Bottom of the spectrum of complete noncompact Kähler manifolds

Abstract: We present a survey on the bottom of the spectrum of the Hodge Laplacian on complete noncompact Kähler manifolds, with particular emphasis on Kähler hyperbolic manifolds and bounded symmetric domains. We also discuss theorems regarding the upper bounds for the bottom of the spectrum under Ricci and bisectional curvature assumptions, along with rigidity results for manifolds attaining the maximal bottom of the spectrum. Throughout the article, we propose several open problems.

• The paper establishes sharp spectral lower bounds for the Hodge Laplacian by leveraging Kähler hyperbolicity and geometric invariants in complete noncompact settings.
• It presents explicit formulas for spectral estimates in bounded symmetric domains with Kähler-Einstein metrics and discusses rigidity phenomena for extremal spectral values.
• Several open problems are posed, including Laplacian comparisons and rigidity extensions, providing clear directions for future research in complex geometry and spectral theory.

Bottom of the Spectrum of Complete Noncompact Kähler Manifolds

Overview

The paper "Bottom of the spectrum of complete noncompact Kähler manifolds" (2606.18740) provides a comprehensive survey of recent advances in understanding the bottom of the spectrum for the Hodge Laplacian on complete noncompact Kähler manifolds. Emphasis is placed on Kähler hyperbolic manifolds and bounded symmetric domains, addressing both lower and upper bounds for the spectrum under various geometric and curvature constraints, as well as rigidity phenomena for manifolds attaining extremal spectral values. Several open problems are posed throughout, reflecting ongoing challenges at the intersection of complex geometry, spectral theory, and global analysis.

Formal Definition and Isometry Invariance

Given a complete Riemannian manifold $(X, g)$, the bottom of the spectrum $\lambda_0(X, g)$ is defined as the infimum of the spectrum of the Hodge Laplacian $\Delta$ acting on $L^2$ functions. Explicitly,

$$\lambda_0(X, g) := \inf_{\lambda \in \sigma(\Delta)} \lambda,$$

which admits a Rayleigh quotient formulation. $\lambda_0$ is an isometry invariant—its value encodes global geometric features beyond local considerations. On manifolds of infinite volume, positivity of $\lambda_0$ strongly correlates with hyperbolic geometric behavior.

Hyperbolic and Kähler Hyperbolic Geometry

Historically, McKean's result establishes a strictly positive lower bound for $\lambda_0$ on simply-connected Riemannian manifolds whose sectional curvature is bounded above by a negative constant:

$\lambda_0(X, g) \geq \frac{(n-1)^2}{4}k,$

where $-k < 0$ is the upper bound for sectional curvature. This bound is sharp for real hyperbolic spaces.

The extension to Kähler geometry is made via the notion of "Kähler hyperbolicity" (Gromov). A complete noncompact Kähler manifold $\lambda_0(X, g)$0 is Kähler hyperbolic if $\lambda_0(X, g)$1 for a globally defined 1-form $\lambda_0(X, g)$2 satisfying a uniform bound on its pointwise norm. This structural property enables sharp spectral lower bounds for $\lambda_0(X, g)$3 in terms of geometric invariants and properties of the underlying domain.

Kähler Hyperbolic Manifolds and Spectral Lower Bounds

Gromov's foundational estimate yields:

$\lambda_0(X, g)$4

for a Kähler hyperbolic manifold $\lambda_0(X, g)$5 of complex dimension $\lambda_0(X, g)$6. Explicit formulas for the constant in this inequality (e.g., $\lambda_0(X, g)$7) have recently been established, showing sharpness for certain classes of manifolds [Cho_Choi_Lee2026].

Notable classes of Kähler hyperbolic manifolds include bounded homogeneous domains with complete Kähler-Einstein metrics, hyperconvex domains, strictly pseudoconvex domains, and Teichmüller spaces. The existence of a constant gradient length Kähler potential leads to optimal control of the hyperbolicity modulus and maximal spectral bounds.

Minimal Constant Gradient Length Potentials and Bounded Symmetric Domains

On bounded symmetric domains equipped with complete Kähler-Einstein metrics, the existence of a global Kähler potential with constant gradient length is significant. The minimal value of $\lambda_0(X, g)$8 for forms $\lambda_0(X, g)$9 with $\Delta$0 equals $\Delta$1, where $\Delta$2 is the supremum of the holomorphic sectional curvature. The paper proves:

$\Delta$3

where the lower bound is sharp, achieved on the unit ball and polydisc [Cho_Choi_Lee2026].

Upper Bounds and Rigidity Results

A parallel line of research considers upper bounds for $\Delta$4 under lower bounds on Ricci and bisectional curvature. Cheng's theorem gives:

$\Delta$5

for Ricci curvature bounded below by $\Delta$6.

Kähler analogues (Li-Wang, Munteanu) refine these bounds using holomorphic bisectional or Ricci curvature constraints. Notably, manifolds attaining maximal values for $\Delta$7 are rigorously classified—complete manifolds with maximal bottom are isometrically warped products $\Delta$8, with $\Delta$9 compact, and, in the Kähler setting, are biholomorphic to noncompact quotients of complex hyperbolic space (under further assumptions).

Open Problems and Future Directions

The paper identifies several unresolved issues:

• Whether a Laplacian comparison theorem analogous to McKean's result can be established for bounded symmetric domains in complex geometry.
• Potential formulation of a Kähler version of the Rauch comparison theorem for sharper estimates.
• Whether rigidity results can be extended by relaxing end conditions or curvature boundedness in characterizing noncompact ball quotients.
• Whether the Kähler-Einstein metric uniquely maximizes $L^2$0 over appropriate metric classes on noncompact Kähler manifolds.

Implications and Outlook

The results surveyed bridge classical spectral geometry and modern complex geometry, enabling sharper characterization of global geometric and analytic invariants for Kähler manifolds. The identification of sharp spectral bounds and rigidity phenomena deepens understanding of the interplay between curvature, topology, and spectrum. The posed open problems suggest directions for future research—including analytic and geometric comparison theorems and refined moduli characterizations—which carry implications for spectral theory, Kähler geometry, and mathematical physics.

Conclusion

This survey delivers a rigorous synthesis of the bottom of the spectrum for complete noncompact Kähler manifolds, connecting hyperbolic geometry, spectral theory, and complex analysis. The sharp bounds, structural classifications, and open problems delineated enrich the theoretical landscape and furnish compelling avenues for further investigation within global differential geometry and spectral analysis.