Papers

We formalize the smooth three-dimensional Poincaré conjecture, together with the Moise smoothing theorem, yielding the topological three-dimensional Poincaré conjecture. The smooth proof follows the Hamilton–Perelman route through Ricci flow with surgery and finite-time extinction.

Following a similar path as the recent construction of a complex structure on $S^6$ we construct another such family, disjoint from the known ones.

Let $M$ be a closed manifold with a bracket-generating distribution $H$. We study the Morse index of exponentially subelliptic harmonic maps. After establishing finiteness, we estimate the index through two test spaces. The extrinsic one averages the index form over projected ambient parallel frames, reducing the estimate to a trace identity involving an extrinsic scalar with a quartic exponential term. The intrinsic one uses parallel sections of the pullback bundle, on which the index form descends to a coupled quadratic form that is exact when a global parallel frame exists. The bounds apply to round spheres, products of spheres, and convex hypersurfaces. On a closed Heisenberg nilmanifold, the intrinsic reduction yields a spectral formula for the index.

We construct a two-parameter family of complex structures on the standard six-sphere, starting from a rational elliptic surface with singular fibers $III^*,I_1,I_1,I_1$. The resulting torus fibration over $\mathbb{P}^1$ has one multiple fiber of multiplicity four, one hexagonal fiber, and two rank-one degenerations. These complex structures are not biholomorphic to the original Alpöge–Claude examples.

Given a compact complex surface with a nonflat Ricci-flat Kähler metric, we show that its holomorphic tangent bundle admits an RC-positive Hermitian metric. The proof relies on a characterization of RC-positivity through the anti-self-dual part of the Weyl operator, the Weitzenböck formula on the four dimensional Einstein manifold, and a conformal perturbation on the Ricci-flat metric. As a consequence, we prove that RC-positivity is not preserved after taking tensor, exterior, or symmetric power. Moreover, we show that RC-positivity is a strictly weaker notion than uniform RC-positivity, and that RC-positivity of holomorphic tangent bundle does not necessarily imply rational connectedness of the base manifold.

We give a short, self-contained proof of Bernstein's theorem: every entire minimal graph in $R^3$ is a plane. The only global tool is the Omori-Yau maximum principle, which holds on entire minimal graphs with their induced metric. We apply it exactly once to a single explicit function $F$ built from the surface's angle function and principal curvatures. A pointwise differential inequality for $F$, combined with the Omori-Yau principle, forces it to attain its minimum value everywhere, which yields the planarity of the surface and hence Bernstein's theorem. As a by-product of our approach, we show that the same method, with a different function and with some extra work, also proves in a simple and self-contained way the generalization of Bernstein's theorem given by Osserman that a complete minimal surface in $R^3$ whose normals omit a neighbourhood of some direction must be a plane.

For every integer $\nu\geq 2$, we construct a $(\nu+1)$-dimensional, locally compact, geodesically complete CAT(0) space of asymptotic rank $\nu$ which does not satisfy a linear isoperimetric inequality for integral $\nu$-cycles. Its filling function is bounded below by $c v\log v$ for all sufficiently large $v$.

Let $X$ be a fixed complex K3 surface and let $0\ne\alpha\in\overline{Kah_X}$ be a $(1,1)$-class on the boundary of the Kähler cone with $\alpha^2=0$. We build a novel connection between two seemingly unrelated, intensively studied problems: the rigidity of a closed positive current in the class $\alpha$ and the collapse geometry of a sequence of Ricci-flat Kähler metrics in the Kähler class $\alpha+t\kappa$ as $t\to 0^+$. More precisely, we prove that $\alpha$ contains a unique closed positive $(1,1)$-current if and only if the Ricci-flat metrics in class $\alpha+t\kappa$ collapse to a point in the Gromov–Hausdorff sense. Then we exhibit the first example of a nonrigid irrational nef class $\alpha$ on a Kummer surface, which answers a question of Filip–Tosatti and Sibony–Soldatenkov–Verbitsky. We also construct many new examples of rigid nef classes on hyperkähler manifolds, which extend the work of Filip–Tosatti and Sibony–Soldatenkov–Verbitsky.

