Let alpha∈(0,1) and let S⊂R^3 be a compact, connected, embedded surface without boundary of class C^3,alpha which is convex, i.e. S bounds a convex body in R^3. Let nu colon S→S^2 be a unit normal vector field on S, and for p∈ S let A_pcolon T_pS→ T_pS denote the associated shape operator, whose eigenvalues kappa_1(p) and kappa_2(p) are the principal curvatures of S at p. A point p∈ S is called umbilic if kappa_1(p)=kappa_2(p), equivalently if A_p is a multiple of the identity on T_pS. Then #p∈ S: p is umbilic ≥ 2 .

Carathéodory conjecture

About 102 years old · traced to 1924

Let $\alpha\in(0,1)$ and let $S\subset\mathbb{R}^{3}$ be a compact, connected, embedded surface without boundary of class $C^{3,\alpha}$ which is convex, i.e. $S$ bounds a convex body in $\mathbb{R}^{3}$ . Let $\nu\colon S\to\mathbb{S}^{2}$ be a unit normal vector field on $S$ , and for $p\in S$ let $A_{p}\colon T_{p}S\to T_{p}S$ denote the associated shape operator, whose eigenvalues $\kappa_{1}(p)$ and $\kappa_{2}(p)$ are the principal curvatures of $S$ at $p$ . A point $p\in S$ is called umbilic if

\kappa_{1}(p)=\kappa_{2}(p),

equivalently if $A_{p}$ is a multiple of the identity on $T_{p}S$ . Then

\#\{p\in S:\ p \text{ is umbilic}\}\ \geq\ 2 .

References

Primary source

Wikipedia

Additional references

Wikipedia, Carathéodory conjecture, the article this problem comes from.

geometry unsolved-problem

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A published proof covers the stated case, but a new AI-assisted claim of a counterexample has not yet been independently checked.

The conjecture asserts that every sufficiently smooth convex closed surface in three-dimensional Euclidean space has at least two umbilic points. It is attributed to Constantin Carathéodory, with an early formulation by Hans Hamburger in 1924.

Known results

Guilfoyle and Klingenberg (2008) report a proof for $C^{3+\alpha}$ strictly convex surfaces, via complex points of Lagrangian sections of $TS^{2}$ .
An index-bound paper (2012) states that isolated umbilic indices are less than $2$ and presents the $C^{3+\alpha}$ case as resolved.
A 2025 exposition presents the same Lagrangian-section argument as proving the conjecture.

August 2026 AI-assisted counterexample claim

A circulating write-up, prepared with help from Claude, claims to disprove the conjecture by exploiting a distinction between smooth and analytic surfaces. No proof or counterexample has yet been independently verified in the retrieved sources, so this challenges but does not overturn the published result.

Current status (as of August 2026): The $C^{3,\alpha}$ Euclidean case has a published proof, while a newly circulated AI-assisted counterexample claim remains unverified.

Sources

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