On the singular set of stationary harmonic maps

Singularities of harmonic maps

“…Thus the small energy regularity theorem of [7] and the higher regularity theory give the partial regularity result of Bethuel Is this the optimal estimate for stationary maps? In 1984, R. Schoen [96] showed that the singular set of a limit of a convergent sequence of smooth harmonic maps has locally finite H n−2 measure.…”

“…Evans [37] established it for a stationary harmonic map to the standard sphere by exploiting the particular structure of (2), motivated by Hélein's proof [67], and the smallness of the BMO norm. Finally, in 1993, F. Bethuel [7] established small energy regularity for a general stationary harmonic map. He cleverly employs a special frame for the tangent bundle of N along u, as in [68], which gives reformulation of the harmonic map equation using Jacobian expressions that admit compensation properties discovered in [25].…”

“…Lin [81] recently derived this estimate for a general stationary harmonic map. His argument involves Bethuel's small energy regularity theorem [7] and consideration of scaling and limit properties of the measure |∇u| 2 dx. See the discussion after Theorem 7 below.…”

Monotonicity formulae for variational problems

“…whenever B(0, R) ⊂ U and 0 < r < R. Consult my old paper [6] for the use of the monotonicity formula (2.10) to prove partial regularity of stationary harmonic maps into spheres and see Bethuel [7] for the generalization to stationary harmonic maps into general target manifolds.…”

Subelliptic p -harmonic maps into spheres and the ghost of Hardy spaces

“…Moreover the number of those indices i ∈ I for which x i ∈ B (y, 2 k −1 ) \ B (y, 2 k −2 ) is bounded by a constant depending on the doubling constant only (because the balls 1 6 B i are pairwise disjoint and their radii are comparable to the diameter of the annulus B (y, 2 k −1 )\ B (y, 2 k −2 ).) Finally, note that when k is large,…”