Compact surfaces with no Bonnet mate
This note gives sufficient conditions (isothermic or totally nonisothermic) for an immersion of a compact surface to have no Bonnet mate.
math.DG↗
arXiv subjects
Gary R. Jensen
Publications and source records attributed to Gary R. Jensen.
The geometric Cauchy problem for the membrane shape equation
We address the geometric Cauchy problem for surfaces associated to the membrane shape equation describing equilibrium configurations of vesicles formed by lipid bilayers. This is the Euler-Lagrange equation of the Canham-Helfrich-Evans elastic curvature energy subject to constraints on the enclosed volume and the surface area. Our approach uses the method of moving frames and techniques from the theory of exterior differential systems.
math.DG↗
Dupin hypersurfaces in Lie sphere geometry
The method of moving frames in Lie sphere geometry has produced significant results in the classification of Dupin hypersurfaces in spheres. What is the secret of its effectiveness? The answer emerges in the classification of nonumbilic isoparametric surfaces in the space form geometries. Using the method of moving frames, the proof of this classification is an elementary exercise. The same proof classifies the cyclides of Dupin in Möbius geometry and finally in Lie sphere geometry, where all nonumbilic Dupin immersions are Lie sphere congruent to each other.
math.DG↗