For $3\le n\le6$, we determine the sharp isoperimetric bound for the mass-normalized first weighted Neumann eigenvalue on bounded Euclidean domains. For every finite nonnegative nonatomic measure $\mu$ on $\overline\Omega$, \begin{equation*} \overline\lambda_1^N(\Omega,\mu)\le\frac{n(n-1)}{n-2}\omega_n\sin^2\vartheta_n\left(\frac{|\Omega|}{\omega_n}\right)^{(n-2)/n}, \end{equation*} where $\omega_n$ is the volume of the unit ball and $\vartheta_n\in(\pi/2,\pi)$ is the angle at the first stationary radius of the regular rotational harmonic-map profile. The bound is attained on balls by a smooth positive radial density. In fact, we affirmatively resolve Question~1.12 posed by Vinokurov \cite{Vinokurov2026}. Furthermore, for admissible domains with finite boundary measure, we also prove the sharp strict Steklov bound \begin{equation*} \sigma_1(\Omega)|\partial\Omega|\,|\Omega|^{(2-n)/n}<\frac{n(n-1)}{n-2}\omega_n^{2/n}\sin^2\vartheta_n, \end{equation*} whose constant is approached by $C^1$ perforated domains.

Let $M^n\to\mathbb{S}^{n+q}(1)$ be a closed connected minimal immersion, where $n\ge3$, and set $Q=S+\lambda_2$, with $S=|h|^2$ and $\lambda_2$ the second largest eigenvalue of Lu's fundamental matrix. We determine the sharp codimension range for Lu's second-gap conjecture. For every $2\le q\le n$, there exists $\gamma_{n,q}>0$ such that, if $Q$ is constant and $Q>n$, then $Q\ge n+\gamma_{n,q}$. Conversely, for every $q\ge n+1$, we construct closed connected homogeneous minimal embeddings, followed when necessary by totally geodesic inclusions, with constant scalar curvature and constant $Q$-values dense in $(n,2n)$. Thus, in every dimension $n\ge3$, Lu's conjecture holds precisely for $q\le n$. Combined with the theorem of Peng-Terng for hypersurfaces and the recent resolution of the two-dimensional case, this gives a complete resolution of Lu's second-gap conjecture: for every $n\ge2$, the conjecture holds exactly when $q\le n$ and fails when $q\ge n+1$. In codimension two we further obtain the explicit admissible gap $\gamma_{n,2}=\exp(-10^{16}n^2)$.

The loop group description of surfaces whose Gauss map is Lorentz harmonic is known for timelike surfaces of constant mean curvature in the flat Lorentzian 3-space, and for surfaces of constant Gaussian curvature in the 3-sphere. We carry out the corresponding construction for timelike surfaces in anti-de Sitter 3-space $\mathbb{H}^3_1\cong SL(2,\mathbb{R})$: those of constant mean curvature, whose Gauss map is harmonic for the conformal structure of the first fundamental form, and those of constant Gaussian curvature $K>-1$, $K\neq0$, whose Gauss map is then an immersion and is harmonic for the second fundamental form. For both classes we prove the harmonicity characterization, recover the surfaces from extended frames by Sym-Bobenko-type formulas, and solve the geometric Cauchy problem by the generalized DPW method, with potentials written explicitly in terms of the data and no normalization of the Gauss map. We then show that the two constructions differ only by a normalization: after the substitution $\lambda=\zeta^2$ and a constant gauge, a constant mean curvature extended frame $\hat F$ is an extended frame of its Gauss map, and the maps $\hat F|_{\lambda=e^{4\theta}}\exp(\theta e_3)\hat F|_{\lambda=1}^{-1}$ and $e_3\hat F|_{\lambda=e^{4\theta}}e_3\exp(\theta e_3)\hat F|_{\lambda=1}^{-1}$, with $e_3=\mathrm{diag}(-1,1)$, are timelike surfaces of constant Gaussian curvature $1/\sinh^2\theta$ and $-1/\cosh^2\theta$ wherever the constant mean curvature surface is not flat. The first is the classical parallel surface, the second its polar surface. Explicit families, with extended frames and surfaces in closed form, illustrate the constructions.

