相关论文: Dirac and Plateau Billiards in Domains with Corner…

We consider classical billiards on surfaces of constant curvature, where the charged billiard ball is exposed to a homogeneous, stationary magnetic field perpendicular to the surface. We establish sufficient conditions for hyperbolicity of…

We point out how some recent developments in the theory of constant scalar curvature K\"ahler metrics can be used to clarify the existence issue for such metrics in the special case of geometrically ruled complex surfaces.

Studies of the formation of Landau levels based on the Schr\"odinger equation for electrons constrained to curved surfaces have a long history. These include as prime examples surfaces with constant positive and negative curvature, the…

We consider positivity properties of vector bundles over conic neighbourhoods of sections of submersions over $1$-convex manifolds.

In this paper we extend the local scalar curvature rigidity result in [6] to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in…

We study the deep interplay between geometry of quadrics in d-dimensional space and the dynamics of related integrable billiard systems. Various generalizations of Poncelet theorem are reviewed. The corresponding analytic conditions of…

We consider dynamics of the FRW Universe with a scalar field. Using Maupertuis principle we find a curvature of geodesics flow and show that zones of positive curvature exist for all considered types of scalar field potential. Usually,…

By exploiting the conformality of a warped product metric with a direct product metric, we develop a new connection on a twisted spinor bundle and its associated Dirac operator. We obtain a Llarull type scalar curvature rigidity for a…

Domains and more generally complex manifolds whose Bergman metrics have constant holomorphic sectional curvature are characterized. Our approach is to treat the Bergman metrics as the pull-back by the Bergman-Bochner maps of the…

We make a systematic investigation of quadrature properties for quadrics, namely integration of holomorphic functions over planar domains bounded by second degree curves. A full understanding requires extending traditional settings by…

In this note we show that the space of all metrics of positive Ricci curvature on a spin manifold of dimension 4k-1 for k at least 2 has infinitely many path components provided that the manifold admits a very particular kind of metric.

We obtain two types of results on positive scalar curvature metrics for compact spin manifolds that are even dimensional. The first type of result are obstructions to the existence of positive scalar curvature metrics on such manifolds,…

On one hand, we study the class of graphs on surfaces, satisfying tessellation properties, with positive Forman curvature on each edge. Via medial graphs, we provide a new proof for the finiteness of the class, and give a complete…

We explore the dynamical behaviour of cosmological models involving a scalar field (with an exponential potential and a canonical kinetic term) and a matter fluid with spatial curvature included in the equations of motion. Using…

The billiard problem of statistical physics is considered in a new geometric approach with a symmetric phase space. The structure and topological features of typical billiard phase portrait are defined. The connection between geometric,…

We use the Dirac operator method to prove a scalar-mean curvature comparison theorem for spin manifolds which carry iterated conical singularities. Our approach is to study the index theory of a twisted Dirac operator on such singular…

For Riemannian metrics of constant positive curvature on a punctured sphere with conic singularities at the punctures and co-axial monodromy of the developing map, possible angles at the singularities are completely described. This…

On a compact Riemannian manifold with boundary, we study the set of conformal metrics of negative constant scalar curvature in the interior and positive constant mean curvature on the boundary. Working in the case of positive Yamabe…

We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows…

This paper investigates conformal deformations of the scalar curvature and mean curvature on complete Riemannian manifolds with boundary. We establish sufficient conditions for the existence of conformal deformations to complete metrics…