Two-dimensional curvature functionals with superquadratic growth

Abstract

For two-dimensional, immersed closed surfaces f : Σ → R n , we study the curvature functionals E p (f ) and W p (f ) with integrands (1+|A| 2 ) p/2 and (1 + |H| 2 ) p/2 , respectively. Here A is the second fundamental form, H is the mean curvature and we assume p > 2. Our main result asserts that W 2,p critical points are smooth in both cases. We also prove a compactness theorem for W p -bounded sequences. In the case of E p this is just Langer's theorem [14], while for W p we have to impose a bound for the Willmore energy strictly below 8π as an additional condition. Finally, we establish versions of the Palais-Smale condition for both functionals.

Lower Semi-continuity of the Index in the Viscosity Method for Minimal Surfaces

“…For k large enough both Φ k and Ξ k are included in O Φ∞ . Because of the continuity of the map w Φ∞ we have respectively From [6] (see also an alternative approach in [1]) we know that under the assumptions that Φ is a critical point of F , it defines a smooth immersion in conformal coordinates. We shall be working in the chart in the neighborhood of [ Φ] in M(Σ g , M m ) given by w Φ from theorem I.2.…”

A viscosity method in the min-max theory of minimal surfaces

“…where g Φ and I Φ are respectively the first and second fundamental forms of Φ(Σ) in N n . Unlike previous existing viscous relaxations for min-max problems in the literature, the energy A σ is intrinsic in the sense that it is invariant under re-parametrization of Φ : A σ ( Φ) = A σ ( Φ • Ψ) for any smooth diffeomorphism Ψ of Σ. Modulo a choice of parametrization it is proved in [24] and [22] that for a fixed σ = 0 the Lagrangian A σ satisfies the Palais-Smale condition. Hence we can consider applying the mountain path lemma to this Lagrangian.…”

Uniform regularity results for critical and subcritical surface energies

“…A proof of this result has been established in the critical case F (s) = s in [13] while a proof of theorem I.1 can be found in [9] in the subcritical case for F (s) = (1 + s) q with q > 1. In order to establish theorem I.…”

“…1 we are adopting the parametric approach of [13] which is based on the local existence of isothermal coordinates (a fact which holds "uniformly" for p ≥ 2). In contrast, the proof in [9] is based on the fact that, for p > 2, any weak W 2,p immersion is obviously locally a graph. This fails for p = 2, and the proof in [9] "blows-up" as p → 2 .…”

“…In contrast, the proof in [9] is based on the fact that, for p > 2, any weak W 2,p immersion is obviously locally a graph. This fails for p = 2, and the proof in [9] "blows-up" as p → 2 . Our next goal is to merge the case p = 2 with the case p > 2 in a single proof by establishing estimates which remain controlled as p → 2.…”