Research

The Carathéodory conjecture, first reported in 1924, states that any closed convex surface in 3-dimensional Euclidean space must have at least 2 umbilic points (points where the surface curves equally in all directions). Background and further references can be found on Wikipedia.

With Wilhelm Klingenberg (University of Durham, England), we have proven the Carathéodory conjecture utilizing PDE in a geometric setting that we have spent the last decade exploring. The complete proof of the Caratheodory Conjecture has now appeared:

  • B. Guilfoyle and W. Klingenberg, Higher codimensional mean curvature flow of compact spacelike submanifolds, Trans. Amer. Math. Soc. 372.9 (2019) 6263-6281. CLICK HERE ,
  • B. Guilfoyle and W. Klingenberg, Fredholm-regularity of holomorphic discs in plane bundles over compact surfaces, Ann. Fac. Sci. Toulouse Math. Serie 6, 29.3 (2020) 565-576. CLICK HERE,
  • B. Guilfoyle and W. Klingenberg, Proof of the Toponogov Conjecture on complete surfaces, J. Gokova Geom. Topol. GGT 17 (2024) 1-50. (Link to follow).

These are essentially the same as the arxiv post of the proof from 2011 - there have been no major changes. Our main innovations have been:
  • reformulation of the conjecture in terms of complex points on Lagrangian surfaces in a neutral Kaehler 4-manifold,
  • application of mean curvature flow with boundary in higher codimension,
  • development sub-manifold theory in neutral Kaehler 4-manifolds,
  • establishing Fredholm regularity of an associated elliptic boundary value problem in the case of a single umbilic point.

We have also extended our proof from the global conjecture to a local index bound for umbilics on smooth convex surfaces. To help explain our methods I have put together a couple of expository youtube video clips that goes through the proof. Below is the introduction video:



In an interesting twist, the smooth bound obtained, which we claim is sharp, is weaker than Hamburger's famous result in the real analytic case. Thus, we predict the existence of "exotic" umbilic points of index 3/2, which are contained on smooth but non-real analytic surfaces.

I also gave the Perspectives in Geometry Lecture Series, at the University of Texas at Austin. The videos of the four lectures can be found by clicking here.

A new insight into the Conjecture (and why it is true) has also recently been provided by the construction of counter-examples in Riemannian spaces arbitrarily close to Euclidean 3-space. The details of this can be found in the paper

  • B. Guilfoyle, On isolated umbilic points, Comm. Anal. Geom. 28.8 (2020) 2005-2018. CLICK HERE.
The paper shows that an arbitrarily small perturbation of the Euclidean metric does not have to satisfy the Caratheodory Conjecture (or Hamburger's umbilic index bound). Here's a short video explainer:



Consequences of Hamburger's local index bound for isolated umbilic points applied to the zeros of holomorphic polynomials are explored in the new paper:

  • B. Guilfoyle and W. Klingenberg, Roots of polynomials and umbilics of surfaces, Results in Math. 78 (2023) 229-247. CLICK HERE.

Two further recent papers on the behaviour of umbilic points at infinity hsave salso appeared in print:

  • B. Guilfoyle, A note on umbilic points at infinity, Beitraege Algebra Geom. (2024) CLICK HERE,
  • B. Guilfoyle and A. Ortiz-Rodriguez, Umbilic points on the finite and infinite parts of certain algebraic surfaces, Math. Proc. R. Ir. Acad. 123A.2 (2023) 63-94. CLICK HERE.