The isoperimetric inequality for minimal surfaces in a Riemannian manifold

1999 ◽  
Vol 1999 (506) ◽  
pp. 205-214 ◽  
Author(s):  
Jaigyoung Choe
Keyword(s):  
Riemannian Manifold ◽  
Minimal Surface ◽  
Sectional Curvature ◽  
Isoperimetric Inequality ◽  
Gaussian Curvature ◽  
Minimal Surfaces ◽  
Constant Gaussian Curvature ◽  
Simply Connected

Abstract It is proved that every minimal surface with one or two boundary components in a simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant K satisfies the sharp isoperimetric inequality 4π A ≦ L2 + K A2. Here equality holds if and only if the minimal surface is a geodesic disk in a surface of constant Gaussian curvature K.

scholarly journals The double cover relative to a convex domain and the relative isoperimetric inequality

2006 ◽  
Vol 80 (3) ◽  
pp. 375-382 ◽  
Author(s):  
Jaigyoung Choe
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Isoperimetric Inequality ◽  
Convex Domain ◽  
Double Cover ◽  
Nonpositive Sectional Curvature ◽  
Relative Isoperimetric Inequality ◽  
Simply Connected

AbstractWe prove that a domain Ω in the exterior of a convex domain C in a four-dimensional simply connected Riemannian manifold of nonpositive sectional curvature satisfies the relative isoperimetric inequality 64π2 Vol(Ω)3 < Vol(∂Ω ~ ∂C)4. Equality holds if and only if Ω is an Euclidean half ball and ∂Ω ~ ∂C is a hemisphere.

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.

scholarly journals AN INTRINSIC PROOF OF GROMOLL–GROVE DIAMETER RIGIDITY THEOREM

2007 ◽  
Vol 09 (03) ◽  
pp. 401-419 ◽  
Author(s):  
JIANGUO CAO ◽  
HONGYAN TANG
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Projective Spaces ◽  
Rigidity Theorem ◽  
Simply Connected

Using the spherical trip theorem, we present a new intrinsic proof of Gromoll–Grove diameter rigidity theorem: "If a simply-connected Riemannian manifold has sectional curvature ≥ 1 and diameter [Formula: see text], then either it is homeomorphic to a sphere, or it is isometric to one of classic projective spaces".

scholarly journals An integral formula for hypersurfaces in space forms

1995 ◽  
Vol 37 (3) ◽  
pp. 337-341 ◽  
Author(s):  
Theodoros Vlachos
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Distance Function ◽  
Integral Formula ◽  
Constant Sectional Curvature ◽  
Position Vector ◽  
Space Forms ◽  
Image Position ◽  
Simply Connected

Let be an n+ 1-dimensional, complete simply connected Riemannian manifold of constant sectional curvature c and We consider the function r(·) = d(·, P0) where d stands for the distance function in and we denote by grad r the gradient of The position vector (see [1]) with origin P0 is defined as where ϕ(r)equalsr, if c = 0, c< 0 or c <0 respectively.

scholarly journals Biharmonic Maps and Laguerre Minimal Surfaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Yusuf Abu Muhanna ◽  
Rosihan M. Ali
Keyword(s):  
Minimal Surface ◽  
Gaussian Curvature ◽  
Minimal Surfaces ◽  
Complete Characterization ◽  
Necessary Condition ◽  
Biharmonic Maps ◽  
Laguerre Minimal Surfaces ◽  
Laguerre Minimal Surface ◽  
Sufficient And Necessary Condition

A Laguerre surface is known to be minimal if and only if its corresponding isotropic map is biharmonic. For every Laguerre surfaceΦis its associated surfaceΨ=1+u2Φ, whereulies in the unit disk. In this paper, the projection of the surfaceΨassociated to a Laguerre minimal surface is shown to be biharmonic. A complete characterization ofΨis obtained under the assumption that the corresponding isotropic map of the Laguerre minimal surface is harmonic. A sufficient and necessary condition is also derived forΨto be a graph. Estimates of the Gaussian curvature to the Laguerre minimal surface are obtained, and several illustrative examples are given.

ON ALGEBRAIC MINIMAL SURFACES IN ℝ3 DERIVING FROM CHARGE 2 MONOPOLE SPECTRAL CURVES

2005 ◽  
Vol 16 (02) ◽  
pp. 173-180 ◽  
Author(s):  
ANTHONY SMALL
Keyword(s):  
Line Bundle ◽  
Minimal Surface ◽  
Elliptic Curves ◽  
Gaussian Curvature ◽  
Minimal Surfaces ◽  
Gauss Map ◽  
Spectral Curves

We give formulae for minimal surfaces in ℝ3 deriving, via classical osculation duality, from elliptic curves in a line bundle over ℙ1. Specialising to the case of charge 2 monopole spectral curves we find that the distribution of Gaussian curvature on the auxiliary minimal surface reflects the monopole's structure. This is elucidated by the behaviour of the surface's Gauss map.

scholarly journals Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature” by A. A. Elmandouh

2021 ◽  
Author(s):  
Wojciech Szumiński ◽  
Andrzej J. Maciejewski
Keyword(s):  
Hamiltonian Systems ◽  
Gaussian Curvature ◽  
Necessary Conditions ◽  
Constant Gaussian Curvature ◽  
Local Coordinates ◽  
Hamiltonian Functions

AbstractIn the paper [1], the author formulates in Theorem 2 necessary conditions for integrability of a certain class of Hamiltonian systems with non-constant Gaussian curvature, which depends on local coordinates. We give a counterexample to show that this theorem is not correct in general. This contradiction is explained in some extent. However, the main result of this note is our theorem that gives new simple and easy to check necessary conditions to integrability of the system considered in [1]. We present several examples, which show that the obtained conditions are effective. Moreover, we justify that our criterion can be extended to wider class of systems, which are given by non-meromorphic Hamiltonian functions.

scholarly journals Some results on stable p-harmonic maps

1994 ◽  
Vol 36 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Complete Riemannian Manifold ◽  
Image Position ◽  
Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

Description of a 3-periodic minimal surface family with trigonal symmetry

1994 ◽  
Vol 209 (1) ◽  
Author(s):  
A. Fogden
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Weierstrass Representation ◽  
Systematic Analysis ◽  
Trigonal Symmetry ◽  
Triply Periodic Minimal Surfaces ◽  
Periodic Minimal Surface ◽  
Triply Periodic ◽  
The Family

AbstractA systematic analysis of a family of triply periodic minimal surfaces of genus seven and trigonal symmetry is given. The family is found to contain five such surfaces free from self-intersections, three of which are previously unknown. Exact parametrisations of all surfaces are provided using the Weierstrass representation.

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