Calabi Flow, Affine Bernstein Problem and 7 More in Differential Geometry

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 ℂℙ², 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. (Lean)

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.

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.

Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth U(10)-invariant Kähler metric on ℂℙ¹⁰ 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.

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. (Lean)

Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support RCD ((n-1)κ,n) condition with reference measure ℋⁿ and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced. (Lean)

Proves that every regular point of a noncollapsed RCD (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 ℋⁿ, and the chart compares ambient distances with a point-dependent finite constant.

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. (Lean)

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. (Lean)