Academic literature on the topic 'Euclidean mean curvature operator'

<abstract><p>A nondegenerate hypersurface in a pseudo-Euclidean space $ \mathbb{E}^{n+1}_{s} $ is called to have proper mean curvature vector if its mean curvature $ \vec{H} $ satisfies $ \Delta \vec{H} = \lambda \vec{H} $ for a constant $ \lambda $. In 2013, Arvanitoyeorgos and Kaimakamis conjectured ^{[<xref ref-type="bibr" rid="b1">1</xref>]}: any hypersurface satisfying $ \Delta \vec{H} = \lambda \vec{H} $ in pseudo-Euclidean space $ \mathbb E_{s}^{n+1} $ has constant mean curvature. This paper will give further support evidences to this conjecture by proving that a linear Weingarten hypersurface $ M^{n}_{r} $ in $ \mathbb E^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda \vec{H} $ has constant mean curvature if $ M^{n}_{r} $ has diagonalizable shape operator with less than seven distinct principal curvatures.</p></abstract>

A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

According to the well-known Nash's theorem, every Riemannian n-manifold admits an isometric immersion into the Euclidean space En(n+1)(3n+11)/2. In general, there exist enormously many isometric immersions from a Riemannian manifold into Euclidean spaces if no restriction on the codimension is made. For a submanifold of a Riemannian manifold there are associated several extrinsic invariants beside its intrinsic invariants. Among the extrinsic invariants, the mean curvature function and shape operator are the most fundamental ones.

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.

We study and examine the rotational hypersurface and its Gauss map in Euclidean four-space E 4 . We calculate the Gauss map, the mean curvature and the Gaussian curvature of the rotational hypersurface and obtain some results. Then, we introduce the third Laplace–Beltrami operator. Moreover, we calculate the third Laplace–Beltrami operator of the rotational hypersurface in E 4 . We also draw some figures of the rotational hypersurface.

AbstractFor a bounded domain Ω in a complete Riemannian manifold M, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator Δ2 − aΔ + b of the Laplacian Δ, where a and b are the nonnegative constants. We obtain an inequality for eigenvalues which contains a constant that only depends on the mean curvature of M. It yields an upper bound of the (k + 1)th eigenvalue Λk + 1. As their applications, some inequalities and bounds of eigenvalues on a complete minimal submanifold in a Euclidean space and a unit sphere are obtained.

AbstractWe define generalized null 2-type submanifolds in them-dimensional Euclidean space 𝔼m. Generalized null 2-type submanifolds are a generalization of null 2-type submanifolds defined by B.-Y. Chen satisfying the conditionΔ H=f H+gCfor some smooth functionsf,gand a constant vectorCin 𝔼m, whereΔandHdenote the Laplace operator and the mean curvature vector of a submanifold, respectively. We study developable surfaces in 𝔼3and investigate developable surfaces of generalized null 2-type surfaces. As a result, all cylindrical surfaces are proved to be of generalized null 2-type. Also, we show that planes are the only tangent developable surfaces which are of generalized null 2-type. Finally, we characterize generalized null 2-type conical surfaces.

In this paper, we study eigenvalues of the closed eigenvalue problem of the differential operator [Formula: see text], which is introduced by Colding and Minicozzi in [Generic mean curvature flow I; generic singularities, Ann. Math.175 (2012) 755–833], on an n-dimensional compact self-shrinker in R n+p. Estimates for eigenvalues of the differential operator [Formula: see text] are obtained. Our estimates for eigenvalues of the differential operator [Formula: see text] are sharp. Furthermore, we also study the Dirichlet eigenvalue problem of the differential operator [Formula: see text] on a bounded domain with a piecewise smooth boundary in an n-dimensional complete self-shrinker in R n+p. For Euclidean space R n, the differential operator [Formula: see text] becomes the Ornstein–Uhlenbeck operator in stochastic analysis. Hence, we also give estimates for eigenvalues of the Ornstein–Uhlenbeck operator.

