The Carathéodory conjecture, dating from the early 1920's, 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). This has remained a conjecture for 80 years, with fitful, occasionally intense, work on the special case where the surface is real analytic.

Background and further references can be found on Wikipedia.

In fact, with Wilhelm Klingenberg (University of Durham, England), we have proved the Carathéodory conjecture utilizing PDE in a geometric setting that we have spent the last decade exploring. While our proof has been posted on the internet since September 2008, it will take a while for the proof of such a long-standing conjecture to be accepted. Following feedback, a new version of proof of the global conjecture has been posted here.

Direct applications of the conjecture are thin on the ground, although the methods we use may prove to be useful in a wide variety of mathematical settings. 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.

Recently, we have 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.

Update (June 2022):

Well it took ten years to prove it and now, ten years to prove we proved it! The first two parts of the Caratheodory Conjecture proof have appeared:

These are essentially Sections 2.2 and 3 of the arxiv post of the proof from 2011 - there have been no major changes. The final part of the proof is currently under review.

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 On Isolated Umbilic Points. 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 preprint:

Finally, the same method of holomorphic discs can be applied to prove a conjecture of Toponogov on umbilic points at infinity. The preprint can be found here: