For a finite dimensional vector space equipped with a C-algebra structure, one can define rational maps using the algebraic structure. In this paper, we describe the growth of the degree sequences for this type of rational maps. [ABSTRACT FROM AUTHOR]
This paper begins the study of relations between Riemannian geometry and contact topology on (2n + 1)-manifolds and continues this study on 3-manifolds. Specifically we provide a lower bound for the radius of a geodesic ball in a contact (2n+ 1)-manifold (M, ξ) that can be embedded in the standard contact structure on R2n+1, that is, on the size of a Darboux ball. The bound is established with respect to a Riemannian metric compatible with an associated contact form a for ξ. In dimension 3, this further leads us to an estimate of the size for a standard neighborhood of a closed Reeb orbit. The main tools are classical comparison theorems in Riemannian geometry. In the same context, we also use holomorphic curve techniques to provide a lower bound for the radius of a PS-tight ball. [ABSTRACT FROM AUTHOR]
The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three- dimensional generic shrinking Ricci soliton is given by quotients of either S³, RS² or R³ under some very weak conditions on the vector field X generating the soliton structure. In doing so we introduce analytical tools that could be useful in other settings; for instance we prove that the Omori-Yau maximum principle holds for the X-Laplacian on every generic Ricci soliton without any assumption on X. [ABSTRACT FROM AUTHOR]