TY - JOUR
AU - Guckenheimer, John
PY - 2021/09/19
Y2 - 2022/05/20
TI - Principal Foliations of Surfaces near Ellipsoids
JF - New Zealand Journal of Mathematics
JA - NZ J Math
VL - 52
IS -
SE - Vaughan Jones Memorial Special Issue
DO - 10.53733/126
UR - https://nzjmath.org/index.php/NZJMATH/article/view/126
SP - 361--379
AB - <p>The lines of curvature of a surface embedded in $\R^3$ comprise its principal foliations. Principal foliations of surfaces embedded in $\R^3$ resemble phase portraits of two dimensional vector fields, but there are significant differences in their geometry because principal foliations are not orientable. The Poincar\'e-Bendixson Theorem precludes flows on the two sphere $S^2$ with recurrent trajectories larger than a periodic orbit, but there are convex surfaces whose principal foliations are closely related to non-vanishing vector fields on the torus $T^2$. This paper investigates families of such surfaces that have dense lines of curvature at a Cantor set $C$ of parameters. It introduces discrete one dimensional return maps of a cross-section whose trajectories are the intersections of a line of curvature with the cross-section. The main result proved here is that the return map of a generic surface has \emph{breaks}; i.e., jump discontinuities of its derivative. Khanin and Vul discovered a qualitative difference between one parameter families of smooth diffeomorphisms of the circle and those with breaks: smooth families have positive Lebesgue measure sets of parameters with irrational rotation number and dense trajectories while families of diffeomorphisms with a single break do not. This paper discusses whether Lebesgue almost all parameters yield closed lines of curvature in families of embedded surfaces.</p>
ER -