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Path selection and an elliptic inequality on degenerating Ricci-flat trees

OpenAI · September 24, 2026

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Abstract

We prove a sequential elliptic inequality for the renormalized Einstein–Hilbert functional on a fixed finite tree of four-dimensional Ricci-flat spaces. As the joining lengths diverge and the scale-neutral weighted Ricci error M tends to zero, the functional satisfies $E=o(M)$. The root end has decay exponent greater than one, and every joined quotient group is nontrivial. We also prove the multivariable path-selection theorem for asymptotic expansions used to obtain this estimate.

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Possibly relevant literature (found by the agent, not checked)

  • Tristan Ozuch, Noncollapsed degeneration of Einstein 4-manifolds I (arXiv:1909.12957; Geom. Topol. 26 (2022), 1483–1528). Shows that Einstein 4-manifolds Gromov–Hausdorff close to an Einstein orbifold are close to a gluing of Ricci-flat ALE trees in weighted norms; related to the paper's comparison of degenerating metrics with its family $h_*(L,z)$, while the paper cites only Part II.

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Cite

@misc{OAI:Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026,
  author = {{OpenAI}},
  title = {{Path selection and an elliptic inequality on degenerating Ricci-flat trees}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf}{OAI:Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026}},
  year = {2026}
}

Family 351: Scalar curvature and finite-time Ricci-flow singularities

Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.

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