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Remark on a geometric inequality for closed hypersurfaces in weighted manifolds

Adam Rudnik

TL;DR

This work extends Willmore-type geometric inequalities to noncompact smooth metric measure spaces with nonnegative Bakry-Émery curvature. By leveraging $f$-volume comparison and tubular-volume analysis, it derives two sharp inequalities under $\text{Ric}_f^N\ge0$ and $\text{Ric}_f\ge0$, with equality cases yielding rigid, explicit model geometries: a warped-product exterior in the finite-$N$ case and a twisted-product exterior in the $N=\infty$ case, together with precise forms of the density $f$ outside $\Omega$. The results hinge on the $f$-asymptotic volume ratio $f$-AVR$(g)$ and the assumption of large weighted volume growth $f$-AVR$(g)>0$, connecting boundary geometry to global volume behavior. These findings generalize Willmore-type rigidity from classical nonnegative Ricci curvature to spaces with density and provide nonexistence results for closed $f$-minimal hypersurfaces under positive $f$-AVR. The work thus deepens the interplay between curvature-dimension conditions, boundary geometry, and global volume growth in weighted manifolds, with potential applications to conformal deformations and transport-based geometries.

Abstract

In this paper we consider noncompact smooth metric measure spaces $(M, g,e^{-f}dvol_{g})$ of nonnegative Bakry-Émery Ricci curvature, i.e. $Ric + D^{2}f - \frac{1}{N}df \otimes df \geq 0$, for $0< N \leq \infty$, in order to obtain geometric inequalities for the boundary of a given open and bounded set $Ω\subset M$, with regular boundary $\partial Ω$. Our inequalities are sharp for both the cases $N< \infty$ and $N= \infty$, provided that the underlying ambient space has large weighted volume growth. The rigidity obtained for the $N=\infty$ case holds true precisely when $M \setminus Ω$ is isometric to a twisted product metric and, as such, is a generalization of the Willmore-type inequality for nonnegative Ricci curvature from Agostiniani, Fagagnolo and Mazzieri to the context of weighted manifolds.

Remark on a geometric inequality for closed hypersurfaces in weighted manifolds

TL;DR

This work extends Willmore-type geometric inequalities to noncompact smooth metric measure spaces with nonnegative Bakry-Émery curvature. By leveraging -volume comparison and tubular-volume analysis, it derives two sharp inequalities under and , with equality cases yielding rigid, explicit model geometries: a warped-product exterior in the finite- case and a twisted-product exterior in the case, together with precise forms of the density outside . The results hinge on the -asymptotic volume ratio -AVR and the assumption of large weighted volume growth -AVR, connecting boundary geometry to global volume behavior. These findings generalize Willmore-type rigidity from classical nonnegative Ricci curvature to spaces with density and provide nonexistence results for closed -minimal hypersurfaces under positive -AVR. The work thus deepens the interplay between curvature-dimension conditions, boundary geometry, and global volume growth in weighted manifolds, with potential applications to conformal deformations and transport-based geometries.

Abstract

In this paper we consider noncompact smooth metric measure spaces of nonnegative Bakry-Émery Ricci curvature, i.e. , for , in order to obtain geometric inequalities for the boundary of a given open and bounded set , with regular boundary . Our inequalities are sharp for both the cases and , provided that the underlying ambient space has large weighted volume growth. The rigidity obtained for the case holds true precisely when is isometric to a twisted product metric and, as such, is a generalization of the Willmore-type inequality for nonnegative Ricci curvature from Agostiniani, Fagagnolo and Mazzieri to the context of weighted manifolds.

Paper Structure

This paper contains 9 sections, 8 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notions and notations
  4. Mean curvature and volume comparison
  5. The $f$-asymptotic volume ratio
  6. Proof of Theorems \ref{['main_theo_N_bakry-emery']} and \ref{['main_theo_infty_bakry-emery']}
  7. A sharp geometric inequality for $\mathrm{R}\space\mathrm{i}\space\mathrm{c}_{f}^{N} \geq 0$
  8. A sharp geometric inequality for $\mathrm{R}\space\mathrm{i}\space\mathrm{c}_{f} \geq 0$
  9. Acknowledgements

Key Result

Theorem 1.1

$[$Willmore-type inequality for nonnegative N-Bakry-Émery$]$ Let $(\textsl{M},\textsl{g}, e^{-f}\textit{dvol}_{\textsl{g}})$ be a noncompact, $n$-dimensional, SMMS with nonnegative N-Bakry-Émery Ricci curvature, $\mathrm{R}\space\mathrm{i}\space\mathrm{c}_{f}^{N} \geq 0$. Let $\Omega$ be an open and where $\mathrm{H}_f = \mathrm{H} - \langle \nu, \text{grad}_{\textsl{g}}\space f \rangle$ is the we

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Corollary 1.9
  • Lemma 2.1
  • ...and 4 more