Differential Geometry
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Paper 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…
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Paper 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…
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Paper Gromov’s integral scalar-curvature bound for simplicial volume
Proves \intM(\mathrm{Scal}g^-)^{n/2}\,dVg\ge an\lVert M\rVert for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with an\gt 0 depending only on dimension. Here…
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Paper 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…
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Paper Nonnegative-curvature Einstein classification and an L^{2} 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,…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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.…
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Paper 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…
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Paper A smooth surface metric with no local isometric immersion in \mathbb{R}^{3}
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…
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Paper 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…
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Paper 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…
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Paper 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 R1,\ldots,Rk embed disjointly into an open ball of capacity R exactly…
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Paper 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.…
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Paper Smooth isometric immersions of surfaces into \mathbb{R}^{4}
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…
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Paper 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…
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Paper 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,…
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Paper 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…
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Paper 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. Manuscript September 23, 2026 · · cite as…
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Paper 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…
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