Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time

Given a control region $\mathrm{\Omega }$ 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 stirred to 0 in an arbitrarily small time $T$ by applying a suitable control $g$ in $L^2([0,T]xOmega)$, and, as $T$ tends to 0, the norm of $g$ grows like $e^{(}C\mathrm{/}T)$ times the norm of the data. We investigate how $C$ depends on the geometry of $Omega$. We prove $C\ge d^2\mathrm{/}4$ where $d$ is the largest distance of a point in $M$ from $\mathrm{\Omega }$. When $M$ is a segment of length $L$ controlled at one end, we prove $C\le alpha{L}^2$ for some $alpha<2$. Moreover, this bound implies $C\leq alpha L_{Omega}^2$where$L_{Omega}$ is the length of the longest generalized geodesic in $M$ which does not intersect $\mathrm{\Omega }$. The control transmutation method used in proving this last result is of a broader interest.