A smooth surface metric with no local isometric immersion in ℝ³

The explanation

Picture the saddle surface z = xy, shaped like a riding saddle. An ant living on it can measure distances along the surface. From distances alone, it can compute the Gaussian curvature [a number saying how the surface bends near a point]. Negative means saddle-shaped, positive means dome-shaped, and zero means flat like paper. On this saddle the curvature is −1 at the center. Above the point (1, 1) it is −1/9. Above (2, 0) it is −1/25.

Now run the question backwards. Start with only a metric [a rule giving the length of every short path in a flat region]. Can you always build a surface in ordinary three-dimensional space with exactly those lengths? Ask only for a small piece around one point. Such a piece is a local isometric immersion [a smooth map into space that keeps every length]. An immersion may cross itself, but near a single point it cannot. So allowing or forbidding self-crossings gives the same local question.

Much was known. Take analytic metrics [ones given by convergent power series]. Janet and Cartan showed in 1926 and 1927 that the answer is yes. For smooth metrics [ones with derivatives of every order], the answer is yes where curvature is not zero. Trouble starts where the curvature hits zero. There the governing equation can change character. On one side it behaves like a stretched membrane, on the other like waves. Nakamura and Maeda in 1989 handled a clean sign change, where curvature crosses zero at a nonzero slope.

Smoothness matters at both ends. Nash and Kuiper showed that with only one derivative required, a local surface always exists. In 1971 Pogorelov built a metric with limited smoothness and no local surface with two derivatives. Nadirashvili and Yuan published a refinement in 2008 with curvature never negative. The fully smooth question stayed open. Yau discussed it, and Ghomi's 2019 survey lists it as a problem. A 2002 preprint by Nadirashvili and Yuan claimed a stronger counterexample, but their later paper calls the question open. This paper leaves that discrepancy unresolved and does not use the preprint's claimSource 1,Source 2.

The paper proves the answer is no. It builds a smooth metric on the open square from −1 to 1 in both directions. No neighborhood of the center, however small, has a smooth surface with those lengthsSource 3. The paper also claims the metric can match the flat metric to every order at the centerSource 4. A flat plane realizes the flat metric. So the metric's derivatives at that one point cannot decide the questionSource 5.

One subtle point is that smoothness must hold on a single fixed piece. Having a solution with any finite number of derivatives is not enough. Those solutions may live on pieces that shrink as the number growsSource 6. The paper must rule out every smooth candidate on every neighborhood at once.

The construction places a chain of tiny disks marching toward the center. Disk number j is centered at distance 1/2^j, with radius 1/2^(j+4)Source 7. Disk 1 sits at 0.5 with radius 1/32, about 0.031. Disk 10 sits at about 0.00098 with radius about 0.000061. Each radius is 1/16 of its distance from the center. Inside each disk the curvature is negative, and on its rim it is zeroSource 8. Any surface built near the center would contain one whole disk. A short argument from the 2002 preprint, re-proved here, then forces a special one-direction bending somewhere on that rimSource 9.

Just outside each disk, the paper places tiny square patches that crowd toward every rim point, tilted at many anglesSource 10. Each patch gets a carefully perturbed metric. A height function records how far the surface rises in one fixed direction. On each patch, no height of the kind that bending produces can solve an equation every such surface obeys. So the surface cannot exist. The paper gets suitable patch metrics from Baire category, a general existence argumentSource 11. It writes no formula for them. Its curvature also takes both signsSource 12. So it does not settle the smooth question when curvature must be never negative, or never positiveSource 13.

How sure is this? A computer proof checker called Lean has checked a formal version of the main statement. It says no neighborhood of the center has a smooth surface with these lengths. The Lean scope note does not mention the flat-matching refinement, so that part sits outside the check. No specialist review has been published.

What it doesn't show. This does not settle metrics whose curvature is never negative, or never positive. The example uses both signs.