Differential geometry

Every closed smooth Riemannian surface admits a smooth isometric immersion into Euclidean four-space, without an orientability assumption. This resolves the closed-surface form of the classical four-dimensional isometric-immersion problem.

We construct a smooth positive-definite Riemannian metric on \((-1,1)^2\) that agrees with the Euclidean metric to every order at the origin, yet no neighborhood of the origin admits a smooth isometric immersion into Euclidean three-space. This gives a negative answer to the unrestricted smooth local isometric realization problem for surfaces.

We prove the integral scalar-curvature inequality proposed by Gromov. For every dimension n ≥ 3, there is a constant \(a_n\gt 0\), depending only on n, such that \(\displaystyle \int_M(\mathop{\mathrm{Scal}}\nolimits _g^-)^{n/2}\,dV_g\ge a_n\|M\|\) for every closed connected oriented smooth n-manifold M and every smooth Riemannian metric g. Here \(\mathop{\mathrm{Scal}}\nolimits _g^-:=\max\{0,-\mathop{\mathrm{Scal}}\nolimits _g\}\), and \(\|M\|\) is real simplicial volume. The proof uses the nonnegative-scalar-curvature vanishing theorem of the companion paper on rational inessentiality.

We prove that every closed connected oriented smooth manifold admitting strictly positive scalar curvature is rationally inessential: its rational fundamental class maps to zero under the classifying map. No spin or fundamental-group hypothesis is needed, and there is no upper dimension bound. This proves the Gromov–Lawson aspherical conjecture; moreover, every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat. We also show that every closed oriented smooth manifold of positive dimension with nonnegative scalar curvature has zero real simplicial volume, proving the qualitative vanishing consequence of Gromov's conjectural comparison.

For every n ≥ 4, a complete connected smooth Riemannian n-manifold without boundary satisfying \(-4\Delta+\mathop{\mathrm{Scal}}\nolimits \ge1\) 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 in terms of n. The bound is measured in the original metric. This extends the uniform Urysohn width theorem from a pointwise scalar-curvature lower bound to a spectral lower bound.

Every connected complete smooth Riemannian three-manifold without boundary satisfying \(-4\Delta+\mathop{\mathrm{Scal}}\nolimits \ge\lambda\gt 0\) as a quadratic-form inequality admits a continuous map to a graph whose entire fibers have diameter at most \(500/\sqrt\lambda\) in the original metric. No orientability, spin, compactness, or bounded-geometry assumption is required.

We prove the quantitative continuous form of Gromov's scalar-curvature conjecture in every dimension n ≥ 4. Every complete connected smooth boundaryless n-manifold with scalar curvature at least one admits a continuous map to a simplicial complex of dimension at most \(n-2\) whose entire fibers have diameter bounded only in terms of n. We also obtain the continuous macroscopic-dimension conclusion for universal covers of closed manifolds with positive scalar curvature in every dimension n ≥ 2.

Every compactly supported integral n-cycle, n ≥ 2, in a proper CAT(0) space bounds a compactly supported integral current with the sharp Euclidean mass bound. The theorem allows arbitrary integer multiplicities and unrestricted ambient dimension. In particular, we prove the Euclidean Cartan–Hadamard isoperimetric conjecture in dimensions at least three.

We resolve the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most κ ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature κ. For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.

We prove that every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space. This resolves Yau's uniformization conjecture positively in every complex dimension.

For every closed connected smooth Riemannian manifold of dimension at least three with strictly negative sectional curvature, we prove that normalized Liouville measure has maximal entropy for the unit-speed geodesic flow if and only if the metric is locally symmetric. This resolves Katok's entropy rigidity conjecture positively in these dimensions, including all rank-one symmetric types at arbitrary scale.

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

Every smooth normalized positive triple of closed two-forms on a closed connected oriented four-manifold admits a smooth deformation, with each cohomology class fixed, to a hyperkähler triple. This resolves Donaldson's hypersymplectic deformation conjecture.

