  We consider the renormalization of d-dimensional hypersurfaces (branes)
embedded in flat (d+1)-dimensional space. We parametrize the truncated
effective action in terms of geometric invariants built from the extrinsic and
intrinsic curvatures. We study the renormalization-group running of the
couplings and explore the fixed-point structure. We find evidence for an
ultraviolet fixed point similar to the one underlying the asymptotic-safety
scenario of gravity. We also examine whether the structure of the Galileon
theory, which can be reproduced in the nonrelativistic limit, is preserved at
the quantum level.
