Extension into a Complete Hermitian Ball

Definitions and mathematical statement

Let $n\geq2$, let $\Delta\subset\mathbb C^n$ be a ball containing the origin, and let $(F,g)$ be a complete Hermitian manifold. Completeness means completeness of the distance defined by the underlying Riemannian metric of $g$. The source asks whether every holomorphic map \[ f:\Delta\setminus\{0\}\longrightarrow F \] has a holomorphic extension $\widetilde f:\Delta\to F$ satisfying $\widetilde f|_{\Delta\setminus\{0\}}=f$.

This is the unqualified statement supplied in the record: it assumes completeness but imposes no curvature bound on $F$. The accompanying research summary instead describes an affirmative extension theorem with the additional hypothesis of nonpositive holomorphic sectional curvature, attributed to Siu and Shiffman, and also alludes to compact analytic exceptional sets without specifying their conditions. That stronger target hypothesis is not inserted into the original question. A theorem requiring it does not by itself establish the record's affirmative claim for arbitrary complete Hermitian targets.

Short English statement

Must a holomorphic map from a punctured complex ball of dimension at least two into any complete Hermitian manifold extend across the puncture? The supplied resolution summary adds a curvature assumption absent from the actual question.

Sources

• Shiing-Shen Chern, Open Problems in Differential Geometry (1970), Problem 10, supplied scan PDF page 11, https://people.math.harvard.edu/~hirolee/pdfs/2014-fall-230a-icm1970-chern-differential-geometry.pdf; accessed 2026-07-29, PDF-text-normalized extraction. The provenance identifies a public scan of ICM proceedings.
• Siu and Shiffman, extension-theorem attributions in the supplied research summary; no titles, dates, theorem numbers, or precise compact-analytic-set hypotheses are provided.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $5600010$, source code catalogue entry, Extension into a Complete Hermitian Ball. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.