geometry-topology / Differential geometry
Carathéodory's conjecture, Problem 8.1 of Ghomi's list and traceable to 1922, asks whether every closed convex surface in $\mathbb{R}^3$ has at least two umbilic points. Hamburger settled the real-analytic case in 1940-41 and it stands. The $C^\infty$ case is false: an explicit support function gives a smoothly embedded two-sphere bounding a convex body with exactly one umbilic point. The same family disproves the smooth Loewner conjecture, whose member at $k=1$ has an isolated trace-free Hessian zero of winding number three.
Temporal state
No reconciled state yet.
Append-only history
Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a $C^\infty$ object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.
Research memory
Carathéodory's conjecture, Problem 8.1 of Ghomi's list and traceable to 1922, asks whether every closed convex surface in $\mathbb{R}^3$ has at least two umbilic points. Hamburger settled the real-analytic case in 1940-41 and it stands. The $C^\infty$ case is false: an explicit support function gives a smoothly embedded two-sphere bounding a convex body with exactly one umbilic point. The same family disproves the smooth Loewner conjecture, whose member at $k=1$ has an isolated trace-free Hessian zero of winding number three.
Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a $C^\infty$ object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.
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