5. FHST-Meeting on Geometry and Analysis

Anna Dall'Acqua: On the Willmore flow of tori or revolution

In this talk we present a stricking relationship between Willmore surfaces of revolution and elastic curves in hyperbolic half-space. Here the term elastic curve refer to a critical point of the energy given by the integral of the curvature squared. In the talk we will discuss this relationship and use it to study long-time existence and asymptotic behavior for the L^2-gradient flow of the Willmore energy, under the condition that the initial datum is a torus of revolution. As in the case of Willmore flow of spheres, we show that if an initial datum has Willmore energy below 8 \pi then the solution of the Willmore flow converges to the Clifford Torus, possibly rescaled and translated. The energy threshold of 8 \pi turns out to be optimal for such a convergence result. The lecture is based on joint work with M. Müller (Univ. Leipzig), R. Schätzle (Univ. Tübingen) and A. Spener (Univ. Ulm).

Marius Müller: Embeddedness-breaking of elastic flows

This talk is based on a joint work with T.Miura (Tokyo) and F. Rupp (Vienna). We study the qualitative behavior of elastic flows of closed curves, i.e. L2-gradient evolutions of the Euler-Bernoulli elastic energy. We are interested in the question of embeddedness-preservation -- Can the evolution of an embedded curve develop self-intersections? In general the answer is 'no', as shown by S. Blatt (2010) for a large class of fourth order geometric flows. We can however expose an (optimal) energy threshold under which evolutions still preserve embeddedness. The optimal threshold has a geometric significance. To understand it we will enter the fantastic world of Euler's elastic curves.

Alex Waldron: Finite-time singularities of 2D harmonic map flow

I'll discuss recent work on continuity of the body map at finite-time singularities of 2D harmonic map flow, assuming the initial data is almost-holomorphic (in the energy sense) or the blowup is "strictly type-II." This is relevant to a conjecture of Topping. Time permitting, I'll also discuss uniqueness of subsequential limits at infinite time.

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