arXiv subjects
Publications and source records attributed to Shing-Tung Yau.
We prove that a smooth homotopy $9$-sphere admits a Riemannian metric of positive sectional curvature if and only if its Hitchin $α$-invariant vanishes, or equivalently if and only if it admits a Riemannian metric of positive scalar curvature.
Coarse PDE observations omit spatial detail and intermediate states of evolving fields. We present Structured Latent Dynamics (SLD), a framework that reconstructs high-resolution trajectories from spatially and temporally downsampled simulations. SLD interpolates the input in time, adds a learned temporal correction and uses the resulting sequence to guide a hidden state on the coarse grid. Each update of the latent state combines a learned gated response with horizontal and vertical central differences and a five-point Laplacian of the state. The three spatial terms have learned channel-wise coefficients. Two Haar wavelet synthesis branches map the hidden state and refined coarse input to the high-resolution field. We evaluate SLD on two-dimensional Gray--Scott, complex Ginzburg--Landau and viscous Burgers systems, reconstructing 100 output frames from 25 input frames with eightfold refinement in each spatial direction. SLD achieves lower mean relative $L_2$ error and mean absolute error than unstructured latent dynamics and ConvLSTM baselines with comparable parameter counts, both on held-out trajectories and on later windows of trajectories used during training. SLD also reduces errors in regions with large reference gradients and across most or all of the evaluated later time intervals. A representative Gray--Scott comparison shows lower spectral error over most intermediate and high wavenumbers. Removing the fixed spatial response increased reconstruction error, supporting its contribution to the SLD architecture.
We introduce the roots of supercritical Leung--Yau--Zaslow pairs and prove pointwise strong positivity of the associated Schur forms. The resulting LYZ cones contain the pliant cone on smooth projective threefolds, with strict inclusion on a Fujino--Sato threefold. We also establish Hard Lefschetz and Hodge--Riemann relations for the associated Chern classes on compact Kähler manifolds in bidegrees $1\le p+q\le\dim X$ with $\min\{p,q\}\le1$.
This paper studies time-dependent semi-similar Prandtl flows with outer flow $U(x)=x^m$, where $m\geq1$. The normal coordinate is scaled by the local boundary-layer thickness, and time by the associated diffusion scale. Every time-monotone solution in the Oleinik class extends globally in time. The Falkner--Skan boundary layer globally attracts every such unsteady semi-similar profile. The convergence rate is exponential for $m=1$ and algebraic for $m>1$. We also prove nonlinear stability of each admissible time-dependent semi-similar trajectory under small compatible perturbations that break the semi-similar ansatz. Together, these results provide a dynamical justification of the time-dependent semi-similar class within the full Prandtl theory. For semi-similar flows, the Crocco transformation reduces the problem to a one-dimensional degenerate parabolic equation. Combined with weighted barriers, comparison principles propagate time monotonicity and lead to global continuation. Zero-order convergence and a one-sided gradient estimate then produce uniform strict concavity after a transient period; this concavity supplies the damping needed for derivative decay. The same damping mechanism extends to the full ultraparabolic perturbation equation, allowing us to close the nonlinear stability estimates.
We give a convergent explicit representation of the Ricci-flat Kähler metrics supplied by the Calabi--Yau theorem in a one-parameter family of classes on a fixed Kummer K3 surface. An explicit radial background has normalized Monge--Ampère residual \(O(a^4)\). Its scalar Green operator \(G_a\) is represented by periodic Ewald kernels, separated radial kernels, finite-dimensional Schur complements, and convergent Neumann series. The coefficients are defined by \(U_{a,1}=-G_af_a\) and \(U_{a,n}=G_a\sum_{j=1}^{n-1}Q_a(U_{a,j},U_{a,n-j})\), where \(Q_a\) is the polarized quadratic Monge--Ampère term. A Green estimate of order \(a^{-1}\) gives a first correction of order \(a^3\) and a convergence parameter of order \(a^2\). A Catalan majorant proves absolute convergence on the entire smooth surface for every sufficiently small fixed \(a\). The sum defines a positive form solving the volume equation; comparison identifies the sum with the smooth normalized Calabi--Yau potential. We also record sufficient analytic conditions for the same recursion on collapsing elliptic K3 surfaces.
We give an explicit convergent representation of the Ricci-flat Kähler metric in a one-parameter family of classes on a fixed special Schoen threefold. The threefold resolves a shifted square-torus quotient with sixteen disjoint elliptic singular curves. Radial Eguchi--Hanson product metrics and fixed flat collars define a positive background with an exact $a^4$ volume defect. Spherical harmonics, tangential Fourier modes, Frobenius recurrences, radial ordered integrals, and the flat quotient heat kernel specify the inverse of the background scalar linearized operator. A boundary transmission series has contraction factor $1/15$. Independent weighted zeroth-order and six-dimensional Schauder estimates give the inverse bound needed for the quadratic and cubic Monge--Ampère terms. The resulting source-degree recursion converges for $0<a\le2^{-2260}$, with explicit coefficient, potential-tail, and relative metric-tail bounds. The representation uses specified kernels, prescribed integrals, and convergent series; its parameter interval is a sufficient quantitative bound on this fixed model.
