Laguerre minimal surfaces in ℝ3

Song, Yu; Wang, Chang Ping

doi:10.1007/s10114-008-7117-0

Cited by 9 publications

(5 citation statements)

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“…In [1], the Laguerre minimal surfaces are studied from a global point of view. This theory has receveid much attention (see notable works such as [12], [8], [23], [26] and [28]), mostly from a local point of view. Hence, our novel local Weierstrass representation for Laguerre minimal surfaces, derived from that for H 2 -surfaces, represents a significant contribution to this field.…”

Section: Introductionmentioning

confidence: 99%

Weingarten Surfaces Associated to Laguerre Minimal Surfaces

Santos,

Corro

2024

Preprint

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In the work \cite{Laredo}, the author shows that every hypersurface in Euclidean space is locally associated to the unit sphere by a sphere congruence, whose radius function $R$ is a geometric invariant of hypersurface. In this paper, we define the spherical mean curvature $H_S$ for any surface $\Sigma$, which depends on the principal curvatures of $\Sigma$ and the radius function $R$. We then explore two classes of surfaces: those with $H_S = 0$, referred to as $H_1$-surfaces, and the surfaces with spherical mean curvature of harmonic type, denoted as $H_2$-surfaces. We provide a Weierstrass-type representation for each of these classes depending on three holomorphic functions. We prove that the $H_1$-surfaces are associated to the minimal surfaces, whereas the $H_2$-surfaces are related to Laguerre minimal surfaces. As an application, we present a new Weierstrass-type representation for Laguerre minimal surfaces, and specifically for minimal surfaces. In this way, the same holomorphic data can be used to provide examples in $H_1$-surface/minimal surface classes or in $H_2$-surface/Laguerre minimal surface classes. We also characterize the rotational cases, allowing us to find a complete rotational Laguerre minimal surface.

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Section: Introductionmentioning

confidence: 99%

Weingarten Surfaces Associated to Laguerre Minimal Surfaces

Santos,

Corro

2024

Preprint

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“…Como cosh z, sinh z, e o produto são holomorfos, temos para todo caminho fechado γ que γ α k = 0, isto é, α k não tem períodos reais. Logo da equação (1.14) temos [6], [3], [4], [5]) e no último século retomou grande interesse ( [1], [13], [15], [16], [20], [12], [21]).…”

Section: Representação De Weierstrassunclassified

“…A geometria das superfícies de Laguerre em R 3 foi desenvolvida por Blaschke e sua escola (ver [6]). Nos últimos anos houve um grande interesse no estudo da geometria das superfícies de Laguerre (ver [15], [16]), principalmente desde o ponto de vista local e foi apresentada mais de uma forma de representar as superfícies mínimas de Laguerre em R 3 (ver [20], [21]).…”

Section: Introductionunclassified

“…O presente trabalho baseia-se principalmente em [13], [21] e [1], os dois últimos apresentam representações para superfícies mínimas e mínimas generalizadas de Laguerre, respectivamente. No primeiro caso em um aspecto local e no segundo em um aspecto global.…”

Section: Introductionunclassified

“…Em seguida, apresentamos uma maneira de representar as superfícies-ML em R 3 , que são pontos críticos do funcional de Laguerre, de forma global usando três dados: uma função meromorfa, uma forma holomorfa e uma função real harmônica.Esta representação permitiu construir vários exemplos de superfícies mínimas de Laguerre generalizadas, isto é, aquelas superfícies mínimas de Laguerre que possuem curvatura zero em pontos isolados.Por outro lado, ainda existem trabalhos nesta linha de pesquisa que obtém resultados envolvendo superfícies-ML. A saber, no artigo[21], cada superfície-ML x : M → R 3 com Ω * ≡ 0 (ver[21], 2) é associada injetivamente a uma solução complexa da equação de Liouville em um aberto U , ∆f = e f , f :U ⊂ M → C.E já que as soluções da equação de Liouville são dadas porf = log 8φ z ψ z (φ + ψ) 2 ,temos que qualquer superfície-ML em R 3 com Ω * ≡ 0 é determinada por duas funções holomorfas.Além dessas representações, no artigo [20] é apresentada uma maneira de construir uma superfície-ML usando congruências de retas, citamos o Teorema central do artigo Teorema 4.1. Seja Σ uma congruência de retas dada por (ξ, F (ξ)).…”

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Geometria de Laguerre e representação para superfícies mínimas generalizadas de Laguerre

Vargas¹

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In this work, we present a study of Laguerre geometry in Euclidean space, presenting the geometry of oriented spheres and planes, as well as the Laguerre transformations. Through this study, we present the Laguerre surfaces and the Laguerre metric , whose volume element is known as the Laguerre functional. Then, study generalized Laguerre minimal surfaces, i.e. surfaces which are critical points of this functional and which allow isolated points with curvature zero. Similarly to the Weierstrass representation for minimal surfaces, we present a kind of Weierstrass representation that allows describe globally generalized Laguerre minimal surfaces using three data: a meromorphic function, a holomorphic form and a real harmonic function. This representation is called conformal representation and coincides with the Weierstrass representation when the harmonic real function is zero.

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Isoparametric Hypersurfaces

Cecil¹,

Ryan²

2015

Springer Monographs in Mathematics

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The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Münzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles.In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expansion formula for the Cartan-Münzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.1991 Mathematics Subject Classification. Primary 53C40.

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Laguerre minimal surfaces in ℝ3

Cited by 9 publications

References 9 publications

Weingarten Surfaces Associated to Laguerre Minimal Surfaces

Weingarten Surfaces Associated to Laguerre Minimal Surfaces

Geometria de Laguerre e representação para superfícies mínimas generalizadas de Laguerre

Isoparametric Hypersurfaces

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