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Conjecture

Caratheodory Conjecture

Geometry

The Caratheodory conjecture, in differential geometry, is attributed to Constantin Caratheodory by Hans Ludwig Hamburger. It states that any convex, closed and sufficiently smooth surface in three dimensional Euclidean space must admit at least two umbilic points. Much of the early work on the conjecture concerned the real-analytic case, where several proofs and revisions followed Hamburger's work, from Gerrit Bol, Tilla Klotz and Charles J. Titus; later reviewers found gaps in each of those proofs. In the more general smooth case, an explicit counterexample was announced by Levent Alpoge in August 2026. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Statement
Any convex, closed and sufficiently smooth surface in three dimensional Euclidean space must admit at least two umbilic points. 1
Proposed Year
1924 1
Progress Toward Resolution
Proved by Hamburger in 1940 for the real analytic case (with later gaps found in various proofs). In August 2026 Levent Alpoge announced an explicit smooth counterexample with exactly one umbilic point. 1
Classification
Resolution Status
Partially Resolved 1
Prize Status
Prize Status (category)
No Prize Offered 1
Connections

In Branch

Differential Geometry, Branches of Mathematics
Source Carathéodory conjecture (Wikipedia)
Sources
1. Caratheodory Conjecture (Wikipedia)
Wikimedia Foundation
  • Lead section
    attributed to Constantin Carathéodory by Hans Ludwig Hamburger, stating that any convex, closed and sufficiently smooth surface in three dimensional Euclidean space must admit at least two umbilic points
  • Statement section
    any convex, closed and sufficiently smooth surface in three dimensional Euclidean space must admit at least two umbilic points
  • General smooth case section
    an explicit C∞ counterexample
  • Inception section
    Hamburger attributed the conjecture to Caratheodory in a session of the Berlin Mathematical Society in 1924.
View the Source
Carathéodory conjecture (Wikipedia)
In Branch: Differential Geometry, Lead sentenceView the Source
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