Subjects

Differential geometry

29 results from OpenAI’s math release. 29 explained so far. OpenAI has published Lean proofs for 15: 10 of the main theorem, 5 of part of the claim.

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• 333 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.

  ▶Episode, 3:30 OpenAI published a Lean proof of the main theorem.

• 334 A smooth surface metric with no local isometric immersion in ℝ³

  Constructs a smooth positive-definite metric on $(-1,1)^2$ for which no neighborhood of the origin admits a smooth isometric immersion into ℝ³. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.

  ▶Episode, 4:18 OpenAI published a Lean proof of the main theorem.

• 335 Gromov’s integral scalar-curvature bound for simplicial volume

  Proves $\int_M(\mathrm{Scal}_g^-)^{n/2}\,dV_g\ge a_n\lVert M\rVert$ for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with $a_n\gt 0$ depending only on dimension. Here $\mathrm{Scal}_g^-=\max\{0,-\mathrm{Scal}_g\}$ and $\lVert M\rVert$ 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.

  2 papers ▶Episode, 4:18

• 336 Spectral scalar curvature, Urysohn width, and macroscopic dimension

  Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying $-4\Delta+\mathrm{Scal}\ge1$ as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most $n-2$ 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 $n-2$ for every n ≥ 2.

  3 papers ▶Episode, 5:01

• 337 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$(0)$ spaces.

  2 papers ▶Episode, 4:28 OpenAI published a Lean proof of part of the claim.

• 338 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.

  ▶Episode, 4:26

• 339 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.

  ▶Episode, 3:55

• 340 A counterexample to the nearby Lagrangian conjecture

  Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even N, constructs a closed exact embedded Lagrangian in $T^*(S^9\times S^{N-1})$ that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.

  ▶Episode, 3:59

• 341 Donaldson's hypersymplectic deformation conjecture

  Proves Donaldson's hypersymplectic deformation conjecture in a cohomology-preserving form. Every positive triple of smooth closed two-forms on a closed connected oriented four-manifold, normalized by $\int\omega_i\wedge\omega_j=\delta_{ij}$, deforms through positive closed triples to a hyperkähler triple while preserving all three cohomology classes. Any positive triple can first be normalized by a constant linear change.

  ▶Episode, 4:00

• 342 Donaldson's tamed-to-compatible conjecture

  Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change.

  ▶Episode, 3:51 OpenAI published a Lean proof of the main theorem.

• 343 Symplectic ball packing in higher dimensions

  Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension $2n\ge6$. Finitely many closed symplectic balls of capacities $R_1,\ldots,R_k$ embed disjointly into an open ball of capacity R exactly when $\sum_iR_i^n\lt R^n$ and $R_i+R_j\lt R$ for every distinct pair i, j.

  ▶Episode, 3:27 OpenAI published a Lean proof of the main theorem.

• 344 The metric Blaschke conjecture

  Proves the metric Blaschke conjecture: every closed connected Riemannian manifold of positive dimension whose injectivity radius equals its diameter is, up to scaling, a standard compact rank-one symmetric space.

  ▶Episode, 4:02

• 345 Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds

  Proves that every smooth Riemannian metric on Sⁿ, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The same conclusion holds on every closed manifold admitting a finite smooth spherical cover and on every closed three-manifold, without orientability or nondegeneracy restrictions.

  ▶Episode, 4:26

• 346 Sharp singular-set bounds for stationary integral varifolds

  Proves that every stationary integral m-varifold in a Euclidean open set has singular set of Hausdorff dimension at most $m-1$, a sharp bound in every positive dimension and codimension. On round spheres, the family also proves almost-everywhere regularity: the singular set has zero m-dimensional Hausdorff measure.

  2 papers ▶Episode, 4:37

• 347 Counterexamples to stable-Morse and strong Arnold fixed-point bounds

  Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension 22, the deficit below the stable Morse number is unbounded. A separate Hamiltonian diffeomorphism of the complex quadric threefold has exactly three fixed points, fewer than the four critical points required of every smooth function.

