35 research outputs found
Distortion Minimal Morphing I: The Theory For Stretching
We consider the problem of distortion minimal morphing of -dimensional
compact connected oriented smooth manifolds without boundary embedded in
. Distortion involves bending and stretching. In this paper, minimal
distortion (with respect to stretching) is defined as the infinitesimal
relative change in volume. The existence of minimal distortion diffeomorphisms
between diffeomorphic manifolds is proved. A definition of minimal distortion
morphing between two isotopic manifolds is given, and the existence of minimal
distortion morphs between every pair of isotopic embedded manifolds is proved
The Chazy XII Equation and Schwarz Triangle Functions
Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348]
showed that the Chazy XII equation , , is equivalent to a projective-invariant equation for an affine
connection on a one-dimensional complex manifold with projective structure. By
exploiting this geometric connection it is shown that the Chazy XII solution,
for certain values of , can be expressed as where
solve the generalized Darboux-Halphen system. This relationship holds
only for certain values of the coefficients and the
Darboux-Halphen parameters , which are enumerated in
Table 2. Consequently, the Chazy XII solution is parametrized by a
particular class of Schwarz triangle functions
which are used to represent the solutions of the Darboux-Halphen system.
The paper only considers the case where . The associated
triangle functions are related among themselves via rational maps that are
derived from the classical algebraic transformations of hypergeometric
functions. The Chazy XII equation is also shown to be equivalent to a
Ramanujan-type differential system for a triple