On the L^p spectrum
Montag, 13.1.25, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
In this talk we will consider the \(L^p\) spectrum of the Laplacian on differential forms. In particular, we will show that the resolvent set of the Laplacian on \(L^p\) integrable \(k\)-forms lies outside a parabola whenever the volume of the manifold has an exponential volume growth rate, removing the requirement on the manifold to be of bounded geometry. Moreover, we find sufficient conditions on an open Riemannian manifold so that a Weyl criterion holds for the \(L^p\)-spectrum of the Laplacian on \(k\)-forms, and we provide a detailed description of the \(L^p\) spectrum of the Laplacian on \(k\)-forms over hyperbolic space. The above results are joint work with Zhiqin Lu.
A sharp isoperimetric gap theorem in non-positive curvature
Montag, 27.1.25, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
In joint work with Cornelia Drutu, Panos Papasoglu, and Stephan Stadler, we\nstudy isoperimetric\ninequalities for null-homotopies of Lipschitz 2-spheres in Hadamard manifolds\nor, more generally,\nproper CAT(0) spaces. In one dimension less, for fillings of circles by discs,\nit is known that a \nquadratic inequality with a constant smaller than the sharp threshold\n\(1/(4\bpi)\) implies that the \nunderlying space is Gromov hyperbolic and satisfies a linear inequality. Our\nmain result is a first \nanalogous gap theorem in higher dimensions, yielding exponents arbitrarily\nclose to 1. Towards \nthis we prove a Euclidean isoperimetric inequality for null-homotopies of\n2-spheres, apparently \nmissing in the literature, and introduce so-called minimal tetrahedra, which we\ndemonstrate satisfy \na linear inequality.