Math Conjectures

Carath\u00e9odory Umbilic Conjecture

OPENMajorConjectureProposed 1924 · Standard version

Canonical statement

Every closed C2C^2, strictly convex surface embedded in R3\mathbb R^3 has at least two umbilic points, where the two principal curvatures are equal.
View source LaTeX
Every closed \(C^2\), strictly convex surface embedded in \(\mathbb R^3\) has at least two umbilic points, where the two principal curvatures are equal.
GEO-DIFF-021:[Hamburger1940Caratheodory][GuilfoyleKlingenberg2008Caratheodory][GuilfoyleKlingenberg2025Caratheodory]
Geometry and topologyPartial differential equations

Notes

The Carathéodory conjecture asserts that a closed strictly convex surface has at least two umbilics. Hamburger established the real-analytic case [Hamburger1940Caratheodory]. Guilfoyle and Klingenberg's multipart program claims a higher-regularity synthesis [GuilfoyleKlingenberg2008Caratheodory] [GuilfoyleKlingenberg2025Caratheodory], but independent validation is pending and the canonical C2C^2 threshold is outside its stated scope. The catalog therefore retains the full problem and links the narrower claim.

Proof-claim watch (1)

Carathéodory umbilic conjecture

Grade C

Substantive higher-regularity full claim; independent acceptance remains pending and C^2 remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.