Passing through nondegenerate singularities in mean curvature flows (2501.16678v1)

Abstract: In this paper, we study the properties of nondegenerate cylindrical singularities of mean curvature flow. We prove they are isolated in spacetime and provide a complete description of the geometry and topology change of the flow passing through the singularities. Particularly, the topology change agrees with the level sets change near a critical point of a Morse function, which is the same as performing surgery. The proof is based on a new $L^2$-distance monotonicity formula, which allows us to derive a discrete almost monotonicity of the ``decay order", a discrete mean curvature flow analog to Almgren's frequency function.

• The paper provides a detailed characterization of isolated nondegenerate cylindrical singularities in mean curvature flows using a rescaled analysis approach.
• It introduces an L2-distance monotonicity framework and decay order concept to capture the exponential decay behavior near singularities.
• The results bridge geometric topology with Morse theory, offering insights that may extend singularity analysis to higher-dimensional flows.

An Analysis of Nondegenerate Singularities in Mean Curvature Flow

The paper under consideration thoroughly investigates the behavior of nondegenerate cylindrical singularities in mean curvature flows (MCFs). This type of singularity, often encountered in geometric analysis, suggests key implications for the progression and topology of the flow. By providing a comprehensive framework, the authors describe the significance of nondegenerate singularities and demonstrate their isolated nature in spacetime.

Key Contributions

The primary contribution of the paper is the detailed examination and characterization of nondegenerate singularities within MCFs. Nondegeneracy refers to situations where singularities exhibit a specific form of stability, marked by the absence of degenerate behavior that might otherwise complicate the flow. The authors build on the rescaled mean curvature flow (RMCF) approach to effectively analyze singularities and their spatio-temporal evolution.

Singular Behavior and Pseudolocality

The paper accomplishes several important tasks:

1. Characterization of Nondegenerate Cylindrical Singularities: The paper establishes that these singularities are isolated in spacetime, meaning that nearby times and regions exhibit different topological characteristics.
2. Geometric and Topological Descriptions: Systematic breakdown of how flows transition through these singularities. The results align with Morse theory, implicating that the level sets change is equivalent to performing surgery near the critical points.
3. Monotonicity Formulation: Introduction of a new $L^2$-distance monotonicity framework and the concept of decay order. This approach provides a robust means to analyze the exponential decay behavior of singularity-governed flows.

Analytical Framework

A significant development in the paper is the derivation of a monotonicity formula specific to nondegenerate cylindrical singularities. The $L^2$-distance, effectively capturing the discrepancy between the evolving hypersurface and a reference cylinder, exhibits nonconcentration near infinity. This principle ensures that a large portion of the $L^2$ mass remains nearby, a feature that is crucial in understanding the local dynamics of the singular flow.

Additionally, the introduction of decay order serves as a discrete parabolic analogue to classical frequency functions. Monitoring the decay order offers insight into the speed at which flows resolve toward or diverge from cylindrical structures. This alignment with Almgren's frequency function reveals deeper analytical parallels.

Implications and Future Directions

Given that this paper touches on fundamental elements of geometric flows, its findings resonate well with ongoing research aimed at understanding similar phenomena in more expansive geometrical settings. The isolation and characterization of nondegenerate singularities illuminate broader questions of stability and topology, providing groundwork for future exploration into higher-dimensional settings and the interaction of complex singularities.

Potential Applications:

• Refinement of MCFs: Potentially incorporate the analytical techniques proposed here into more generalized frameworks.
• Geometric Topology: The investigation offers tools to better understand evolutionary processes on manifolds with complex topologies.
• Theory Extension: Laying the groundwork for extending these principles beyond hypersurfaces into higher-dimensional and non-Euclidean spaces.

Conclusion

This paper offers an in-depth portrayal of MCFs through the lens of nondegenerate singularity behavior. Its robust analytical methods and novel paradigms such as the decay order analysis suggest substantial implications for understanding singular behaviors in geometric flows. As such, the insights gained from this research could serve as foundational pillars supporting advancements in geometric and topological theories broadly.