# Curvature and geometric flows > 17 AI-made results on curvature and geometric flows, each explained in plain English. How curved a space or surface is, and what happens when a shape is… Canonical: https://closedform.nishachay.tech/topic/curvature-and-geometric-flows/ Last updated: 2026-10-14 How curved a space or surface is, and what happens when a shape is smoothed over time by a flow. 17 AI-made results on curvature and geometric flows, from 36 papers in the OpenAI math collection, each explained in plain English. 9 have a formal Lean proof listed; 8 are claimed in a paper and not yet machine-checked. They span 3 fields, led by differential geometry (14), algebraic and complex geometry (2) and dynamical systems and ergodic theory (1). The oldest question here dates back to 1926: [Sharp Cartan–Hadamard isoperimetry and rigidity](https://closedform.nishachay.tech/e/openai-math-2026-337/). Results rated Landmark include [Gromov’s integral scalar-curvature bound for simplicial volume](https://closedform.nishachay.tech/e/openai-math-2026-335/) and [Yau's uniformization conjecture](https://closedform.nishachay.tech/e/openai-math-2026-338/). ## Results (17) - [A counterexample to Griffiths’ positivity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-050/) - Entry 050 · Algebraic positivity need not come with positive curvature, even on the simple surface built from two spheres. (Formal proof listed) - [Anticanonical nonvanishing in every dimension](https://closedform.nishachay.tech/e/openai-math-2026-068/) - Entry 068 · A claimed proof that a smooth nonnegatively curved anticanonical bundle always has a section of some power. (Claimed, not checked) - [The near-boundary Birkhoff conjecture](https://closedform.nishachay.tech/e/openai-math-2026-147/) - Entry 147 · Two papers claim that a smooth billiard table with an unbroken band of caustics along its edge must be an ellipse. (Claimed, not checked) - [Smooth isometric immersions of surfaces into ℝ 4](https://closedform.nishachay.tech/e/openai-math-2026-333/) - Entry 333 · Proves that every closed smooth curved surface can be placed in four-dimensional space with all its distances kept. (Formal proof listed) - [Gromov’s integral scalar-curvature bound for simplicial volume](https://closedform.nishachay.tech/e/openai-math-2026-335/) - Entry 335 · Claims Gromov's integral scalar-curvature bound for simplicial volume and the Gromov–Lawson aspherical conjecture. (Claimed, not checked) - [Spectral scalar curvature, Urysohn width, and macroscopic dimension](https://closedform.nishachay.tech/e/openai-math-2026-336/) - Entry 336 · Claims Gromov's codimension-two width bound for positive scalar curvature in every dimension, and a spectral version. (Claimed, not checked) - [Sharp Cartan–Hadamard isoperimetry and rigidity](https://closedform.nishachay.tech/e/openai-math-2026-337/) - Entry 337 · Proves sharp Euclidean fillings in CAT(0) spaces and claims the generalized Cartan–Hadamard isoperimetric inequality. (Formal proof listed) - [Yau's uniformization conjecture](https://closedform.nishachay.tech/e/openai-math-2026-338/) - Entry 338 · Claims to prove Yau's uniformization conjecture: positive bisectional curvature makes a complete Kähler manifold ℂⁿ. (Claimed, not checked) - [Katok's entropy rigidity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-339/) - Entry 339 · Claims to prove Katok's entropy rigidity conjecture for negatively curved manifolds of dimension three and higher. (Claimed, not checked) - [Nonnegative-curvature Einstein classification and an L 2 topological gap](https://closedform.nishachay.tech/e/openai-math-2026-348/) - Entry 348 · Classifies positive Einstein four-manifolds with nonnegative curvature, and claims a small-energy topological gap. (Formal proof listed) - [Scalar curvature and finite-time Ricci-flow singularities](https://closedform.nishachay.tech/e/openai-math-2026-351/) - Entry 351 · Claims four-dimensional Ricci flow continues while scalar curvature is bounded, but not in some high dimension. (Claimed, not checked) - [A finite-time singularity of Calabi flow](https://closedform.nishachay.tech/e/openai-math-2026-352/) - Entry 352 · Claims a smooth starting metric on complex projective 10-space whose Calabi flow breaks down in finite time. (Claimed, not checked) - [Gigli’s characterization of Alexandrov curvature](https://closedform.nishachay.tech/e/openai-math-2026-356/) - Entry 356 · Claims Gigli's conjecture: Alexandrov curvature bounds match RCD spaces with a distributional sectional curvature bound. (Formal proof listed) - [A three-manifold without conjugate points or nonpositive curvature](https://closedform.nishachay.tech/e/openai-math-2026-358/) - Entry 358 · Proves that a closed three-manifold can carry a metric without conjugate points yet no metric of nonpositive curvature. (Formal proof listed) - [Negative Kähler curvature without bounded holomorphic coordinates](https://closedform.nishachay.tech/e/openai-math-2026-359/) - Entry 359 · Proves a complete Kähler threefold with pinched negative curvature can lack bounded holomorphic coordinates. (Formal proof listed) - [Weak MTW curvature gives convexity and regular optimal transport](https://closedform.nishachay.tech/e/openai-math-2026-360/) - Entry 360 · Proves weak MTW curvature makes every injectivity domain convex, as Villani conjectured, and transport maps Hölder. (Formal proof listed) - [Failure of integer-degree harmonic dimension comparison](https://closedform.nishachay.tech/e/openai-math-2026-361/) - Entry 361 · Proves Yau's Euclidean bound on polynomial-growth harmonic functions fails under nonnegative Ricci curvature. 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