Blow up criteria for geometric flows on surfaces
Tuesday, 23.10.18, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
I will present a scheme to prove blow up criteria for various (intrinsic) geometric flows on closed surfaces. \nThe uniformization theorem allow us to split a curve of Riemannian metrics into a curve of constant curvature metrics and conformal factors. A further refinement, due to Buzano, Rupflin and Topping, allows us to view the evolution of the curve of constant curvature metrics as a finite dimensional dynamical system.\n\nCombining this splitting with a compactness theorem adapted to the situation allows us to apply standard PDE techniques to rule out blow up under certain conditions, depending on the flow. I will discuss the approach for the specific example of the harmonic Ricci flow.\n
Curve flows with a global forcing term
Tuesday, 23.10.18, 17:30-18:30, Raum 404, Ernst-Zermelo-Str. 1
We study curve shortening flow with global forcing terms for embedded, closed, smooth curves in the plane. We derive an analogue to Huisken's distance comparison principle for curve shortening flow for initial curves whose local total curvature does not lie below \(-\bpi\) and show that this condition is sharp. For bounded forcing terms, this excludes singularities in finite time. For immortal flows whose forcing terms provide non-vanishing enclosed area and bounded length, we prove convexity in finite time and smooth and exponential convergence to a circle. In particular, the above holds for the area preserving curve shortening flow.
On the Plateau-Douglas problem for the Willmore energy
Tuesday, 4.12.18, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
In this talk we will introduce the Willmore energy of surfaces in the three-dimensional Euclidean space, which is the surface integral of the squared mean curvature. For a smooth closed embedded planar curve, we will consider the minimization of the Willmore energy among immersed surfaces of a prescribed genus having the given curve as boundary. Such problem can be seen as a generalization of the classical Plateau-Douglas problem, which is immediately trivial in the case of planar boundary curves. Exploiting the conformal properties of the functional and tools from the theory of varifolds with boundary, we will see that the problem does not reduce to a minimal surfaces problem and we will present some recent explicit results both of existence and non-existence of minimizers, depending on the prescribed boundary curve.
Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case
Tuesday, 29.1.19, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
In this talk, we shall present a new result about prescribing\nGaussian curvature on a closed Riemann surface with conical\nsingularities in the negative case. This is a joint work Prof. Yunyan Yang.