Translating surfaces under flows by sub-affine-critical powers of Gauss curvature
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866913435773566976 |
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| author | Choi, Beomjun Choi, Kyeongsu Kim, Soojung |
| author_facet | Choi, Beomjun Choi, Kyeongsu Kim, Soojung |
| contents | We classify the surfaces translating under the flows by sub-affine-critical powers of the Gauss curvature. This, in particular, lists all translating solitons possibly model Type II singularities for convex closed solutions in all positive powers. The surfaces are entire graphs, and therefore our result corresponds to the Liouville theorem for the degenerate Monge--Ampère equations $\det D^2 u=(1+|Du|^2)^{2-\frac{1}{2α}}$ on $\mathbb{R}^2$ in the range $0<α<1/4$. The result also reveals that the moduli spaces of solutions are homeomorphic to either Euclidean spaces or cylinders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_13186 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Translating surfaces under flows by sub-affine-critical powers of Gauss curvature Choi, Beomjun Choi, Kyeongsu Kim, Soojung Differential Geometry Analysis of PDEs 53E99, 35J96 We classify the surfaces translating under the flows by sub-affine-critical powers of the Gauss curvature. This, in particular, lists all translating solitons possibly model Type II singularities for convex closed solutions in all positive powers. The surfaces are entire graphs, and therefore our result corresponds to the Liouville theorem for the degenerate Monge--Ampère equations $\det D^2 u=(1+|Du|^2)^{2-\frac{1}{2α}}$ on $\mathbb{R}^2$ in the range $0<α<1/4$. The result also reveals that the moduli spaces of solutions are homeomorphic to either Euclidean spaces or cylinders. |
| title | Translating surfaces under flows by sub-affine-critical powers of Gauss curvature |
| topic | Differential Geometry Analysis of PDEs 53E99, 35J96 |
| url | https://arxiv.org/abs/2104.13186 |