Macroscopic scalar curvature bounds on surfaces
We prove the generalized Geroch conjecture, and the corresponding hyperbolic conjecture, in dimension two.
math.DG↗
arXiv subjects
Noa Vikman
Publications and source records attributed to Noa Vikman.
Systolic inequalities for metric surfaces via filling minimality
We prove optimal systolic inequalities for length spaces homeomorphic to a torus of genus one or a real projective plane. In both cases, the optimal constant coincides with the constant from the (reversible) Finsler setting. This generalizes the classical results for Riemannian and Finsler surfaces. The proof of the inequality for the torus relies on an analysis of the asymptotic volume growth of the universal cover, together with a strengthening of the previously known area minimality of two-dimensional normed planes. For the inequality of the real projective plane, we similarly extend a minimality result of hemispheres. These results build upon works of Burago-Ivanov and Ivanov, respectively. In their proofs we apply recent uniformization theorems for metric disks and the theory of area-minimizing disks in metric spaces due to Lytchak-Wenger.
math.MG↗
Monotone Sobolev extensions in metric surfaces and applications to uniformization
We prove a monotone Sobolev extension theorem for maps to Jordan domains with rectifiable boundary in metric surfaces of locally finite Hausdorff 2-measure. This is then used to prove a uniformization result for compact metric surfaces by minimizing energy in the class of monotone Sobolev maps.
math.MG↗
Energy minimizing harmonic 2-spheres in metric spaces
In their seminal 1981 article, Sacks-Uhlenbeck famously proved the existence of non-trivial harmonic 2-spheres in every closed Riemannian manifold with non-zero second homotopy group. Their arguments heavily rely on PDE techniques. The purpose of the present paper is to develop a conceptually simple metric approach to the existence of harmonic spheres. This allows us to generalize the Sacks-Uhlenbeck result to a large class of compact metric spaces.
math.DG↗