scholarly journals On minimal surfaces on two-step Carnot groups

2019 ◽  
Vol 485 (4) ◽  
pp. 410-414
Author(s):  
M. B. Karmanova
Keyword(s):  
Minimal Surfaces ◽  
Mean Curvature ◽  
Carnot Groups ◽  
Correct Formulation ◽  
Area Functional ◽  
Suitable Notion ◽  
Minimality Conditions

For graph mappings constructed from contact mappings of arbitrary two-step Carnot groups, conditions for the correct formulation of minimal surfaces’ problem are found. A suitable notion of the (sub-Riemannian) area functional increment is introduced, differentiability of this functional is proved, and necessary minimality conditions are deduced. They are also expressed in terms of sub-Riemaninan mean curvature.  

Integral criteria for closed surfaces with constant mean curvature

2007 ◽  
Vol 463 (2081) ◽  
pp. 1199-1210
Author(s):  
Luca Guzzardi ◽  
Epifanio G Virga
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Closed Surfaces ◽  
Area Functional ◽  
Symmetric Surface ◽  
The Second Variation ◽  
Integral Identities

We propose three integral criteria that must be satisfied by all closed surfaces with constant mean curvature immersed in the three-dimensional Euclidean space. These criteria are integral identities that follow from requiring the second variation of the area functional to be invariant under rigid displacements. We obtain from them a new proof of the old result by Delaunay, to the effect that the sphere is the only closed axis-symmetric surface.

scholarly journals Self θ-congruent minimal surfaces in ℝ3

2000 ◽  
Vol 69 (2) ◽  
pp. 229-244
Author(s):  
Weihuan Chen ◽  
Yi Fang
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rigid Motions ◽  
Necessary And Sufficient ◽  
Weierstrass Pair

AbstractA minimal surface is a surface with vanishing mean curvature. In this paper we study self θ -congruent minimal surfaces, that is, surfaces which are congruent to their θ-associates under rigid motions in R3 for 0 ≤ θ < 2π. We give necessary and sufficient conditions in terms of its Weierstrass pair for a surface to be self θ-congruent. We also construct some examples and give an application.

scholarly journals On the existence of min-max minimal surface of genus g ≥ 2

2017 ◽  
Vol 19 (04) ◽  
pp. 1750041 ◽  
Author(s):  
Xin Zhou
Keyword(s):  
Variational Method ◽  
Strong Convergence ◽  
Convergence Theorem ◽  
Minimal Surfaces ◽  
Strong Convergence Theorem ◽  
Plateau Problem ◽  
Mountain Pass Lemma ◽  
Area Functional ◽  
Genus Formula ◽  
Direct Variational Method

In this paper, we establish a min-max theory for minimal surfaces using sweepouts of surfaces of genus [Formula: see text]. We develop a direct variational method similar to the proof of the famous Plateau problem by Douglas [Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263–321] and Rado [On Plateau’s problem, Ann. Math. 31 (1930) 457–469]. As a result, we show that the min-max value for the area functional can be achieved by a bubble tree limit consisting of branched genus-[Formula: see text] minimal surfaces with nodes, and possibly finitely many branched minimal spheres. We also prove a Colding–Minicozzi type strong convergence theorem similar to the classical mountain pass lemma. Our results extend the min-max theory by Colding–Minicozzi and the author to all genera.

scholarly journals Foliations by minimal surfaces and contact structures on certain closed3-manifolds

2003 ◽  
Vol 2003 (21) ◽  
pp. 1323-1330
Author(s):  
Richard H. Escobales
Keyword(s):  
Vector Field ◽  
Fundamental Group ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Vector Fields ◽  
Contact Structure ◽  
Contact Structures ◽  
Volume Form ◽  
Basic Vector ◽  
Transverse Volume

Let(M,g)be a closed, connected, orientedC∞Riemannian 3-manifold with tangentially oriented flowF. Suppose thatFadmits a basic transverse volume formμand mean curvature one-formκwhich is horizontally closed. Let{X,Y}be any pair of basic vector fields, soμ(X,Y)=1. Suppose further that the globally defined vector𝒱[X,Y]tangent to the flow satisfies[Z.𝒱[X,Y]]=fZ𝒱[X,Y]for any basic vector fieldZand for some functionfZdepending onZ. Then,𝒱[X,Y]is either always zero andH, the distribution orthogonal to the flow inT(M), is integrable with minimal leaves, or𝒱[X,Y]never vanishes andHis a contact structure. If additionally,Mhas a finite-fundamental group, then𝒱[X,Y]never vanishes onM, by the above together with a theorem of Sullivan (1979). In this caseHis always a contact structure. We conclude with some simple examples.

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