A hypersurface $x : M^n\rightarrow\mathbb{E}^{n+1}$ is said to be biharmonic if $\Delta^2x=0$, where $\Delta$ is the Laplace operator of $M^n$. Based on a well-known conjecture of Bang-Yen Chen, the only biharmonic hypersurfaces in $E^{n+1}$ are the minimal ones. In this paper, we study an extension of biharmonic hypersurfaces in 4-dimentional Euclidean space $\mathbb{E}^4$. A hypersurface $x : M^n\rightarrow\mathbb{E}^{n+1}$ is called $L_r$-biharmonic if $L_r^2x=0$, where $L_r$ is the linearized opereator of $(r + 1)$th mean curvature of $M^n$. Since $L_0=\Delta$, the subject of $L_r$-biharmonic hypersurface is an extension of biharmonic ones. We prove that any $L_2$-biharmonic hypersurface in $\mathbb{E}^4$ with constant $2$-th mean curvature is $2$-minimal. We also prove that any $L_1$-biharmonic hypersurfaces in $\mathbb{E}^4$ with constant mean curvature is $1$-minimal.

In this paper, we establish a unilateral global bifurcation result for half-linear perturbation problems with mean curvature operator in Minkowski space. As applications of the abovementioned result, we shall prove the existence of nodal solutions for the following problem −div∇v/1−∇v2=αxv++βxv−+λaxfv, in BR0,vx=0, on ∂BR0, where λ ≠ 0 is a parameter, R is a positive constant, and BR0=x∈ℝN:x<R is the standard open ball in the Euclidean space ℝNN≥1 which is centered at the origin and has radius R. a(|x|) ∈ C[0, R] is positive, v+ = max{v, 0}, v− = −min{v, 0}, α(|x|), β(|x|) ∈ C[0, R]; f∈Cℝ,ℝ, s f (s) > 0 for s ≠ 0, and f0 ∈ [0, ∞], where f0 = lim|s|⟶0 f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

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

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico
CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior
Neste trabalho, provaremos alguns resultados sobre espectro essencial de uma classe de variedades Riemannianas, nÃo necessariamente completas, com condiÃÃes de curvatura na vizinhanÃa de um raio. Sobre essas condiÃÃes obtemos que o espectro essencial do operador de Laplace contÃm um intervalo. Como aplicaÃÃo, obteremos o espectro do operador de Laplace de regiÃes ilimitadas dos espaÃos formas, tais como a horobola do espaÃo hiperbÃlico e cones do espaÃo Euclidiano. Construiremos tambÃm um exemplo que indica a necessidade das condiÃÃes globais sobre o supremo das curvaturas seccionais fora de uma bola para que a variedade nÃo tenha espectro essencial.
In this thesis we consider a family of Riemannian manifolds, not necessarily complete, with curvature conditions in a neighborhood of a ray. Under these conditions we obtain that the essential spectrum of the Laplace operator contains an interval. The results presented in this thesis allow to determine the spectrum of the Laplace operator on unlimited regions of space forms, such as horoball in hyperbolic space and cones in Euclidean space. Also construct an example that shows the need of global conditions on the supreme sectional curvature outside a ball, so that the variety has no essential spectrum.

Two main results are proved. The first is for the maximal graph system in semi-Euclidean spaces. Existence of smooth solutions to the Dirichlet problem is proved, under certain assumptions on the boundary data. These assumptions allow the application of standard elliptic PDE methods by providing sufficiently strong a priori gradient estimates. The second result is a version of Brian White’s local regularity theorem, but now for the spacelike mean curvature flow system in semi-Euclidean spaces. This is proved using a version of Huisken’s monotonicity formula. Under the assumption of a suitable gradient bound, this theorem will give a priori estimates that allow such flows to be smoothly extended locally.

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.

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.

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.

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).