On isolated umbilic points

@article{Guilfoyle2018OnIU,
  title={On isolated umbilic points},
  author={Brendan Guilfoyle},
  journal={arXiv: Differential Geometry},
  year={2018}
}
  • B. Guilfoyle
  • Published 9 December 2018
  • Mathematics
  • arXiv: Differential Geometry
Counter-examples to the famous conjecture of Caratheodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close to the flat metric on Euclidean 3-space. In particular, Riemannian metrics with a smooth strictly convex 2-sphere containing a single umbilic point are constructed explicitly, in contradiction with any direct extension of Caratheodory's conjecture. Additionally, a Riemannian metric with an embedded… 
Roots of Polynomials and Umbilics of Surfaces
. For certain polynomials we relate the number of roots inside the unit circle of with the index of a non-degenerate isolated umbilic point on a real analytic surface in Euclidean 3-space. In
A Note on Umbilic Points at Infinity
In this note a definition of umbilic point at infinity is proposed, at least for surfaces that are homogeneous polynomial graphs over a plane in Euclidean 3-space. This is a stronger definition than

References

SHOWING 1-10 OF 32 REFERENCES
From Global to Local: an index bound for umbilic points on smooth convex surfaces
We prove that the half-integer valued index of an isolated umbilic point on a smooth convex surface in Euclidean 3-space is less than 2. This follows from a localization of the authors' proof of the
Normal curvatures of asymptotically constant graphs and Carathéodory’s conjecture
We show that Caratheodory’s conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f : R → R whose
SOME USES OF THE SECOND CONFORMAL STRUCTURE ON STRICTLY CONVEX SURFACES
1. An oriented surface 5 immersed smoothly in E3 has a conformai structure imposed upon it by the metric of the surrounding space. Thus S may be viewed as a Riemann surface Pi. But if S is strictly
An umbilical point on a non-real-analytic surface
Let F be a smooth function of two variables which is zero at ð0; 0Þ and positive on a punctured neighborhood of ð0; 0Þ. Then the function expð� 1=F Þ is smoothly extended to ð0; 0Þ and then the
The Analytic Caratheodory Conjecture
The aim of this article is to provide the reader with a real possibility of becoming confident that the index of an isolated umbilic point of an analytic surface is never greater than one. For a
Evolving to non-round Weingarten spheres: integer linear Hopf flows
In the 1950's Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper we
Eigenvalue Estimates and the Index of Hessian Fields
The local Caratheodory conjecture on the index of an isolated singularity of the principal foliations in surface theory is equivalent to a conjecture of Loewner on the index of the isolated
A Topological Characterization of a Class of Integral Operators
x = x(t) (2) y = y(t) may be interpreted as the parametric representation of a closed oriented curve in the x-y-plane. Any curve obtained in this way will be said to be 'generated by the kernel
On a Carathéodory’s Conjecture on Umbilics: Representing Ovaloids.
L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation
C1-umbilics with arbitrarily high indices
In this paper, the existence of C^1-umbilics with arbitrarily high indices is shown. This implies that more than C^1-regularity is required to prove Loewner's conjecture.
...
...