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Gálvez, José A.; Martínez, Ana Cristina Lahuerta; Teruel, José L.

doi:10.1016/j.jmaa.2015.05.049

Cited by 7 publications

(7 citation statements)

References 12 publications

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“…They prove that such a surface is topologically a finitely punctured compact surface, its surface area is finite, and each puncture is a cusp extending to infinity, asymptotic to a ray. Similar results on complete surfaces in non-Euclidean space forms are obtained in Gálvez et al (2015b).…”

Section: Results Motivated By the Theory Of Surfacessupporting

confidence: 81%

Around Efimov’s differential test for homeomorphism

Alexandrov

2020

Beitr Algebra Geom

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In 1968, Efimov proved the following remarkable theorem: Let f : R 2 → R 2 ∈ C 1 be such that det f (x) < 0 for all x ∈ R 2 and let there exist a function a(x) > 0 and constantsThen f (R 2 ) is a convex domain and f maps R 2 onto f (R 2 ) homeomorphically. Here curl f (x) stands for the curl of f at x ∈ R 2 . This article is an overview of analogues of this theorem, its generalizations and applications in the theory of surfaces, theory of global inverse functions, as well as in the study of the Jacobian Conjecture and the global asymptotic stability of dynamical systems.

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Section: Results Motivated By the Theory Of Surfacessupporting

confidence: 81%

Around Efimov’s differential test for homeomorphism

Alexandrov

2020

Beitr Algebra Geom

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“…A small step toward the solution of the above conjecture was obtained by Smyth-Xavier [25], where a consequence of their principal curvature theorem is that any complete surface of nonpositive Gauss curvature in ℝ 3 with one of its principal curvatures bounded away from zero must be a cylinder. Milnor-type results were also obtained for surfaces of ℍ 3 and 3 by Gálvez-Martínez-Teruel [17], where the authors use an abstract result about Codazzi pairs to prove that no complete surface can be immersed into ℍ 3 or 3 with nonpositive extrinsic or intrinsic sectional curvature, respectively, and one of its principal curvatures bounded away from zero.…”

Section: Milnor's Conjecturementioning

confidence: 90%

Complete submanifolds with relative nullity in space forms

et al. 2020

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We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong consequences in hyperbolic and Euclidean spaces. Among the consequences of our main theorem are results on submanifolds with sufficiently high index of relative nullity, submanifolds with nonpositive extrinsic curvature and submanifolds with integrable relative conullity. We show that no complete submanifold of hyperbolic space with sufficiently high index of relative nullity has extrinsic geometry bounded away from zero. As an application of these results, we derive an interesting corollary for complete submanifolds of hyperbolic space with nonpositive extrinsic curvature and discourse on their relation to Milnor's conjecture about complete surfaces with second fundamental form bounded away from zero. Finally, we also prove that every complete Euclidean submanifold with integrable relative conullity is a cylinder over the relative conullity.

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“…Inspirados pelos trabalhos de José Gálvez, Antonio Martínez e José Teruel em [21] e [20], estudaremos pares de Codazzi em superfícies e seus invariantes associados, tais como as curvaturas principais, a curvatura média, a curvatura extrínseca e a diferencial de Hopf. Veremos a partir destes trabalhos que a teoria de pares de Codazzi é uma ferramenta vantajosa, uma vez que, a partir de um resultado abstrato para pares de Codazzi em superfícies, serão apresentados resultados tipos Milnor e Emov para superfícies em H 3 e S 3 .…”

Section: Agradecimentosunclassified

“…O capitulo 2 é dedicado ao estudo dos trabalhos [21] e [20]. Instigado pelos trabalhos de Hilbert, apresentaremos inicialmente o Teorema de Emov, no qual nos diz que nenhuma superfície pode ser imersa no espaço Euclidiano tridimensional, tal que na métrica induzida, seja completa e tenha curvatura Gaussiana K ≤ const < 0.…”

Section: Agradecimentosunclassified

“…Neste capítulo, baseado nos estudos de José Galvez, Antônio Martínez e José Teruel, em [21] e [20], utilizaremos a técnica de pares de Codazzi para demonstrar um resultado abstrato para pares de Codazzi em superfícies de curvatura extrínseca não-positiva e o aplicaremos para estudar uma solução parcial para conjectura de Milnor e resultados do tipo Emov nas formas espaciais não-Euclidianas H 3 e S 3 .…”

Section: Analogamente Seunclassified

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Pares de codazzi em superfícies de variedades homogêneas

Gimarez¹

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Neste trabalho apresentamos um estudo de pares de Codazzi em superfícies de variedades homogêneas tridimensionais. Inicialmente, apresentamos um resultado abstrato para pares de Codazzi em superfícies completas com curvatura Gaussiana não-positiva e o aplicamos para obter resultados do tipo Emov e Milnor para superfícies completas nas formas espaciais não-Euclidianas. Para superfícies de espaços produto, a técnica de pares de Codazzi é utilizada na apresentação de um resultado do tipo Liebmann para superfícies completas com curvatura Gaussiana constante. Nos espaços homogêneos E(κ, τ ), com τ = 0, apresentamos um par de Codazzi denido sobre superfícies de curvatura média constante, cuja sua (2, 0)-parte é a diferencial de Abresch-Rosenberg.Palavras-Chaves: pares de Codazzi; variedades homogêneas; conjectura de Milnor; curvatura Gaussiana limitada; curvatura Gaussiana constante; curvatura média constante.

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Complete surfaces with non-positive extrinsic curvature in H3 and S3

Abstract: In this paper we give Efimov and Milnor's type results on complete surfaces in non-Euclidean space forms which satisfy that outside a compact set they have nonpositive Gauss curvature and the square of a principal curvature function is bounded from below by a positive constant.

Cited by 7 publications

References 12 publications

Around Efimov’s differential test for homeomorphism

Around Efimov’s differential test for homeomorphism

Complete submanifolds with relative nullity in space forms

Pares de codazzi em superfícies de variedades homogêneas

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