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We investigate point singularities of Willmore surfaces, which for example appear as blowups of the Willmore flow near singularities, and prove that closed Willmore surfaces with one unit density point singularity are smooth in codimension one. As applications we get in codimension one that the Willmore flow of spheres with energy less than 8π exists for all time and converges to a round sphere and further that the set of Willmore tori with energy less than 8π − δ is compact up to Möbius transformations.

We consider two-dimensional, compact immersed surfaces in R n moving by the gradient of the L 2 integral of their curvature. It is not known whether solutions to this fourth-order, geometric evolution equation may develop singularities in finite time. We give a lower bound on the lifespan of a smooth solution, which depends only on how much the curvature of the initial surface is concentrated in space.

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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.

We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface into R n , n = 3, 4, if there is one conformal immersion with Willmore energy smaller than a certain bound W n,p depending on codimension and genus p of the Riemann surface. For tori in codimension 1, we know W 3,1 = 8π . Let Σ be a closed orientable surface of genus p ≥ 1 with smooth metric g satisfyingas in the situation of Corollary 7.3 where W n,p is defined in (1.2) above and n = 3, 4 . To get a minimizer, we consider a minimizing sequence of immersions f m : Σ → R n conformal to g in the senseAfter applying suitable Möbius transformations according to [KuSch06] Theorem 4.1 we will be able to estimate f m in W 2,2 (Σ) and W 1,∞ (Σ) , see below, and after passing to a subsequence, we get a limit f ∈ W 2,2 (Σ) ∩ W 1,∞ (Σ) which is an immersion in a weak sense, see (2.6) below. To prove that f is smooth, which implies that it is a minimizer, and that f satisfies the Euler-Lagrange equation in Corollary 7.3 we will consider variations, say of the form f + V . In general, these are not conformal to g anymore, and we want to correct it by f + V + λ r V r for suitable selected variations V r . Now even these are not conformal to g since the set of conformal metrics is quite small in the set of all metrics. To increase the set of admissible pull-back metrics, we observe that it suffices for (f + V + λ r V r ) • φ being conformal to g for some diffeomorphism φ of Σ . In other words, the pullback metric (f + V + λ r V r ) * g euc need not be conformal to g , but has to coincide only in the modul space. Actually, we will consider the Teichmüller space, which is coarser than the modul space, but is instead a smooth open manifold, and the bundle projection π : M → T of the sets of metrics M into the Teichmüller space T , see [FiTr84], [Tr]. Clearly W(Σ, g, n) depends only on the conformal structure defined by g , in particular descends to Teichmüller space and leads to the following definition. Definition 2.1 We define M p,n : T → [0, ∞] for p ≥ 1, n ≥ 3, by selecting a closed, orientable surface Σ of genus p and M p,n (τ ) := inf{W(f ) | f : Σ → R n smooth immersion, π(f * g euc ) = τ }. ✷We see M p,n (π(g)) = W(Σ, g, n).Next, inf τ M p,n (τ ) = β n p for the infimum under fixed genus defined in (1.3), and, as the minimum is attained and 4π < β n p < 8π , see [Sim93] and [BaKu03], 4π < min τ ∈T M p,n (τ ) = β n p < 8π.In the following proposition, we consider a slightly more general situation than above.Proposition 2.2 Let f m : Σ → R n , n = 3, 4, be smooth immersions of a closed, orientable surface Σ of genus p ≥ 1 satisfyingand π(f * m g euc ) → τ 0 in T . (2.2) Then replacing f m by Φ m • f m • φ m for suitable Möbius transformations Φ m and diffeomorphisms φ m of Σ homotopic to the identity, we get lim sup m→∞ f m W 2,2 (Σ) ≤ C(p, δ, τ 0 ) (2.3) and f * m g euc = e 2um g poin,m for some unit volume constant curvature metrics g poin,m with u m L ∞ (Σ) , ∇u m L 2 (Σ,g poin,m ) ≤ C...

We consider a free boundary problem for the Willmore functional W(f ) = 1 4 Σ H 2 dµ f . Given a smooth bounded domain Ω ⊂ R 3 , we construct Willmore disks which are critical in the class of surfaces meeting ∂Ω at a right angle along their boundary and having small prescribed area. Using rescaling and the implicit function theorem, we first obtain constrained solutions with prescribed barycenter on ∂Ω. We then study the variation of that barycenter.

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