In this article we give necessary and sufficient conditions for an irreducible K¨ahler C-space with b2 = 1 to have nonnegative
or positive quadratic bisectional curvature, assuming the space is not Hermitian symmetric. In the cases of the five exceptional Lie
groups E6,E7,E8, F4,G2, the computer package MAPLE is used to assist our calculations. The results are related to two conjectures
We show that any star-shaped convex hypersurface with constantWeingarten curvature in the deSitter-Schwarzschildmanifold
is a sphere of symmetry.Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold.
We prove the existence of an isoperimetric surface for any value of the enclosed volume, and we completely describe the isoperimetric
surfaces for very large enclosed volume. This complements work in H. Bray’s thesis, where isoperimetric surfaces homologous to the
horizon are studied.
This work uncovers the tropical analogue, for measured laminations, of the convex hull construction in decorated Teichm¨uller
theory; namely, it is a study in coordinates of geometric degeneration to a point of Thurston’s boundary for Teichm¨uller space. This
may offer a paradigm for the extension of the basic cell decomposition of Riemann’s moduli space to other contexts for general
moduli spaces of flat connections on a surface. In any case, this discussion drastically simplifies aspects of previous related studies
as is explained. Furthermore, a new class of measured laminations relative to an ideal cell decomposition of a surface is discovered
in the limit. Finally, the tropical analogue of the convex hull construction inMinkowski space is formulated as an explicit algorithm
that serially simplifies a triangulation with respect to a fixed lamination and has its own independent interest.
We give a complete classification of irreducible symmetric spaces for which there exist proper SL(2,R)-actions as isometries, using
the criterion for proper actions by T. Kobayashi [Math. Ann. ’89] and combinatorial techniques of nilpotent orbits. In particular, we
classify irreducible symmetric spaces that admit surface groups as discontinuous groups, combining this with Benoist’s theorem
[Ann. Math. ’96].