Weighted volume comparison and monotonicity for $L^p$-bound of Bakry-Émery Ricci curvature

Abstract: We prove a relative volume comparison theorem of Petersen-Wei for both $L<sup>P$-bound of Bakry-Émery Ricci curvature and gradient of potential function. As an application, we give a modified proof for a volume comparison and monotonicity of Kähler-Ricci flow established in a recent work of Tian-Zhang-Zhang-Zhu-Zhu.

• The paper establishes an L^p-based weighted relative volume comparison theorem for Bakry–Émery Ricci curvature, showing near-monotonicity of normalized volume ratios as the radius approaches zero.
• It utilizes Riccati-type differential inequalities, weighted Bochner formulas, and delicate interpolation inequalities to control curvature errors and the gradient of the potential function.
• The results extend comparison geometry to settings such as Kähler–Ricci flows and singularity analysis by replacing strict pointwise bounds on |∇f| with L^p and L^q integrability conditions.

Weighted Volume Comparison and Monotonicity for $L^p$-Bounds of Bakry-Émery Ricci Curvature

Introduction and Context

The paper addresses the extension of relative volume comparison theorems—classically centered on Ricci curvature lower bounds—to the context of the Bakry-Émery (BE) Ricci tensor under $L^p$-integral curvature assumptions. The framework provides analytic tools applicable to weighted Riemannian manifolds $(M^n,g,e^{-f}\mathrm{dvol})$, encompassing cases where the potential function $f$ exhibits only controlled $L^q$-growth in its gradient, rather than uniform pointwise bounds. The analysis is motivated by, and makes connections to, singularity and limit space studies for geometric flows such as the Kähler–Ricci flow, and addresses technical gaps in prior extension works that required restrictive bounds on $|\nabla f|$.

Main Theoretical Contribution

The main result is an $L^p$-based weighted relative volume comparison theorem (Theorem 1), which generalizes the classical Petersen–Wei volume comparison result for integral Ricci curvature bounds to the Bakry–Émery setting with only local $L^p$ and $L^q$ control of the BE Ricci tensor and the gradient of the potential function:

• Hypotheses: A weighted Riemannian manifold $(M^n,g,e^{-f}d\mathrm{vol})$ with local weighted volume bound $L^p$0 for $L^p$1 small, and control over the $L^p$2-norm of the negative part of the smallest BE Ricci eigenvalue and over the $L^p$3-norm of the excess $L^p$4 (the deviation from being bounded by $L^p$5).
• Conclusion: For $L^p$6 and $L^p$7, the normalized weighted volume ratios $L^p$8 satisfy a monotonicity deficit estimate controlled by the $L^p$9 and $(M^n,g,e^{-f}\mathrm{dvol})$0 quantities. As $(M^n,g,e^{-f}\mathrm{dvol})$1, the volume ratio is nearly monotone, quantifying how close the geometry is to that of a model space of constant curvature.

The proof involves deriving a Riccati-type differential inequality for the mean curvature error between the weighted manifold and the model space, leveraging Bochner formulas and comparison geometry, and then establishing sharp $(M^n,g,e^{-f}\mathrm{dvol})$2 estimates for this error. The requirement for only $(M^n,g,e^{-f}\mathrm{dvol})$3-integrability of $(M^n,g,e^{-f}\mathrm{dvol})$4 (rather than pointwise bounds) marks a significant relaxation, allowing the inclusion of important geometric situations, specifically for gradient Ricci solitons, where $(M^n,g,e^{-f}\mathrm{dvol})$5 often grows like the distance function.

Technical Developments

Key technical components include:

• Weighted Bochner and Laplacian Comparison: The authors adapt the Bochner formula to the weighted Laplacian $(M^n,g,e^{-f}\mathrm{dvol})$6 and track the contributions of $(M^n,g,e^{-f}\mathrm{dvol})$7 to the mean curvature evolution.
• Integral Control via Riccati-Type Inequalities: The mean curvature error is subjected to a precise Riccati-type differential inequality, where the main new analytic hurdle is the presence of terms scaling with $(M^n,g,e^{-f}\mathrm{dvol})$8—these encode how far the weighted geometry deviates from models with bounded $(M^n,g,e^{-f}\mathrm{dvol})$9.
• Sophisticated $f$0 and $f$1 Interpolation and Power Integral Estimates: The authors systematically control all arising error terms, employing delicate interpolation inequalities, and volume comparison arguments, resolving the analytic challenges posed by only having integral (not pointwise) bounds.

The constants in all main inequalities are given explicitly and shown to depend optimally on parameters such as $f$2, and the geometry of the reference model space.

Application to Kähler–Ricci Flow and Hamilton–Tian Conjecture

As an application, the theorem is used to reprove and strengthen recent monotonicity and volume comparison results for the Kähler–Ricci flow on Fano manifolds, as established in [Tian, Zhang, Zhang, Zhu, Zhu, (Tian et al., 18 Sep 2025)], directly linking the $f$3-control of BE Ricci curvature and gradient norm to the volume monotonicity formula of the evolving metrics. The primary corollary provides uniform control of the normalized volume difference $f$4 in terms of $f$5 for appropriate $f$6, where $f$7 is the complex dimension. This forms a crucial ingredient in regularity theory for the singularity formation in Kähler–Ricci flow and in the verification of the Hamilton–Tian conjecture for three-dimensional Fano manifolds.

Moreover, the techniques circumvent the need for pointwise control on the potential function, as previously required, allowing instead for $f$8-control over the weighted potential gradient and accommodating known growth properties of $f$9 under the Ricci flow.

Theoretical and Practical Implications

The development has several broad implications:

• Extension of Comparison Geometry: The result effectively closes the gap in comparison geometry for manifolds with only $L^q$0-bounded Bakry–Émery Ricci curvature, providing tools for integral geometry settings where volume control under tight curvature conditions is sought.
• Applications to Flow Compactness and Limit Spaces: The new volume comparison handles situations with low regularity in the potential function, as in Ricci limit theory and singularity models for geometric flows.
• Broader Reach in Kähler–Ricci Flow Analysis: By removing the requirement of bounded $L^q$1, the results apply uniformly across the Kähler–Ricci flow, facilitating progress on precise regularity estimates and compactness theorems.

Future Directions

This work suggests several avenues for further investigation:

• Sharper Volume Comparison for $L^q$2 Lower Bounds: Extending these techniques to cases where only lower $L^q$3-bounds are available (rather than boundedness of the negative part) could yield further generalizations.
• Analysis on Noncompact and Singular Spaces: The robust analytic framework for weighted comparison may be instrumental in the study of limit spaces arising in geometric flows, particularly with singularity formation.
• Interplay with Optimal Transport and Diffusion: Since the BE Ricci curvature arises naturally in diffusion and optimal transport, these tools have potential applications in metric measure space theory and geometric analysis beyond smooth manifolds.

Conclusion

This paper represents an advancement in the theory of comparison geometry under Bakry–Émery Ricci curvature by establishing a strong, quantitatively explicit relative volume comparison and monotonicity theorem for the $L^q$4-normed setting, together with $L^q$5-control on the potential gradient. The removal of strong pointwise restrictions on $L^q$6 and the application to Kähler–Ricci flow demonstrate both theoretical robustness and practical applicability, positioning the result as a foundational tool for future geometric and analytic studies of spaces with integral curvature constraints (2604.17367).