Fourier restriction for positively curved surfaces

What OpenAI claims

Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface Σ ⊂ ℝ³, including surfaces with boundary, the extension operator is bounded from L^p(Σ) to L^p(ℝ³) for every p > 3.

Papers (2)

1. Elliptic capacity propagation and Fourier restriction to the sphere

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove the bounded-data Fourier restriction conjecture for the sphere in three dimensions: the Fourier extension operator maps L^∞(S²) boundedly into L^p(ℝ³) for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp.

2. Diagonal Fourier extension for positively curved surfaces in three dimensions

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove the diagonal Fourier extension conjecture for compact smooth positively curved surfaces Σ ⊂ ℝ³, including surfaces with smooth boundary. For every 3 < p < ∞, the extension operator maps L^p(Σ) boundedly into L^p(ℝ³). The compact paraboloid case also yields free Schrödinger local smoothing in two spatial dimensions for every 3 < p < ∞ and Sobolev order s > 2-6/p.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.