We study the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain in the Euclidean plane. Under certain conditions on the initial curve the flow does not develop any singularity, and it subconverges smoothly to an arc of a circle sitting outside of the given fixed domain and enclosing the same area as the initial curve.

We consider unbranched Willmore surfaces in the Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index -the number of linearly independent variational directions that locally decrease the Willmore energy. We in particular compute the Index of a Willmore sphere in the three-space. This Index is m − d, where m is the number of ends of the corresponding complete minimal surface and d is the dimension of the span of the normals at the m-fold point. The dimension d is either two or three. For m = 4 we prove that d = 3. In general, we show that there is a strong connection of the Morse Index to the number of logarithmically growing Jacobi fields on the corresponding minimal surface.

Abstract. Let K = RP 2 ♯RP 2 be a Klein bottle. We show that the infimum of the Willmore energy among all immersed Klein bottles f : K → R n , n ≥ 4, is attained by a smooth embedded Klein bottle. We know from [21,9] that there are three distinct regular homotopy classes of immersions f : K → R 4 each one containing an embedding. One is characterized by the property that it contains the minimizer just mentioned. For the other two regular homotopy classes we show W( f ) ≥ 8π. We give a classification of the minimizers of these two regular homotopy classes. In particular, we prove the existence of infinitely many distinct embedded Klein bottles in R 4 that have Euler normal number −4 or +4 and Willmore energy 8π. The surfaces are distinct even when we allow conformal transformations of R 4 . As they are all minimizers in their regular homotopy class they are Willmore surfaces.

We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial curves are convex, then an appropriate rescaling at the finite maximal time of existence yields a grim reaper or half a grim reaper as limit flow. We construct examples of initial curves satisfying the mentioned criterion.

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