Area-Nonincreasing Smoothing of Lipschitz Maps

Definitions and mathematical statement

The supplied reconstructed statement concerns a Lipschitz map $f\colon M\to S^n$, a metric $g$ on $M$, and the sphere metric $g_{S^n}$. To interpret its differential inequality, regard $(M,g)$ as a smooth Riemannian manifold and $g_{S^n}$ as the intended Riemannian metric on the sphere. These are interpretive prerequisites: the record does not specify the dimension, regularity, compactness, completeness, or boundary conditions of $M$, the range of $n$, or the normalization of $g_{S^n}$.

For a differentiable map $h\colon M\to S^n$, its differential induces \[ \Lambda^2{\,\mathrm{d}} h_x\colon \Lambda^2T_xM\longrightarrow\Lambda^2T_{h(x)}S^n, \qquad u\wedge v\longmapsto{\,\mathrm{d}} h_x(u)\wedge{\,\mathrm{d}} h_x(v). \] The induced metric on an exterior square is determined by \[ (\Lambda^2g)(u\wedge v,a\wedge b) =g(u,a)g(v,b)-g(u,b)g(v,a). \] Under this interpretation, the notation $\Lambda^2h^*g_{S^n}\leq\Lambda^2g$ means \[ \|\Lambda^2{\,\mathrm{d}} h_x(\xi)\|_{\Lambda^2g_{S^n}} \leq \|\xi\|_{\Lambda^2g} \qquad\text{for every }\xi\in\Lambda^2T_xM. \] For Lipschitz $f$, the differential and this inequality are interpreted almost everywhere, with respect to Riemannian volume.

Question as reconstructed in the supplied record. Suppose that $f\colon M\to S^n$ is Lipschitz and \[ \|\Lambda^2{\,\mathrm{d}} f_x(\xi)\|_{\Lambda^2g_{S^n}} \leq\|\xi\|_{\Lambda^2g} \quad\text{for almost every }x\in M \text{ and every }\xi\in\Lambda^2T_xM. \] Can $f$ be smoothed to a smooth map $F\colon M\to S^n$ satisfying \[ \|\Lambda^2{\,\mathrm{d}} F_x(\xi)\|_{\Lambda^2g_{S^n}} \leq\|\xi\|_{\Lambda^2g} \quad\text{for every }x\in M \text{ and every }\xi\in\Lambda^2T_xM? \] Unresolved formulation. The record does not define the required relationship between $F$ and $f$: it specifies neither an approximation topology and accuracy nor a homotopy or relative condition. Consequently, the displayed target inequality does not by itself define a precise smoothing assertion. No additional quantifier over approximation tolerances or additional domain hypothesis is imposed here. The reconstructed statement is supplied with a verification claim, but the supplied original excerpt contains no mathematical question, and the dated literature assessment describes the question as missing.

Short English statement

The reconstructed question asks whether a Lipschitz map to a sphere that does not increase two-dimensional area almost everywhere can be smoothed while preserving the exact area inequality everywhere. The domain hypotheses and the meaning of smoothing are unspecified. The supplied reconstruction and the dated assessment of the missing original question require reconciliation; this is a statement-only draft for human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005858$, title Area-Nonincreasing Smoothing of Lipschitz Maps, source code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies the proposed year $2024$ and names no proposer.
• Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2024), Oberwolfach Reports, pp. 483--566, DOI 10.4171/owr/2024/9: https://doi.org/10.4171/owr/2024/9. This is also the supplied literature-source link labelled "original OWR report." The supplied original excerpt contains only an introduction to Question 1 and a report header with page number $537$; it does not contain the question itself.
• Supplied literature assessment, checked on 2026-08-22: the question was described as missing and its mathematical content as unclear, with status normalized to open. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The record labels its statement corrected_verified and says that it was reconstructed and checked against the official report. Its supplied update timestamp is 2026-09-16T12:06:53Z. These are source-record claims, not verification performed for this draft. The supplied open label is likewise retained as a registry classification rather than an independently established mathematical status.