[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"P379nbOXxW":3},"# OpenGALib\n\n[![Prove2Me](https://img.shields.io/badge/Prove2Me-Poincar%C3%A9_Conjecture-2b7489)](https://prove2.me/missions/Formalization%20of%20the%20Poincar%C3%A9%20Conjecture)\n\nGeometric analysis and geometric measure theory, formalized in Lean 4 on top\nof Mathlib. OpenGA develops reusable definitions and proofs, and reviews and\nadapts results from other formalization projects with their attribution and\nmathematical hypotheses preserved.\n\n## Library contents\n\n| Area | Available foundations |\n| --- | --- |\n| [Geometric measure theory](OpenGALib/GeometricMeasureTheory) | Compact unoriented Grassmannians; Euclidean varifolds as Radon measures; weight measures, mass and support; weak-* convergence; weighted and atomic constructions; varifolds of finite-area `C¹` parametrizations. |\n| [Area and energy](OpenGALib/Analysis/AreaEnergy.lean) | The area–energy inequality and its equality criterion; orthonormal-frame invariance; bounds on parametrized varifold mass by energy. |\n| [Riemannian surfaces](OpenGALib/Riemannian/Surface) | Induced metrics, global area and coordinate compatibility; tangent and normal projections; geometric Dirichlet energy; smooth sweepouts and their homotopy width. |\n| [Comparison geometry](OpenGALib/ComparisonGeometry) | Metric balls, local Ricci bounds, volume measures and surgery volume budgets; [Bishop–Gromov comparison](OpenGALib/Interoperability/BishopGromov.lean) for complete manifolds with nonpositive model curvature, reused from DifferentialGeometry. |\n| [Ricci flow and extinction](OpenGALib/Interoperability/RicciFlow) | Scalar lower bounds; local and closed-surface area variation; [width comparison](OpenGALib/Analysis/WidthComparison.lean) and [finite-time extinction](OpenGALib/Analysis/WidthExtinction.lean) under explicit analytic hypotheses. |\n| [Geometry](OpenGALib/Riemannian) and [topology](OpenGALib/Topology) | Riemannian metrics, geodesic and exponential-map foundations; covering spaces, connected sums and the topological extinction endgame under explicit reconstruction hypotheses. |\n\nThe library is under active development. The smooth-sweepout width is a\nprecursor to the Colding–Minicozzi Sobolev width; identifying the two and\nconstructing the required uniform area-evolution data remain open work.\nThese foundations do not yet constitute a proof of the Poincaré conjecture.\n\n## Collaborative formalization\n\nOpen tasks are coordinated in the [Prove2Me Poincaré mission](https://prove2.me/missions/Formalization%20of%20the%20Poincar%C3%A9%20Conjecture).\nThe [PoincareConjecture workspace](PoincareConjecture) prepares submissions,\nmirrors platform definitions and proofs, and records source reviews and\nverification receipts. Its open problems and proof sketches are separate\nfrom the reusable library in `OpenGALib/`.\n\nWe thank the authors and contributors of\n[DifferentialGeometry](https://github.com/qinz1yang/differential-geometry),\n[Poincare-Conjecture](https://github.com/frenzymath/Poincare-Conjecture),\nMathlib and Lean, and the Prove2Me team. Source-specific attribution and\ndependency revisions are recorded alongside the reused results.\n\n## Use\n\nAdd to your `lakefile.lean`:\n\n```lean\nrequire OpenGALib from git \"https://github.com/MathNetwork/OpenGA.git\" @ \"main\"\n```\n\nThen:\n\n```sh\nlake exe cache get\nlake build\n```\n\nPins Lean v4.33.1 and Mathlib at\n`0df444a360eaa60ab8c11dca51a86af692955474`, matching Prove2Me's\nLean v4.33.1 verification environment. Dependency revisions are recorded\nin `lake-manifest.json`.\n\nOpenGA also pins DifferentialGeometry v0.1.2. Public interfaces to the\nreviewed upstream results live in `OpenGALib/Interoperability`.\n\nImport the whole library or a focused entry point:\n\n```lean\nimport OpenGALib\n-- Or:\nimport OpenGALib.GeometricMeasureTheory\nimport OpenGALib.Riemannian.Surface\n```\n\n## Contributing and verification\n\nSee [CONTRIBUTING.md](CONTRIBUTING.md) and the [review guide](.github/REVIEW.md).\nThe `main` branch requires a successful build and no `sorry` in `OpenGALib/`.\nCI also checks source-review receipts and the axioms of curated declarations.\n\n```sh\nlake build\npython3 -m unittest discover -s PoincareConjecture -p 'test_*.py'\npython3 PoincareConjecture/check_curation.py\n```\n\n## License\n\nApache 2.0 — see LICENSE.\n",1791148328034]