Integral geometry of complex space forms

We show how Alesker's theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this approach yields a new formula expressing the volumes of the tubes about a totally real submanifold in terms of its intrinsic Riemannian structure. We also show by direct calculation that the Lipschitz-Killing valuations stabilize the subspace of invariant angular curvature measures, suggesting the possibility that a similar phenomenon holds for all Riemannian manifolds. We conclude with a number of open questions and conjectures.Comment: 68 pages; minor change

Uniform convergence of discrete curvatures from nets of curvature lines

We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.Comment: 21 pages, 8 figure

Fluorescent T7 display phages obtained by translational frameshift

Lytic phages form a powerful platform for the display of large cDNA libraries and offer the possibility to screen for interactions with almost any substrate. To visualize these interactions directly by fluorescence microscopy, we constructed fluorescent T7 phages by exploiting the flexibility of phages to incorporate modified versions of its capsid protein. By applying translational frameshift sequences, helper plasmids were constructed that expressed a fixed ratio of both wild-type capsid protein (gp10) and capsid protein fused to enhanced yellow fluorescent protein (EYFP). The frameshift sequences were inserted between the 3′ end of the capsid gene and the sequence encoding EYFP. Fluorescent fusion proteins are only formed when the ribosome makes a −1 shift in reading frame during translation. Using standard fluorescence microscopy, we could sensitively monitor the enrichment of specific binders in a cDNA library displayed on fluorescent T7 phages. The perspectives of fluorescent display phages in the fast emerging field of single molecule detection and sorting technologies are discussed