Local smooth isometric immersion of surface metrics into R³
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Does every smooth positive-definite surface metric admit a smooth isometric immersion into R³ near each point? The target counterexample is a C∞ metric on (−1,1)² with no such immersion on any neighborhood of the origin.
Why it matters and what counts as progress
Why it matters: A local obstruction identifies a limit of smooth surface realization in three-dimensional Euclidean space.
What counts as progress: A no-local-immersion proof for a smooth positive metric, or a general realization theorem; agreement with the Euclidean metric to every order is a separate stronger refinement.
How the problem developed
- 2000 · Open problem [1]
Shing-Tung Yau Identified smooth local isometric realization of surface metrics in R³ as an open problem.
- 2019 · Problem formulation [2]
Mohammad Ghomi Formulated the local question explicitly: whether every smooth surface metric has a neighborhood isometrically embedded in R³.
- 24 Sept 2026 · Counterexample claim [3]
OpenAI Claimed a smooth positive-definite metric on a square with no smooth local isometric immersion into R³ near the origin, even though its full Taylor jet there agrees with the Euclidean metric.
AI contribution
A Smooth Metric with No Local Isometric Immersion into Three-Space
OpenAI’s Theorem A constructs a C∞ positive-definite metric g on (-1,1)^2 such that no open neighborhood of the origin admits a C∞ isometric immersion into R³. The paper also says g can agree with the Euclidean metric to every order at the origin. The Lean endpoint directly states the smooth positive metric and no-local-immersion claim; the flat-jet refinement is not part of that endpoint.
Sources
Website timeline
Publication dates and editorial changes.
OpenAI gives a smooth metric with no local immersion in three-space