Degree theorems and Lipschitz simplicial volume for non-positively curved manifolds of finite volume

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively curved manifolds or locally symmetric spaces of non-compact type. In contrast, we provide vanishing results for the ordinary simplicial volume; for instance, we show that the ordinary simplicial volume of non-compact locally symmetric spaces with finite volume of Q-rank at least 3 is zero.Comment: 33 pages; corrected the vanishing result (and adapted Section 5 accordingly), minor expository changes in the introductio

Pluripotential-theoretic methods in K-stability and the space of K\ue4hler metrics

It is a natural problem, dating back to Calabi, to find canonical metrics on complex manifolds. In the case of polarized compact K\ue4hler manifolds, a natural candidate is a metric with constant scalar curvature (cscK metric).Since the 80s, Yau, Tian, Donaldson among others proposed that the existence of these special metrics are equivalent to an algebrico-geometric notion of K-stability. There are several known approaches to the study of K-stability and canonical metrics, using various tools from the theory of PDEs, algebraic geometry and non-Archimedean geometry for example. In this thesis, we study a different approach, based on pluripotential theory. In geometric terms, pluripotential theory is the study of positively curved metrics on vector bundles. For the purpose of K-stability, we only need pluripotential theory on an ample line bundle. In this case, pluripotential theory can be identified with the study of quasi-plurisubharmonic functions on the manifold. The application of pluripotential theory in K-stability is not completely new, but previously, people are principally interested in the regular (or mildly singular) quasi-plurisubharmonic functions. In this thesis, we put more emphasis on the role of singular\ua0quasi-plurisubharmonic functions and their singularities. In Paper 1 and Paper 2, we prove a criterion for the existence of canonical metrics on Fano manifolds in terms of quasi-plurisubharmonic functions. In Paper 3, we are concerned with the case when there are no canonical metrics, we prove that there is always an optimal destabilizer to the K-stability