[Paper Review] Isoparametric functions and exotic spheres

Jianquan Ge, Zizhou Tang|arXiv (Cornell University)|Mar 1, 2010

Geometric Analysis and Curvature Flows42 references3 citations

TL;DR

This paper advances the theory of isoparametric functions on general Riemannian manifolds, proving that no properly transnormal function exists on any hypothetical exotic 4-sphere. It further constructs a properly transnormal but non-isoparametric function on the Gromoll-Meyer sphere via $S^3$-invariant projection, with focal sets at two points, demonstrating that the Laplacian of the function is not a function of the function itself, thus violating the isoparametric condition.

ABSTRACT

The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on "exotic" 4-spheres (if exist) and the Gromoll-Meyer sphere. In particular, as one of main results we prove: there exists no properly transnormal function on any exotic 4-sphere if it exists. Furthermore, by projecting an $S^3$-invariant isoparametric function on $Sp(2)$, we construct a properly transnormal but not an isoparametric function on the Gromoll-Meyer sphere with two points as the focal varieties.

Motivation & Objective

• To extend the theory of isoparametric functions beyond space forms to general Riemannian manifolds.
• To investigate the existence of transnormal and isoparametric functions on exotic spheres, particularly exotic 4-spheres and the Gromoll-Meyer sphere.
• To determine whether isoparametric functions can exist on exotic 4-spheres, given the open smooth Poincaré conjecture in dimension four.
• To analyze the geometric and topological properties of level sets of transnormal functions, especially their focal varieties and minimality.

Proposed method

• Improves foundational theory of isoparametric functions on general Riemannian manifolds using Wang’s result on singular level sets being submanifolds.
• Applies three constructive methods: conformal deformation, cohomogeneity one actions, and Riemannian submersions to generate isoparametric functions.
• Constructs an $S^3$-invariant isoparametric function on $Sp(2)$ and projects it to the Gromoll-Meyer sphere via the quotient map $\pi: Sp(2) \to \Sigma^7$.
• Computes the Laplacian of the projected function $f$ on the Gromoll-Meyer sphere using the formula $\triangle f = -7f + \phi$, where $\phi = \langle H, \nabla F \rangle$ is the projected mean curvature term.
• Uses $S^3$-invariance to simplify calculations by choosing representative points in orbits, reducing the computation to matrices with specific forms.
• Employs quaternions and frame calculations to derive the metric components $g_{\alpha\beta}$ and their inverses $g^{\alpha\beta}$, enabling explicit computation of $\phi$ on $f^{-1}(0)$.

Experimental results

Research questions

• RQ1 Does there exist a properly transnormal function on any exotic 4-sphere, assuming such a manifold exists?
• RQ2 Can an isoparametric function be constructed on the Gromoll-Meyer sphere via $S^3$-invariant projection from $Sp(2)$?
• RQ3 Is the projected function on the Gromoll-Meyer sphere isoparametric, or does it fail due to non-constant $\triangle f$?
• RQ4 What is the geometric nature of the focal varieties of level sets of transnormal functions on exotic spheres?
• RQ5 How does the Laplacian of a projected function behave when the original function is isoparametric but the quotient space has non-trivial geometry?

Key findings

• There exists no properly transnormal function on any exotic 4-sphere, assuming such a manifold exists.
• The projected function $f$ on the Gromoll-Meyer sphere is properly transnormal but not isoparametric, as $\triangle f$ is not a function of $f$ alone.
• The function $\phi = \langle H, \nabla F \rangle$ is non-constant on $f^{-1}(0)$, explicitly shown via formula (23), proving $\triangle f$ is not a function of $f$.
• The focal varieties of the level sets are exactly two points: $f^{-1}(\pm 1)$, which are isolated and of codimension 7.
• The mean curvature of $f^{-1}(0)$ is non-constant, as derived from the non-constant $\phi$, confirming that the level set does not have constant mean curvature.
• The explicit expression for $\phi([Q])$ on $f^{-1}(0)$ is $\frac{8a_{1}b_{1}b_{0}}{EF}(E - 8a_{2}^{2}b_{1}^{2})$, which depends on the coordinates $a_1, a_2, b_0, b_1$, proving non-constancy.