Lifespan Theorem for simple constrained surface diffusion flows |
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Authors: | Glen Wheeler |
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Institution: | School of Mathematics and Applied Statistics, University of Wollongong, Northfields Ave., Wollongong, NSW 2500, Australia |
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Abstract: | We consider closed immersed hypersurfaces in R3 and R4 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 L2 for M2 immersed in R3 and additionally on the concentration in L3 for M3 immersed in R4. 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. |
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Keywords: | Global differential geometry Fourth order Geometric analysis Parabolic partial differential equations |
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