Rolling construction for anisotropic Delaunay surfaces

Koiso, Miyuki; Palmer, Bennett

doi:10.2140/pjm.2008.234.345

Cited by 14 publications

(3 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We refer the reader to [4] for details. A consequence of the previous formula, which we will use later, is that the third coordinate of χ is…”

Section: Preliminariesmentioning

confidence: 99%

A version of Krust’s theorem for anisotropic minimal surfaces

Palmer

2023

Proc. Amer. Math. Soc. Ser. B

Self Cite

View full text Add to dashboard Cite

We generalize Krust’s theorem to an anisotropic setting by showing the following. If Σ \Sigma is an anisotropic minimal surface in an axially symmetric normed linear space which is a graph over a convex domain contained in a plane orthogonal to the axis of symmetry, then its conjugate anisotropic minimal surface must also be a graph. We also generalize a reflection principle of Lawson relating symmetries of an anisotropic minimal surface with symmetries of its conjugate surface.

show abstract

“…We refer the reader to [4] for details. A consequence of the previous formula, which we will use later, is that the third coordinate of χ is…”

Section: Preliminariesmentioning

confidence: 99%

A version of Krust’s theorem for anisotropic minimal surfaces

Palmer

2023

Proc. Amer. Math. Soc. Ser. B

Self Cite

View full text Add to dashboard Cite

show abstract

“…Bennett Palmer and Álvaro Pámpano ask (and answer) the question of how Euler's classification of elasticae (including lemniscates) is affected if the rod's energy is anisotropic in Classification of planar anisotropic elasticae. Previously, a similar question of extending CMC surfaces to the anisotropic case, led to constant anisotropic mean curvature (CAMC) surfaces [5].…”

Section: Editorialmentioning

confidence: 99%

Editorial

Gielis¹,

Goemans²

2020

GANDF

View full text Add to dashboard Cite

“…When υ is the constant function 1, the anisotropic mean curvature agrees with the mean curvature. Delaunay's result for constant mean curvature of revolution was extended to anisotropic constant mean curvature surfaces of revolution by Miyuki and Palmer in 2008, [5]. Likewise, the dynamical interpretation for helicoidal constant mean curvature, [6], was extended to anisotropic constant mean curvature surfaces by Khuns and Palmer [4].…”

Section: Introductionmentioning

confidence: 97%

The treadmillsled of a curve

Perdomo

2011

Preprint

View full text Add to dashboard Cite

In [6], the author introduced the notion of TreadmillSled of a curve, which is an operator that takes regular curves in R 2 to curves in R 2 . This operator turned out to be very useful to describe helicoidal surfaces, for example, it provides an interpretation for the profile curve of helicoidal surfaces with constant mean curvature similar to the well known interpretation of the profile curve of Delaunay's surfaces using conics, see [6]. In [4] the authors used the TreadmillSled to classify all helicoidal surfaces with constant anisotropic mean curvature coming from axially symmetric anisotropic energy density. Also, in [6], the author proves that an helicoidal surface different from a cylinder has constant Gauss curvature if and only if the TreadmillSled of its profile curve lies in a vertical semi line contained in the lower or upper half plane and not contained in the y-axis... Why not the whole vertical line? and why the semi-line cannot be contained in the y-axis? In this paper we provide several properties of the TreadmillSled operator, in particular we will answer the questions in the previous sentence. Finally, we prove that the TreadmillSled of the profile curve of a minimal helicoidal surface is either a hyperbola or a the x-axis. The latter case occurs only when the surface is a helicoid.

show abstract

Rolling construction for anisotropic Delaunay surfaces

Cited by 14 publications

References 8 publications

A version of Krust’s theorem for anisotropic minimal surfaces

A version of Krust’s theorem for anisotropic minimal surfaces

Editorial

The treadmillsled of a curve

Contact Info

Product

Resources

About