First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds

2012 ◽  
Vol 47 (1-2) ◽  
pp. 117-157 ◽  
Author(s):  
Matteo Galli
Keyword(s):  
Three Dimensional ◽  
Second Variation ◽  
Hermitian Manifolds ◽  
First And Second Variation

Harmonic G-structures

2009 ◽  
Vol 146 (2) ◽  
pp. 435-459 ◽  
Author(s):  
J. C. GONZÁLEZ–DÁVILA ◽  
F. MARTÍN CABRERA
Keyword(s):  
Riemannian Manifold ◽  
Critical Points ◽  
Compact Riemannian Manifold ◽  
Harmonic Maps ◽  
Energy Functional ◽  
Second Variation ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Intrinsic Torsion ◽  
First And Second Variation

AbstractFor closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold (M, 〈⋅, ⋅〉), where G-structures are considered as sections of the quotient bundle (M)/G. We deduce the corresponding first and second variation formulae and the characterising conditions for critical points by means of tools closely related to the study of G-structures. In this direction, we show the rôle in the energy functional played by the intrinsic torsion of the G-structure. Moreover, we analyse the particular case G=U(n) for 2n-dimensional manifolds. This leads to the study of harmonic almost Hermitian manifolds and harmonic maps from M into (M)/U(n).

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 THE FIRST AND SECOND VARIATION OF THE TOTAL ENERGY OF CLOSED DUPLEX DNA IN PLANAR CASE

2010 ◽  
Vol 24 (05) ◽  
pp. 587-597 ◽  
Author(s):  
XIAO-HUA ZHOU
Keyword(s):  
Total Energy ◽  
Equilibrium Shape ◽  
Variation Method ◽  
Second Variation ◽  
General Shape ◽  
Planar Case ◽  
The First Variation ◽  
The Second Variation ◽  
First And Second Variation ◽  
Classical Variation

DNA's shape mostly lies on its total energy F. Its corresponding equilibrium shape equations can be obtained by classical variation method: letting the first energy variation δ(1)F = 0. Here, we not only provide the first variation δ(1)F but also give the second variation δ(2)F in planar case. Moreover, the general shape equations of DNA are abstained and a mistake in Zhang et al., [Phys. Rev. E70, 051902 (2004)] is pointed out.

scholarly journals Second order infinitesimal bending of curves

Filomat ◽  
2017 ◽  
Vol 31 (13) ◽  
pp. 4127-4137 ◽  
Author(s):  
Marija Najdanovic ◽  
Ljubica Velimirovic
Keyword(s):  
Vector Fields ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Second Order ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Explicit Formulas ◽  
Infinitesimal Bending ◽  
Necessary And Sufficient ◽  
The Second Variation

We investigate a second order infinitesimal bending of curves in a three-dimensional Euclidean space in this paper. We give the necessary and sufficient conditions for the vector fields to be infinitesimal bending fields of the corresponding order, as well as explicit formulas which determine these fields. We examine the first and the second variation of some geometric magnitudes which describe a curve, specially a change of the curvature. Two illustrative examples (a circle and a helix) are studied not only analytically but also by drawing curves using computer program Mathematica.

scholarly journals Stability of equilibrium configurations for elastic films in two and three dimensions

2015 ◽  
Vol 8 (2) ◽  
pp. 117-153 ◽  
Author(s):  
Marco Bonacini
Keyword(s):  
Epitaxial Growth ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Variational Model ◽  
Second Variation ◽  
Strict Positivity ◽  
Equilibrium Configurations ◽  
Local Minimality ◽  
Nonlinear Elastic ◽  
The Second Variation

AbstractWe establish a local minimality sufficiency criterion, based on the strict positivity of the second variation, in the context of a variational model for the epitaxial growth of elastic films. Our result holds also in the three-dimensional case and for a general class of nonlinear elastic energies. Applications to the study of the local minimality of flat morphologies are also shown.

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