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math.DG — Differential Geometry

A contractible open four-manifold with a complete metric of uniformly positive scalar curvature

Contributed by Jiangcheng You, Heng Zhang

We construct a smooth contractible open four-manifold that is not homeomorphic to $\R^4$ and admits a complete Riemannian metric with scalar curvature at least one. This gives a negative answer to a question raised by Chang--Weinberger--Yu and Chodosh--M\'aximo--Mukherjee.

A mix of human-written and AI-generated textHuman understanding: all partsBaumslag–Solitar groupsGromov–Lawson surgerycontractible four-manifoldsgeometric topologyopen four-manifoldspositive scalar curvaturetopology at infinity

Oct 01, 2026PDF ↗View work →

hexagon:2609.00144v1

math.PR — Probability

On the Asymptotic $W_1$ Cost of Two-Dimensional Semi-Discrete Matching

Contributed by Heng Ma

Let $X_1,X_2,\ldots$ be independent uniform points in the unit square and let $\lambda$ denote Lebesgue probability measure. We prove that the normalized expected $W_1$ distance between the empirical measure and Lebesgue measure converges to a positive constant: \[ \lim_{N\to\infty} \sqrt{\frac{N}{\log N}}\, \mathbb E W_1\!\left(\frac1N\sum_{i=1}^N\delta_{X_i},\lambda\right) =c_\star. \] The constant is characterized by the convex viscosity equation \[ \partial_tU=\frac1{4\pi}\sqrt{\det D^2U}, \qquad U(0,q)=|q|. \] Its solution $U$ is unique under certain growth and regularity conditions, and the limiting constant $c_\star=U(1,0)/\sqrt2$ .

Primarily AI-generated textHuman understanding: no parts

Sep 30, 2026PDF ↗View work →

math.AC — Commutative Algebra
math.AG — Algebraic Geometry
math.AP — Analysis of PDEs
math.AT — Algebraic Topology
math.CA — Classical Analysis and ODEs
math.CO — Combinatorics
math.CT — Category Theory
math.CV — Complex Variables
math.DG — Differential Geometry
math.DS — Dynamical Systems
math.FA — Functional Analysis
math.GM — General Mathematics
math.GN — General Topology
math.GR — Group Theory
math.GT — Geometric Topology
math.HO — History and Overview
math.IT — Information Theory
math.KT — K-Theory and Homology
math.LO — Logic
math.MG — Metric Geometry
math.NA — Numerical Analysis
math.NT — Number Theory
math.OA — Operator Algebras
math.OC — Optimization and Control
math.PR — Probability
math.QA — Quantum Algebra
math.RA — Rings and Algebras
math.RT — Representation Theory
math.SG — Symplectic Geometry
math.SP — Spectral Theory
math.ST — Statistics Theory
cs.CC — Computational Complexity
cs.CG — Computational Geometry
cs.DM — Discrete Mathematics
cs.DS — Data Structures and Algorithms
cs.FL — Formal Languages and Automata Theory
cs.IT — Information Theory
cs.LO — Logic in Computer Science
cs.MS — Mathematical Software
cs.NA — Numerical Analysis
cs.SC — Symbolic Computation
math-ph — Mathematical Physics
quant-ph — Quantum Physics
stat.TH — Statistics Theory

Matching works

A contractible open four-manifold with a complete metric of uniformly positive scalar curvature

On the Asymptotic W1W_1W1​ Cost of Two-Dimensional Semi-Discrete Matching

On the Asymptotic $W_1$ Cost of Two-Dimensional Semi-Discrete Matching