Comparing Corresponding Dihedral Angles on Classical Geometric Simplices
In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.
math.DG↗
arXiv subjects
Thomas Kwok-keung Au
Publications and source records attributed to Thomas Kwok-keung Au.
Images of Harmonic Maps with Symmetry
We show that under certain symmetry, the images of complete harmonic embeddings from the complex plane into the hyperbolic plane is completely determined by the geometric information of the vertical measured foliation and is independent of the horizontal measured foliation of the corresponding Hopf differentials.
math.DG↗
An Analysis on the Shape Equation for Biconcave Axisymmetric Vesicles
We study the conditions on the physical parameters in the Helfrich bending energy of lipid bilayer vesicles. Among embedded surfaces with a biconcave axisymmetric shape, the variation equation is analyzed in detail. This leads to simple conditions which guarantee the solution the information about the geometry.
math.AP↗
Hopf Differentials and the Images of Harmonic Maps
Non-polynomial growth harmonic maps from the complex plane to the hyperbolic space are studied. Some non-surjectivity results are obtained. Moreover, images of such harmonic maps are investigated with reference to their Hopf differentials.
math.DG↗
On the Saddle Point Property of Abresch-Langer curves under the Curve Shortening Flow
In the study of the curve shortening flow on general closed curves, Abresch and Langer posed a conjecture that the homothetic curves can be regarded as saddle points between multi-folded circles and some singular curves. In other words, these homothetic curves are the watershed between curves with a nonsingular future and those with singular future along the flow. In this article, we provide an affirmitive proof to this conjecture.
math.AP↗
Off-center Reflections: Caustics and Chaos
In this article, we attempt to study the possible link between the dynamics of a circle map and the caustics of its iterations. The attention is on a geometrically defined off-center reflections, which, coincidentally, is also a generalization of standard perturbation of rotation. Some results of the caustics and dynamics of the family of maps are given. The method of symplectic topology is particularly exploited to study the caustics because of its potential in further development. The mode-locking behavior and width of resonance zone are also studied. Furthermore, there are interesting problems arised from experimental observations.
math.DS↗
The Dynamics of Off-center Reflection
The dynamical properties, especially the symmetric orbits, of the 2-parameter family of circle maps called off-center reflection is studied.
math.DS↗
Physical Parameters for Biconcave Shape Vesicles
The Helfrich's shape equation of axisymmetric vesicles is studied. A sufficient condition on the physical parameters and some geometric properties are discovered for the biconcave shape.
physics.bio-ph↗
Conditions for the Formation of Axisymmetric Biconcave Vesicles
In this article, the formation of biconcave vesicle according to the minimization bending energy (Helfrich's model) is studied. We present that in order to have biconcave shape vesicle, there are necessary conditions on the parameters of the energy functional and there are implications on the geometric quantities of the vesicle. Furthermore, we also give a clear sufficient condition on the parameters to have the formation of biconcave shape vesicle.
physics.bio-ph↗