A long-standing conjecture in non-Kähler geometry states that a compact Hermitian manifold with constant Chern holomorphic sectional curvature should be Kähler when the constant is nonzero and Chern flat when the constant is zero. In this article, we study the corresponding problem for a Hermitian connection $\nabla$ with $\nabla T=0$. We first show that if the $\nabla$-holomorphic sectional curvature is constant, then the $(1,1)$-part of the curvature is $\nabla$-parallel. Our main result states that a nonzero constant forces $T=0$; consequently, the metric is Kähler and locally a complex space form. The proof is pointwise and requires neither compactness nor completeness. We also obtain applications to the Chern and Bismut connections.

For complex Hessian equations on compact Kähler manifolds, we give an alternative proof of the gradient estimate without using blow-up argument, and clarify its dependence on the right-hand side. In addition, using the similar approach, the case of Kähler manifolds with boundary is also investigated.

Borel's theorem presents the flag manifold cohomology as the coinvariant algebra $H^*(K/T; \mathbb{C}) \cong S(\mathfrak{h}^\vee)/I_W$, establishing the vanishing of characteristic forms $p_j(F_\mathfrak{t}) = \mathrm{tr}(F_\mathfrak{t}^j)$ non-constructively; projection from the Weil algebra $W(\mathfrak{k})$ yields closed primitives on $K$ failing to descend. We construct relative transgression primitives $\alpha_j \in (\Lambda^{2j-1}\mathfrak{m}^\vee)^T \cong {}^K\Omega^{2j-1}(K/T)$ satisfying $d_\mathfrak{m} \alpha_j = -p_j(F_\mathfrak{t})$ on $K/T$. Via connections $\theta_s = \theta_\mathfrak{t} + s\theta_\mathfrak{m}$, the curvature factors as $F_s = (1-s)(F_\mathfrak{t} - s\theta_\mathfrak{m}^2)$; expanding via non-commutative word sums and Euler's Beta integral yields a universal closed formula for all $j \ge 1$. These primitives exhibit a structural transition: $\alpha_2 = -\frac{1}{3}\mathrm{tr}(\theta_\mathfrak{m}^3)$ is curvature-free, $\alpha_3$ is linear in $F_\mathfrak{t}$, while at $j = 4$, $[F_\mathfrak{t}, \theta_\mathfrak{m}^2] \neq 0$ shatters cyclic trace symmetry into independent necklaces, producing the irreducible invariant $\mathrm{tr}(\theta_\mathfrak{m} \wedge F_\mathfrak{t} \wedge \theta_\mathfrak{m}^2 \wedge F_\mathfrak{t})$. On the $A_2$ root lattice ($\mathfrak{sl}_3(\mathbb{C})$), $T$-invariant cochains correspond to closed loops, $\alpha_2 = \Omega_- - \Omega_+$ is the oriented difference of triangles, and $d_\mathfrak{m}$ acts as an edge-fracturing operator witnessing the exactness of $c_2(E)$. While character theory encounters a multiplicity barrier at $k=4$ in Specht decompositions, a minimal Koszul DGA resolution $K_\bullet(p_2, \dots, p_k)$ resolves the coinvariant algebra, recovering the Poincaré polynomial $P_{K/T}(t) = \prod_{j=2}^k \frac{1-t^{2j}}{1-t^2}$ and Euler characteristic $\chi(K/T) = k!$ without matrix-rank ambiguities.

We calculate the exact values of the relative isoperimetric profile for a unit square with a central square removed. In other words, given a removed central square of a fixed size, we provide a function which produces the smallest relative perimeter which bounds any possible area within this shape.

We study stable solutions of the Allen-Cahn equation with homogeneous Neumann boundary condition on Riemannian manifolds with boundary in the regime where the transition layers meet the boundary orthogonally. Under an a priori bound for the enhanced second fundamental form in the transition region, we establish uniform second-order Hölder estimates and quantitative mean curvature decay for the nodal hypersurfaces up to the boundary in ambient dimensions at most 10. The proof adapts the technique pioneered by Wang-Wei, adding a further correction determined by the geometry of the ambient boundary. Its leading contribution to mean curvature cancels on the nodal set but survives on non-zero level sets in the transition region, whose mean curvature need not decay uniformly. This phenomenon has no interior analogue and is exhibited by stable solutions constructed by Kowalczyk.