We prove that every smooth almost complex structure on a closed four-manifold which is tamed by a symplectic form is compatible with a symplectic form. This gives a positive solution to Donaldson's tamed-to-compatible conjecture.

We prove that, for all integers n ≥ 3 and k ≥ 1 and all positive real capacities \(R_1,\ldots,R_k\), the closed standard symplectic \(2n\)-balls of these capacities embed disjointly into the interior of a ball of capacity R > 0 if and only if \(\displaystyle \sum_{i=1}^k R_i^n\lt R^n, \qquad R_i+R_j\lt R\quad(i\ne j).\) Capacity is π times the squared Euclidean radius, and each embedding is defined on a neighborhood of its closed source ball. This proves Siegel and Yao's Conjecture A.

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

We resolve the sphere case of the closed-geodesic infinitude problem: every smooth Riemannian metric on the standard sphere Sⁿ, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The two-sphere case is classical; the result in higher dimensions includes degenerate metrics. Using finite covers and theorems of Perelman and Rademacher–Taimanov, we also obtain the same conclusion for every smooth Riemannian metric on a nonempty closed smooth three-manifold.

The singular set of every stationary integral m-varifold in an open Euclidean set has Hausdorff dimension at most \(m-1\), in every positive dimension and codimension. This sharp bound proves the Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis.

We resolve the almost-everywhere regularity conjecture of Brena, Decio, and De Lellis for stationary integral varifolds of arbitrary positive dimension and codimension in Euclidean open sets. The singular set has zero measure in the dimension of the varifold: near almost every support point, the varifold is a constant positive integer multiple of a smooth embedded minimal submanifold. The corresponding statement also holds on round spheres.

We construct a simply connected closed Kähler manifold of real dimension 3332 and minimal Chern number one whose Hamiltonian fixed-point count attains the cyclic integral Floer bound while falling below the stable Morse number. The Hamiltonian diffeomorphism has exactly \(1\,872\,232\) fixed points, all nondegenerate and with contractible orbit loops, whereas the stable Morse number is \(1\,872\,264\). Thus the stronger stable Morse bound fails by exactly 32, even when the cyclic integral bound is sharp.

We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points with examples in fixed real dimension twenty-two. For every integer m ≥ 1, we construct a simply connected closed Kähler manifold with stable Morse number \(80+1968m\) and a Hamiltonian diffeomorphism with exactly \(80+1952m\) fixed points, all nondegenerate and with contractible orbit loops. The deficit is therefore unbounded, and the fixed-point count is at most 127/128 of the stable Morse number.

We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points by constructing a simply connected closed Kähler manifold with sixteen fewer fixed points than its stable Morse number. All the corresponding periodic orbits are contractible. The construction combines a Hamiltonian involution with integral homology torsion at two different primes; its proof uses finite-dimensional Morse theory and complex blowups.

We construct a smooth Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, at least one of which is degenerate. The critical number and the unit-inclusive rational cup length of this manifold are both four. Thus the example disproves the unrestricted critical-number and rational cup-length forms of the Arnold conjecture.

We construct a smooth one-periodic Hamiltonian on a closed symplectic twelve-manifold whose time-one map has fewer fixed points than the manifold's ordinary Morse number. Every fixed point is nondegenerate and has a contractible Hamiltonian trajectory. This disproves the Morse-number form of the Arnold conjecture. The deficit can be arbitrarily large among twelve-dimensional examples.

We prove a universal, scale-invariant L² gap for the trace-free Ricci tensor on simply connected closed four-manifolds with nonnegative sectional curvature. If the trace-free Ricci energy is sufficiently small, the manifold is diffeomorphic to S⁴, \(\mathbb{CP}^2\), or \(S^2\times S^2\). The proof uses the classification of positive-Einstein, nonnegatively curved four-manifolds supplied by the companion zero-plane rigidity theorem, stated explicitly as the classification premise of our result. No auxiliary curvature, volume, diameter, injectivity-radius, or Sobolev bound is required. The conclusion concerns the smooth manifold, not an isometry of the original metric.

