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Glued spaces and lower Ricci curvature bounds

Christian Ketterer

TL;DR

This work develops a gluing theory for weighted Riemannian manifolds under Bakry–Émery curvature bounds, showing that gluing along isometric boundary components preserves a synthetic lower Ricci bound in the CD$(K,N)$ sense. The authors prove that, under seam inequalities for the second fundamental forms and a boundary-derivative condition on the weights, the glued space carries a CD$(K,\lceil N\rceil)$ structure and arises as a collapsed GM-H limit of smooth manifolds with Ricci lower bounds. Conversely, they establish necessary seam conditions for any glued space to satisfy CD$(K,N)$, strengthening the link between geometric glueing and synthetic curvature bounds. The approach blends warped-product constructions, Kosovskiĭ-type gluing, and 1D localisation to translate higher-dimensional curvature questions into 1D analyses, with corollaries for noncollapsed and intermediate-curvature settings. The results extend classical gluing theorems to the synthetic Ricci framework and provide insights into the stability and limits of curvature-dimension bounds under gluing and collapsing operations.

Abstract

We consider Riemannian manifolds $M_i$, ${i=0,1}$, with boundary and $Φ_i\in C^{\infty}(M_i)$ non-negative such that the pair $(M_i, Φ_i)$ admits Bakry-Emery $N$-Ricci curvature bounded from below by $K$. Let $Y_0$ and $Y_1$ be isometric, compact components of the boundary of $M_0$ and $M_1$ respectively and assume $Φ_0=Φ_1$ on $Y_0\simeq Y_1$. We assume that $Π_0+Π_1=Π\geq 0$ (*), and $dΦ_0(ν_0)+ dΦ_1(ν_1)\leq \mbox{tr}Π$ on $Y_0\simeq Y_1$ (**) where $Π_i$ is the second fundamental form and $ν_i$ is inner unit normal field along $\partial M_i$. We show that the metric glued space $M=M_0\cup_{\mathcal I}M_1$ together with the measure $Φd\mathcal H^n$ satisfies the curvature-dimension condition $CD(K,\lceil N \rceil)$ where $Φ: M\rightarrow [0,\infty)$ arises tautologically from $Φ_1$ and $Φ_2$. Moreover, $(M, Φd\mathcal H^n)$ is the collapsed Gromov-Hausdorff limit of smooth, $\lceil N \rceil$-dimensional Riemannian manifolds with Ricci curvature bounded from below by $K- ε$ and is also the measured Gromov-Hausdorff limit of smooth, weighted Riemannian manifolds such that the Bakry-Emery $\lceil N \rceil$-Ricci curvature is bounded from below by $K-ε$. On the other hand we show that given a glued manifold as described it satisfies the curvature-dimension condition $CD(K,N)$ only if the condition (*) and (**) hold. The latter statement generalizes a theorem of Kosovskiĭ for sectional lower curvature bounds and especially applies for the unweighted case where a lower Ricci curvature bound and $\dim_{M_i}\leq N$ replaces a lower Bakry-Emery $N$-Ricci curvature bound.

Glued spaces and lower Ricci curvature bounds

TL;DR

This work develops a gluing theory for weighted Riemannian manifolds under Bakry–Émery curvature bounds, showing that gluing along isometric boundary components preserves a synthetic lower Ricci bound in the CD sense. The authors prove that, under seam inequalities for the second fundamental forms and a boundary-derivative condition on the weights, the glued space carries a CD structure and arises as a collapsed GM-H limit of smooth manifolds with Ricci lower bounds. Conversely, they establish necessary seam conditions for any glued space to satisfy CD, strengthening the link between geometric glueing and synthetic curvature bounds. The approach blends warped-product constructions, Kosovskiĭ-type gluing, and 1D localisation to translate higher-dimensional curvature questions into 1D analyses, with corollaries for noncollapsed and intermediate-curvature settings. The results extend classical gluing theorems to the synthetic Ricci framework and provide insights into the stability and limits of curvature-dimension bounds under gluing and collapsing operations.

Abstract

We consider Riemannian manifolds , , with boundary and non-negative such that the pair admits Bakry-Emery -Ricci curvature bounded from below by . Let and be isometric, compact components of the boundary of and respectively and assume on . We assume that (*), and on (**) where is the second fundamental form and is inner unit normal field along . We show that the metric glued space together with the measure satisfies the curvature-dimension condition where arises tautologically from and . Moreover, is the collapsed Gromov-Hausdorff limit of smooth, -dimensional Riemannian manifolds with Ricci curvature bounded from below by and is also the measured Gromov-Hausdorff limit of smooth, weighted Riemannian manifolds such that the Bakry-Emery -Ricci curvature is bounded from below by . On the other hand we show that given a glued manifold as described it satisfies the curvature-dimension condition only if the condition (*) and (**) hold. The latter statement generalizes a theorem of Kosovskiĭ for sectional lower curvature bounds and especially applies for the unweighted case where a lower Ricci curvature bound and replaces a lower Bakry-Emery -Ricci curvature bound.
Paper Structure (17 sections, 37 theorems, 126 equations)

This paper contains 17 sections, 37 theorems, 126 equations.

Table of Contents

  1. Introduction and Statement of Main Results
  2. Preliminaries
  3. Bakry-Emery curvature condition
  4. Curvature-dimension condition
  5. Semiconcave functions
  6. Alexandrov spaces
  7. Semiconcave functions on Alexandrov spaces
  8. $1D$ localisation
  9. Characterization of curvature bounds via $1D$ localisation
  10. Glued spaces
  11. Proof of Theorem \ref{['th:glue2']}
  12. Glued spaces with weights
  13. Proof of Theorem 1.1 and Theorem 1.2
  14. Warped products over Riemann manifolds
  15. Proof of Corollary 1.3
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $K\in \R$ and $N\in [1,\infty)$. Assume for $i=0,1$ that $X_i=\emptyset$, $\mathop{\mathrm{ric}}\nolimits_{g_i}^{\Phi_i, N}\geq K$ and There exist $\lceil N \rceil$-dimensional Riemannian manifolds $(M_i, g_i)_{i\in \N}$ with boundary such that $\mathop{\mathrm{ric}}\nolimits_{g_i}\geq K-\epsilon(i)$ with $\epsilon(i)\downarrow 0$ for $i\rightarrow \infty$ converging in measured Gromov-Hausdo

Theorems & Definitions (88)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Corollary 1.7
  • Definition 2.1: baem
  • proof
  • ...and 78 more