[Paper Review] Inductive Analysis on Singular Minimal Hypersurfaces

Joachim Lohkamp|ArXiv.org|Aug 14, 2008

Geometric Analysis and Curvature Flows17 references3 citations

TL;DR

This paper establishes a framework for analyzing the conformal Laplacian and its first eigenfunctions on singular minimal hypersurfaces by introducing 'cone reducible functors' that ensure compatibility between the asymptotic behavior of these operators near singularities and their limit models—tangent cones. The key result is that Perron solutions on all tangent cones at a singular point share identical growth rates, enabling a unified treatment of scalar curvature geometry in both smooth and singular settings.

ABSTRACT

The present paper describes a way to relate Martin boundaries on spaces of varying topology. This enables us to approach some detailed inductive analysis of the eigenfunctions of conformal Laplacians on minimal hypersurfaces near their singularities. This can directly be used resp. translated to understand the way how such minimal hypersurfaces inherit positive scalar curvature from their ambience resp. how to smooth singular minimal hypersurfaces to regular hypersurfaces with positive mean curvature.

Motivation & Objective

• To develop a geometric analysis framework for singular minimal hypersurfaces that extends the classical theory of conformal Laplacians from smooth compact manifolds.
• To address the challenge of transferring scalar curvature and spectral information from singular minimal hypersurfaces to their tangent cones via a functorial structure.
• To establish that the first eigenfunctions of the conformal Laplacian on singular minimal hypersurfaces behave consistently under cone reduction, ensuring compatibility with the limit geometry.
• To enable the construction of scalar curvature positive metrics on singular minimal hypersurfaces through stratified surgeries, thereby unifying smooth and singular cases in scalar curvature geometry.

Proposed method

• Introduces the concept of 'cone reducible functors' to formalize the compatibility of geometric objects—such as eigenfunctions of the conformal Laplacian—between a singular minimal hypersurface and its tangent cones under rescaling.
• Uses the flat norm convergence of rescaled hypersurfaces to tangent cones to ensure that the asymptotic behavior of eigenfunctions on the hypersurface matches their behavior on the limit cone.
• Applies a telescope argument across a sequence of approximating tangent cones to track the growth rate of solutions to the conformal Laplacian, leveraging compactness of the space of tangent cones.
• Employs the freezing effect—where the variation of tangent cones slows near the singular set—to ensure that recovery of the correct growth rate dominates over perturbations during cone transitions.
• Utilizes the Perron solution construction on cones to define a canonical positive solution with controlled growth, which is then compared across different tangent cones.
• Establishes that the growth rate of the Perron solution is invariant across all tangent cones at a given singular point, using iterative approximation and uniform bounds from compactness.

Experimental results

Research questions

• RQ1 Can the conformal Laplacian and its first eigenfunctions on singular minimal hypersurfaces be analyzed in a way compatible with tangent cone reductions?
• RQ2 Do all tangent cones at a singular point of a minimal hypersurface admit Perron solutions with identical growth rates?
• RQ3 How can scalar curvature information from a singular minimal hypersurface be transferred to its tangent cones in a stable and functorial manner?
• RQ4 To what extent can the spectral theory of the conformal Laplacian on singular minimal hypersurfaces be reduced to the analysis of model cones?
• RQ5 Can the theory of conformal deformations on singular minimal hypersurfaces be used to construct scalar curvature positive metrics via stratified surgery?

Key findings

• The assignment of first eigenfunctions of the conformal Laplacian to singular minimal hypersurfaces is cone reducible, meaning their asymptotic behavior near a singular point is governed by the limit cone's eigenfunctions.
• All Perron solutions on tangent cones at a given singular point $ p otin ext{Reg}(H) $ have the same growth rate, a consequence of the telescope argument and compactness of the tangent cone space.
• The growth rate of the Perron solution on any tangent cone is preserved under cone transitions, with recovery processes dominating perturbations due to the freezing effect near the singular set.
• The conformal Laplacian's eigenfunctions on singular minimal hypersurfaces can be controlled uniformly near the singular set using this cone-reduction framework.
• The theory enables the construction of scalar curvature positive metrics on singular minimal hypersurfaces that are amenable to stratified surgeries, providing a lossless method to eliminate singularities.
• The framework unifies the treatment of smooth and singular minimal hypersurfaces in scalar curvature geometry by extending the classical conformal deformation method to the singular case.