The Growth of the Morse Index of Minimal Hypersurfaces in Four-Dimensional Space Forms

Core Concepts

In four-dimensional space forms, the Morse index of minimal hypersurfaces can grow linearly with certain geometric parameters, contrasting with the three-dimensional case where index growth is primarily determined by topology.

Abstract

Carlotto, A., Schulz, M. B., & Wiygul, D. (2024). Index growth not imputable to topology. arXiv preprint arXiv:2407.11147v2.

This research paper investigates the growth of the Morse index of minimal hypersurfaces in four-dimensional space forms, specifically the round four-sphere (S4) and the Euclidean four-ball (B4). The authors aim to demonstrate that, unlike in three dimensions, topological complexity is not the sole factor driving index growth in higher dimensions.

Key Insights Distilled From

Index growth not imputable to topology

by Alessandro C... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2407.11147.pdf

Deeper Inquiries

How might the insights gained from studying the Morse index of minimal hypersurfaces in four-dimensional space forms inform our understanding of similar problems in higher dimensions?

Answer: The study of minimal hypersurfaces in four-dimensional space forms, specifically the discovery of index growth not solely attributable to topology, provides crucial insights that can be extrapolated to higher dimensions. Here's how:

Challenging Existing Conjectures: The Schoen-Marques-Neves conjecture, positing a linear relationship between the Morse index and the first Betti number, holds true in three dimensions. However, the examples presented in the paper, like the Hsiang hypersurfaces in S⁴ and the O(2) x O(2)-invariant examples in B⁴, demonstrate that this conjecture might not extend directly to higher dimensions. This suggests the need for a more nuanced understanding of the factors influencing index growth.

Identifying New Geometric Factors: The paper highlights the role of symmetries and group actions in influencing the Morse index. The examples studied exhibit high degrees of symmetry, which contribute to the presence of degenerate eigenspaces and, consequently, higher index values. This suggests that group actions and the geometry they induce on the hypersurface play a significant role in determining the index, a factor that might be even more pronounced in higher dimensions.

Developing New Techniques: The paper utilizes partitioning methods, inspired by Montiel-Ros, adapted for general actions of compact Lie groups. These techniques prove effective in analyzing the index of the specific examples and hold promise for broader applicability in higher-dimensional settings. Further development and refinement of such methods could be instrumental in tackling similar problems in more general ambient manifolds.

Exploring New Topological Invariants: The limitations of the first Betti number in predicting index growth in higher dimensions necessitate the exploration of alternative topological invariants. Higher homotopy groups, homology groups with different coefficients, or more specialized invariants sensitive to the geometry of the embedding might provide a more accurate measure of topological complexity relevant to the Morse index.
In essence, the insights gained from the four-dimensional case provide a roadmap for future research, prompting the investigation of new geometric and topological factors, the development of advanced techniques, and the search for more refined topological invariants to understand the Morse index of minimal hypersurfaces in higher dimensions.

Could there be alternative explanations for the observed index growth, perhaps related to subtle topological invariants not captured by the first Betti number or other geometric features not yet considered?

Answer: Absolutely, the observed index growth beyond what the first Betti number can explain strongly suggests alternative explanations are at play. Here are some potential avenues for investigation:

Higher Homotopy Groups: While the first Betti number captures information about the "holes" in a space, higher homotopy groups, like the fundamental group (π₁) for loops or π₂ for spheres, describe more complex topological features. Minimal hypersurfaces in higher dimensions could possess intricate structures detectable by these higher groups, potentially correlating with index growth.

Torsion in Homology: The first Betti number only reflects the free part of the first homology group. Torsion subgroups, present in more general homology groups, capture "holes" that are "filled in" in a finite way. These subtle topological features might contribute to the index but remain undetected by the first Betti number alone.

Geometric Features of the Embedding: The specific way a minimal hypersurface embeds into the ambient manifold can significantly impact its index. Factors to consider include:

Curvature Concentration: Regions of high curvature on the hypersurface or in the ambient space might lead to higher index values.
Index of Intersection: For hypersurfaces with multiple components or self-intersections, the index of these intersections could contribute to the overall index growth.
Asymptotic Behavior: The asymptotic behavior of non-compact minimal hypersurfaces, such as their rate of approach to a limiting cone or hyperplane, might also influence the index.

Spectral Properties of the Jacobi Operator: The Morse index is fundamentally linked to the spectrum of the Jacobi operator. Investigating the distribution of eigenvalues, the presence of spectral gaps, or the behavior of the eigenfunctions could reveal deeper connections between the geometry of the hypersurface and its index.
In summary, the search for alternative explanations for index growth should encompass a broader exploration of topological invariants, a deeper understanding of the geometric interplay between the hypersurface and its ambient space, and a more refined analysis of the spectral properties of the Jacobi operator.

If we consider the space of all minimal hypersurfaces within a given ambient manifold as a kind of landscape, what does the Morse index tell us about the "shape" or "topology" of this landscape, and how does this relate to the geometry of the ambient space itself?

Answer: Thinking of the space of all minimal hypersurfaces as a landscape is a powerful analogy. In this landscape:

Minimal hypersurfaces are the "critical points": Just as hills, valleys, and mountain passes are critical points in a topographic map, minimal hypersurfaces represent equilibrium points within the space of all possible hypersurfaces.

The Morse index measures "instability": The Morse index, indicating the number of independent directions in which a hypersurface can be deformed to decrease its area (volume), corresponds to the "instability" of a critical point. A high index suggests a "mountain pass" or a "saddle point" – a critical point easily perturbed. Conversely, a low index suggests a more stable configuration, like a valley or a peak.

The landscape's shape reflects the ambient geometry: The overall "shape" or "topology" of this landscape is intricately linked to the geometry of the ambient manifold. For instance:

Positive curvature tends to create "peaks": In ambient spaces with positive curvature, like the sphere, minimal hypersurfaces are often "stable" (low index), analogous to peaks in the landscape. This is because positive curvature tends to "push" hypersurfaces outward, making it harder to find area-decreasing deformations.
Negative curvature favors "saddle points": In spaces with negative curvature, like hyperbolic space, minimal hypersurfaces are often "unstable" (high index), resembling saddle points. Negative curvature tends to "pull" hypersurfaces inward, making them more susceptible to area-decreasing deformations.

Index growth suggests a complex landscape: The discovery of index growth not solely explained by topology suggests that the landscape of minimal hypersurfaces, at least in dimensions four and higher, is far more complex than initially thought. It implies the existence of many "mountain passes" and "saddle points," indicating a rich and intricate structure within this space.
In conclusion, the Morse index provides a valuable tool for understanding the "shape" and "topology" of the space of minimal hypersurfaces. It reveals the stability of individual minimal hypersurfaces and provides insights into how the geometry of the ambient space influences the overall structure of this landscape. The observed index growth challenges our previous understanding and motivates further exploration of this intricate mathematical landscape.