We consider singular Liouville systems on closed Riemann surfaces with a common sign-changing coefficient. Direct continuation from a positive coefficient is obstructed by the unavoidable appearance of degenerate zeroes along such a homotopy. We instead identify an intrinsic ray in the parameter space on which the system admits synchronized solutions and reduces exactly to a scalar sign-changing mean-field equation. Combining this reduction with the variational existence theory of De Marchis–López-Soriano–Ruiz and the nodal singularity compactness theorem of Wu, we obtain existence for every noncritical parameter on this ray. Positive singular sources are allowed on the nodal curve. In particular, when the surface has nonpositive Euler characteristic and every negative component of the coefficient is a disk, the topological hypothesis required by the scalar min–max construction is automatic.

We study the construction of positively curved Riemannian metrics by non-isometric circle actions. For a family of homotopy eleven-spheres whose Eells–Kuiper invariants form the even subgroup of $\mathbb{Z}/992$, we construct, on each member, a smooth background metric $q$ and three effective circle actions with generators $W_1,W_2,W_3$ such that the metric determined by $g^{-1}=q^{-1}+\sum_{a=1}^3W_a\otimes W_a$ has positive sectional curvature. Each action is non-isometric for every partial metric, including its incoming metric and the final metric. We first describe the sphere by gauge transformations of the quaternionic Hopf bundle. We then construct compatible metrics on two disks and smooth their inverse metrics while preserving the action formula. A local conjugation makes the circle actions non-isometric. For each fixed member, we obtain an explicit positive lower bound $2^{-54}(1+M_{0,k}+M_{1,k})^{-28}$, where $M_{0,k}$ and $M_{1,k}$ are norms of the curvature and its first covariant derivative for its fixed connection. The bound may depend on the member of the family.

The compact four-dimensional manifold $\mathbb{CP}_2\# 3\overline{\mathbb{CP}_2}$ is known to admit a toric Kähler-Einstein metric $g_{\text{KE}}$, but the metric is not known in closed form, which makes it difficult to draw conclusions about its geometry. In this article, we use a combination of analytic and computer-assisted techniques to produce an approximate Einstein metric $g$ described explicitly herein, and also prove that the true Einstein metric $g_{\text{KE}}$ is close to $g$, where both the closeness and the topology are described explicitly. As an application, we prove bounds on the first invariant eigenvalue of the Laplace-Beltrami operator, and prove that this Kähler-Einstein metric does not have positive holomorphic sectional curvature everywhere.

Motivated by recent developments in generalized Kähler geometry, we study morphisms of double Lie groupoids from a simplicial viewpoint. We show that the codiagonal functor $\Wbar$ extends from objects to horizontal and vertical principal bibundles and to square-shaped $(1,1)$-morphisms. The latter determine 2-dimensional morphisms of square and globular shape between anafunctors of Lie 2-groupoids. We illustrate these constructions through two applications: First, we revisit the equivalence between two models of nonabelian gerbes: groupoid bundle gerbes and principal 2-bundles. We show that a principal 2-bundle and its associated groupoid bundle gerbe determine equivalent anafunctors from a manifold to the corresponding structure Lie 2-group. Second, we refine the integration of Manin triples in transitive Courant algebroids. We show that integrations associated with different choices of suitably transverse Manin triples in the same Courant algebroid are related by symplectic Morita equivalences. Consequently, the resulting integration of the background Courant algebroid is well defined up to symplectic Morita equivalence.

