scholarly journals Hamiltonian stationary Lagrangian surfaces in $\mathbb{C}^2$

2002 ◽  
Vol 10 (1) ◽  
pp. 79-126 ◽  
Author(s):  
Frédéric Hélein ◽  
Pascal Romon
Keyword(s):  
Lagrangian Surfaces ◽  
Hamiltonian Stationary

Hamiltonian-stationary Lagrangian surfaces of constant curvature in complex space form

2009 ◽  
Vol 71 (7-8) ◽  
pp. 2640-2659 ◽  
Author(s):  
Bang-Yen Chen ◽  
Oscar J. Garay ◽  
Zhengfang Zhou
Keyword(s):  
Constant Curvature ◽  
Complex Space ◽  
Space Form ◽  
Complex Space Form ◽  
Lagrangian Surfaces ◽  
Hamiltonian Stationary

scholarly journals Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

2011 ◽  
Vol 61 (1) ◽  
pp. 237-247 ◽  
Author(s):  
Henri Anciaux ◽  
Brendan Guilfoyle ◽  
Pascal Romon
Keyword(s):  
Tangent Bundle ◽  
Riemannian Surface ◽  
Lagrangian Surfaces ◽  
Minimal Lagrangian Surfaces

scholarly journals On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions

2010 ◽  
Vol 661 ◽  
pp. 446-481 ◽  
Author(s):  
YUE YANG ◽  
D. I. PULLIN
Keyword(s):  
Incompressible Flows ◽  
Initial Conditions ◽  
Scalar Fields ◽  
Local Geometry ◽  
Optimization Approach ◽  
Vortex Reconnection ◽  
Barotropic Flow ◽  
Lagrangian Surfaces ◽  
Definition Of ◽  
Surface Fields

For a strictly inviscid barotropic flow with conservative body forces, the Helmholtz vorticity theorem shows that material or Lagrangian surfaces which are vortex surfaces at time t = 0 remain so for t > 0. In this study, a systematic methodology is developed for constructing smooth scalar fields φ(x, y, z, t = 0) for Taylor–Green and Kida–Pelz velocity fields, which, at t = 0, satisfy ω·∇φ = 0. We refer to such fields as vortex-surface fields. Then, for some constant C, iso-surfaces φ = C define vortex surfaces. It is shown that, given the vorticity, our definition of a vortex-surface field admits non-uniqueness, and this is presently resolved numerically using an optimization approach. Additionally, relations between vortex-surface fields and the classical Clebsch representation are discussed for flows with zero helicity. Equations describing the evolution of vortex-surface fields are then obtained for both inviscid and viscous incompressible flows. Both uniqueness and the distinction separating the evolution of vortex-surface fields and Lagrangian fields are discussed. By tracking φ as a Lagrangian field in slightly viscous flows, we show that the well-defined evolution of Lagrangian surfaces that are initially vortex surfaces can be a good approximation to vortex surfaces at later times prior to vortex reconnection. In the evolution of such Lagrangian fields, we observe that initially blob-like vortex surfaces are progressively stretched to sheet-like shapes so that neighbouring portions approach each other, with subsequent rolling up of structures near the interface, which reveals more information on dynamics than the iso-surfaces of vorticity magnitude. The non-local geometry in the evolution is quantified by two differential geometry properties. Rolled-up local shapes are found in the Lagrangian structures that were initially vortex surfaces close to the time of vortex reconnection. It is hypothesized that this is related to the formation of the very high vorticity regions.

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