Optimal regularity of stable harmonic maps to spheres

Abstract: In this paper, we show that the codimension of the singular set of a stable stationary harmonic map to a round $k$-sphere is at least $k+1$ when $k$ is between 3 and 6, and is at least 7 when $k$ is at least 7. The result is sharp in the sense that there exist energy minimizing 0-homogeneous maps in the aforementioned critical dimensions. We also establish non-trivial index bounds for the non-constant harmonic maps from $n$-spheres to $k$-spheres, provided that $n$ is less than $k$.

• The paper proves that stable stationary harmonic maps into S^k are smooth outside a singular set of Hausdorff dimension at most n−k−1 for 3≤k≤6 and n−7 for k≥7, with both bounds sharp.
• The method uses geometry-adapted directional test fields and spectral analysis of the first spherical-harmonic coefficient matrix to rule out nonconstant regular 0-homogeneous stable tangent maps, including the critical (n,k)=(5,6) case.
• The paper also shows that nonconstant harmonic maps S^n→S^k with 6≤n≤k−1 have Jacobi index at least n+2, improving the classical n+1 lower bound.

This paper by Xuanyu Li resolves the remaining dimensional gaps in the regularity theory of stable stationary harmonic maps into round spheres. The central result is a sharp bound on the Hausdorff dimension of the singular set, obtained by proving that no non-constant regular 0-homogeneous stable stationary harmonic maps exist in the previously unresolved critical dimensions.

Main theorem

Let $u: M^n \to S^k$ be a stable stationary harmonic map. Then $u$ is smooth away from a relatively closed singular set with

$$\dim_H \operatorname{sing} u \leqslant \begin{cases} n-k-1, & 3 \leqslant k \leqslant 6, \ n-7, & k \geqslant 7. \end{cases}$$

The result is sharp: for $3 \leqslant k \leqslant 6$, the radial projection $x \mapsto x/|x|$ is an energy-minimizing 0-homogeneous map $R^{k+1} \to S^k$ with a singularity at the origin; for $k \geqslant 7$, Schoen and Uhlenbeck showed the equatorial map $x \mapsto (x/|x|,0,\dots,0)$ is minimizing. This closes the qualitative gap left by Lin and Wang, where for maps from a six-dimensional domain into $S^k$ with $6 \leqslant k \leqslant 9$ neither regularity nor singular examples were known. Beyond intrinsic interest, the improved regularity feeds directly into the partial regularity theory of harmonic maps constructed variationally by Karpukhin and Stern.

Reduction to stable tangent maps

For $u$0 the sphere admits no stable harmonic 2-sphere. By Hsu's observation (elaborated also by Hsu–Li and Karpukhin–Stern) that energy defects in convergence of stable stationary harmonic maps are carried precisely by stable harmonic 2-spheres, bubbling is ruled out. Federer-type dimension reduction then reduces the main theorem to:

Theorem (nonexistence of stable tangent maps). If $u$1 is a regular 0-homogeneous stable stationary harmonic map with $u$2 (for $u$3) or $u$4 (for $u$5), then $u$6 is constant.

Equivalently, via restriction to $u$7: if $u$8 is harmonic with stable homogeneous extension, then $u$9 is constant whenever $$\dim_H \operatorname{sing} u \leqslant \begin{cases} n-k-1, & 3 \leqslant k \leqslant 6, \ n-7, & k \geqslant 7. \end{cases}$$0, $$\dim_H \operatorname{sing} u \leqslant \begin{cases} n-k-1, & 3 \leqslant k \leqslant 6, \ n-7, & k \geqslant 7. \end{cases}$$1, except possibly $$\dim_H \operatorname{sing} u \leqslant \begin{cases} n-k-1, & 3 \leqslant k \leqslant 6, \ n-7, & k \geqslant 7. \end{cases}$$2 — which the paper handles separately.

Method: directional test fields

Prior arguments inserted all conformal vector fields $$\dim_H \operatorname{sing} u \leqslant \begin{cases} n-k-1, & 3 \leqslant k \leqslant 6, \ n-7, & k \geqslant 7. \end{cases}$$3 into the second-variation form $$\dim_H \operatorname{sing} u \leqslant \begin{cases} n-k-1, & 3 \leqslant k \leqslant 6, \ n-7, & k \geqslant 7. \end{cases}$$4 and summed over $$\dim_H \operatorname{sing} u \leqslant \begin{cases} n-k-1, & 3 \leqslant k \leqslant 6, \ n-7, & k \geqslant 7. \end{cases}$$5, yielding isotropic estimates well adapted to the fully symmetric model $$\dim_H \operatorname{sing} u \leqslant \begin{cases} n-k-1, & 3 \leqslant k \leqslant 6, \ n-7, & k \geqslant 7. \end{cases}$$6 but not to equatorial models when $$\dim_H \operatorname{sing} u \leqslant \begin{cases} n-k-1, & 3 \leqslant k \leqslant 6, \ n-7, & k \geqslant 7. \end{cases}$$7. The paper instead uses geometrically distinguished directions, in the spirit of Simons' analysis of stable codimension-one minimal cones, and avoids the Kato-type inequality for $$\dim_H \operatorname{sing} u \leqslant \begin{cases} n-k-1, & 3 \leqslant k \leqslant 6, \ n-7, & k \geqslant 7. \end{cases}$$8.