Difference-in-means steering requires activations recorded while a model shows the desired behavior, which a sandbagging model withholds by deliberately underperforming. We introduce Activation Flow (ActFlow), which manufactures these activations from $k$ correct labels without fine-tuning. ActFlow sets target logits that rank each labeled item's correct answer first, and moves the logits toward them by adding one vector $x$ to all $k$ residual streams at one layer. ActFlow is a family of ordinary differential equations for $x$, one for each rule that maps the required logit change to the velocity of $x$. The smallest-norm rule lands exactly on the targets, while the others keep only the top singular directions of the Jacobian. We test ActFlow on three instruction-tuned models, each locked by a sandbagging prompt and by a password-locked LoRA. At $k=40$, ActFlow keeping five singular directions raises the mean held-out ARC-Easy accuracy over the six locked models from $0.05$ to $0.85$, against $0.88$ for rank-$16$ LoRA fine-tuning, which trains about $3{,}000$ times as many parameters, and $0.92$ for the honest models. Furthermore, it scores higher than the smallest-norm rule in 16 of the 18 combinations of locked model and $k$, and its steering direction is nearly orthogonal to the honest difference-in-means direction. It also unlocks two LoRA locks where the honest direction fails. Surprisingly, with one labeled item, the single-step smallest-norm rule raises the Qwen2.5-7B prompt lock from $0.04$ to $0.59$.

We give a complete classification of proper biharmonic Lagrangian $H$-umbilical submanifolds in complex space forms. Such submanifolds exist only in complex projective space and are locally congruent to an explicit model obtained via the Hopf fibration. In particular, their nonexistence in complex Euclidean space provides a partial affirmative answer to Chen's conjecture.

We construct a sequence of AE Kähler metrics on C^2 with nonnegative and integrable scalar curvature whose ADM mass tends to zero but which, for a suitable choice of basepoints, fails to converge to Euclidean space in the pointed Gromov-Hausdorff sense. This shows that, without any additional assumptions, the excision in Klemmensen's stability result for the positive mass theorem for Kähler manifolds is crucial, which aligns with the findings in the (3-dimensional) Riemannian case.

Let $A$ be a commutative associative unital algebra over a field $K$ of characteristic zero, and let $(G,\rho)$ be a Lie–Rinehart algebra over $A$. We revisit the Gerstenhaber structure carried by the exterior algebra $\bigwedge_A G$ from the point of view of graded derivations. Particular attention is paid to the well-definedness of the Schouten–Nijenhuis extension over the coefficient algebra $A$: the Lie–Rinehart identity is shown to provide exactly the correction terms needed for the bracket to descend to the $A$-balanced exterior algebra. After establishing graded antisymmetry, the Leibniz rule and the graded Jacobi identity without circularity, we study the inner graded derivations $\adSN(P)=[P,\cdot]_{SN}$ and prove \[ [\adSN(P),\adSN(Q)]=\adSN([P,Q]_{SN}). \] Finally, we give a complete proof of the converse construction: Gerstenhaber brackets on $\bigwedge_A G$ are in one-to-one correspondence with Lie–Rinehart structures on $G$.

Let $(\G,\rho,\omega)$ be a symplectic Lie–Rinehart–Jacobi algebra over a commutative algebra $A$, equivalently a Lie–Rinehart algebra $(\G,\bar\rho)$ endowed with a $1$-cocycle $\chi$ and a nondegenerate $2$-form closed for the twisted differential $d_\rho=d_{\bar\rho}+\chi\wedge$. Starting from the standard extension by the $2$-cocycle $\omega$, we study the strict lifting problem for its canonical $1$-form $\eta$ in the LRJ setting. We prove that the strict symmetries of $\eta$ are precisely the elements $X_f+fe$ and that $f\mapsto X_f+fe$ identifies the Jacobi Lie algebra $(A,\{\,,\})$ with this Lie algebra of strict symmetries. For locally Hamiltonian elements, the obstruction to a strict lift is the class $[i_x\omega]\in H^1_\rho(\G,A)$, yielding the exact sequence $0\to\ham\to\hamloc\to H^1_\rho(\G,A)\to0$. We also relate Okassa's symplectic curl to the variation of $d\eta$ and establish a conformal triviality criterion. In the smooth case the framework reduces to locally conformally symplectic geometry, and on a closed surface of genus $g\geq2$ we obtain structures that are not conformally trivial and whose obstruction space has dimension $2g-2$.

