Curvature and geometric flows
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. Results rated Landmark include Gromov’s integral scalar-curvature bound for simplicial volume and Yau's uniformization conjecture.
Algebraic and complex geometry2
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A counterexample to Griffiths’ positivity conjecture
Constructs ample rank-two bundles on with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.
Formal proof listed -
Anticanonical nonvanishing in every dimension
If X is a smooth connected complex projective variety and admits a smooth Hermitian metric with nonnegative curvature, then for some m > 0. Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.
Claimed, not checked
Dynamical systems and ergodic theory1
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The near-boundary Birkhoff conjecture
Resolves the near-boundary Birkhoff conjecture for smooth strictly convex planar billiards of positive curvature. Such a billiard is an ellipse whenever a full grazing annulus is continuously foliated by individually invariant essential curves. A continuous physical collar of smooth closed convex caustics also suffices.
Claimed, not checked
Differential geometry14
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Smooth isometric immersions of surfaces into ℝ4
Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ4, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature.
Formal proof listed -
Gromov’s integral scalar-curvature bound for simplicial volume
Proves for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with depending only on dimension. Here and is real simplicial volume. Also proves rational inessentiality under positive scalar curvature, resolving the Gromov–Lawson conjecture; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.
Claimed, not checked -
Spectral scalar curvature, Urysohn width, and macroscopic dimension
Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most whose entire fibers have diameter bounded only by n in the original metric. This strengthens Gromov's width conclusion to spectral scalar curvature. Universal covers of closed positive-scalar-curvature manifolds also have continuous macroscopic dimension at most for every n ≥ 2.
Claimed, not checked -
Sharp Cartan–Hadamard isoperimetry and rigidity
Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most κ ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for κ = 0 are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral n-cycles, n ≥ 2, in arbitrary proper CAT spaces.
Formal proof listed -
Yau's uniformization conjecture
Proves Yau's uniformization conjecture: every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to ℂn.
Claimed, not checked -
Katok's entropy rigidity conjecture
Proves Katok's entropy rigidity conjecture for closed connected Riemannian manifolds of dimension at least three with strictly negative sectional curvature: normalized Liouville measure maximizes entropy for the geodesic flow if and only if the metric is locally symmetric.
Claimed, not checked -
Nonnegative-curvature Einstein classification and an L2 topological gap
Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S4, Fubini–Study , or a product of equal round two-spheres. A closed simply connected nonnegatively curved four-manifold is diffeomorphic to one of these whenever the scale-invariant L2 norm of its trace-free Ricci curvature lies below a universal positive constant.
Formal proof listed -
Scalar curvature and finite-time Ricci-flow singularities
Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.
Claimed, not checked -
A finite-time singularity of Calabi flow
Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth -invariant Kähler metric on whose flow develops a finite-time singularity. The metric lies in the Fubini–Study class, so the failure occurs even in a class containing a constant-scalar-curvature metric.
Claimed, not checked -
Gigli’s characterization of Alexandrov curvature
Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support condition with reference measure and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced.
Formal proof listed -
A three-manifold without conjugate points or nonpositive curvature
Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether the first metric-existence property implies the second.
Formal proof listed -
Negative Kähler curvature without bounded holomorphic coordinates
Constructs a contractible domain in ℂ3 with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below.
Formal proof listed -
Weak MTW curvature gives convexity and regular optimal transport
On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous.
Formal proof listed -
Failure of integer-degree harmonic dimension comparison
Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer k, a complete smooth metric on ℝ3 has at least independent harmonic functions of growth at most k, exceeding the Euclidean count . The metric may depend on k.
Formal proof listed