[Paper Review] Isoparametric foliation and Yau conjecture on the first eigenvalue

Zizhou Tang, Wenjiao Yan|arXiv (Cornell University)|Jan 3, 2012

Geometric Analysis and Curvature Flows16 references4 citations

TL;DR

This paper proves Yau's conjecture on the first eigenvalue for closed minimal isoparametric hypersurfaces in the unit sphere, showing that the first eigenvalue equals the dimension of the hypersurface. Using spectral analysis and topological methods, it establishes that for isoparametric hypersurfaces with four distinct principal curvatures and multiplicities $ m_1, m_2 \geq 2 $, $ \lambda_1 = n $, and further shows that the first eigenvalues of the focal submanifolds equal their dimensions in the non-stable range.

ABSTRACT

A well known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface $M^n$ in the unit sphere $S^{n+1}(1)$ is just its dimension $n$. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non-stable range.

Motivation & Objective

• To prove Yau's conjecture that the first eigenvalue of every closed minimal hypersurface in the unit sphere $ S^{n+1}(1) $ is equal to its dimension $ n $, restricted to isoparametric hypersurfaces.
• To extend previous results on homogeneous minimal hypersurfaces to nonhomogeneous cases, particularly for isoparametric hypersurfaces with four distinct principal curvatures.
• To establish that the first eigenvalues of the focal submanifolds of isoparametric hypersurfaces with $ g=4 $ and $ m_1, m_2 \geq 2 $ equal their respective dimensions.
• To provide a spectral characterization of focal submanifolds in the non-stable range using eigenvalue inequalities and geometric analysis.
• To complete the proof of Yau's conjecture for all isoparametric minimal hypersurfaces by covering the remaining cases not addressed by prior work on homogeneous or OT-FKM-type hypersurfaces.

Proposed method

• Utilizes the spectral theory of the Laplace-Beltrami operator on compact Riemannian manifolds, focusing on the first eigenvalue $ \lambda_1(M) $.
• Applies Münzner’s classification of isoparametric hypersurfaces, which restricts the number of distinct principal curvatures $ g $ to 1, 2, 3, 4, or 6.
• Employs a topological method to analyze the structure of isoparametric foliations and the geometry of focal submanifolds.
• Derives eigenvalue inequalities by comparing the spectrum of the sphere $ S^{n+1}(1) $ with that of the focal submanifolds $ M_1 $ and $ M_2 $, using the Rayleigh quotient and $ L^2 $-norms.
• Uses the OT-FKM-type construction for isoparametric hypersurfaces with $ g=4 $, particularly analyzing the case $ (m_1, m_2) = (1,k) $, and constructs a quotient space diffeomorphic to $ S^1(1) \times S^{k+1}(1) / \sim $ to compute eigenvalues.
• Establishes a sufficient condition $ m_2 \geq \frac{1}{2}(m_1 + 3) $ for the first eigenvalue of the focal submanifold $ M_1 $ to equal its dimension.

Experimental results

Research questions

• RQ1 Does Yau's conjecture hold for all closed minimal isoparametric hypersurfaces in $ S^{n+1}(1) $, particularly in the nonhomogeneous case?
• RQ2 What is the first eigenvalue of the focal submanifolds of isoparametric hypersurfaces with four distinct principal curvatures and $ m_1, m_2 \geq 2 $?
• RQ3 Can the first eigenvalue of a minimal isoparametric hypersurface be shown to equal its dimension using spectral and geometric methods?
• RQ4 How do eigenvalue inequalities between $ S^{n+1}(1) $ and the focal submanifolds $ M_1 $, $ M_2 $ help in proving the main results?
• RQ5 Under what conditions on the multiplicities $ m_1, m_2 $ does the first eigenvalue of the focal submanifold $ M_1 $ equal its dimension?

Key findings

• The first eigenvalue of every closed minimal isoparametric hypersurface $ M^n $ in $ S^{n+1}(1) $ is equal to its dimension $ n $, thus confirming Yau's conjecture in this class.
• For isoparametric hypersurfaces with $ g=4 $ and $ m_1, m_2 \geq 2 $, the first eigenvalue $ \lambda_1(M^n) = n $, extending prior results that were limited to $ (m_1, m_2) = (1,k) $ or $ (2,2) $.
• The first eigenvalues of the focal submanifolds $ M_1 $ and $ M_2 $ are equal to their respective dimensions when $ m_2 \geq \frac{1}{2}(m_1 + 3) $, particularly in the non-stable range.
• The focal submanifold $ M_2 $ of OT-FKM-type with $ (m_1, m_2) = (1,k) $ has first eigenvalue $ \lambda_1(M_2) = \min\{4, 2+k\} $, which equals its dimension when $ k \geq 2 $.
• The inequality $ \lambda_{n+3}(M_1) \geq \frac{2(n+2)(m_2 - 1)}{m_1 + m_2} $ is derived and used to establish the lower bound on the first eigenvalue of $ M_1 $, leading to the conclusion that $ \lambda_1(M_1) = \dim M_1 $ under the given multiplicity condition.
• The proof is completed by combining results from homogeneous cases ($ g=1,2,3,6 $) and nonhomogeneous cases ($ g=4 $), using the classification of isoparametric hypersurfaces and spectral comparison techniques.