scholarly journals Mean curvature one surfaces in hyperbolic space, and their relationship to minimal surfaces in Euclidean space

2001 ◽  
Vol 11 (4) ◽  
pp. 669-692 ◽  
Author(s):  
Wayne Rossman
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Surfaces In Euclidean Space

scholarly journals Construction of constant mean curvature n-noids using the DPW method

2020 ◽  
Vol 2020 (763) ◽  
pp. 223-249 ◽  
Author(s):  
Martin Traizet
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Genus Zero ◽  
Constant Mean Curvature Surfaces ◽  
Surfaces In Euclidean Space

AbstractWe construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.

scholarly journals Offset Ruled Surface in Euclidean Space with Density

2021 ◽  
Vol 29 (1) ◽  
pp. 219-233
Author(s):  
Neslihan Ulucan ◽  
Mahmut Akyigit
Keyword(s):  
Euclidean Space ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Surfaces In Euclidean Space ◽  
The Mean

Abstract In this paper, offset ruled surfaces in these spaces are defined by using the geometry of ruled surfaces in Euclidean space with density. The mean curvature and Gaussian curvature of these surfaces are studied. In addition, the relationships between the mean curvature and mean curvature with density, and the Gaussian curvature and the Gaussian curvature with density of the offset ruled surfaces in E 3 with density e z and e − x 2− y 2 are given.

scholarly journals Minimal surfaces in euclidean 3-space and their mean curvature 1 cousins in hyperbolic 3-space

2003 ◽  
Vol 75 (3) ◽  
pp. 271-278
Author(s):  
Shoichi Fujimori
Keyword(s):  
Minimal Surface ◽  
Hyperbolic Space ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
3 Dimensional

We show that the Hopf differentials of a pair of isometric cousin surfaces, a minimal surface in euclidean 3-space and a constant mean curvature (CMC) one surface in the 3-dimensional hyperbolic space, with properly embedded annular ends, extend holomorphically to each end. Using this result, we derive conditions for when the pair must be a plane and a horosphere.

scholarly journals Quasi-minimal rotational surfaces in pseudo-Euclidean four-dimensional space

2014 ◽  
Vol 12 (10) ◽  
Author(s):  
Georgi Ganchev ◽  
Velichka Milousheva
Keyword(s):  
Vector Field ◽  
Euclidean Space ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Dimensional Space ◽  
Curvature Vector ◽  
Parabolic Type ◽  
Mean Curvature Vector ◽  
Two Dimensional ◽  
Rotational Surfaces

AbstractIn the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis — rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all rotational quasi-minimal surfaces of elliptic, hyperbolic and parabolic type, respectively.

Horospherical convex hypersurfaces contracting of the hyperbolic space by functions of the mean curvature

2019 ◽  
Vol 30 (08) ◽  
pp. 1950039
Author(s):  
Shunzi Guo
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Single Point ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Final Time ◽  
Maximal Interval ◽  
The Mean ◽  
Convex Hypersurfaces

This paper concerns the evolution of a closed hypersurface of the hyperbolic space, convex by horospheres, in direction of its inner unit normal vector, where the speed equals a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity by horospheres is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This generalizes the previous result [S. Guo, Convex hypersurfaces evolving by functions of the mean curvature, preprint (2016), arXiv:1610.08214 ] for convex hypersurfaces in the Euclidean space by the author to the setting in the hyperbolic space for the same class of flows.

scholarly journals $d$-Minimal Surfaces in Three-Dimensional Singular Semi-Euclidean Space $\mathbb{R}^{0,2,1}$

2021 ◽  
Vol 52 (1) ◽  
pp. 37-67
Author(s):  
Yuichiro Sato
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Three Dimensional ◽  
Representation Formula ◽  
Degenerate Metric ◽  
Dimensional Minkowski Space ◽  
One To One ◽  
Zero Mean Curvature

In this paper, we investigate surfaces in singular semi-Euclidean space $\mathbb{R}^{0,2,1}$ endowed with a degenerate metric. We define $d$-minimal surfaces, and give a representation formula of Weierstrass type. Moreover, we prove that $d$-minimal surfaces in $\mathbb{R}^{0,2,1}$ and spacelike flat zero mean curvature (ZMC) surfaces in four-dimensional Minkowski space $\mathbb{R}^{4}_{1}$ are in one-to-one correspondence.

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