If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f)$ of $f$ is nonzero (in a suitable domain). If the metric $g$ has nonzero Gaussian curvature, then $f$ is nonconstant and therefore $Var(f)$ is nonzero. Can this conclusion be quantified? Namely, assume $g$ has curvature bounded away from zero on a suitable disk. Can one get a lower bound for $Var(f)$? This has immediate applications to a stronger version of Loewner's torus inequality with isosystolic defect term a la Bonnesen, see http://arxiv.org/abs/arXiv:0803.0690 and http://arxiv.org/abs/1105.0553

Note 1. The Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is given by $K=\frac{-1}{2f^2}\left(\frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}\right)\log f$, so that the problem involves partial differential inequalities for the Laplacian.

Note 2. To restate the curvature hypothesis more carefully: we have a metric $r$-disk (for the metric $g$) where the curvature is bounded below by an $\epsilon>0$. The problem is to obtain a lower bound for $Var(f)$ in terms of $r$ and $\epsilon$ (certainly the estimate will become weaker as $r$ gets smaller).

Note 3 (reformulation along the lines of Robert's suggestion). On a fixed domain $\Omega$ of unit area in the $xy$-plane, we consider metrics $g=f^2(dx^2+dy^2)$ with the property that $\Omega$ contains a subdomain (say, a disk) $D$ on which Gaussian curvature $K\geq C>0$ and such that the $g$-area of $D$, i.e., $\int_D f^2 dxdy$, is at least $A>0$. We want to know whether there is a lower bound for $Var_\Omega(f)$ in terms of the constants $C$ and $A$. Is there an optimal lower bound, and if so does a rotationally symmetric metric on a disk $D$ attain it? A similar question for $K$ negative and bounded away from $0$.