Eigenvalue comparison theorems for the Witten-Laplacian and the weighted $p$-Laplacian on complete manifolds with a modified Ricci curvature bounded from below

Abstract: In this paper, for complete manifolds with a modified Ricci curvature bounded from below, we can successfully set up Cheng-type eigenvalue comparison theorems for the first Dirichlet eigenvalues of the Witten-Laplacian and the weighted $p$-Laplacian on geodesic balls of these manifolds.

• The paper establishes sharp Cheng-type eigenvalue comparison theorems for both the Witten-Laplacian and the weighted p-Laplacian under modified Ricci curvature lower bounds.
• It employs variational methods, Jacobi field analysis, and model domain techniques to derive explicit upper bounds and rigidity conditions.
• The results extend classical sphere comparisons to nonlinear settings, with practical implications for diffusion models and geometric analysis in weighted spaces.

Eigenvalue Comparison Theorems for the Witten-Laplacian and Weighted $p$-Laplacian under Modified Ricci Curvature Lower Bounds

Introduction

This work presents Cheng-type eigenvalue comparison theorems for the first Dirichlet eigenvalues of the Witten-Laplacian and weighted $p$-Laplacian on geodesic balls of complete Riemannian manifolds, with the curvature assumption formulated via a modified Ricci curvature bound. The focus is the establishment of sharp upper bounds under non-trivial geometric and analytic configurations, extending the classical sphere and model space comparisons to the nonlinear and weighted settings.

Theoretical Framework

A central element is the use of a smooth, strictly positive weight function $w$ on a Riemannian manifold $(M^n,g)$, giving rise to the Witten-Laplacian: $$\mathcal{L} = -\Delta - \langle \nabla \log w, \nabla \cdot \rangle,$$ and a corresponding modified Ricci curvature: $$\mathrm{Ric}_w = \mathrm{Ric} - w^{-1}\mathrm{Hess}(w).$$ For $1 < p < \infty$, the weighted $p$-Laplacian is introduced: $$\mathcal{L}_p = -\mathrm{div}(w |\nabla \cdot|^{p-2}\nabla \cdot).$$ The eigenvalue problems for these operators under Dirichlet conditions are analyzed using variational characterizations, with the first Dirichlet eigenvalues denoted $A_{1,w}$ and $p$0, respectively.

Main Results

Cheng-Type Eigenvalue Comparisons

The core contributions are stated as:

• If the modified Ricci curvature $p$1 satisfies a lower bound $p$2 along all radial directions from a basepoint $p$3, where $p$4 is effectively the Ricci curvature model of the comparison manifold, then for all geodesic balls $p$5,

$p$6

where $p$7 is the geodesic ball of the same radius in an $p$8-dimensional spherically symmetric model space with warping determined by $p$9.

• Analogously, for the weighted $w$0-Laplacian,

$w$1

The results hold without the need for the weight $w$2 to be radially symmetric—a significant technical relaxation compared to prior work.

Analytical Tools

A substantial technical step is the generalization of Bishop’s volume comparison theorem to the setting of the modified Ricci curvature. The argument leverages Jacobi field analysis in geodesic polar coordinates and careful control of the divergence terms arising from the weight.

Trial functions for the eigenvalue estimates are constructed via the eigenfunctions of the model space, and the spherically symmetric structure is crucial for both the upper bounds and the cases of equality (rigidity). The variational approach is robust to both the linear ($w$3) and nonlinear ($w$4) cases.

Numerical and Rigidity Implications

The theorems provide explicit, computable upper bounds for the first Dirichlet eigenvalues in terms of model spaces with prescribed curvature and weight structure. For fixed curvature data, the bounds reduce to a computation involving Bessel functions or the solution of analogous ODEs for warping functions, thereby connecting geometric data with spectral invariants.

A noteworthy aspect is that equality is attained if and only if the geodesic ball isometric to the model domain, thus providing optimality and rigidity characteristics. The relaxation of weight symmetry assumptions broadens the applicability to more general geometric flows, non-gradient Ricci solitons, and spaces naturally endowed with non-radial densities.

Theoretical and Practical Implications

The results deepen the understanding of how weighted, nonlinear elliptic operators capture geometric information encoded in Ricci-type curvature lower bounds, especially in weighted Riemannian geometry (Bakry-Émery theory). The extension to the weighted $w$5-Laplacian directly interfaces with questions in nonlinear potential theory, analysis of diffusion processes with drift, and the study of stability and concentration in measure-metric spaces.

Practically, these comparison theorems yield estimates for the speed of diffusion and wave propagation in physical and stochastic models with spatial heterogeneities described by weights. The approach informs geometric analysis on manifolds with density, relevant in optimal transport, isoperimetric inequalities, and the theory of metric measure spaces.

Future Directions

Potential developments include extension to higher eigenvalues, heat kernel estimates, and comparison theorems for other nonlinear or non-local elliptic operators. The flexibility regarding the non-radiality of weights positions these theorems as foundational for weighted comparison geometry, with implications for singular space limits, Ricci flow with density, and functional inequalities under measure concentration frameworks.

Conclusion

This paper establishes sharp eigenvalue comparison theorems for the first Dirichlet eigenvalues of the Witten-Laplacian and weighted $w$6-Laplacian under lower bounds on a modified Ricci curvature, significantly relaxing symmetry constraints on the underlying geometric data. The results represent a substantive advance in the spectral analysis of weighted manifolds and provide both theoretical and practical tools for further study in geometry, analysis, and mathematical physics (2607.04588).