Imaginary-Axis Geodesics in Period-Matrix Loci

Definitions and mathematical statement

For the period-matrix locus $H^+(U)$, is the imaginary axis the image of a Teichmüller geodesic?

The supplied assessment places this question in a special infinite-genus Teichmüller construction. The parameter $U$, locus $H^+(U)$, underlying surface, marking conventions, and period map are not defined in the extraction. In particular, no finite-dimensional Siegel-space model or finite-genus period theorem is substituted.

If $\Phi$ denotes the intended period map from the relevant Teichmüller space, the requested existence would amount to a geodesic $\gamma:I\to\mathcal T$ whose set of period images is exactly the source's imaginary axis: \[ \Phi(\gamma(I))=\{\text{points on the imaginary axis intended in }H^+(U)\}. \] This formula records the source's image requirement, not a definition of the missing axis or map. In the usual metric sense an arclength-parametrized Teichmüller geodesic satisfies $d_{\mathcal T}(\gamma(s),\gamma(t))=|s-t|$. The record does not specify the parameter interval, whether a complete line or a ray is intended, or whether only an unparametrized image is requested.

For matrix or operator period coordinates, "imaginary axis" could mean different subsets, and the source does not provide a one-parameter expression or explain how such an axis lies in the locus. The question asks for an image of a geodesic, not that the period map be an isometry or that every path in a purely imaginary subspace lift geodesically. A precise statement requires recovery of the special construction and its coordinates.

Short English statement

The question asks whether a particular imaginary axis in an infinite-genus period-matrix construction is exactly the period image of a Teichmüller geodesic. The locus, axis, surface, period map, and geodesic range are undefined in the supplied fragment.

Sources

• Local UnsolvedMath registry, permanent ID 30000508, code catalogue entry, "Imaginary-Axis Geodesics in Period-Matrix Loci." Proposed year 2006; proposer unspecified.
• Teichmüller Space (Classical and Quantum) (2007), Oberwolfach Reports, pp. 1537--1614, DOI 10.4171/owr/2006/26; https://doi.org/10.4171/owr/2006/26. The displayed report year 2007 differs from the DOI and proposed year 2006.
• Statement status corrected_verified; the supplied verification says official-report content was checked and notation, context, and question form normalized without changing the source claim.
• Literature triage checked 2026-08-21: open. It describes a special infinite-genus construction, says the fragment is not a complete theorem, and reports no located source settling its general assertion. Created 2026-08-25; updated 2026-09-16. Time-sensitive status and source claims require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.