Abstract
We consider a flow by powers of Gauss curvature under the obstruction that the flow cannot penetrate a prescribed region, so called an obstacle. For all dimensions and positive powers, we prove the optimal curvature bounds of solutions and all time existence with its long time behavior. We also prove the \(C^1\) regularity of free boundaries under a uniform thickness assumption.
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Acknowledgements
We are grateful to the referee for helpful comments. Ki-Ahm Lee was supported by NRF grant NRF-2020R1A2C1A01006256 funded by the Korean government (MSIP). Taehun Lee was supported by the NRF grant RS-2023-00211258 and KIAS Individual Grant MG079501.
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Lee, KA., Lee, T. Gauss curvature flow with shrinking obstacle. Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02739-y
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DOI: https://doi.org/10.1007/s00208-023-02739-y