Continuous family of surfaces translating by powers of Gauss curvature
| Autor: | Choi, Beomjun, Choi, Kyeongsu, Kim, Soojung |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | This paper shows the existence of convex translating surfaces under the flow by the $\alpha$-th power of Gauss curvature for the sub-affine-critical regime $ 0 < \alpha < 1/4$. The key aspect of our study is that our ansatz at infinity is the graph of homogeneous functions whose level sets are closed curves shrinking under the flow by the $\frac{\alpha}{1-\alpha}$-th power of curvature. For each ansatz, we construct a family of translating surfaces generated by the Jacobi fields with effective growth rates. Moreover, the construction shows quantitative estimate on the rate of convergence between different translators to each other, which is required to show the continuity of the family. As a result, the family is regarded as a topological manifold. The construction in this paper will become the ground of forthcoming research, where we aim to prove that every translating surface must correspond to one of the solutions obtained herein, classifying translating surfaces and identifying the topology of the moduli space. Comment: A part of this work was introduced in arXiv:2104.13186v1; however, we have separated and further elaborated on this existence theory. See footnote 3 on page 3 of the manuscript. In v2, we added Corollary A.1 |
| Databáze: | arXiv |
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