| Abstract: |
| Chen`s operator for a submanifold is the twice iterated Laplacian on the pullback bundle, sometimes known as the rough Laplacian in the literature. Chen`s conjecture is that if Chen`s operator applied to the immersion map vanishes, then the submanifold is minimal. In the last few years, work has progressed on the parabolic flow with velocity corresponding to Chen`s operator applied to the immersion. This parabolic flow has an interesting variational characterisation, being between the biharmonic map heat flow and the Willmore flow. Algebraically, the Chen flow sits instead between surface diffusion and Willmore flow. Qualitatively its behaviour is much closer to the mean curvature flow. In particular, spheres shrink to points in finite time. In this talk we describe the Chen operator, Chen's conjecture, and some recent work on Chen's flow in two and four dimensions. |
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