Geometric bounds on the growth rate of nullcontrollability cost for the heat equation in small time

Given a control region (omega) on a compact Riemannian manifold M, we consider the heat equation with a source term g localized in (omega). It is known that any initial data in L^2(M) can be steered to 0 in an arbitrarily small time T by applying a suitable control g in L^2([0,T]×(omega)), and, as T tends to 0, the norm of g grows like exp(C/T) times the norm of the data. We investigate how C depends on the geometry of (omega). We prove C>=d^2/4 where d is the largest distance of a point in M from (omega). When M is a segment of length L controlled at one end, we prove C=<(alpha)*L^2 for some (alpha)* <2. Moreover, this bound implies C=<(alpha)*L^2(omega) where L(omega) is the length of the longest generalized geodesic in M which does not intersect (omega). The control transmutation method used in proving this last result is of a broader interest.