Abstract
We study the evolution of strictly mean-convex entire graphs over \({{\mathbb {R}}}^n\) by Inverse Mean Curvature flow. First we establish the global existence of starshaped entire graphs with superlinear growth at infinity. The main result in this work concerns the critical case of asymptotically conical entire convex graphs. In this case we show that there exists a time \( T < +\infty \), which depends on the growth at infinity of the initial data, such that the unique solution of the flow exists for all \(t < T\). Moreover, as \(t \rightarrow T\) the solution converges to a flat plane. Our techniques exploit the ultra-fast diffusion character of the fully-nonlinear flow, a property that implies that the asymptotic behavior at spatial infinity of our solution plays a crucial influence on the maximal time of existence, as such behavior propagates infinitely fast towards the interior.
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Acknowledgements
P. Daskalopoulos has been partially supported by NSF Grant DMS-1600658. She also wishes to thank University of Tübingen and Oberwolfach Research Institute for Mathematics for their hospitality during the preparation of this work. G. Huisken wishes to thank Columbia University and the Institute for Theoretical Studies ITS at ETH Zürich for their hospitality during the preparation of this work.
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Communicated by L. Ambrosio.
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