Topology of
Umbilics
In [11] it was shown that an
isolated umbilic whose index has absolute value at least one must be a critical
point for both the mean and Gaussian curvatures, H and K. Furthermore, the
3-jet of H^2 - K must be zero at the umbilic. In particular, one has the following
elegant theorem of classical differential geometry: "Let M
be a smoothly immersed sphere in Euclidean 3-space and suppose that the critical
sets of the mean and Gaussian curvatures are disjoint. Then M has at least four umbilics". This
result "explains" the four umbilics in the classical example of the
asymmetric ellipsoid. In a 1996
Much of the study of the umbilical points has been dominated by the classical conjecture of Caratheodory:" any smooth immersion of the 2-sphere in Euclidean 3-space has at least two umbilics". To date, none of the published proofs (all in the analytic case) has been accepted without reservations. Our work centers on the smooth case. The general question is the following: What germs of singularities of line fields appear as the Gaussian images of fields of principal directions near isolated umbilics? A natural approach to the Caratheodory conjecture is to show that the index of an isolated umbilic cannot be bigger than one. In [14] we show that this condition on the index is not the only obstruction for a singularity to be umbilical. We explicitly construct germs of line fields in the plane, with every index, that cannot be realized as the Gaussian image of the principal line fields near an isolated umbilic of positive curvature on any surface. The examples include the standard dipole foliations and controlled distortions of it. The geometric results of [15] are a consequence of our work on the (real) solvability of the operator obtained by squaring the Cauchy-Riemann operator. In [20] we work on a problem of Loewner.
Related Publications
[3] F. Xavier (with L. P.Jorge), An inequality between the exterior diameter and the mean curvature of bounded immersions, Mathematische Zeitschrift, 178, 77-82, (1981).
[7] F. Xavier, A non-immersion theorem for hyperbolic manifolds, Comm. Math.Helv. 60, 280-283, (1985).
[9] F. Xavier (with B. Smyth), Efimov's theorem in dimension greater than two, Invent. Math. 90, 443-450, (l987).
[11] F. Xavier (with B. Smyth), A Sharp geometric estimate for the index of an umbilic on a smooth
surface, , Bull.
[14] F. Xavier (with B. Smyth) Real solvability of the equation $\partial_{\overline z}^2 w = g$ and the topology of isolated umbilics, Journal of Geometric Analysis, 8, 656-671.
[20] F. Xavier (with B. Smyth), Eigenvalue
estimates and the index of Hessian fields, Bull.