Constrained Willmore tori in S^3
Dienstag, 8.5.12, 16:15-17:15, Raum 404, Eckerstr. 1
Numerical Ricci-DeTurck flow
Dienstag, 22.5.12, 16:15-17:15, Raum 404, Eckerstr. 1
We present a numerical method for the computation of\nn-dimensional Ricci-DeTurck flow.\nRicci flow is a geometric flow deforming a time-dependent\nmetric on a Riemannian manifold\nproportional to its Ricci curvature. Ricci-DeTurck flow is\na reparametrization of this flow\nusing the harmonic map flow in order to get a strictly\nparabolic PDE.\nOur numerical method is based on the assumption\nthat the manifold is orientable and differentiably\nembeddable in \bR^{n+1}.\nBy this means, it is possible to do computations in the\nbasis of the ambient space.\nA weak formulation of Ricci-DeTurck flow is derived such\nthat it only contains tangential gradients.\nA spatial discretization of this formulation with finite\nelements on polyhedral hypersurfaces\nand an implicit time discretization lead to an algorithm\nfor computing Ricci-DeTurck flow.\nWe have performed numerical tests for two and three\ndimensional hypersurfaces\nusing piecewise linear finite elements.\nThe generalization to non-orientable hypersurfaces of\nhigher codimensions is still open.\n
The fundamental gap conjecture (Paper by Andrews/Clutterbuck)
Dienstag, 19.6.12, 16:15-17:15, Raum 404, Eckerstr. 1
Constant Mean Curvature Tori in S^3
Dienstag, 19.6.12, 18:00-19:00, Raum 404, Eckerstr. 1
A new conformal invariant of 3-dimensional manifolds (1/2)(Paper by Ge/Wang)
Dienstag, 26.6.12, 16:15-17:15, Raum 404, Eckerstr. 1
A new conformal invariant of 3-dimensional manifolds (2/2)(Paper by Ge/Wang)
Mittwoch, 27.6.12, 18:00-19:00, Raum 127, Eckerstr. 1
1st Steklov eigenvalue of embedded minimal surfaces w/ free boundary 1 (by Fraser and Schoen)
Dienstag, 3.7.12, 16:15-17:15, Raum 404, Eckerstr. 1
1st Steklov eigenvalue of embedded minimal surfaces w/ free boundary 2 (by Fraser and Schoen)
Dienstag, 10.7.12, 16:15-17:15, Raum 404, Eckerstr. 1
Finite-time singularities in Mean Curvature Flow and Ricci flow
Mittwoch, 11.7.12, 16:15-17:15, Hörsaal II, Albertstr. 23b
In this talk, we will study finite-time singularities of two\ngeometric flows. I will start with the Mean Curvature Flow\nfor which I will draw a lot of pictures and recall (without proofs) some of the most important results about the singularity formation. \n\nI will then show how these results should be translated to corresponding statements for finite-time Ricci flow singularities. Some of these statements have recently been proved in joint work with various collaborators, some of them are work in progress and others are completely open conjectures.\n\n
Point-singularities of Willmore Surfaces [new and augmented version]
Dienstag, 17.7.12, 16:15-17:15, Raum 404, Eckerstr. 1
We consider a branched Willmore surface immersed in \(\bR^{m\bge3}\) with square-integrable second fundamental form. We develop around each branch point local asymptotics for the immersion, its first, and its second derivatives. These expansions are given in terms of a first residue" (constant vector in $\bR^m$) and in terms ofsecond residues" (integer-valued). These residues are computed as circulation integrals around each branch point. We then deduce explicit point removability conditions in terms of the residues, ensuring that the (branched) immersion is smooth across its branch points. Do residues pass through the weak limit? We'll see...\n[Talk based on a joint-work with Tristan Rivière]
Global analysis of the generalised Helfrich flow of curves immersed in IR^n
Mittwoch, 25.7.12, 16:15-17:15, Hörsaal II, Albertstr. 23b
The Helfrich energy is a measure of the elastic bending energy of a manifold, and for surfaces has been in the mind of researchers in one form or another since Poisson's treatise on elasticity in 1812. In 1973 Helfrich used the theory of elastic lipid bilayers to motivate the specific form of the functional which is common today. In its full generality, the functional incorporates an ambient "spontaneous curvature". The presence of even a trivial (constant) spontaneous curvature has historically resisted analysis. In this talk we consider the gradient flow of the functional defined on an immersed curve, with arbitrary codimension. We shall prove that under mild assumptions on the spontaneous curvature the flow exists for all time for initial data with arbitrarily high energy, subconverging to a critical point of a limiting functional. Asymptotic analysis is made particularly difficult by the presence of the spontaneous curvature: we shall present explicit examples where the flow exists for all time but does not converge. Nevertheless (time permitting), following an idea of Ben Andrews, we shall present three conditions under which it is possible to obtain full convergence of the flow. One of these conditions includes the case of the Willmore flow, strengthening a well-known result of Dziuk, Kuwert, and Schaetzle.
The Scalar curvature flow
Mittwoch, 26.9.12, 16:15-17:15, Hörsaal II, Albertstr. 23b
In this talk, I will survey some recent progress on the conformal curvature flow for various prescribed curvature problems, including the perturbation theory for the prescribed scalar curvature problem on the unit sphere, prescribed Q-curvature problem and the prescribed mean curvature on the boundary of the unit ball.