Fillability of Bartnik Boundary Data Under Scalar Curvature Bounds

Definitions and mathematical statement

Let $(Y,h)$ be a closed Riemannian manifold of dimension $n-1$, where closed means compact without boundary, and let $\varphi\in C^\infty(Y,{\mathbb{R}})$ be prescribed boundary mean-curvature data. Let $\sigma$ denote the prescribed scalar-curvature lower bound. The source does not explicitly specify the admissible dimensions or the domain of $\sigma$; interpreting $\sigma$ as a real constant is a qualification requiring confirmation.

For a smooth Riemannian $n$-manifold $(X,g)$ with boundary, let $g^\top$ denote the restriction of $g$ to tangent vectors along $\partial X$, let $R_g$ denote its scalar curvature, and let $H_g$ denote its boundary mean curvature. The source does not specify the normal direction, sign convention, or whether mean curvature is the trace or the normalized trace of the second fundamental form. Thus the following existence problem depends on fixing that convention.

Problem. Find necessary and sufficient conditions on $(Y,h,\varphi,\sigma)$ for the existence of a complete smooth Riemannian $n$-manifold $(X,g)$ and a diffeomorphism $\iota:Y\longrightarrow\partial X$ such that \[ R_g(x)\geq\sigma\quad\text{for every }x\in X, \qquad \iota^*(g^\top)=h, \qquad H_g\circ\iota=\varphi. \] The pair $(g^\top,H_g)$ is the Bartnik boundary data. The identification requires $Y$ to represent the entire boundary. No compactness assumption on $X$ is supplied. For a manifold with boundary, completeness can be interpreted as completeness of the intrinsic Riemannian length metric on $X$, including its boundary; the source does not explicitly confirm this interpretation. The dimension range, meaning of $\sigma$, completeness interpretation, and mean-curvature convention remain unresolved qualifications of this draft.

Short English statement

Characterize which prescribed boundary metrics and mean curvatures occur on the entire boundary of a complete Riemannian manifold with scalar curvature bounded below by $\sigma$. The source leaves the dimension range, the domain of $\sigma$, the completeness interpretation, and the mean-curvature convention unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $30004719$, source problem code catalogue entry, title: Fillability of Bartnik Boundary Data Under Scalar Curvature Bounds. The supplied record labels the statement corrected_verified and says that the Bartnik boundary data and scalar-curvature notation were normalized without changing the existence problem. These are supplied registry assertions, not independent verification.
• Supplied report attribution: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2022), Oberwolfach Reports, pp. 1579--1649, DOI 10.4171/owr/2021/30. https://doi.org/10.4171/owr/2021/30. The collection describes its problems as extracted from workshop discussions. The supplied proposed year is 2021; no individual proposer is identified.
• Supplied literature reference: Shi, Wang, Wei and Zhu, "fill-ins with scalar bounds." https://arxiv.org/abs/1907.12173.
• Supplied literature reference: "Mathematische Annalen 2021." https://doi.org/10.1007/s00208-020-02087-1.
• Supplied literature reference: "Fill-ins with scalar curvature lower bounds 2024." https://doi.org/10.1007/s10455-024-09956-x.
• Supplied literature assessment checked on 2026-08-22: status partially_solved. The supplied assessment reports existence and nonexistence criteria in several regimes, particularly large mean curvature and low dimensions, and reports that no general characterization of prescribed Bartnik data is known. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.