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Статті в журналах з теми "Euclidean mean curvature operator"
Yang, Dan, Jinchao Yu, Jingjing Zhang, and Xiaoying Zhu. "A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $." AIMS Mathematics 7, no. 1 (2022): 39–53. http://dx.doi.org/10.3934/math.2022003.
Повний текст джерелаPashaie, Firooz. "On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$." Tamkang Journal of Mathematics 51, no. 4 (November 1, 2020): 313–32. http://dx.doi.org/10.5556/j.tkjm.51.2020.3188.
Повний текст джерелаChen, Bang-Yen. "Mean curvature and shape operator of isometric immersions in real-space-forms." Glasgow Mathematical Journal 38, no. 1 (January 1996): 87–97. http://dx.doi.org/10.1017/s001708950003130x.
Повний текст джерелаPashaie, Firooz. "Weakly convex hypersurfaces of pseudo-Euclidean spaces satisfying the condition LkHk+1 = λHk+1". Proyecciones (Antofagasta) 40, № 3 (1 червня 2021): 711–19. http://dx.doi.org/10.22199/issn.0717-6279-3584.
Повний текст джерелаGüler, Erhan, Hasan Hacısalihoğlu, and Young Kim. "The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space." Symmetry 10, no. 9 (September 12, 2018): 398. http://dx.doi.org/10.3390/sym10090398.
Повний текст джерелаHEJUN, SUN, and QI XUERONG. "EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN." Glasgow Mathematical Journal 53, no. 2 (December 8, 2010): 321–32. http://dx.doi.org/10.1017/s0017089510000728.
Повний текст джерелаLee, Jae Won, Dong-Soo Kim, Young Ho Kim, and Dae Won Yoon. "Generalized null 2-type immersions in Euclidean space." Advances in Geometry 18, no. 1 (January 26, 2018): 27–36. http://dx.doi.org/10.1515/advgeom-2017-0029.
Повний текст джерелаCHENG, QING-MING, and YEJUAN PENG. "ESTIMATES FOR EIGENVALUES OF $\mathfrak L$ OPERATOR ON SELF-SHRINKERS." Communications in Contemporary Mathematics 15, no. 06 (November 19, 2013): 1350011. http://dx.doi.org/10.1142/s0219199713500119.
Повний текст джерелаMohammadpouri, Akram, and Firooz Pashaei. "$L_r$-biharmonic hypersurfaces in $\mathbb{E}^4$." Boletim da Sociedade Paranaense de Matemática 38, no. 5 (March 31, 2019): 9–18. http://dx.doi.org/10.5269/bspm.v38i5.38484.
Повний текст джерелаShen, Wenguo. "Nodal Solutions for Problems with Mean Curvature Operator in Minkowski Space with Nonlinearity Jumping Only at the Origin." Journal of Function Spaces 2020 (April 13, 2020): 1–11. http://dx.doi.org/10.1155/2020/9801931.
Повний текст джерелаДисертації з теми "Euclidean mean curvature operator"
Corsato, Chiara. "Mathematical analysis of some differential models involving the Euclidean or the Minkowski mean curvature operator." Doctoral thesis, Università degli studi di Trieste, 2015. http://hdl.handle.net/10077/11127.
Повний текст джерелаQuesta 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
Monte, Luiz AntÃnio Caetano. "Espectro essencial de uma classe de variedades riemannianas." Universidade Federal do CearÃ, 2012. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=9185.
Повний текст джерелаCoordenaÃÃo 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.
Thorpe, Benjamin Stuart. "Maximal graphs and spacelike mean curvature flows in semi-Euclidean spaces." Thesis, Durham University, 2011. http://etheses.dur.ac.uk/711/.
Повний текст джерелаHalldórsson, Höskuldur Pétur. "Self-similar solutions to the mean curvature flow in Euclidean and Minkowski space." Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/83693.
Повний текст джерелаCataloged 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.
Mantegazza, Carlo. "Smooth geometric evolutions of hypersurfaces and singular approximation of mean curvature flow." Doctoral thesis, Scuola Normale Superiore, 2014. http://hdl.handle.net/11384/85686.
Повний текст джерелаRostirolla, Adames Márcio [Verfasser]. "Spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-euclidean spaces / Márcio Rostirolla Adames." Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2012. http://d-nb.info/1024815757/34.
Повний текст джерелаCOLOMBO, GIULIO. "GLOBAL GRADIENT BOUNDS FOR SOLUTIONS OF PRESCRIBED MEAN CURVATURE EQUATIONS ON RIEMANNIAN MANIFOLDS." Doctoral thesis, Università degli Studi di Milano, 2021. http://hdl.handle.net/2434/813095.
Повний текст джерелаDerlet, Ann. "Eigenvalues of the p-Laplacian in population dynamics and nodal solutions of a prescribed mean curvature problem." Doctoral thesis, Universite Libre de Bruxelles, 2011. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/209932.
