Existence of boundary points of the systole function on Teichmüller space of closed surfaces

Determine whether boundary points of the systole function f_sys exist in Teichmüller space T_g of closed, connected, orientable hyperbolic surfaces of genus g≥2; equivalently, ascertain whether there exists any point x in T_g at which the cone of increase of f_sys is not full (i.e., there is no open cone of directions in which f_sys increases to first order), or, in Schmutz’s terminology, whether there is any point where Sys(C) intersects ∂Min(C) for some filling set of curves C.

Background

The systole function f_sys assigns to each point in Teichmüller space the length of the systoles (shortest geodesics) of the corresponding hyperbolic surface and is a piecewise-smooth topological Morse function. Because f_sys is not smooth, regular points can, in principle, have no open cone of directions along which f_sys increases to first order; such points are called boundary points of the systole function.

Schmutz provided an equivalent characterisation: boundary points of f_sys are points where the stratum Sys(C) of systoles intersects the boundary ∂Min(C) of the set of minima associated with a filling set of curves C. Boundary points (as well as critical points) are known to be isolated, and explicit examples of boundary points are known for surfaces with boundary. However, for closed surfaces (without boundary), it remains unresolved whether any such points occur at all.

Establishing the existence or nonexistence of boundary points for closed surfaces would clarify the local and global structure of the Thurston spine and the behaviour of f_sys as a topological Morse function, with implications for Morse–Smale-type properties and related geometric constructions.