Minimal Translation Hypersurfaces in Euclidean Space

Munteanu, Marian Ioan; Palmas, Oscar; Ruiz-Hernández, Gabriel

doi:10.1007/s00009-015-0645-9

Cited by 17 publications

(9 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the literature, other works have appeared on the study of translation surfaces with constant mean curvature, also in other ambient spaces: we refer [2,4,5,7,10,11], without to be a complete list. However, in all these works, the translation surface is of planar type, so the problem of finding such surfaces reduces into a problem of solving a PDE by separation of variables.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Classification and Construction of Minimal Translation Surfaces in Euclidean Space

Hasanis

López

2019

Results Math

View full text Add to dashboard Cite

A translation surface of Euclidean space R 3 is the sum of two regular curves α and β, called the generating curves. In this paper we classify the minimal translation surfaces of R 3 and we give a method of construction of explicit examples. Besides the plane and the minimal surfaces of Scherk type, it is proved that up to reparameterizations of the generating curves, any minimal translation surface is described as Ψ(s, t) = α(s) + α(t), where α is a curve parameterized by arc length s, its curvature κ is a positive solution of the autonomous ODE (y ) 2 + y 4 + c 3 y 2 + c 2 1 y −2 + c 1 c 2 = 0 and its torsion is τ (s) = c 1 /κ(s) 2 . Here c 1 = 0, c 2 and c 3 are constants such that the cubic equation −λ 3 + c 2 λ 2 − c 3 λ + c 1 = 0 has three real roots λ 1 , λ 2 and λ 3 .

show abstract

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Classification and Construction of Minimal Translation Surfaces in Euclidean Space

Hasanis

López

2019

Results Math

View full text Add to dashboard Cite

show abstract

“…In the literature, there are many works on the study of translation (hyper) surfaces of plane type in different ambient spaces and different conditions on the curvatures, where the problem of finding such surfaces reduces into a problem of solving a PDE by separation of variables: without to be a complete list, we refer: [4,6,8,10,11,13].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Translation surfaces in Euclidean space with constant Gaussian curvature

Hasanis¹,

López²

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…Inoguchi, Lopez, and Munteanu [12,13] made important contributions to minimal translation surfaces in ambient spaces Nil 3 and Sol 3 . For more recent results and progress on translation surfaces or translation hypersurfaces, see [14][15][16][17][18][19].…”

Section: Theoremmentioning

confidence: 99%

A Note on Minimal Translation Graphs in Euclidean Space

Yang

Zhang

2019

Mathematics

View full text Add to dashboard Cite

In this note, we give a characterization of a class of minimal translation graphs generated by planar curves. Precisely, we prove that a hypersurface that can be written as the sum of n planar curves is either a hyperplane or a cylinder on the generalized Scherk surface. This result can be considered as a generalization of the results on minimal translation hypersurfaces due to Dillen–Verstraelen–Zafindratafa in 1991 and minimal translation surfaces due to Liu–Yu in 2013.

show abstract

Minimal Translation Hypersurfaces in Euclidean Space

Cited by 17 publications

References 8 publications

Classification and Construction of Minimal Translation Surfaces in Euclidean Space

Classification and Construction of Minimal Translation Surfaces in Euclidean Space

Translation surfaces in Euclidean space with constant Gaussian curvature

A Note on Minimal Translation Graphs in Euclidean Space

Contact Info

Product

Resources

About