Geometry & Topology
29 results · 49 papers · 15 with a Lean formalization
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.
1 paperLeanConstructs 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.
1 paperLeanProves \(\int_M(\mathrm{Scal}_g^-)^{n/2}\,dV_g\ge a_n\lVert M\rVert\) 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 papersEvery 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 papersProves 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 papersLeanProves Yau's uniformization conjecture: every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to ℂn.
1 paperProves 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.
1 paperDisproves 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.
1 paperProves 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.
1 paperProves 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.
1 paperLeanResolves 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.
1 paperLeanProves 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.
1 paperProves 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.
1 paperProves 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 papersDisproves 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 papersLeanClassifies 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 papersLeanProves 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.
1 paperProves 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 papersLeanProves 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 papersDisproves 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.
1 paperProves 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 papersLeanDetermines 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.
1 paperLeanProves 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.
1 paperProves 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 papersLeanProves 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.
1 paperConstructs 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.
1 paperLeanConstructs 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 papersLeanOn 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 papersLeanDisproves 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 papersLeanNo result matches this filter yet.