scholarly journals Uniqueness of Clifford torus with prescribed isoperimetric ratio

2021 ◽  
Author(s):  
Thomas Yu ◽  
Jingmin Chen
Keyword(s):  
Clifford Torus ◽  
Isoperimetric Ratio

A characterization of the Clifford Torus

1999 ◽  
Vol 48 (3) ◽  
pp. 537-540 ◽  
Author(s):  
I. Guadalupe ◽  
Aldir Brasil Junior ◽  
J. A. Delgado
Keyword(s):  
Clifford Torus

Basic Geometric Constructions

2021 ◽  
pp. 217-226
Author(s):  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Dimensional Space ◽  
Intersection Point ◽  
Important Application ◽  
Geometric Constructions ◽  
Clifford Torus ◽  
The Cost ◽  
Regular Homotopy ◽  
Associated Surfaces

Basic geometric constructions, including tubing, boundary twisting, pushing down intersections, and contraction followed by push-off are presented. These moves are used repeatedly later in the proof. New, detailed pictures illustrating these constructions are provided. The Clifford torus at an intersection point between two surfaces in 4-dimensional space is described. The chapter closes with an important application of some of these moves called the Geometric Casson Lemma. This lemma upgrades algebraically dual spheres to geometrically dual spheres, at the cost of introducing more self-intersections. It is also shown that an immersed Whitney move is a regular homotopy of the associated surfaces.

scholarly journals A characterization of complete Riemannian manifolds minimally immersed in the unit sphere

1993 ◽  
Vol 131 ◽  
pp. 127-133 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Second Fundamental Form ◽  
Clifford Torus ◽  
Veronese Surface ◽  
The Second Fundamental Form ◽  
Complete Riemannian Manifolds

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totallygeodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

scholarly journals Characterization of Clifford Torus in Three-Spheres

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 718
Author(s):  
Dong-Soo Kim ◽  
Young Ho Kim ◽  
Jinhua Qian
Keyword(s):  
Laplace Operator ◽  
Unit Sphere ◽  
Three Dimensional ◽  
Gauss Map ◽  
Dimensional Sphere ◽  
Clifford Torus ◽  
Closed Surfaces ◽  
Dimensional Unit ◽  
The Laplace Operator

We characterize spheres and the tori, the product of the two plane circles immersed in the three-dimensional unit sphere, which are associated with the Laplace operator and the Gauss map defined by the elliptic linear Weingarten metric defined on closed surfaces in the three-dimensional sphere.

scholarly journals Remarks on the self-shrinking Clifford torus

2020 ◽  
Vol 2020 (765) ◽  
pp. 139-170
Author(s):  
Christopher G. Evans ◽  
Jason D. Lotay ◽  
Felix Schulze
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
The Self ◽  
The Other ◽  
Local Entropy ◽  
Clifford Torus ◽  
Infinitesimal Deformations ◽  
Rigid Motions ◽  
The One

AbstractOn the one hand, we prove that the Clifford torus in {\mathbb{C}^{2}} is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian F-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is rigid: it is locally unique as a self-shrinker for mean curvature flow, despite having infinitesimal deformations which do not arise from rigid motions. The proofs rely on analysing higher order phenomena: specifically, showing that the Clifford torus is not a local entropy minimiser even under Hamiltonian variations, and demonstrating that infinitesimal deformations which do not generate rigid motions are genuinely obstructed.

Rigidity theorems of Clifford torus

2010 ◽  
Vol 30 (3) ◽  
pp. 890-896 ◽  
Author(s):  
Zhang Yuntao
Keyword(s):  
Clifford Torus ◽  
Rigidity Theorems

A characterization of the Clifford torus

2005 ◽  
Vol 85 (2) ◽  
pp. 175-182 ◽  
Author(s):  
Theodoros Vlachos
Keyword(s):  
Clifford Torus

New examples of conformally constrained Willmore minimizers of explicit type

2015 ◽  
Vol 8 (4) ◽  
Author(s):  
Cheikh Birahim Ndiaye ◽  
Reiner Michael Schätzle
Keyword(s):  
Weak Convergence ◽  
Clifford Torus ◽  
Smooth Convergence

AbstractWe develop a convergence procedure to improve weak convergence of conformally constrained Willmore immersions to smooth convergence and apply this to get new examples of conformally constrained Willmore minimizers of explicit type. In fact by estimates of Li–Yau in [Inventiones Mathematicae 69 (1982), 269–291] and Montiel–Ros in [Inventiones Mathematicae 83 (1986), 153–166], the Clifford torus

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