  We consider closed immersed hypersurfaces in $\R^3$ and $\R^4$ evolving by a
special class of constrained surface diffusion flows. This class of constrained
flows includes the classical surface diffusion flow. In this paper we present a
Lifespan Theorem for these flows, which gives a positive lower bound on the
time for which a smooth solution exists, and a small upper bound on the total
curvature during this time. The hypothesis of the theorem is that the surface
is not already singular in terms of concentration of curvature. This turns out
to be a deep property of the initial manifold, as the lower bound on maximal
time obtained depends precisely upon the concentration of curvature of the
initial manifold in $L^2$ for $M^2$ immersed in $R^3$ and additionally on the
concentration in $L^3$ for $M^3$ immersed in $R^4$. This is stronger than a
previous result on a different class of constrained surface diffusion flows, as
here we obtain an improved lower bound on maximal time, a better estimate
during this period, and eliminate any assumption on the area of the evolving
hypersurface.