Повний текст джерелаLa 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
Ramos, Álvaro Krüger. "Constant mean curvature hypersurfaces on symmetric spaces, minimal graphs on semidirect products and properly embedded surfaces in hyperbolic 3-manifolds." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2015. http://hdl.handle.net/10183/118222.
Повний текст джерелаWe 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.
Cárdenas, Carlos Wilson Rodríguez. "Genericity of bumpy metrics, bifurcation and stability in free boundary CMC hypersurfaces." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-15022019-111803/.
Повний текст джерелаNesta 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).
Книги з теми "Euclidean mean curvature operator"
Nonlinear elliptic equations of the second order. Providence, Rhode Island: American Mathematical Society, 2016.
Знайти повний текст джерелаЧастини книг з теми "Euclidean mean curvature operator"
Kapouleas, Nikolaos. "Constant Mean Curvature Surfaces in Euclidean Spaces." In Proceedings of the International Congress of Mathematicians, 481–90. Basel: Birkhäuser Basel, 1995. http://dx.doi.org/10.1007/978-3-0348-9078-6_41.
Повний текст джерелаRitoré, Manuel, and Carlo Sinestrari. "The classical isoperimetric inequality in Euclidean space." In Mean Curvature Flow and Isoperimetric Inequalities, 53–67. Basel: Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-0346-0213-6_12.
Повний текст джерелаBertsch, Michiel, and Roberta Dal Passo. "A Parabolic Equation with a Mean-Curvature Type Operator." In Nonlinear Diffusion Equations and Their Equilibrium States, 3, 89–97. Boston, MA: Birkhäuser Boston, 1992. http://dx.doi.org/10.1007/978-1-4612-0393-3_6.
Повний текст джерелаAlencar, Hilario, Manfredo do Carmo, and Maria Fernanda Elbert. "Stability of hypersurfaces with vanishing r-mean curvature in euclidean space." In Manfredo P. do Carmo – Selected Papers, 425–40. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-25588-5_31.
Повний текст джерелаBereanu, Cristian, Petru Jebelean, and Jean Mawhin. "Multiple Radial Solutions at Resonance for Neumann Problems Involving the Mean Extrinsic Curvature Operator." In Analysis and Topology in Nonlinear Differential Equations, 87–101. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-04214-5_5.
Повний текст джерелаBishop, R. L. "A Relation Between Volume, Mean Curvature and Diameter." In Euclidean Quantum Gravity, 161. WORLD SCIENTIFIC, 1993. http://dx.doi.org/10.1142/9789814539395_0009.
Повний текст джерелаТези доповідей конференцій з теми "Euclidean mean curvature operator"
Bereanu, Cristian, Petru Jebelean, Jean Mawhin, Alberto Cabada, Eduardo Liz, and Juan J. Nieto. "Radial solutions for systems involving mean curvature operators in Euclidean and Minkowski spaces." In MATHEMATICAL MODELS IN ENGINEERING, BIOLOGY AND MEDICINE: International Conference on Boundary Value Problems: Mathematical Models in Engineering, Biology and Medicine. AIP, 2009. http://dx.doi.org/10.1063/1.3142953.
Повний текст джерелаALÍAS, LUIS J., and J. MIGUEL MALACARNE. "HYPERSURFACES WITH CONSTANT HIGHER ORDER MEAN CURVATURE IN EUCLIDEAN SPACE." In Proceedings of the International Conference held to honour the 60th Birthday of A M Naveira. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812777751_0003.
Повний текст джерелаBereanu, Cristian, Petru Jebelean, and Călin Şerban. "Dirichlet problems with mean curvature operator in Minkowski space." In 8th Congress of Romanian Mathematicians. WORLD SCIENTIFIC, 2016. http://dx.doi.org/10.1142/9789813142862_0001.
Повний текст джерелаЗвіти організацій з теми "Euclidean mean curvature operator"
Brander, David, and Wayne Rossman. Constant Mean Curvature Surfaces in Euclidean and Minkowski Three-Spaces. GIQ, 2012. http://dx.doi.org/10.7546/giq-10-2009-133-142.
Повний текст джерелаBrander, David, and Wayne Rossman. Constant Mean Curvature Surfaces in Euclidean and Minkowski 3-spaces. Journal of Geometry and Symmetry in Physics, 2012. http://dx.doi.org/10.7546/jgsp-12-2008-15-26.
Повний текст джерелаFetcu, Dorel. Integral Submanifolds in Three-Sasakian Manifolds Whose Mean Curvature Vector Fields are Eigenvectors of the Laplace Operator. GIQ, 2012. http://dx.doi.org/10.7546/giq-9-2008-210-223.
Повний текст джерела