We prove that a closed Einstein four-manifold with positive Einstein constant, nonnegative sectional curvature, and a zero-curvature plane has universal Riemannian cover isometric to a product of two round two-spheres. Under the normalization \(\mathop{\mathrm{Ric}}\nolimits =3g\), both spheres have radius \(1/\sqrt3\). The proof extends coupled estimates for the two Weyl curvature blocks to the boundary of the sectional-curvature cone and determines their equality case.

We prove the classification conjecture for connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini–Study metric, or real projective four-space with its round metric. No orientability assumption is needed.

We prove the Solomon–Yau least-volume conjecture for minimal hypersurfaces of unit round spheres: in every dimension m ≥ 2, a closed connected minimal immersion with non-totally-geodesic image has volume at least the smallest m-dimensional minimal Clifford product. Volume is measured on the domain, so covering multiplicities are included.

We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on \(S^2\times\mathbb T^2\) for which sequences of exact real Laplace eigenfunctions have unbounded nodal measure divided by the square root of the eigenvalue. Each sequence belongs to one fixed metric. Thus the upper-bound part of Yau's nodal conjecture fails for smooth metrics in dimensions three and four.

We prove that a nonzero real Laplace eigenfunction with eigenvalue λ > 0 on a fixed smooth closed connected Riemannian surface has nodal length at most \(C\sqrt\lambda\). Together with the known lower bound, this proves Yau's conjecture in this setting.

We construct a smooth Riemannian metric on \(S^4\times S^1\) and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than \(\lambda^{1/2+\epsilon_0}\) for one fixed \(\epsilon_0\gt 0\). This disproves the smooth upper-bound assertion in Yau's nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.

We prove a sequential elliptic inequality for the renormalized Einstein–Hilbert functional on a fixed finite tree of four-dimensional Ricci-flat spaces. As the joining lengths diverge and the scale-neutral weighted Ricci error M tends to zero, the functional satisfies \(E=o(M)\). The root end has decay exponent greater than one, and every joined quotient group is nontrivial. We also prove the multivariable path-selection theorem for asymptotic expansions used to obtain this estimate.

We prove that a smooth Ricci flow on a closed real four-manifold extends on the same manifold through every finite time at which its scalar curvature remains uniformly bounded. This resolves the scalar-curvature extension problem in dimension four.

We disprove the scalar-curvature extension conjecture in its unrestricted all-dimensions form. In sufficiently high dimension, we construct a Ricci flow on a closed manifold whose scalar curvature remains uniformly bounded while full curvature diverges at a finite maximal time. In one fixed sufficiently high dimension, the examples have two-sided power-law curvature blowup with arbitrarily large exponents.

We construct a smooth Kähler metric in the Fubini–Study class of \(\mathbb{CP}^{10}\) whose Calabi flow develops unbounded scalar curvature in finite time. This disproves Chen's smooth long-time existence conjecture, even in a class containing a constant-scalar-curvature metric.

We construct a smooth nonquadratic entire graph in dimension ten that solves the classical affine maximal equation and has positive-definite Hessian everywhere. This gives a smooth counterexample to the entire-graph affine Bernstein assertion in dimension ten. Hessian positivity is pointwise; no global uniform lower bound or completeness of the Berwald–Blaschke metric is asserted.

We prove the Euclidean-complete affine Bernstein conjecture in dimensions three through nine: a smooth locally uniformly convex affine maximal graph is an elliptic paraboloid whenever its induced Euclidean metric is complete. The same conclusion holds, without an initial graph assumption, for connected smooth open (noncompact and without boundary) affine-complete locally uniformly convex immersed hypersurfaces that are classically affine maximal in these dimensions.

We prove the isoperimetric conjecture for the cubic flat three-torus and classify all minimizers, including the equality cases at the transition volumes \(4\pi/81\) and \(1/\pi\). The minimizing regions are balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements.