We study GLMY path homology of circulant digraphs through step-word complexes and infinite cyclic lifts. We first prove chain retractions for interval connection sets. For a fixed finite positive connection set $S$ containing $1$, the infinite lift yields a split exact sequence and all-degree stability over every field. We then encode the short additive relations of $S$ in a lattice lift and obtain exterior homology in arbitrarily high degrees for collision-free step sets. Under an exact Freiman admissibility condition, we classify all connection sets with three positive steps. For arbitrarily many steps, we compute rectangular additive configurations and arithmetic progressions over every field, and doubling strings in characteristic zero, including their boundary ranks.
We develop an equivariant K-theoretic vertex formalism for a colored, abelianized higher-rank sector on smooth projective toric threefolds $X$. Nonzero first Chern classes are encoded as magnetic fluxes and absorbed into line-bundle twists, after which the virtual character is redistributed into finite perturbative, vertex, and edge contributions. We compute the resulting (unreduced and trace-free) virtual dimensions and analyze the interaction terms relevant to a possible reduction to rank one. We emphasize that, on a compact toric threefold, colored partitions do not classify all torus-fixed stable higher-rank sheaves: the resulting series is therefore a proposed abelianized/framed vertex series unless an additional comparison theorem with the full stable-sheaf moduli problem is supplied. When the threefold obstruction theory is not available in the desired form, we pass to the local Calabi--Yau fourfold $Y=\operatorname{Tot}(K_X)$ and formulate the corresponding colored DT4 vertex. A $K_X$-twisted tautological insertion removes fixed points with nontrivial fiber layers, reducing solid partitions to plane partitions at the level of the local vertex. In rank one we give a scheme-theoretic explanation: the Hilbert functor of subschemes of $X$ is canonically isomorphic to the constrained Hilbert functor of subschemes of $Y$ lying scheme-theoretically in the fixed zero section. Under the hypotheses of the DT4 K-theoretic Lefschetz/virtual-pullback formalism, the associated support insertion reduces the rank-one fourfold theory to the threefold theory. We keep this global rank-one statement logically distinct from the higher-rank colored-vertex reduction.
In this article, we study the Bridgeland stability of the cotangent bundle on K3 surfaces and surfaces with ample canonical class. On a surface $X$ with ample canonical class $K$, we find values $t_0>0$ such that the cotangent bundle is semistable with respect to the Bridgeland stability $σ_{tK}$ for all $t > t_0$. We also study special cases such as the fake projective planes and the fake quadrics.
We propose a unified framework for bijective parameterizations of $n$-dimensional manifolds that jointly control angular and volumetric distortion through a weighted combination of conformal and volume-preserving energies. A logarithmic barrier based on signed simplex Jacobians prevents element inversion and degeneration during energy minimization. On oriented simplicial manifolds, the gradients of all three discrete energies admit a unified cotangent Laplacian-type representation that preserves the sparsity of the mesh connectivity. Based on these formulations, we develop a three-stage algorithm that constructs an initial parameterization, repairs foldings to restore strict feasibility, and subsequently minimizes the weighted distortion energy while maintaining local injectivity. The framework is formulated in a general setting, with particular attention to parameterizations onto the unit sphere $\mathbb S^n$ and the unit ball $\mathbb B^n$. We derive sufficient conditions for discrete spherical maps to have and retain degree one, and apply classical topological criteria to establish global bijectivity. Numerical experiments demonstrate that the proposed framework achieves simultaneous control of angular and volumetric distortion, while maintaining bijectivity and eliminating surface and volumetric folds. In a brain-tumor label-transfer study, the method improves tumor-label reconstruction compared with a non-bijective ablation.
We construct a smooth Riemannian metric with strictly positive sectional curvature on every smooth homotopy seven-sphere. For each smooth structure, we construct positively curved metrics on two seven-dimensional disks whose induced boundary metrics agree under the prescribed attaching map. Under this identification, a gluing theorem then yields the required smooth metric on the closed manifold. The construction applies to all 28 oriented diffeomorphism classes.
We show that genus-0, $n$ Neveu-Schwarz marked, super Gromov-Witten invariants of a convex, toric variety $X$ can be defined and computed using tropical geometry. When $X$ is a point, the tropical, super Gromov-Witten invariants of $X$ are descendant invariants on the moduli space of tropical curves. When $X$ is a general convex, toric variety, we define a procedure that computes the tropical, inverse Euler class of the SUSY normal bundle $\overline{N}_{n, β} \rightarrow \overline{\mathcal{M}}_{0,n}(X, β)$, under the assumption that $\overline{N}_{n, β}$ is in some sense locally tropicalizable. We define the tropical, genus-0, $n$ Neveu-Schwarz marked, super Gromov-Witten invariants of $X$, and show that the definition recovers the tropical, super Gromov-Witten invariants of a point. We compute a tropical, super Gromov-Witten invariant of $\mathbb{P}^1$.
