On the PPW Conjecture for Hopf-Symmetric Sets in Non-compact Rank One Symmetric Space

Key Points

• This paper aims to establish a relationship between the second Dirichlet eigenvalue of bounded Hopf-symmetric domains and geodesic balls in non-compact rank one symmetric spaces.
• Proving eigenvalue inequalities for bounded Hopf-symmetric domains
• Comparing eigenvalues of domains with corresponding geodesic balls
• Extending previous work on eigenvalues in constant curvature spaces
• The second Dirichlet eigenvalue for bounded Hopf-symmetric domains is less than or equal to that of geodesic balls.
• The first Dirichlet eigenvalues for both domains are equal.
• The findings generalize previous results in geometry, indicating a deeper connection in eigenvalue theory.