Product Information
Description
An annual survey of the math.DG arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 132 papers and organised into 37 chapters, it maps the year by the field's research sub-directions.
Book framework (share of the year's output):
- Hyperbolic and Constant Curvature Space Forms (14.9%) — Investigates geometric properties, harmonic maps, and spectral inequalities in hyperbolic and constant curvature space forms.
- Riemannian and Kähler Manifold Rigidity (13.1%) — Focuses on rigidity theorems, geometric inequalities, and stability properties of compact and complete Riemannian and Kähler manifolds.
- Minimal and Free Boundary Surfaces (10.0%) — Studies the existence, regularity, and spectral properties of minimal surfaces and free boundary problems in various ambient geometries.
- Symmetric Spaces and Moduli Geometry (9.9%) — Explores the geometry of symmetric spaces, homogeneous spaces, and their associated moduli spaces and metric structures.
- Ricci Flow and Geometric Solitons (9.0%) — Analyzes the evolution of metrics under Ricci flow, focusing on soliton solutions, curvature estimates, and convergence properties.
- Einstein and Kähler-Einstein Metrics (8.9%) — Examines the existence, uniqueness, and classification of Einstein and Kähler-Einstein metrics on complex and real manifolds.
- Positive Mass and Geometric Rigidity Theorems (8.2%) — Develops and applies positive mass theorems and rigidity results in geometric analysis, particularly in Lorentzian and Riemannian contexts.
- Symplectic, Poisson, and Finsler Geometry (7.8%) — Investigates foundational aspects and applications of symplectic, Poisson, and Finsler geometries, including noncommutative extensions.
- Lie Groups, Algebras, and Invariant Structures (7.7%) — Studies Lie groups, Lie algebras, and their invariant structures, including sub-Riemannian and contact geometries.
- Scalar Curvature and Spectral Geometry (7.3%) — Focuses on scalar curvature constraints, spectral properties of Dirac operators, and curvature-induced phenomena in geometric analysis.
- Mean Curvature Flow and Singularities (3.3%) — Analyzes mean curvature flow, constant mean curvature surfaces, and the formation of singularities in geometric evolution equations.
Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.
