Geometric flows and space-periodic solitons on the light-cone
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909485640974336 |
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| author | Yang, Yun |
| author_facet | Yang, Yun |
| contents | This paper investigates curve flows on the light-cone in the 3-dimensional Minkowski space. We derive the Harnack inequality for the heat flow and present a detailed classification of space-periodic solitons for a third-order curvature flow. The nontrivial periodic solutions to this flow are expressed in terms of the Jacobi elliptic sine function. Additionally, the closed soliton solutions form a family of transcendental curves, denoted by $\mathrm{x}_{p,q}$, which are characterized by a rotation index $p$ and close after $q$ periods of their curvature functions. The ratio $p/q$ satisfies $p/q \in (\sqrt{2/3}, 1)$, where $p$ and $q$ are relatively prime positive integers. Guided by the classification process, we obtain the analytic solutions to a second-order nonlinear ordinary differential equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06035 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric flows and space-periodic solitons on the light-cone Yang, Yun Differential Geometry This paper investigates curve flows on the light-cone in the 3-dimensional Minkowski space. We derive the Harnack inequality for the heat flow and present a detailed classification of space-periodic solitons for a third-order curvature flow. The nontrivial periodic solutions to this flow are expressed in terms of the Jacobi elliptic sine function. Additionally, the closed soliton solutions form a family of transcendental curves, denoted by $\mathrm{x}_{p,q}$, which are characterized by a rotation index $p$ and close after $q$ periods of their curvature functions. The ratio $p/q$ satisfies $p/q \in (\sqrt{2/3}, 1)$, where $p$ and $q$ are relatively prime positive integers. Guided by the classification process, we obtain the analytic solutions to a second-order nonlinear ordinary differential equation. |
| title | Geometric flows and space-periodic solitons on the light-cone |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2502.06035 |