α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
We consider the α\alpha -mean curvature flow for convex graphs in Euclidean space. Given a smooth, complete, strictly convex, non-compact initial hypersurface over a strictly convex projected domain, we derive uniform curvature bounds, which are independent of the height of a graph, to give C2{C}^{2...
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| Format: | Article |
| Language: | English |
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De Gruyter
2025-08-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2025-0101 |
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| author | Kang Hyunsuk Lee Ki-Ahm Lee Taehun |
| author_facet | Kang Hyunsuk Lee Ki-Ahm Lee Taehun |
| author_sort | Kang Hyunsuk |
| collection | DOAJ |
| description | We consider the α\alpha -mean curvature flow for convex graphs in Euclidean space. Given a smooth, complete, strictly convex, non-compact initial hypersurface over a strictly convex projected domain, we derive uniform curvature bounds, which are independent of the height of a graph, to give C2{C}^{2}-estimates for convex graphs. Consequently, these height-independent estimates imply that all the derivatives for level sets converge uniformly. Furthermore, with these estimates on level sets, the boundary of the domain of a graph, which demonstrates the behavior of level sets as the height tends to infinity, is shown to be a smooth solution for the α\alpha -mean curvature flow of codimension two in the classical sense. |
| format | Article |
| id | doaj-art-cc647b05f43141629cce83b51abacb3f |
| institution | Kabale University |
| issn | 2191-950X |
| language | English |
| publishDate | 2025-08-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj-art-cc647b05f43141629cce83b51abacb3f2025-08-25T06:10:01ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2025-08-0114115117110.1515/anona-2025-0101α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level setsKang Hyunsuk0Lee Ki-Ahm1Lee Taehun2Department of Mathematical Sciences, Gwangju Institute of Science and Technology, Gwangju 500-712, KoreaDepartment of Mathematical Sciences, Seoul National University, Seoul 151-747, KoreaDepartment of Mathematics, Konkuk University, Seoul 05029, KoreaWe consider the α\alpha -mean curvature flow for convex graphs in Euclidean space. Given a smooth, complete, strictly convex, non-compact initial hypersurface over a strictly convex projected domain, we derive uniform curvature bounds, which are independent of the height of a graph, to give C2{C}^{2}-estimates for convex graphs. Consequently, these height-independent estimates imply that all the derivatives for level sets converge uniformly. Furthermore, with these estimates on level sets, the boundary of the domain of a graph, which demonstrates the behavior of level sets as the height tends to infinity, is shown to be a smooth solution for the α\alpha -mean curvature flow of codimension two in the classical sense.https://doi.org/10.1515/anona-2025-0101mean curvature flowgraph53e1053c21 |
| spellingShingle | Kang Hyunsuk Lee Ki-Ahm Lee Taehun α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets Advances in Nonlinear Analysis mean curvature flow graph 53e10 53c21 |
| title | α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets |
| title_full | α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets |
| title_fullStr | α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets |
| title_full_unstemmed | α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets |
| title_short | α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets |
| title_sort | α mean curvature flow of non compact complete convex hypersurfaces and the evolution of level sets |
| topic | mean curvature flow graph 53e10 53c21 |
| url | https://doi.org/10.1515/anona-2025-0101 |
| work_keys_str_mv | AT kanghyunsuk ameancurvatureflowofnoncompactcompleteconvexhypersurfacesandtheevolutionoflevelsets AT leekiahm ameancurvatureflowofnoncompactcompleteconvexhypersurfacesandtheevolutionoflevelsets AT leetaehun ameancurvatureflowofnoncompactcompleteconvexhypersurfacesandtheevolutionoflevelsets |