$\varepsilon$-Regularity Estimate for Donaldson's Equation

Definitions and mathematical statement

Let $M$ denote the underlying manifold in the intended setting of Donaldson's almost-complex Calabi--Yau equation, let $g$ denote the background metric, and let $\widetilde g$ denote a metric associated with a solution. Write \[ B_g(p,r)=\{x\in M:d_g(p,x)<r\},\qquad f=\operatorname{tr}_g(\widetilde g), \] where $d_g$ is the distance induced by $g$, and let $dV_g$ be its volume measure. For real Riemannian metrics the relative trace is $g^{ij}\widetilde g_{ij}$; the supplied record does not specify whether this convention or a Hermitian trace convention is intended.

The proposed $\varepsilon$-regularity assertion asks whether there exist uniform constants $\varepsilon>0$ and $C>0$ such that, for every solution metric $\widetilde g$, every center $p$, and every radius $r>0$ admitted by the intended setting, \[ r^{-2}\int_{B_g(p,r)} f\,dV_g\leq\varepsilon \quad\Longrightarrow\quad \sup_{B_g(p,r/2)} f \leq C r^{-4}\int_{B_g(p,r)} f\,dV_g. \] The same constants must apply throughout the intended range of solutions, centers, and radii. However, the supplied record does not state the equation, the manifold dimension, the geometric hypotheses defining that range, restrictions on radii or boundary behavior, or the background data on which the constants may depend. Consequently this is the supplied estimate with its unresolved setting made explicit, rather than a fully specified analytic conjecture.

Short English statement

The proposed estimate says that sufficiently small scaled integral of the relative metric trace on a ball controls its maximum on the concentric half-radius ball. The geometric setting, admissible radii, trace convention, and permitted dependence of the uniform constants require clarification.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30001491$, exact source problem code catalogue entry, title "$\varepsilon$-Regularity Estimate for Donaldson's Equation," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2010 and identifies no individual proposer.
• Supplied report citation: Geometrie (2010), Oberwolfach Reports, pp. 1535--1572, DOI 10.4171/owr/2010/26. This is also the supplied literature source labeled "original OWR Report 26/2010": https://doi.org/10.4171/owr/2010/26.
• Supplied literature source labeled "Tosatti--Weinkove--Yau partial estimates": https://doi.org/10.1112/plms/pdn008. No further bibliographic details or precise results from this reference are supplied.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem open and reports that the proposed estimate remains part of the unresolved general analytic problem, with no general proof of the displayed estimate identified. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a check against the official Oberwolfach report. These are supplied verification claims; no source was independently consulted for this draft.