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Volume growth with nonnegative Ricci and positive scalar curvature

Does every complete connected n-dimensional Riemannian manifold with Ric≥0 and scalar curvature at least one satisfy Vol B_R(p)≤C(n)R^(n−2) in the stated large-radius regime?

Why it matters and what counts as progress

Why it matters: The codimension-two bound links a scalar-curvature lower bound to global volume growth under nonnegative Ricci curvature.

What counts as progress: A uniform dimension-dependent growth estimate, an alternative proof under the same assumptions, or a manifold violating it.

How the problem developed

  1. 1986 · Conjecture [1]

    Mikhael Gromov Posed the codimension-two volume-growth question: under nonnegative Ricci curvature and scalar curvature at least one, should balls have volume bounded by C_n R^(n−2)?

  2. 13 Aug 2026 · Reported claim [2]

    Jian Ge Reported an affirmative proof of Gromov's codimension-two volume-growth conjecture using heat-kernel geometry.

  3. 14 Aug 2026 · Reported claim [3]

    Bochao Kong and Xingyu Zhu Reported sharp codimension-one and codimension-two volume-growth estimates for complete manifolds with nonnegative Ricci curvature.

  4. 14 Aug 2026 · Reported claim [4]

    Gioacchino Antonelli Reported a family of volume-growth bounds from positive intermediate curvature, with the codimension-two conjecture as a special case.

AI contribution

Sources

  1. Gromov, Large Riemannian manifolds
  2. Ge, Heat kernel geometry and Gromov's volume growth conjecture
  3. Kong and Zhu, Positive Scalar Curvature and Volume Growth
  4. Antonelli, Universal Volume Growth Bounds from Positive Intermediate Curvature
  5. Universal Volume Growth Bounds from Positive Intermediate Curvature
  6. Ge’s independent volume-growth proof
  7. Kong–Zhu’s separate heat-transport proof
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