The Hijazi inequality on conformally parabolic manifolds

关于Dirac算子的Bär-Hijazi-Lott不变量及Yamabe问题的旋量证明

On the Bär-Hijazi-Lott invariant for the Dirac operator and a spinorial proof of the Yamabe problem

本文证明了非局部共形平坦闭自旋流形上Dirac算子的Bär-Hijazi-Lott不变量严格小于球面情形，作为Aubin估计的旋量类比，解决了该设定下的Yamabe问题。

Parabolic conformally symplectic structures II; parabolic contactification

Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying structure is conformally symplectic, then one obtains a PCS-structure. In the current article, we relate PCS-structures to parabolic contact structures. Starting from a parabolic contact structure with a transversal infinitesimal automorphism, we first construct a natural PCS-structure on any leaf space of the corresponding foliation. Then we develop a parabolic ver…

Geometric inequalities on locally conformally flat manifolds

Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.

Conformally Fedosov manifolds

We introduce the notion of a conformally Fedosov structure and construct an associated Cartan connection. When an appropriate curvature vanishes, this allows us to construct a family of natural differential complexes akin to the BGG complexes from parabolic geometry.

Parabolic conformally symplectic structures III; Invariant differential operators and complexes

This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, the second part focused on the case that the underlying structure is conformally symplectic (PCS-structures). In that case, we obtained a close relation to parabolic contact structures via a concept of parabolic contactification. It was also shown that special symplectic connections (and thus all connections of exotic symplectic holonomy) …

Parabolic conformally symplectic structures I; definition and distinguished connections

We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type $C_n$ and admits a contact grading. We show that a structure of each of these types on a smooth manifold $M$ determines a canonical compatible linear connection on the tangent bundle $TM$. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant's theorem on Lie algebra cohomology. For each type, we give an …