviour is sublinear, linear, superlinear or sub-superlinear at $s=0$. The attention is addressed towards the existence and the multiplicity of positive solutions of the problem. In parallel to the second part of the thesis, this chapter is subdivided into three sections, which are focused on the one-dimensional, the radial and the general $N$-dimensional case, respectively. In the one-dimensional case, a time-map approach is employed for treating a simple autonomous situation. In the radial case in a ball, the technique is variational and the study of the action functional associated with the problem evidences the existence of either one (sublinear or linear cases), or two (superlinear case), or three (sub-superlinear case) non-trivial critical points of the action functional: each of them is a positive solution of the problem. Finally, in the general $N$-dimensional case, we adopt a topological approach which allows to study the non-variational problem, where the function $f$ may also depend on the gradient of the solution. Namely, by a lower and upper solution method specifically developed for this problem, we prove several existence, multiplicity and localization results, in relation to the presence of a single lower solution, or a single upper solution, or a couple of ordered or non-ordered lower and upper solutions of the problem. The Appendix completes this thesis: here several mathematical tools that have been used to prove the results are collected.
XXVI Ciclo
1986

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 99-103).
In the first part of this thesis, we give a classification of all self-similar solutions to the curve shortening flow in the Euclidean plane R² and discuss basic properties of the curves. The problem of finding the curves is reduced to the study of a twodimensional system of ODEs with two parameters that determine the type of the self-similar motion. In the second part, we describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal surfaces that rotate/translate with constant velocity under the flow. We look at their limiting behaviour as the pitch of the helicoidal motion goes to 0 and compare it with the limiting behaviour of the classical helicoidal minimal surfaces. Finally, we give a classification of the immersed cylinders in the family of constant mean curvature helicoidal surfaces. In the third part, we introduce the mean curvature flow of curves in the Minkowski plane R¹,¹ and give a classification of all the self-similar solutions. In addition, we demonstrate five non-self-similar exact solutions to the flow.
by Höskuldur Pétur Halldórsson.
Ph.D.

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In this dissertation we will present a demonstration that a linear Weingarten translation surface in Euclidean space and Lorentz-Minkowski space should have constant mean curvature or constant Gaussian curvature. The work is based on the article "Translation surfaces of linear Weingarten type" Antonio Bueno and Rafael López.
Nesta dissertação apresentaremos uma demonstração de que uma superfície de translação Weingarten linear no espaço euclidiano e no espaço Lorentz- Minkowski deve ter curvatura média constante ou curvatura de Gauss constante. O trabalho é baseado no artigo "Translation surfaces of linear Weingarten type"de Antonio Bueno e Rafael López.

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.

CoordenaÃÃo de AperfeiÃoamento de NÃvel Superior
Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico
Neste trabalho, consideramos funcionais paramÃtricos elÃpticos como generalizaÃÃes naturais para o clÃssico funcional Ãrea. Calculamos a primeira variaÃÃo de tais funcionais e, a partir da equaÃÃo de Euler-Lagrange, definimos a curvatura mÃdia anisotrÃpica de uma hipersuperfÃcie imersa em uma variedade Riemanniana como generalizaÃÃo natural da curvatura mÃdia usual. Em seguida, estabelecemos a fÃrmula da segunda variaÃÃo e classificamos as hipersuperfÃcies rotacionalmente simÃtricas que possuem curvatura mÃdia anisotrÃpica constante. A fim de compreender a estabilidade dos exemplo rotacionais,deduzimos a primeira e a segunda fÃrmulas de Minkowski. AlÃm disso, no contexto anisotrÃpico, apresentamos as equaÃÃes fundamentais de Weingarten, Codazzi e Gauss e, por fim, estudamos a harmonicidade da aplicaÃÃo de Gauss.
It is stated that critical points of a parametric elliptic functional in a Riemannian manifold are hypersurfaces with prescrebed anisotropic mean curvature. We prove that the anisotropic Gauss map of surfaces immersed in Euclidean space with constant anisotropic mean curvature is a harmonic map. In the case of rotatioally invariat functionals in some homogeneous three-dimensional ambients, we present a abridged version of a existence result for constant anisotropic mean curvature surfaces as cylinders, spheres, tori and annuli corresponding to the anisotropic analogs of onduloids and nodoids. In the Euclidean case M = R3, examples of stable critical points are provided by the Wulff shapes associated to functional F. Paralleling the case of constant curvature mean spheres, a characterization of Wulff shapes is provided, which answers affirmatively a question posed by M. Koiso and B. Parmer in [13].