We show that every left-invariant elliptic structure of minimal rank on an odd-dimensional compact Lie group admits an algebraic decomposition analogous to those obtained by Pittie in the complex case and by Charbonnel and Khalgui for left-invariant CR structures of hypersurface type. We then use this decomposition to prove that the cohomology associated with such an elliptic structure can be reduced to a computation on a suitable maximal torus.

Let $S$ be a closed, oriented surface of genus at least $2$, let $h$ be a smooth Riemannian metric on $S$ with curvature $K\in (-1,0]$, and let $c\in \cT_S$ be a conformal structure on $S$. There exists a unique equivariant isometric immersion of $(S,h)$ in $\HH^3$ such that the pull-back by the hyperbolic Gauss map of the conformal structure at infinity is $c$. Dually, if $h^*$ is a smooth metric on $S$ with curvature $K^*\in (-\infty, 0)$ and if $c\in \cT_S$, there exists a unique equivariant immersion of $S$ into $\HH^3$ with third fundamental form $h^*$ and such that the pull-back of the conformal class at infinity by the Gauss map is $c$. Equivalently, given $h$ and $c\in \cT_S$, there is a unique pair $(E,u)$ where $E$ is a hyperbolic end with conformal structure at infinity $c$ and $u$ is an isometric embedding of $(S,h)$ in $E$. Given $h^*$ and $c$, there exists a unique pair $(E,u^*)$, where $E$ is a hyperbolic end with conformal structure at infinity $c$, and $u^*:S\to E$ is an embedding inducing the third fundamental form $h^*$. Those statements can be considered as a smooth counterpart of known or conjectural statements on the grafting map and on circle patterns.

We construct a degeneration of Hitchin's non-standard minitwistor spaces associated with toric gravitational instantons of type A$_{\rm odd}$ to the minitwistor space of a simple hyperbolic orbifold. We show that the corresponding families of minitwistor lines also converge under this degeneration. Consequently, the associated non-standard Einstein–Weyl spaces converge to the hyperbolic orbifold.

Motivated by the systematic study of spiral minimal products, we consider harmonic maps obtained by coupling two spherical eigenmaps through a profile curve in $\mathbb S^3$, with a doubly warped product metric on the source. The source warps are independent of the profile magnitudes. This gives two complementary problems: reconstructing a compatible source metric from prescribed profile or source data, and finding closed profiles when the source metric is fixed. For arbitrary prescribed warps, homogenization of the reduced Lagrangian gives a strongly convex conic Finsler metric whose oriented graph geodesics are exactly the harmonic profiles. In the unit-speed inverse problem, the phase momenta reduce reconstruction to a scalar magnitude equation. We obtain global reconstructions from a prescribed magnitude, one source warp, or a monotone volume factor, together with periodic families in which profiles closing on finite covers are dense. In the fixed-metric problem, a change of time gives an autonomous Neumann system. One fixed source metric then carries continuous families of closed profiles and infinitely many closed graph geodesics. For fixed even $p,q\geq 2$, every fixed metric in this Neumann class on $\mathbb S^1\times\mathbb S^p\times\mathbb S^q$ supports harmonic maps into $\mathbb S^{p+q+1}$ of degree $4m$ for every $m\geq 1$. In the matched constant-warp case, these maps reduce to explicit eigenmaps.

This paper investigates the theory of fractional Sobolev mappings between Carnot groups, with a particular focus on the Korevaar-Schoen energy functional and interpolation properties. We establish that weak contact equations hold for fractional Sobolev mappings on Carnot groups, generalizing classical results from the Euclidean setting to the sub-Riemannian framework. Our main results include: (1) the proof that fractional Sobolev mappings satisfying weak contact equations belong to the horizontal fractional Sobolev space $W_H^{s-\lambda,q}(\Omega;G_2)$ for any sufficiently small $\lambda>0$; (2) interpolation theorems for fractional Sobolev mappings between Carnot groups, demonstrating that the limiting behavior as $s \to 1$ recovers the first-order Sobolev space; (3) a rigorous treatment of the Korevaar-Schoen energy on Carnot groups, including a characterization of its minimizers via horizontal differential forms.