  5 papers ▶Episode, 4:57 OpenAI published a Lean proof of part of the claim.

• 348 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 $\mathbb{CP}^2$, 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.

  3 papers ▶Episode, 4:42 OpenAI published a Lean proof of part of the claim.

• 349 The Solomon–Yau least-volume conjecture

  Proves the Solomon–Yau least-volume conjecture for minimal hypersurfaces of round spheres. For every m ≥ 2, a closed connected minimal immersion into the unit sphere $S^{m+1}$ with non-totally-geodesic image has volume at least that of the smallest minimal Clifford product, counting covering multiplicity.

  ▶Episode, 4:28

• 350 Yau’s nodal bounds: surfaces and higher dimensions

  Proves the sharp $C\sqrt\lambda$ upper bound for nodal length on every fixed smooth closed surface, completing Yau's conjecture there. The upper bound fails for fixed smooth metrics in dimensions three and four, including metrics on S³ arbitrarily close to round. In dimension five, nodal measure can grow faster than $\lambda^{1/2+\varepsilon_0}$ for some fixed $\varepsilon_0\gt 0$, ruling out even arbitrarily small power losses.

  3 papers ▶Episode, 4:09 OpenAI published a Lean proof of the main theorem.

• 351 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.

  3 papers ▶Episode, 4:11

• 352 A finite-time singularity of Calabi flow

  Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth $U(10)$-invariant Kähler metric on $\mathbb{CP}^{10}$ 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.

  ▶Episode, 4:04

• 353 Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample

  Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in dimension ten makes this range sharp. In dimensions three through nine, the paraboloid classification also holds for connected open locally uniformly convex affine-maximal hypersurfaces complete for the affine Berwald–Blaschke metric.

  2 papers ▶Episode, 4:15 OpenAI published a Lean proof of the main theorem.

• 354 The isoperimetric profile of the cubic three-torus

  Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are $4\pi/81$ and $1/\pi$, with exactly the adjacent two types minimizing at each transition.

  ▶Episode, 3:51 OpenAI published a Lean proof of the main theorem.

• 355 Unique tangent flows at the first surface singularity

  Proves the first-singular-time case of tangent-flow uniqueness for smooth compact connected embedded surfaces without boundary in ℝ³. At every singular point, all fixed-center backward tangent flows agree as area measures at every negative time in the original ambient coordinates, without mean-convexity or a prescribed tangent model.

  ▶Episode, 4:18

• 356 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 $\mathop{\mathrm{RCD}}\nolimits ((n-1)\kappa,n)$ condition with reference measure $\mathcal H^n$ and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced.

  2 papers ▶Episode, 4:15 OpenAI published a Lean proof of part of the claim.

• 357 Bi-Lipschitz coordinates at every regular RCD point

  Proves that every regular point of a noncollapsed $\mathop{\mathrm{RCD}}\nolimits (K,n)$ space, for K ∈ ℝ and integer n ≥ 2, has an open neighborhood bi-Lipschitz to an open subset of ℝⁿ. Regularity requires all pointed tangents to be Euclidean, the reference measure is exactly $\mathcal H^n$, and the chart compares ambient distances with a point-dependent finite constant.

  ▶Episode, 4:10

• 358 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.

  ▶Episode, 3:50 OpenAI published a Lean proof of the main theorem.

• 359 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.

  2 papers ▶Episode, 4:29 OpenAI published a Lean proof of the main theorem.

• 360 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.

  2 papers ▶Episode, 4:08 OpenAI published a Lean proof of the main theorem.

• 361 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 $(k+2)^2$ independent harmonic functions of growth at most k, exceeding the Euclidean count $(k+1)^2$. The metric may depend on k.

  2 papers ▶Episode, 3:52 OpenAI published a Lean proof of part of the claim.

Summary and paper titles from OpenAI’s catalog, used under the Apache License 2.0. Last checked October 19, 2026.