Extensions of CMC Gluing Constructions

Definitions and mathematical statement

Let $(M^{n+1},g)$ be a Riemannian manifold and let $\Sigma^n\subset M$ be a hypersurface, with dimension and immersion or embedding requirements left unspecified by the supplied record. Where a unit normal $\nu$ is chosen, write $A$ for its second fundamental form and $H=\frac1n\operatorname{tr}_{\Sigma}A$ for its mean curvature. The hypersurface has constant mean curvature (CMC) if $H\equiv H_0$ for some constant $H_0$. The record does not specify $H_0$, orientability, or a global choice of normal.

The construction under discussion glues spherical and Delaunay pieces according to a graph $\Gamma$, described in the supplied statement as balanced. Delaunay pieces refer to pieces of the classical rotationally symmetric CMC model hypersurfaces. The record does not specify their parameters, the ambient space of the starting construction, the graph data, or the balancing equations; these remain part of the missing construction-specific hypotheses.

Questions. For this particular gluing construction, determine:

1. What hypotheses on $\Gamma$ and what geometric or analytic restrictions on $(M,g)$ permit the construction of the intended CMC hypersurfaces, including extensions beyond the original ambient setting?
2. Under what hypotheses can the construction produce hypersurfaces of infinite topology? The intended meaning of infinite topology, and any completeness, properness, or end conditions, are not specified in the record.
3. Are the hypersurfaces produced by the construction nondegenerate in the appropriate deformation problem? Under what additional hypotheses do admissible nearby graph deformations give smooth curves in a smooth moduli space of CMC hypersurfaces?

For a two-sided hypersurface, the normal Jacobi differential expression is, with the convention $\Delta_{\Sigma}=\operatorname{div}_{\Sigma}\nabla_{\Sigma}$, \[ L_{\Sigma}u=\Delta_{\Sigma}u+ \left(|A|^2+\operatorname{Ric}_g(\nu,\nu)\right)u. \] This expression alone does not define the nondegeneracy asked for: one must specify admissible normal variations, function spaces, boundary or asymptotic conditions, any constraints, and whether Jacobi fields induced by ambient symmetries are removed. The record also leaves unspecified whether mean curvature is fixed in the moduli problem and which hypersurfaces are identified. Thus no unconditional implication from nondegeneracy to the asserted moduli-space behavior is formulated here. These are construction-dependent questions, not a claim that every balanced graph or every ambient manifold admits such a gluing.

Short English statement

Determine which graph and ambient-manifold hypotheses allow the supplied CMC gluing construction to extend to broader settings and infinite topology. Determine also the appropriate nondegeneracy and smooth-deformation statements. The starting construction, balancing equations, meaning of infinite topology, and deformation spaces require clarification before a fully quantified formulation is possible.

Sources

• Local UnsolvedMath registry: permanent ID $30002938$, source problem code catalogue entry, title Extensions of CMC Gluing Constructions; supplied complete record from the Oberwolfach Reports Open Problems collection. Project registry locations are data/unsolvedmath/problems.json and catalog/problems.json; their contents were not independently inspected for this draft.
• Supplied report citation: Partial Differential Equations (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/36, pp. 2065--2124. Original report link: https://doi.org/10.4171/owr/2015/36. The record gives proposed year 2015 and no individual proposer. The report title's year 2016 and the supplied 2015 provenance are retained without reconciliation.
• Supplied literature source, labeled "full report text": https://ems.press/content/serial-article-files/46583?nt=1.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the questions open and reports that no definitive general theorem was located for extensions to broader ambient manifolds and infinite topology. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The supplied record labels the statement corrected_verified and reports that the official report passage was checked directly and its question, definitions, and notation restored without strengthening the claim. These are source-record assertions; the report and its mathematical claims were not independently checked in preparing this statement-only draft.