An extension problem for Riemannian metrics
Dienstag, 23.5.17, 16:15-17:15, Raum 404, Eckerstr. 1
The problem of extending a given metric on a finite three\ndimensional domain to one which is asymptotically flat and which satisfies the constraint equations of general relativity was proposed by Robert Bartnik. He proposed that the minimal mass of such an extension would be a measure of the quasi-local mass of the domain and this is called the Bartnik mass. In this talk we will describe this problem and recent results on it including a comparison of the Bartnik mass to other quasi-local mass notions.
An extension problem for Riemannian metrics
Dienstag, 30.5.17, 16:00-17:00, Raum 404, Eckerstr. 1
The problem of extending a given metric on a\nfinite three dimensional domain to one which is\nasymptotically flat and which satisfies the constraint\nequations of general relativity was proposed by Robert\nBartnik. He proposed that the minimal mass of such an\nextension would be a measure of the quasi-local mass of the\ndomain and this is called the Bartnik mass. In this talk we\nwill describe this problem and recent results on it\nincluding a comparison of the Bartnik mass to other\nquasi-local mass notions.\n
Minkowski's formula, Reilly's formula and Alexandrov's theorem
Dienstag, 30.5.17, 16:00-17:00, Raum 404, Eckerstr. 1
In this talk, I first review the method of Reilly and Ros on reproving Alexandrov’s theorem about the rigidity of embedded CMC (constant mean curvature) hypersurfaces in Euclidean space by simply using two integral formulae —Minkowski's formula and Reilly's formula. Then I will introduce our recent result on new Reilly type formula and Minkowski type formula as well as their applications on Alexandrov type theorem in two different settings: (i) the ambient spaces in a sub-static warped product spaces and\n(ii) hypersurfaces with free boundary in unit ball in space forms.\nThis is a report of joint works with Junfang Li, and separately with Guofang Wang.
Minkowski's formula, Reilly's formula and Alexandrov's theorem
Dienstag, 30.5.17, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk, I first review the method of Reilly and Ros on reproving Alexandrov’s theorem about the rigidity of embedded CMC (constant mean curvature) hypersurfaces in Euclidean space by simply using two integral formulae —Minkowski's formula and Reilly's formula. Then I will introduce our recent result on new Reilly type formula and Minkowski type formula as well as their applications on Alexandrov type theorem in two different settings: (i) the ambient spaces in a sub-static warped product spaces and\n(ii) hypersurfaces with free boundary in unit ball in space forms.\nThis is a report of joint works with Junfang Li, and separately with Guofang Wang.
Geometric analysis on stratified spaces
Dienstag, 13.6.17, 16:00-17:00, Raum 404, Eckerstr. 1
Stratified spaces are singular metric spaces that have been studied from a topological and analytical point of view. In this talk we will give an introduction about this singular setting; we will show how Riemannian geometry can be used to study stratified spaces and how one can obtain geometric and analytic results depending on the positivity of the Ricci curvature.
Stability of lower curvature bounds under \(C0\) deformations of the metric
Dienstag, 13.6.17, 17:00-18:00, Raum 404, Eckerstr. 1
If a sequence of Riemannian manifolds with sectional curvature bounded from below Gromov-Hausdorff converges to a smooth limit manifold, then the limit has sectional curvature bounded from below. This comes from the fact that lower bounds on the sectional curvature have a strong geometric meaning in term of « fatness of geodesic triangles » through Toponogov’s theorem. The aim of this talk is to show how one can deal with other kind of curvature bounds which do not have such a strong geometric flavor (like lower bounds on the curvature operator), at the cost of requiring \(C0\) convergence of the metric instead of Gromov-Hausdorff convergence. This builds up on previous works by Koch-Lamm and Bamler.
Ancient pancakes
Dienstag, 11.7.17, 16:00-17:00, Raum 404, Eckerstr. 1
We will show how to construct a compact, convex ancient solution of mean curvature flow which lies in a slab region of \(\bmathbb{R}^3\) (of width \(\bpi\)) and prove unique asymptotics for such solutions: The maximum of the mean curvature is close to one and the `edge' regions are close to grim planes (of width \(\bpi\)) when \(t\) is close to minus infinity. This is joint work with Theodora Bourni (FU Berlin) and Giuseppe Tinaglia (King's College London).
Some maximum principles on complete manifolds and their applications in geometry and analysis
Dienstag, 25.7.17, 17:00-18:00, Raum 404, Eckerstr. 1
The Hopf maximum principle is a fundamental tool for geometry and analysis on compact manifolds. For non-compact complete manifolds, in 1960-70's, H. Omori, S. T. Yau, and S. T. Yau-S. Y. Cheng respectively established maximum principles with the sectional/Ricci curvature bounded below by a constant. This kind of results are called Omori-Yau\nmaximum principles, they provide a powerful tool in the geometry and analysis on non-compact complete manifolds. In this talk, we will present some new Omori-Yau maximum principles and give their applications in submanifold geometry, harmonic maps and holomorphic maps.\n