Bibliographic Information: Han, X., Li, J., Sun, J. (2024). Translating Solitons to a Lagrangian Mean Curvature Flow with Zero Maslov Class. arXiv:2410.17850v1 [math.DG]
Research Objective: This paper investigates whether specific examples of Lagrangian translating solitons, particularly those constructed by Joyce-Lee-Tsui and the Grim Reaper, can arise as blow-up limits of finite-time singularities in Lagrangian mean curvature flows with zero Maslov class.
Methodology: The authors develop a new weighted monotonicity formula for Lagrangian mean curvature flow with zero Maslov class. By applying this formula, they derive a necessary condition for an eternal mean curvature flow to emerge as a blow-up limit of such a flow. They then employ this condition to analyze the specific examples of translating solitons in question.
Key Findings: The authors successfully demonstrate that neither the Joyce-Lee-Tsui translating solitons nor the Grim Reaper can be blow-up limits at a finite time singularity of a Lagrangian mean curvature flow with zero Maslov class. This result is achieved by constructing specific test functions within the derived necessary condition, leading to a contradiction.
Main Conclusions: The paper refutes the possibility of the examined translating solitons being blow-up limits for Lagrangian mean curvature flows with zero Maslov class. This conclusion provides a definitive answer to an open question posed by Joyce-Lee-Tsui and Neves-Tian.
Significance: This research significantly contributes to the understanding of singularity formation in Lagrangian mean curvature flows, particularly in the case of zero Maslov class. It clarifies the limitations on potential blow-up limits and provides valuable insights into the long-term behavior of these flows.
Limitations and Future Research: The paper focuses on specific examples of translating solitons. Further research could explore the broader class of translating solitons or other types of eternal solutions to determine their potential as blow-up limits in this context. Additionally, investigating the implications of these findings for the Thomas-Yau conjecture on the long-time existence and convergence of Lagrangian mean curvature flows remains an open avenue for future investigation.
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by Xiaoli Han, ... at arxiv.org 10-24-2024
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