Density estimates for minimal surfaces and surfaces flowing by mean curvature

2005 ◽  
pp. 129-139
Author(s):  
Robert Gulliver
Keyword(s):  
Minimal Surfaces ◽  
Mean Curvature ◽  
Density Estimates

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.  

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 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.

scholarly journals Black holes, marginally trapped surfaces and quasi-minimal surfaces

2009 ◽  
Vol 40 (4) ◽  
pp. 313-341 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Minimal Surfaces ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Space Time ◽  
Mean Curvature Vector ◽  
Trapped Surfaces ◽  
Marginally Trapped Surface ◽  
Marginally Trapped Surfaces ◽  
Trapped Surface ◽  
Recent Classification

The concept of trapped surfaces introduced by Sir Roger Penrose in [Phys. Rev. Lett. 14 (1965), 57-59] plays an extremely important role in cosmology and general relativity. A black hole is a trapped region in a space-time enclosed by a marginally trapped surface. In term of mean curvature vector, a space-like surface in a space-time is marginally trapped if its mean curvature vector field is light-like at each point. In this article, we survey recent classification results on marginally trapped surfaces from differential geometric viewpoint. Also, we survey recent results on a closely related subject; namely, quasi-minimal surfaces in pseudo-Riemannian manifolds.

scholarly journals Algebraic minimal surfaces in $\mathsf R^4$

2004 ◽  
Vol 94 (1) ◽  
pp. 109 ◽  
Author(s):  
Anthony Small
Keyword(s):  
Minimal Surfaces ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Null Curves

There exists a natural correspondence between null curves in $\mathbf{C}^4$ and "free" curves on $\mathcal O(1)\oplus \mathcal O(1)$; it underlies the existence of "Weierstrass type formulae" for minimal surfaces in $\mathbf{R}^4$. The construction determines correspondences for minimal surfaces in $\mathbf{R}^3$, and constant mean curvature 1 surfaces in $\mathrm{H}^3$; moreover it facilitates the study of symmetric minimal surfaces in $\mathbf{R}^4$.

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