The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of the partial differential operators that are surjective on

The local Phragmén-Lindelöf condition for analytic subvarieties of  at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hörmander has shown. Here, necessary geometric conditions for this Phragmén-Lindelöf condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in  . The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on  .

     

A uniqueness result of Kähler Ricci flow with an application

In this paper, we will study the problem of uniqueness of Kähler Ricci flow on some complete noncompact Kähler manifolds and the convergence of the flow on with the initial metric constructed by Wu and Zheng.

     

Geometric inequalities for a class of exponential measures

Using -ellipsoids we prove versions of the inverse Santaló inequality and the inverse Brunn-Minkowski inequality for a general class of measures replacing the usual volume on . This class contains in particular the Gaussian measure on .

     

Simple birational extensions of the polynomial algebra

The Abhyankar-Sathaye Problem asks whether any biregular embedding can be rectified, that is, whether there exists an automorphism such that is a linear embedding. Here we study this problem for the embeddings whose image is given in by an equation , where and . Under certain additional assumptions we show that, indeed, the polynomial is a variable of the polynomial ring (i.e., a coordinate of a polynomial automorphism of ). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings . Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial as above, a criterion for when .

     

Radix representations, self-affine tiles, and multivariable wavelets

We investigate the connection between radix representations for and self-affine tilings of . We apply our results to show that Haar-like multivariable wavelets exist for all dilation matrices that are sufficiently large.

     

Upper bound for isometric embeddings

The isometric embeddings (, ) over a field are considered, and an upper bound for the minimal is proved. In the commutative case ( ) the bound was obtained by Delbaen, Jarchow and Pełczyński (1998) in a different way.

     

Directional derivative estimates for Berezin's operator calculus

Directional derivative estimates for Berezin symbols of bounded operators on Bergman spaces of arbitrary bounded domains in are obtained. These estimates also hold in the setting of the Segal-Bargmann space on . It is also shown that our estimates are sharp at every point of by exhibiting the optimizers explicitly.

     

Hausdorff dimension of Cantor sets and polynomial hulls

We give examples of Cantor sets in of Hausdorff dimension 1 whose polynomial hulls have non-empty interior.

     

Manifolds with an -action on the tangent bundle

We study manifolds arising as spaces of sections of complex manifolds fibering over with the normal bundle of each section isomorphic to .

     

Every real ellipsoid in admits CR umbilical points

We prove that every real ellipsoid admits at least four umbilical points, which can be compared to the result of Webster that a generic real ellipsoid in with does not admit any umbilical point.