Cette thèse est consacrée à l'étude de plusieurs problèmes d'équations aux dérivées partielles non-linéaires.

La première partie (chapitres 1-2-3) traite d'un problème trouvant son origine en biologie mathématique, à savoir l'étude de la survie à long terme d'une population dont l'évolution est gouvernée par une équation parabolique non-linéaire. Dans le modèle considéré, le mécanisme de diffusion est contrôlé par le p-Laplacien, la non-linéarité est de type logistique et fait intervenir un poids m pouvant changer de signe, et les conditions aux limites sont de flux nul. Le poids m correspond à une répartition des ressources devant permettre la survie de la population. Dans le chapitre 1, nous déterminons entre autres un critère de survie à long terme faisant intervenir la valeur propre principale du p-Laplacien avec poids m. Cette valeur propre apparait, plus précisément, comme la valeur limite d'un paramètre en-dessous de laquelle toute solution positive de l'équation converge vers zéro lorsque t tend vers l'infini. Ceci nous conduit naturellement au problème de minimiser la valeur propre en question lorsque m varie dans une classe adéquate de poids. Dans le chapitre 2, nous prouvons l'existence de minimiseurs et montrons que ces derniers satisfont une propriété de type “bang-bang”. Plusieurs propriétés de montonie sont aussi étudiées dans des situations géométriques particulières, et une caractérisation complète est donnée en dimension 1. Le chapitre 3 est consacré à l'élaboration de simulations numériques, où l'algorithme utilisé combine un méthode de plus grande pente avec une représentation de certains ensembles comme ensembles de niveaux.

La deuxième sujet de cette thèse (chapitre 4) est un problème elliptique faisant intervenir l'opérateur de courbure moyenne. Nous nous intéressons à l'existence et à la multiplicité de solutions nodales de ce problème. Nous montrons que, si un certain paramètre de l'équation est suffisamment grand, il existe une solution nodale qui change de signe exactement deux fois. Nous établissons également l'existence d'un nombre arbitrairement grand de solutions nodales. Enfin, dans le cas particulier où le domaine est une boule, un résultat de brisure de symétrie est obtenu, résultat qui induit l'existence d'au moins deux solutions à deux domaines nodaux.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished

Provamos resultados sobre a geometria de hipersuperfícies em diferentes espaços ambiente. Primeiro, definimos uma aplicação de Gauss generalizada para uma hipersuperfície Mn-1 c/ Nn, onde N é um espaço simétrico de dimensão n ≥ 3. Em particular, generalizamos um resultado de Ruh-Vilms e apresentamos aplicações. Em seguida, estudamos superfícies em espaços de dimensão 3: estudamos a equação da curvatura média em um produto semidireto R2oAR e obtemos estimativas da altura e a existência de gráficos mínimos do tipo Scherk. Finalmente, no espaço ambiente de uma variedade hiperbólica de dimensão 3: nós apresentamos condições suficientes para que um mergulho completo de uma superfície ∑ de topologia finita em N com curvatura média |H∑| ≤ 1 seja próprio.
We prove results concerning the geometry of hypersurfaces on di erent ambient spaces. First, we de ne a generalized Gauss map for a hypersurface Mn-1 c/ Nn, where N is a symmetric space of dimension n ≥ 3. In particular, we generalize a result due to Ruh-Vilms and make some applications. Then, we focus on surfaces on spaces of dimension 3: we study the mean curvature equation of a semidirect product R2 oA R to obtain height estimates and the existence of a Scherk-like minimal graph. Finally, on the ambient space of a hyperbolic manifold N of dimension 3 we give su cient conditions for a complete embedding of a nite topology surface ∑ on N with mean curvature |H∑| ≤ 1 to be proper.