We prove that, at each singular point of the first singular time of the mean-curvature flow of a smooth compact connected embedded surface without boundary in ℝ³, all fixed-center rescalings converge locally smoothly on compact negative-time intervals to one multiplicity-one homothetic self-shrinker flow. The limit is unique in the original ambient coordinates, including its position and axes. No mean-convexity assumption or prescribed tangent model is required.

On a full-support \(\mathrm{RCD}(K,N)\) space with \(1\lt N\lt \infty\), we prove that a bounded globally Lipschitz function whose distributional Hessian is bounded above by a bounded continuous function satisfies the corresponding second-derivative inequality along every minimizing geodesic.

For every integer n ≥ 2 and κ ∈ ℝ, we prove that a complete separable metric space is an n-dimensional Alexandrov space of curvature at least κ if and only if, with reference measure \(\mathcal H^n\), it is a full-support \(\mathrm{RCD}((n-1)\kappa,n)\) space whose distributional sectional curvature is at least κ in Gigli's original global test classes. This resolves Gigli's characterization conjecture in dimensions at least two.

We resolve the regular-point bi-Lipschitz conjecture in the noncollapsed setting. For every integer n ≥ 2 and every real K, every regular point of a noncollapsed \(\mathrm{RCD}(K,n)\) space has an open neighborhood bi-Lipschitz homeomorphic to an open subset of ℝⁿ. The bi-Lipschitz constant depends only on n, and the neighborhood uses the restricted ambient distance.

We construct a closed connected orientable smooth three-manifold that admits a smooth Riemannian metric without conjugate points but admits no smooth Riemannian metric of nonpositive sectional curvature.

We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂ^m diffeomorphic to \(\mathbb R^{2m}\) that admits a complete Kähler metric with real sectional curvature at most −1 and has only constant bounded holomorphic functions. Its sectional curvatures are unbounded below. The construction gives a negative answer to the one-sided bounded-holomorphic-function question.

We construct a contractible domain in complex dimension three with a complete Kähler metric whose real sectional curvatures lie between two finite negative constants. It admits no bounded holomorphic map to ℂ³ with nowhere-vanishing Jacobian and is therefore not biholomorphic to a bounded domain. This gives a negative answer to the negatively pinched Kähler uniformization question.

We prove that the weak Ma–Trudinger–Wang condition on a fixed smooth connected compact boundaryless Riemannian manifold of dimension at least two implies a common Hölder estimate for optimal transport maps and their inverses over the entire class of probability densities with fixed positive upper and lower bounds. The maps have homeomorphic representatives, conjugate cut points are allowed, and no density regularity is assumed.

We prove that weak Ma–Trudinger–Wang curvature on a smooth, connected, compact Riemannian manifold of dimension at least two without boundary implies convexity of every tangent injectivity domain, resolving Villani's conjecture in this setting. Conjugate cut points are allowed. More generally, every ordinary subgradient of a squared-distance cost potential is a minimizing velocity with a global supporting mountain. No density hypothesis is used.

For every \(4/9\lt v\lt 1\) and \(1\lt c\lt 9v/4\), all sufficiently large integers k admit a complete smooth metric on ℝ³ with nonnegative Ricci curvature, asymptotic volume ratio v, and at least \(c(k+1)^2\) linearly independent real harmonic functions of pointwise growth at most k. This answers Yau's integer-degree dimension comparison question negatively in dimension three, with a fixed-factor excess over the Euclidean count. For each \(1\lt c\lt 9/4\), these metrics can be chosen arbitrarily close to the Euclidean metric in global bi-Lipschitz distance. The metric may depend on k.

For some even n ≥ 8 and integer k ≥ 2, we construct a complete smooth metric on ℝⁿ with nonnegative Ricci curvature whose space of real harmonic functions of pointwise polynomial growth at most k has dimension larger than the Euclidean harmonic-polynomial dimension. The metric is Euclidean near the origin, has asymptotic volume ratio strictly between zero and one, and has nonunique tangent cones at infinity. This answers the integer-degree form of Yau's dimension comparison question in the negative.