$W^{2,2}$-conformal immersions of a closed Riemann~surface into $\R^n$

Abstract

We study sequences f k : Σ k → R n of conformally immersed, compact Riemann surfaces with fixed genus and Willmore energy W(f ) ≤ Λ. Assume that Σ k converges to Σ in moduli space, i.e. φ * k (Σ k ) → Σ as complex structures for diffeomorphisms φ k . Then we construct a branched conformal immersion f : Σ → R n and Möbius transformations σ k , such that for a subsequenceloc away from finitely many points. For Λ < 8π the map f is unbranched. If the Σ k diverge in moduli space, then we show lim inf k→∞ W(f k ) ≥ min(8π, ω n p ). Our work generalizes results in [K-S2] to arbitrary codimension.

Weak convergence of branched conformal immersions with uniformly bounded areas and Willmore energies

“…Theorem 1.1) and later, Kuwert and Li proved γ n is the optimal constant(c.f. Theorem 5.1 in [11]); In the case m ≥ 1, the theorem was proved by Lamm and Nguyen (c.f. Theorem 3.2 in [14]).…”

“…Here, we should mention that Kuwert and Li [11] discussed the compactness for a sequence of conformal immersions of a compact Riemann surface. More precisely, they proved that, if f k ∈ W 2,2 conf (Σ k , R n ) are conformal immersions with W (f k ) < Λ < ∞, and Σ k converge to Σ in moduli space, then there exist Möbius transformations σ k and diffeomorphisms φ k , such that σ k • f k • φ k converge to f 0 locally in weak W 2,2 sense on Σ minus finitely many concentration points and the weak limit f 0 is a W 2,2 branched conformal immersion.…”

Existence and Regularity of Spheres Minimising the Canham–Helfrich Energy

“…We adapt the notion of weak immersions which was independently formalized by Rivière [Riv14] and Kuwert and Li [KL12]. Let (Σ, c 0 ) be a smooth closed Riemann surface (in the rest of the paper we will take (Σ, c 0 ) to be the 2-sphere endowed with the standard round metric).…”

Reflection of Willmore Surfaces with Free Boundaries

“…Furthermore the derivatives have traces (∂ i u) ± ∈ L 2 (I). In the following we need the concept of W 2,2 conformal immersions, see [7].…”