Cette thèse porte sur la question de stabilité de certaines surfaces minimales évoluant sous le flot de courbure moyenne nulle dans l'espace de Minkowski. Cette problématique conduit à l'étude d'un système d'équations qui s'avère d'être hyperbolique sous la condition que les surfaces en question restent de type temps.Le travail qu'on présente ici se compose de deux parties. La première partie est liée à la formation de singularités en temps fini pour des surfaces asymptotiques au cône de Simons à l'infini et la seconde partie est consacrée à la stabilité de l'hélicoïde.Dans la première partie de cette thèse, on montre en collaboration avec Hajer Bahouri et Galina Perelman par une approche constructive l'existence d'une famille de surfaces évoluant par le flot de courbure moyenne nulle dans l'espace de Minkowski qui explose lorsque t tend vers 0 vers une surface qui se comporte comme le cône de Simons à l'infini. Ce problème revient à étudier les phénomènes d'explosion pour une équation d'ondes quasi linéaire du second ordre.L'objectif de la seconde partie est d’étudier la stabilité de l'hélicoïde soumise à des perturbations radiales normales. En fait, l'hélicoïde est linéairement instable d'indice 1 et c'est pourquoi on ne peut s'attendre à un résultat de stabilité pour des perturbations arbitraires. Nous montrons dans cette partie que cette instabilité est la seule obstruction pour la stabilité non linéaire globale de l'hélicoïde. Plus précisément, en se plaçant dans le cadre des perturbations radiales normales, on a démontré l'existence d'une variété de codimension 1 constituée de données initiales générant des solutions globales convergeant vers l'hélicoïde à l'infini
This thesis focuses on the stability of some minimal surfaces under the vanishing mean curvature flow in Minkowski space. This issue amounts to investigate a system which turns out to be hyperbolic as long as the involved surfaces are time-like surfaces.The work presented here includes two parts. The first part in joint work with Hajer Bahouri and Galina Perelman is dedicated to the issue of singularity formation in finite time for surfaces asymptotic to the Simons cone at infinity and the second part is devoted to the study of the stability of the helicoid.In the first part of this thesis, we prove by a constructive approach the existence of a family of surfaces which evolve by the vanishing mean curvature flow in Minkowski space and which as t tends to~0 blow up towards a surface which behaves like the Simons cone at infinity. This issue amounts to investigate the singularity formation for a second order quasilinear wave equation.The aim of the second part is to investigate the stability of the helicoid under normal radial perturbations. Actually, the helicoid is linearly unstable of index 1, and that is why we cannot expect to have stability for arbitrary perturbations. In this part, we establish that this instability is the only obstruction to the global nonlinear stability for the helicoid. More precisely, in the framework of normal radial perturbations, we prove the existence of a codimension one set of small initial data generating global solutions converging to the helicoid at infinity

