[Paper Review] Problems related to isoparametric theory

Jianquan Ge|arXiv (Cornell University)|Oct 27, 2019

Analytic and geometric function theory50 references4 citations

TL;DR

This paper surveys and proposes open problems in isoparametric theory, focusing on classification and curvature properties of isoparametric hypersurfaces in unit spheres and general Riemannian manifolds. It introduces conjectures linking minimal hypersurfaces with constant scalar curvature to isoparametricity, spectral properties of minimal isoparametric hypersurfaces, and curvature positivity in isoparametric foliations, particularly in relation to exotic smooth structures and the 4-dimensional Poincaré conjecture.

ABSTRACT

In this note we briefly survey and propose some open problems related to isoparametric theory.

Motivation & Objective

• To identify and formulate key open problems in isoparametric theory, especially concerning classification and curvature properties in Riemannian manifolds.
• To explore the relationship between minimal hypersurfaces with constant scalar curvature and isoparametricity, extending the Chern conjecture.
• To investigate the spectral properties of minimal isoparametric hypersurfaces in spheres, particularly for $g=4$ and $g=6$ cases.
• To examine whether leafwise nonnegative curvature in isoparametric foliations implies global nonnegative curvature on the ambient manifold.
• To analyze the compatibility of nonnegatively curved metrics on leaves with the existence of a smooth family of leafwise metrics in isoparametric foliations.

Proposed method

• Utilizes the framework of isoparametric functions satisfying $|\nabla f|^2 = b(f)$ and $\Delta f = a(f)$, with $f$ real analytic.
• Applies the Cartan-Münzner polynomial formulation for isoparametric hypersurfaces in $\mathbb{S}^{n+1}$, where $F(x)$ is a homogeneous polynomial of degree $g$ satisfying specific gradient and Laplacian conditions.
• Employs the concept of singular Riemannian foliations (SRF) and foliated diffeomorphisms to classify isoparametric foliations up to diffeomorphism, not just isometry.
• Analyzes leafwise curvature via one-parameter families of smooth metrics $\mathbf{g}_t$ on the leaves $M_t$, restricting to the regular part $N' = N \setminus (M_- \cup M_+)$.
• Uses the canonical bundle-like metric $\mathbf{g}^N$ to define a canonical family of smooth leafwise metrics $\mathbf{g}_t^N$ on the regular foliation $\mathfrak{F}'$.
• Applies results from differential topology, such as Zhang’s theorem on positive leafwise scalar curvature implying positive scalar curvature on the ambient manifold, to generalize to singular isoparametric foliations.

Experimental results

Research questions

• RQ1 Does every closed minimal hypersurface with constant scalar curvature in $\mathbb{S}^{n+1}$ necessarily have constant principal curvatures (i.e., is isoparametric), as per the Chern conjecture?
• RQ2 What is the full spectrum of the Laplacian on minimal isoparametric hypersurfaces for $g=4$ and $g=6$, beyond the first eigenvalue?
• RQ3 If all leaves of an isoparametric foliation have nonnegative (or positive) sectional, Ricci, or scalar curvature, does the ambient manifold admit a metric with corresponding global nonnegative curvature?
• RQ4 Can nonnegatively curved metrics on individual isoparametric hypersurfaces and focal submanifolds be extended to a smooth family of leafwise metrics on the entire foliation?
• RQ5 Does the Tang conjecture—that every isoparametric hypersurface and focal submanifold in $\mathbb{S}^n$ admits a metric of nonnegative sectional curvature—hold for homotopy spheres, including Hitchin’s exotic spheres?

Key findings

• The Chern conjecture remains open for $n \geq 4$, though it is known to be true for $n \leq 3$, including the case of constant mean curvature hypersurfaces.
• The spectrum of minimal isoparametric hypersurfaces is fully computed for $g=1$ (spheres), $g=2$ (Clifford tori), and $g=3$ (Veronese-type tubes), but only the first eigenvalue is known for $g=4$ and $g=6$ cases.
• For $g=4$ and $g=6$, the first eigenvalue of the Laplacian on minimal isoparametric hypersurfaces is $n$, confirming Yau’s conjecture in the isoparametric case.
• The Tang conjecture holds for homogeneous isoparametric foliations due to the existence of normal homogeneous metrics with nonnegative sectional curvature.
• The existence of nonnegatively curved metrics on individual leaves does not guarantee a smooth family of leafwise metrics on the entire foliation, which may explain why the Tang conjecture does not contradict Conjecture 5.
• Hitchin’s result on exotic spheres with no positive scalar curvature metrics shows that isoparametric foliations alone do not imply global curvature positivity, highlighting the subtlety of curvature extension in singular foliations.