On the area of constrained polygonal linkages

We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the configuration space. Its critical points are described and Bott-Morse indices are computed. This paper is a generalization of analogous results for polygonal linkages (obtained earlier by G. Khimshiashvili, G. Panina, and A. Zhukova)

Extremal Configuration of Robot Arms in Three Dimensions

We define a volume function for a robot arms in 3-dimensional Euclidean space and give geometric conditions for its critical points. For 3-arms this volume function is an exact topological Morse function on the 3-sphere.Comment: 13 pages; Updated version of sections 6-9 of Oberwolfach preprint 2011-2

T-equivariant disc potentials for toric Calabi-Yau manifolds

We formulate and compute the equivariant disc potentials of immersed SYZ fibers in toric Calabi-Yau manifolds, which are closely related to the open Gromov-Witten invariants of Aganagic-Vafa branes. The main tool is an equivariant version of the gluing method in \cite{CHL-glue,HKL}.First author draf

The injectivity radius of hyperbolic surfaces and some Morse functions over moduli spaces

This article is devoted to the variational study of two functions defined over some Teichmueller spaces of hyperbolic surfaces. One is the systole of geodesic loops based at some fixed point, and the other one is the systole of arcs.\par For each of them we determine all the critical points. It appears that the systole of arcs is a topological Morse function, whereas the systole of geodesic loops have some degenerate critical points. However, these degenerate critical points are in some sense the obvious one, and they do not interfere in the variational study of the function.\par At a nondegenerate critical point, the systolic curves (arcs or loops depending on the function involved) decompose the surface into regular polygons. This enables a complete classification of these points, and some explicit computations. In particular we determine the global maxima of these functions. This generalizes optimal inequalities due to Bavard and Deblois. We also observe that there is only one local maximum, this was already proved in some cases by Deblois.\par Our approach is based on the geometric Vorono\''i theory developed by Bavard. To use this variational framework, one has to show that the length functions (of arcs or loops) have positive definite Hessians with respect to some system of coordinates for the Teichm\''uller space. Following a previous work, we choose Bonahon's shearing coordinates, and we compute explicitly the Hessian of the length functions of geodesic loops. Then we use a characterization of the nondegenerate critical points due to Akrout.Comment: Preliminary version, to be improved shortly.The author is fully supported by the fund FIRB 2010 (RBFR10GHHH003