Regularity and quantification for harmonic maps with free boundary

Abstract

We prove a quantification result for harmonic maps with free boundary from arbitrary Riemannian surfaces into the unit ball of R n+1 with bounded energy. This generalizes results obtained by Da Lio [1] on the disc.Let (M, g) be a smooth Riemannian surface with a smooth nonempty boundary with s connected components. We fix n ≥ 2 and let B n+1 be the unit ball of R n+1 . A map u : (M, g) → B n+1 is a smooth harmonic map with free boundary if it is harmonic and smooth up to the boundary, u(∂M ) ⊂ S n and ∂ ν u is parallel to u, (or ∂ ν u ∈ (T u S n ) ⊥ ). The energy of such a map is defined as

Min-max minimal disks with free boundary in Riemannian manifolds

“…which yields The next result is a refined -regularity for weakly harmonic maps on D + with Dirichlet and free boundaries, c.f. [44,28,24]. This -regularity is crucial to the proof of Theorem 2.2.…”

“…[23,35,12,34,38,4,11]; for quantification results for harmonic maps with free boundary, see e.g. [28,24] . A similar strong convergence property was first proven by Colding-Minicozzi in [6] for the min-max construction of minimal spheres, and it played an essential role in their proof of the finite time extinction for certain 3-dimensional Ricci flow.…”

Existence of min-max free boundary disks realizing the width of a manifold

“…as n → +∞ so that once (5.1) is proved, we get proposition 5.1. Notice that if {u n } is already a sequence of harmonic maps defined on D + with free boundary in M on I, the third assumption can be deleted since (5.2) is already true by a Pohožaev identity, see [27] for the case M = S n which extend straightforward to general target. However, the third assumption is in some sense stronger because in this case, we do not need u n to be defined on D + .…”

“…Thanks to (6.6) and (6.7) and Proposition 5.1 we have the no-neck-energy for the sequence σ n tn . Now we can conclude by the classical bubble tree decomposition to get the W 1,2 -bubble convergence, following for instance verbatim section 3 of [27], Step 1 and Step 2 being consequences of the fact that there is a free boundary harmonic map W 1,2 -close which satisfies the ε-regularity. Finally, in Step 3, Claim 3.1 has to be replace by the no-neck-energy we have just proved.…”

Bubbling analysis near the Dirichlet boundary for approximate harmonic maps from surfaces

“…When M is a compact Riemann surface with smooth boundary, Laurain-Petrides [14] considered a sequence of harmonic maps {u n } from M to the unit ball B n+1 ⊂ R n+1 with free boundary u n (∂M) on S n and with uniformly bounded energy and proved the energy identity. The blow-up theory (including the energy identity and the no neck property) of the more general case of a sequence of maps into a general compact target manifold with free boundary on a general closed supporting submanifold with uniformly L 2 bounded tension fields and with uniformly bounded energy was completed in [11].…”