117 research outputs found
A new construction of Lagrangians in the complex Euclidean plane in terms of planar curves
We introduce a new method to construct a large family of Lagrangian surfaces
in complex Euclidean plane by means of two planar curves making use of their
usual product as complex functions and integrating the Hermitian product of
their position and tangent vectors.
Among this family, we characterize minimal, constant mean curvature,
Hamiltonian stationary, solitons for mean curvature flow and Willmore surfaces
in terms of simple properties of the curvatures of the generating curves. As an
application, we provide explicitly conformal parametrizations of known and new
examples of these classes of Lagrangians in complex Euclidean plane.Comment: 15 pages, 5 figure
Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves
We present a method to construct a large family of Lagrangian surfaces in
complex Euclidean plane by using Legendre curves in the 3-sphere and in the
anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic
curves, respectively. Among this family, we characterize minimal, constant mean
curvature, Hamiltonian-minimal and Willmore surfaces in terms of simple
properties of the curvature of the generating curves. As applications, we
provide explicitly conformal parametrizations of known and new examples of
minimal, constant mean curvature, Hamiltonian-minimal and Willmore surfaces in
complex Euclidean plane.Comment: 16 pages To be published in Tohoku Mathematical Journa
The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow
We provide several rigidity results for the Clifford torus in the class of
compact self-shrinkers for Lagrangian mean curvature flow.Comment: 10 page
On energy gap phenomena of the Whitney sphere and related problems
In this paper, we study Lagrangian submanifolds satisfying introduced by Zhang \cite{Zh} in the complex space forms ,
where and is the Lagrangian trace-free
second fundamental form. We obtain some integral inequalities and rigidity
theorems for such Lagrangian submanifolds. Moreover we study Lagrangian
surfaces in satisfying and introduce a
flow method related to them.Comment: An appendix added and typos corrected; 15 page
- β¦