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In Finsler geometry, we have several volume forms, hence various of mean curvature forms. The two best known volumes forms are the Busemann-Hausdorff and Holmes- Thompson volume form. The minimal surface with respect to these volume forms are called BH-minimal and HT-minimal surface, respectively. Let (R3; eFb) be a Minkowski space of Randers type with eFb = ea+eb; where ea is the Euclidean metric and eb = bdx3; 0 < b < 1: If a connected surface M in (R3; eFb) is minimal with respect to both volume forms Busemann-Hausdorff and Holmes-Thompson, then up to a parallel translation of R3; M is either a piece of plane or a piece of helicoid which is generated by lines screwing along the x3-axis. Furthermore it gives an explicit rotation hypersurfaces BH-minimal and HT-minimal generated by a plane curve around the axis in the direction of eb] in Minkowski (a;b)-space (Vn+1; eFb); where Vn+1 is an (n+1)-dimensional real vector space, eFb = eaf eb ea ; ea is the Euclidean metric, eb is a one form of constant length b = kebkea; eb] is the dual vector of eb with respect to ea: As an application, it give us an explicit expression of surface of rotation “ forward” BH-minimal generated by the rotation around the axis in the direction of eb] in Minkowski space of Randers type (V3; ea+eb):
Na Geometria de Finsler, temos várias formas volume, consequentemente várias formas curvaturas médias. As duas mais conhecidas são as formas de volumes Busemann- Hausdorff e Holmes-Thompson. As superfícies mínimas com respeito a estes são chamados superfícies BH-mínimas e HT-mínimas, respectivamente. Seja (R3; eFb) um espaço de Minkowski do tipo Randers com eFb = ea+eb; onde ea é a métrica Euclidiana e eb = bdx3;0 < b < 1: Uma superfície em (R3; eFb) conexa M é mínima com respeito a ambas formas volumes Busemann-Hausdorff e Holmes-Thompson, então a menos de uma translação paralela de R3; M é parte de um plano ou parte de um helicóide, a qual é gerada pela rotação de uma reta (perpendicular ao eixo x3) ao longo do eixo x3: Ademais podemos obter explicitamente hipersuperfícies de rotação BH-mínima e HT-mínima geradas por uma curva plana em torno do eixo na direção de eb] num espaço (a; b) de Minkowski (Vn+1; eFb); onde Vn+1 é um espaço vetorial de dimensão (n+1); eFb = eaf eb ea ; ea é a métrica Euclidiana, eb é uma 1-forma constante com norma b := kebkea; eb] é o vetor dual de eb com respeito a a: Como aplicação, se dá uma expressão explícita de superfície de rotação completa “forward” BH-mínima gerada pela rotação em torno do eixo na direção de eb] num espaço de Minkowski do tipo Randers (V3; ea+eb):

In this thesis we prove the genericity of the set of metrics on a manifold with boundary M^{n+1}, such that all free boundary constant mean curvature (CMC) embeddings \\varphi: \\Sigma^n \\to M^{n+1}, being \\Sigma a manifold with boundary, are non-degenerate (Bumpy Metrics), (Theorem 2.4.1). We also give sufficient conditions to obtain a free boundary CMC deformation of a CMC inmersion (Theorems 3.2.1 and 3.2.2), and a stability criterion for this type of immersions (Theorem 3.3.3 and Corollary 3.3.5). In addition, given a one-parametric family, {\\varphi _t : \\Sigma \\to M} , of free boundary CMC immersions, we give criteria for the existence of smooth bifurcated branches of free boundary CMC immersions for the family {\\varphi_t}, via the implicit function theorem when the kernel of the Jacobi operator J is non-trivial, (Theorems 4.2.3 and 4.3.2), and we study stability and instability problems for hypersurfaces in this bifurcated branches (Theorems 5.3.1 and 5.3.3).
Nesta tese, provamos a genericidade do conjunto de métricas em uma variedade com fronteira M^{n+1}, de modo que todos os mergulhos de curvatura média constante (CMC) e fronteira livre \\varphi : \\Sigma^n \\to M^{n+1}, sendo \\Sigma uma variedade com fronteira, sejam não-degenerados (Métricas Bumpy), (Teorema 2.4.1). Nós também fornecemos condições suficientes para obter uma deformação CMC e fronteira livre de uma imersão CMC (Teoremas 3.2.1 and 3.2.2), e um critério de estabilidade para este tipo de imersões (Teorema 3.3.3 and Corolario 3.3.5). Além disso, dada uma família 1-paramétrica, {\\varphi _t : \\Sigma \\to M} , de imersões de CMC e fronteira livre, damos os critérios para a existência de ramos de bifurcação suaves de imersões CMC e fronteira livre para a familia {\\varphi_t}, por meio de o teorema da função implícita quando o kernel do operador Jacobi J é não-trivial, (Teoremas 4.2.3 and 4.3.2), e estudamos o problema da estabilidade e instabilidade para hipersuperfícies em naqueles ramos de bifurcação (Teoremas 5.3.1 and 5.3.3).