Classical continuous-time non-linear filtering fails in generic non-linear state spaces due to an infinite Lie algebraic derivative explosion ($\dim(\mathcal{E}) = \infty$), leading to filter divergence. To resolve this four-decade crisis, we introduce the SMG-Yau-Yau filter and lossless data assimilation framework built on Statistical Fiber Theory over an Orlicz manifold $\mathcal{M}$. By equipping $\mathcal{M}$ with a Riemannian submersion and an Ehresmann connection, the unconstrained Duncan-Mortensen-Zakai score velocity field is orthogonally decomposed into Statistically Verifiable Directions ($\text{SVD}χ_f$) and Structural Internal Directions ($\text{SID}_f$). System non-linearities and unclosed Lie commutators are orthogonally quarantined in $\text{SID}_f$, protecting macroscopic base parameters from spatial derivative pollution while preserving total score variance energy. We unify the asymptotics through a Dual-Axis Collapse Mechanism: proving our model identically recovers classical Yau-Yau dynamics when $\dim(\mathcal{E}) < \infty$, while large-sample limits ($N \to \infty$) induce a thermodynamic quench that flattens infinite-dimensional geometry under $\dim(\mathcal{E}) = \infty$. Finally, we formulate the Active Acausal Tension (AAT) functional to monitor accumulated model misspecification online. Exceeding a topological capacity threshold triggers Gauge Symmetry Breaking (GSB), which dynamically expands base coordinates ($d \to d+1$) to ensure non-asymptotic stability and convergence to the exact state density.
We study the supercritical LYZ equation on compact Kähler manifolds at the boundary of stability. Under numerical semistability, we prove a quantitative positivity estimate on the modified nef cone. It implies that the destabilizing prime divisors form a finite exceptional family. Assuming the existence of a smooth semisubsolution, we prove that the associated intersection form is negative definite on the span of classes of these divisors. Using this rigidity, we then construct a logarithmic singular subsolution along their union and obtain a bounded Bedford--Taylor solution which is smooth on its complement.
We define a signed GLMY chain complex over $\mathbb{R}$ for signed digraphs using sheet-labelled regular paths. The complex is naturally isomorphic to the deck anti-invariant subcomplex of the ordinary GLMY complex on the signed double cover. The double-cover realization yields switching invariance and recovers ordinary GLMY homology for switching-balanced signings. Bidirected completion gives an orientation-independent homology theory for signed graphs. For a signed graph, the zero-dimensional homology identifies with the kernel of the signed Laplacian and has dimension equal to the number of balanced connected components. Signed GLMY homology is functorial under signed weak morphisms, which combine vertex maps with switching functions and allow compatible arrow contractions. For signed digraphs, the all-positive reduction retains the orientation sensitivity of ordinary GLMY homology, while explicit computations show additional sensitivity to the arrow signs. For a fixed digraph with five vertices and nine arrows, we classify all 512 arrow signings and obtain exactly four signed Betti vectors. Precisely 16 signings have nonzero second signed GLMY homology.
We prove a purely quasi-local positivity theorem for the Wang--Yau mass for a class of initial data whose Jang deformation, after a boundary-preserving conformal reduction to zero scalar curvature, lies in a sufficiently small transverse--traceless (TT) perturbative neighborhood of a strictly convex Euclidean fill-in.We explicitly construct a nontrivial class of physical initial data whose admissible Jang reductions realize this TT-generated sector. The argument reduces the Wang--Yau energy to the Brown--York mass of the resulting scalar-flat compact metric, together with nonnegative bulk terms determined by the Jang deformation, and establishes strict positivity by computing the second variation of the Brown--York functional at the Euclidean metric in transverse--traceless directions. The proof is entirely confined to the compact fill-in and uses neither an asymptotically flat extension nor the positive mass theorem. This gives a partial answer to a question of R. Schoen concerning a genuinely quasi-local proof of positivity for quasi-local mass.
In an earlier work, Yau and Zheng proposed a Hermitian analogue of a weak form of the Frankel conjecture, which states that any compact Hermitian manifold with positive Strominger-Bismut sectional curvature must be biholomorphic to the complex projective space. In this article, we confirm the conjecture in complex dimension two. This is achieved by a vanishing theorem for anti-self-dual harmonic 2-forms on such surfaces, utilizing a curvature decomposition formula by Ferreira and a refined Kato inequality.
We develop a homotopy theory of directed graphs based on cubical homotopy groups, also known as $A$-groups or reduced GLMY homotopy groups. Localizing the category of directed graphs at morphisms that induce isomorphisms on these groups yields an $\infty$-category, denoted by ${\sf DGra}_\infty$. We prove that ${\sf DGra}_\infty$ is equivalent to the $\infty$-category of spaces, establishing a directed version of the discrete homotopy hypothesis of Carranza and Kapulkin.