Harmonic Two-Spheres in Compact Metric Spaces

Definitions and mathematical statement

Let $(X,d)$ be a compact metric space. To make the homotopy hypothesis explicit, choose a basepoint $x_0\in X$ and suppose that \[ \pi_k(X,x_0)\neq\{0\} \qquad\text{for some integer }k\geq 2. \] Here $\pi_k(X,x_0)$ denotes the group of based homotopy classes of continuous maps $(S^k,s_0)\to(X,x_0)$. No connectedness or local regularity assumption on $X$ is supplied.

Question (source Question 2). What additional conditions on $(X,d)$ guarantee that $X$ admits a nontrivial harmonic two-sphere, that is, a map \[ u:S^2\longrightarrow X \] that is harmonic and nontrivial in the intended metric-space sense?

Unresolved qualifications. The source does not specify an energy functional, admissible map class, or definition of harmonicity for arbitrary metric targets. These must be specified before the question becomes a fully defined existence problem. It also does not define "nontrivial": nonconstancy and nonzero homotopy class are distinct requirements and must not be identified. In particular, the hypothesis concerns some $k\geq2$, not necessarily $k=2$, and the source does not explicitly require the sought sphere to represent a nonzero element of $\pi_2(X,x_0)$. For disconnected $X$, whether the sphere must lie in the component of $x_0$ is also unspecified. The request is to identify sufficient additional conditions; no particular condition or necessary-and-sufficient characterization is supplied.

The motivating passage attributes to Sacks--Uhlenbeck the existence of a nontrivial harmonic two-sphere in every closed Riemannian manifold having a nontrivial $k$th homotopy group for some $k\geq2$, citing its reference [6, Theorem 5.7]. This is retained as supplied attribution and context, not as an independently verified theorem statement.

Short English statement

Determine additional hypotheses under which a compact metric space with $\pi_k(X,x_0)\neq\{0\}$ for some $k\geq2$ admits a nontrivial harmonic map from $S^2$. The metric-space meaning of harmonicity and the meaning of "nontrivial" remain unspecified and require mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30006420$, title "Harmonic Two-Spheres in Compact Metric Spaces," source code catalogue entry; supplied complete source record in the Oberwolfach Reports Open Problems collection. The extracted passage identifies the question as Question 2. No individual proposer is supplied.
• Supplied report citation: Differentialgeometrie im Großen (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/30, pp. 1591--1646. https://doi.org/10.4171/owr/2025/30. The record separately gives proposed year 2025; the differing year indications are retained without resolution.
• Supplied literature source: "Oberwolfach Report 30." https://ems.press/content/serial-article-files/52249.
• Supplied literature source: "Energy minimizing harmonic spheres." https://arxiv.org/abs/2503.08553. Authors and a precise supporting result are not supplied.
• The motivating passage attributes the closed Riemannian manifold result to Sacks--Uhlenbeck and cites [6, Theorem 5.7]. The bibliographic entry for [6] is not included in the supplied record.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the question open, attributes the smooth closed-manifold result to Sacks--Uhlenbeck, describes metric-space existence results as requiring additional hypotheses, and reports that no general compact-metric-space criterion resolving the extracted question is established. The background expressly requests independent verification of time-sensitive status and cited claims.