Uniqueness and flatness of tangent cones in limits under Theorem 1.2 conditions

Determine whether, for any sequence of compact 4-dimensional Kähler manifolds satisfying Ricci curvature lower bound Ric ≥ −3, volume lower bound Vol(M) ≥ v > 0, diameter upper bound diam(M) ≤ D, and scalar curvature L2 bound ∫_M |R|^2 ≤ Λ, every tangent cone of the associated Gromov–Hausdorff limit space is unique and flat.

Background

The paper establishes a finite diffeomorphism theorem for manifolds with Ricci curvature bounded below and bounded L^{n/2}-energy, and proves a corresponding result for 4-dimensional Kähler manifolds under an L2 scalar curvature bound (Theorem 1.2). In the non-Kähler setting, the analysis leverages Reifenberg radius and cone structure of limits; in the Kähler case the authors introduce a modified monotone quantity to control the geometry.

While the main results provide finiteness of diffeomorphism types, the authors explicitly note that under the assumptions of Theorem 1.2 they cannot establish that tangent cones of Gromov–Hausdorff limits are unique and flat. Instead, they prove weaker properties: each tangent cone is a cone with smooth cross section, with uniform control on topology and Reifenberg radius. The unresolved question asks for the stronger conclusion of uniqueness and flatness of tangent cones in this Kähler/L2-scalar-curvature regime.