Intersections of totally real and holomorphic disks

It is shown that a holomorphically embedded open disk in C2 and a totally real embedded open disk which have a common smooth boundary have nontrivial intersection

Single basepoint subdivision schemes for manifold-valued data: Timesymmetry without space-symmetry.

Abstract. This paper establishes smoothness results for a class of nonlinear subdivision schemes, known as the single basepoint manifold-valued subdivision schemes, which shows up in the construction of waveletlike transform for manifold-valued data. This class includes the (single basepoint) Log-Exp subdivision scheme as a special case. In these schemes, the exponential map is replaced by a so-called retraction map f from the tangent bundle of a manifold to the manifold. It is known that any choice of retraction map yields a C 2 scheme, provided the underlying linear scheme is C 2 (this is called "C 2 equivalence"). But when the underlying linear scheme is C 3 , Navayazdani and Yu have shown that to guarantee C 3 equivalence, a certain tensor P f associated to f must vanish. They also show that P f vanishes when the underlying manifold is a symmetric space and f is the exponential map. In the present paper, a geometric interpretation of the tensor P f is given. Associated to the retraction map f is a torsion-free affine connection, which in turn defines an exponential map. The condition P f = 0 is shown to be equivalent to the condition that f agrees with the exponential map of the connection up to the 3rd order. In particular, when f is the exponential map of a connection, one recovers the original connection and P f vanishes. It then follows that the condition P f = 0 is satisfied by a wider class of manifolds than was previously known. Under the additional assumption that the subdivision rule satisfies a time-symmetry, it is shown that the vanishing of P f also guarantees C 4 equivalence. Finally, the analysis in the paper strongly indicates that vanishing curvature of the connection associated to f is a necessary condition for C k equivalence for k ≥ 5

Architecture and utopia, 2015

No abstract available

Contact geometry and CR-structures on spheres

A normal form for small CR-deformations of the standard CR-structure on the (2n+1)-sphere is presented. The space of normal forms is parameterized by a single function on the sphere. For n>1, the normal form is used to obtain explicit embeddings into $ℂ^{n+1}$. For n=1, the cohomological obstruction to embeddability is identified

Intersections of totally real and holomorphic disks

It is shown that a holomorphically embedded open disk in C2 and a totally real embedded open disk which have a common smooth boundary have nontrivial intersection