The key test fields are

$$\dim_H \operatorname{sing} u \leqslant \begin{cases} n-k-1, & 3 \leqslant k \leqslant 6, \ n-7, & k \geqslant 7. \end{cases}$$9

where $3 \leqslant k \leqslant 6$0 is the average of $3 \leqslant k \leqslant 6$1 over $3 \leqslant k \leqslant 6$2 and $3 \leqslant k \leqslant 6$3 is a subspace of $3 \leqslant k \leqslant 6$4. Combining these with the stability inequality of Schoen–Uhlenbeck gives, for any $3 \leqslant k \leqslant 6$5-dimensional $3 \leqslant k \leqslant 6$6,

$3 \leqslant k \leqslant 6$7

The crucial refinement comes from choosing $3 \leqslant k \leqslant 6$8, where $3 \leqslant k \leqslant 6$9 is the first spherical-harmonic coefficient matrix of $x \mapsto x/|x|$0. In those directions $x \mapsto x/|x|$1 has no degree-one component, so the Poincaré constant improves from $x \mapsto x/|x|$2 to $x \mapsto x/|x|$3. Balancing this lower bound on $x \mapsto x/|x|$4 against the upper bound $x \mapsto x/|x|$5 forces $x \mapsto x/|x|$6 for $x \mapsto x/|x|$7 (with $x \mapsto x/|x|$8) and for $x \mapsto x/|x|$9, $R^{k+1} \to S^k$0 (with $R^{k+1} \to S^k$1, using $R^{k+1} \to S^k$2).

The critical case $R^{k+1} \to S^k$3

Here the preceding argument only yields $R^{k+1} \to S^k$4 with $R^{k+1} \to S^k$5, which is insufficient. The proof instead controls the eigenvalues $R^{k+1} \to S^k$6 of $R^{k+1} \to S^k$7 (squared singular values of $R^{k+1} \to S^k$8):

• Lower bound: taking $R^{k+1} \to S^k$9 spanned by the leading eigenvectors, tracking the constants through the Poincaré inequalities, and applying weak supermajorization together with concavity of $k \geqslant 7$0,

$k \geqslant 7$1

• Upper bound: inserting both the derivative fields $k \geqslant 7$2 (eigensections of the Jacobi operator with eigenvalue $k \geqslant 7$3) and the modified conformal fields $k \geqslant 7$4 into the shifted form $k \geqslant 7$5, whose cross-block equals $k \geqslant 7$6, a nuclear-norm estimate gives

$k \geqslant 7$7

Combining the two yields $k \geqslant 7$8, a contradiction since $k \geqslant 7$9. Hence $x \mapsto (x/|x|,0,\dots,0)$0 is constant, completing the sharpness claim of the main theorem.

Index estimates

As a corollary of the same technique, the paper proves a rigidity statement: if $x \mapsto (x/|x|,0,\dots,0)$1 is harmonic with $x \mapsto (x/|x|,0,\dots,0)$2 and the first eigenvalue of its Jacobi operator satisfies $x \mapsto (x/|x|,0,\dots,0)$3, then $x \mapsto (x/|x|,0,\dots,0)$4 is constant. Since Xin and El Soufi showed every non-constant harmonic map $x \mapsto (x/|x|,0,\dots,0)$5 has at least $x \mapsto (x/|x|,0,\dots,0)$6 negative directions with eigenvalue exactly $x \mapsto (x/|x|,0,\dots,0)$7 (given by covariant derivatives along conformal vector fields), it follows that any non-constant such map has index at least $x \mapsto (x/|x|,0,\dots,0)$8 — one strictly more than the classical bound. The proof replaces the factor $x \mapsto (x/|x|,0,\dots,0)$9 by $S^k$0 in the directional estimates and derives incompatible upper and lower bounds on $S^k$1 when $S^k$2.

Scope and limitations

The nonexistence theorem covers only $S^k$3 with $S^k$4; for $S^k$5 or $S^k$6 non-constant stable tangent maps may exist, and indeed do in the minimizing case ($S^k$7 at the critical dimension). The index improvement requires $S^k$8 and $S^k$9, so the low-dimensional index behavior of harmonic spheres remains governed by the classical $6 \leqslant k \leqslant 9$0 bound. The paper does not address whether the general stationary harmonic map singular set must have codimension at least three — a widely questioned open problem that stability was used here to circumvent rather than resolve.

Conclusion

The paper establishes optimal codimension bounds for the singular set of stable stationary harmonic maps into spheres, resolving the previously indeterminate range $6 \leqslant k \leqslant 9$1 for six-dimensional domains, and proves a strict index improvement over the Xin–El Soufi bound. The methodological contribution — selecting test variations adapted to the geometry of the map rather than averaging over all target directions, combined with spectral analysis of the first harmonic coefficient matrix in the borderline case — provides a sharper alternative to the standard Kato-inequality-based second-variation arguments.