Lifespan Theorem for simple constrained surface diffusion flows

We consider closed immersed hypersurfaces in R3 and R4 evolving by a special class of constrainedsurfacediffusionflows. This class of constrainedflows includes the classical surfacediffusionflow. In this paper we present a LifespanTheoremfor 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 L2forM2 immersed in R3 and additionally on the concentration in L3forM3 immersed in R4. This is stronger than a previous result on a different class of constrainedsurfacediffusionflows, 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.