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The C∞C^\infty Carathéodory Conjecture on Umbilic Points

Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a C∞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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geometry-topologyAug 19, 2026Significance 55/100Registry: lean checked

The C∞C^\infty Carathéodory Conjecture on Umbilic Points

Prior state unknown→disproved

Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a C∞C^\infty object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.

Source authenticatedThe C∞C^\infty Carathéodory Conjecture on Umbilic Points →
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Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a C∞C^\infty object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.

Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a C∞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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The $C^\infty$ Carathéodory Conjecture on Umbilic Points — Mathematical Frontier Network