scholarly journals NON-PROPERLY EMBEDDED MINIMAL PLANES IN HYPERBOLIC 3-SPACE

2011 ◽  
Vol 13 (05) ◽  
pp. 727-739 ◽  
Author(s):  
BARIS COSKUNUZER
Keyword(s):  
Minimal Surfaces ◽  
Negative Curvature ◽  
Positive Curvature ◽  
Finite Topology ◽  
Simply Connected

In this paper, we show that there are non-properly embedded minimal surfaces with finite topology in a simply connected Riemannian 3-manifold with non-positive curvature. We show this result by constructing a non-properly embedded minimal plane in H3. Hence, this gives a counterexample to Calabi–Yau conjecture for embedded minimal surfaces in negative curvature case.

scholarly journals AN INFINITE NUMBER OF CLOSED FLRW UNIVERSES FOR ANY VALUE OF THE SPATIAL CURVATURE

2012 ◽  
Vol 18 ◽  
pp. 77-80
Author(s):  
HELIO V. FAGUNDES
Keyword(s):  
Finite Volume ◽  
Infinite Number ◽  
Negative Curvature ◽  
Large Scale ◽  
Positive Curvature ◽  
Cosmological Models ◽  
Infinite Volume ◽  
Spatial Curvature ◽  
Multiply Connected ◽  
Simply Connected

The Friedman-Lemaître-Robertson-Walker cosmological models are based on the assumptions of large-scale homogeneity and isotropy of the distribution of matter and energy. They are usually taken to have spatial sections that are simply connected; they have finite volume in the positive curvature case, and infinite volume in the null and negative curvature ones. I want to call the attention to the existence of an infinite number of models, which are based on these same metrics, but have compact, finite volume, multiply connected spatial sections. Some observational implications are briefly mentioned.

Hypothetical graphite structures with negative gaussian curvature

1993 ◽  
Vol 343 (1667) ◽  
pp. 113-127 ◽  
Keyword(s):  
Constant Curvature ◽  
Gaussian Curvature ◽  
Minimal Surfaces ◽  
Hyperbolic Plane ◽  
Negative Curvature ◽  
Positive Curvature ◽  
Three Dimensions ◽  
Ring Size ◽  
Two Dimensional ◽  
Planar Lattice

We consider the geometries of hypothetical structures, derived from a graphite net by the inclusion of rings of seven or eight bonds, which may be periodic in three dimensions. Just as the positive curvature of fullerene sheets is produced by the presence of pentagons, so negative curvature appears with a mean ring size of more than six. These structures are based on coverings of periodic minimal surfaces, and surfaces parallel to these, which are known as exactly defined mathematical objects. In the same way that the cylindrical and conical structures can be generated (geometrically) by curving flat sheets so that the perimeter of a ring can be identified with a vector in the two-dimensional planar lattice, so these structures can be related to tessellations of the hyperbolic plane. The geometry of transformations at constant curvature relates various surfaces. Some of the proposed structures, which are reviewed here, promise to have lower energies than those of the convex fullerenes

Torus actions, maximality, and non-negative curvature

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Christine Escher ◽  
Catherine Searle
Keyword(s):  
Negative Curvature ◽  
Torus Action ◽  
Maximal Symmetry ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Product Of Spheres ◽  
Rank Conjecture ◽  
Simply Connected ◽  
Symmetry Rank ◽  
Special Case

Abstract Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.

Negative Gaussian curvature induces significant suppression of thermal conduction in carbon crystals

Nanoscale ◽  
2017 ◽  
Vol 9 (37) ◽  
pp. 14208-14214 ◽  
Author(s):  
Zhongwei Zhang ◽  
Jie Chen ◽  
Baowen Li
Keyword(s):  
Gaussian Curvature ◽  
Thermal Conduction ◽  
Negative Curvature ◽  
Positive Curvature ◽  
Significant Suppression ◽  
Carbon Allotropes ◽  
Zero Curvature ◽  
Negative Gaussian Curvature

From the mathematic category of surface Gaussian curvature, carbon allotropes can be classified into three types: zero curvature, positive curvature, and negative curvature.

scholarly journals Embedded minimal surfaces of finite topology

2019 ◽  
Vol 2019 (753) ◽  
pp. 159-191 ◽  
Author(s):  
William H. Meeks III ◽  
Joaquín Pérez
Keyword(s):  
Asymptotic Behavior ◽  
Riemann Surface ◽  
Finite Number ◽  
Minimal Surface ◽  
Minimal Surfaces ◽  
Compact Riemann Surface ◽  
Finite Topology ◽  
Compact Boundary ◽  
Interior Points

AbstractIn this paper we prove that a complete, embedded minimal surface M in {\mathbb{R}^{3}} with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface {\overline{M}} with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic data on its conformal completion {\overline{M}}. In particular, we demonstrate that M is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of M.

scholarly journals C∞-Convergence of Circle Patterns to Minimal Surfaces

2009 ◽  
Vol 194 ◽  
pp. 149-167 ◽  
Author(s):  
Shi-Yi Lan ◽  
Dao-Qing Dai
Keyword(s):  
Minimal Surface ◽  
Existence Theorem ◽  
Minimal Surfaces ◽  
Weierstrass Representation ◽  
Circle Pattern ◽  
Vertex Set ◽  
Circle Patterns ◽  
Simply Connected Region ◽  
Simply Connected ◽  
The Relationship

AbstractGiven a smooth minimal surface F: Ω → ℝ3 defined on a simply connected region Ω in the complex plane ℂ, there is a regular SG circle pattern . By the Weierstrass representation of F and the existence theorem of SG circle patterns, there exists an associated SG circle pattern in ℂ with the combinatoric of . Based on the relationship between the circle pattern and the corresponding discrete minimal surface F∊: → ℝ3 defined on the vertex set of the graph of , we show that there exists a family of discrete minimal surface Γ∊: → ℝ3, which converges in C∞(Ω) to the minimal surface F: Ω → ℝ3 as ∊ → 0.

Sign in / Sign up

Export Citation Format

Share Document