A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces

Let $M$ be a closed hypersurface in a simply connected rank-1 symmetric space $\olm$. In this paper, we give an upper bound for the first eigenvalue of the Laplacian of $M$ in terms of the Ricci curvature of $\olm$ and the square of the length of the second fundamental form of the geodesic spheres with center at the center-of-mass of $M$.

💡 Research Summary

The paper investigates the first non‑zero eigenvalue λ₁ of the Laplace–Beltrami operator on a closed hypersurface M immersed in a simply connected rank‑1 symmetric space ℳ. Rank‑1 symmetric spaces include the round sphere Sⁿ, the complex, quaternionic and octonionic projective spaces ℂPⁿ, ℍPⁿ, 𝕆P², and their non‑compact hyperbolic counterparts. These manifolds possess constant sectional curvature (positive for the compact models, negative for the non‑compact ones) and a Ricci tensor that is a constant multiple of the metric: Ric_ℳ = (n‑1)·c or Ric_ℳ = –(n‑1)·c, where c > 0 denotes the absolute value of the curvature.

The main goal is to bound λ₁(M) from above by quantities intrinsic to the ambient space and to the geometry of geodesic spheres centered at a distinguished point of M. The distinguished point is the center of mass p₀ of M, defined by the condition ∫M x dV = 0 when the ambient space is viewed in its standard Euclidean (or model) coordinates. For each point p ∈ M, let r(p) = dist_ℳ(p, p₀) and consider the geodesic sphere S{r(p)}(p₀). The second fundamental form of this sphere, denoted B_{r(p)}, has a norm that depends only on the radius r(p) because of the high symmetry of ℳ. In compact models the norm is |B_r|² = (n‑1)·c·cot²(√c r), while in the hyperbolic models it is |B_r|² = (n‑1)·c·coth²(√c r).

Theorem (Sharp Upper Bound).
For any closed hypersurface M ⊂ ℳ, \