## Complete convex surfaces: intrinsic and extrinsic properties |

## Wilhelm Klingenberg (Durham university) |

## Thursday, 2017-12-21 17:00 |

We will examine convex surfaces which divide the ambient Euclidean 3-space into two parts. By the Theorem of Gauss-Bonnet such surfaces need to be of the intrinsic type of a cylinder, plane, or sphere. We will then discuss the extrinsic symmetry property of rotational invariance, and its infinitesimal version at a point of the surface. We outline the proof of a global extrinsic conjecture of Victor Andreevich Toponogov : "Any convex plane admits (at least) one point of infinitesimal symmetry, possibly at infinity". The proof, in collaboration with Brendan Guilfoyle, uses complex analysis and a parabolic curvature flow in the space of lines of Euclidean 3-space. |