The second variational formula for Willmore submanifolds in Sn

2001 ◽  
Vol 40 (1-4) ◽  
pp. 205-225 ◽  
Author(s):  
Zhen Guo ◽  
Changping Wang ◽  
Haizhong Li
Keyword(s):  
Variational Formula ◽  
Second Variational Formula

scholarly journals Inequalities on Generalized Sasakian Space Forms

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Najma Abdul Rehman ◽  
Abdul Ghaffar ◽  
Esmaeil Abedi ◽  
Mustafa Inc ◽  
Mohammed K. A. Kaabar
Keyword(s):  
Space Form ◽  
Variational Formula ◽  
Stability Criteria ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Metric Connection ◽  
Generalized Sasakian Space Form ◽  
The Stability ◽  
Second Variational Formula

In this paper, we find the second variational formula for a generalized Sasakian space form admitting a semisymmetric metric connection. Inequalities regarding the stability criteria of a compact generalized Sasakian space form admitting a semisymmetric metric connection are established.

scholarly journals Pansu–Wulff shapes in ℍ1

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián Pozuelo ◽  
Manuel Ritoré
Keyword(s):  
Mean Curvature ◽  
Isoperimetric Problem ◽  
Variational Formula ◽  
Geodesic Curvature ◽  
Vertical Axis ◽  
Singular Part ◽  
Tangent Plane ◽  
Curvature Function ◽  
Invariant Norm ◽  
The First Variation

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

scholarly journals Generalized Variational Principle for Long Water-Wave Equation by He's Semi-Inverse Method

2009 ◽  
Vol 2009 ◽  
pp. 1-5 ◽  
Author(s):  
Weimin Zhang
Keyword(s):  
Water Wave ◽  
Inverse Method ◽  
Variational Principles ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Variational Formula ◽  
Generalized Variational Principle ◽  
Partial Differential ◽  
Water Wave Problem ◽  
Mathematics And Physics

Variational principles for nonlinear partial differential equations have come to play an important role in mathematics and physics. However, it is well known that not every nonlinear partial differential equation admits a variational formula. In this paper, He's semi-inverse method is used to construct a family of variational principles for the long water-wave problem.

scholarly journals Identification of the Polaron measure in strong coupling and the Pekar variational formula

2020 ◽  
Vol 48 (5) ◽  
pp. 2119-2144
Author(s):  
Chiranjib Mukherjee ◽  
S. R. S. Varadhan
Keyword(s):  
Strong Coupling ◽  
Variational Formula

scholarly journals A Variational Formula for Risk-Sensitive Reward

2017 ◽  
Vol 55 (2) ◽  
pp. 961-988 ◽  
Author(s):  
V. Anantharam ◽  
V. S. Borkar
Keyword(s):  
Variational Formula ◽  
Risk Sensitive
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