An arXiv preprint gives sharp upper bounds on the size of the singular set for a class of sphere-valued maps. For a stable stationary harmonic map u:M→S^k from a Riemannian n-manifold, it says the map is smooth outside a relatively closed singular set whose Hausdorff dimension is at most n − k − 1 for 3 ≤ k ≤ 6, and at most n − 7 for k ≥ 7.
For general readers, the central point is that the theorem puts a ceiling on the dimension of the part of the domain where smoothness can fail. The paper describes the estimate as sharp and supplies minimizing 0-homogeneous examples in the critical cases.
The mathematical setting
The work investigates regularity bounds for stable stationary harmonic maps into round spheres. Its main object is a map written as u:M→S^k, where M is a Riemannian n-manifold and S^k is the round sphere serving as the target.
The symbols n and k track the dimensions of the source and target in the theorem. The conclusions apply only to maps satisfying the stated harmonicity, stability, stationarity and dimensional assumptions; they are not blanket statements about every map with a sphere as its target.
The theorem uses Hausdorff dimension to measure the size of the singular set. In practical terms, it separates the regular region, where the map is smooth, from the relatively closed set where smoothness may fail.
Two regimes, with different ceilings
The regularity result has two dimension ranges. When the target dimension k is from 3 through 6, the singular set has Hausdorff dimension no greater than n − k − 1. When k is 7 or higher, the upper bound is n − 7.
The preprint backs up the claim of sharpness with explicit model maps. For 3 ≤ k ≤ 6, it identifies the radial map x ↦ x/|x| as the critical model. For k ≥ 7, it gives the minimizing 0-homogeneous map from R^7 to S^k that sends x to (x/|x|, 0, …, 0).
Those examples explain why the stated ceilings are presented as optimal within the theorem’s settings. They support sharpness of the regularity result without turning the dimension bound into a claim that every map reaches it.
Some tangent maps must be constant
A second theorem addresses regular 0-homogeneous stable stationary harmonic maps u:R^(n+1)→S^k. It says every such map is constant when n + 1 ≤ k for 3 ≤ k ≤ 6, or when n + 1 ≤ 6 for k ≥ 7.
Here, 0-homogeneous refers to the degree-zero condition in the theorem, while a constant map has the same value throughout its domain. Under the listed assumptions, the result rules out a nonconstant regular tangent model in the specified dimension ranges.
The rigidity statement complements the singular-set estimate. The first theorem limits the size of the possible singular region; the second says that, in the covered cases, a regular 0-homogeneous stable stationary model cannot carry nontrivial variation.
A separate bound on variational instability
The paper also studies harmonic maps from one round sphere to another through their Morse index, a measure of how many independent variational directions can lower the relevant energy. This part concerns nonconstant maps u:S^n→S^k rather than the local regularity problem.
For every nonconstant harmonic map in the range 6 ≤ n ≤ k − 1, the theorem gives an index of at least n + 2. The statement is a conditional lower bound for the specified sphere dimensions.
The index argument depends on another rigidity result. When 6 ≤ n ≤ k − 1, a harmonic map whose first Jacobi-operator eigenvalue is at least −(n − 2) must be constant. That eigenvalue threshold is the spectral condition used to support the index conclusion.
The proof’s route through geometry and spectra
The proof uses target directions adapted to the geometry of the map and builds directional test fields. This replaces isotropic averaging over all target directions with a more selective construction tailored to the estimates being proved.
For the spherical part of the argument, the first spherical-harmonic component is represented by a coefficient matrix called H_u. The matrix packages the first harmonic information used in the subsequent inequalities.
The proof then considers directions in (Im H_u)^⊥, the subspace orthogonal to the image of that matrix. In those directions, the first-order spherical-harmonic term is removed, giving the argument a restricted collection of test fields.
One exceptional case, (n,k) = (5,6), is treated separately. The paper sets up a contradiction and analyzes the eigenvalues of H_uH_u^T to handle that dimensional combination.
A result with firm boundaries
The author frames the preprint as filling gaps in earlier regularity work, improving a prior singular-set dimension bound, establishing sharpness and obtaining a nontrivial improvement in the index estimate. Those are the author’s descriptions of the contribution.
The theorems are confined to round-sphere targets and to the dimension ranges written into them. They do not establish the same regularity bounds for arbitrary target manifolds, all weak harmonic maps or dimension regimes outside those treated.
The conclusions should therefore be read as results in harmonic-map theory: they concern the structure, regularity and variational behavior of the specified mathematical map classes.
Preprint status
The supplied document is an arXiv preprint, version v1, dated 20 Aug 2026. It reports support from NSF grant DMS-2404992. The manuscript also discloses AI assistance in developing its strategy and exposition, followed by author-directed revision and independent verification of the mathematical content.
Paper data and sources
Original title: Optimal regularity of stable harmonic maps to spheres
Authors: Xuanyu Li
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text