A Remark on Conformal Anomaly and Extrinsic Geometry of Random Surfaces

In low dimensions, conformal anomaly has profound influence on the critical behavior of random surfaces with extrinsic curvature rigidity $1/\a$. We illustrate this by making a small $D$ expansion of rigid random surfaces, where a non-trivial infra-red fixed point is shown to exist. We speculate on the renormalization group flow diagram in the $(\a,D)$ plane. We argue that the qualitative behavior of numerical simulations in $D=3, 4$ could be understood on the basis of the phase diagram.