Glued spaces and lower Ricci curvature bounds

Christian Ketterer

arxiv: 2308.06848 · v5 · submitted 2023-08-13 · 🧮 math.DG · math.MG

Glued spaces and lower Ricci curvature bounds

Christian Ketterer This is my paper

Pith reviewed 2026-05-24 07:12 UTC · model grok-4.3

classification 🧮 math.DG math.MG

keywords glued manifoldsRicci curvature boundscurvature-dimension conditionBakry-Emery curvaturemetric measure spacesboundary conditionssynthetic geometry

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The pith

Gluing two manifolds with boundary under matching conditions on the second fundamental form and weight derivatives produces a space satisfying the curvature-dimension condition CD(K, ceil(N)).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that if two Riemannian manifolds each have Bakry-Emery N-Ricci curvature at least K and their boundaries are glued isometrically with equal weights, then the resulting space with the induced measure satisfies CD(K, ceil(N)) provided the second fundamental forms sum non-negatively and a related inequality on the derivatives of the weights holds. It also proves the converse: the CD condition forces those boundary conditions. The result extends to limits of approximating smooth spaces with nearly the same bounds and generalizes earlier work on sectional curvature bounds.

Core claim

The metric glued space M = M0 ∪_I M1 together with the measure Φ dH^n satisfies the curvature-dimension condition CD(K, ⌈N⌉) if and only if the boundary conditions Π0 + Π1 = Π ≥ 0 and dΦ0(ν0) + dΦ1(ν1) ≤ tr Π hold on the glued hypersurface, where the manifolds (Mi, Φi) satisfy the Bakry-Emery N-Ricci bound K.

What carries the argument

The curvature-dimension condition CD(K,N) for metric measure spaces, preserved under the isometric gluing of boundaries when the stated matching conditions hold.

If this is right

The glued space is itself a CD(K, ⌈N⌉) space.
It arises as the collapsed Gromov-Hausdorff limit of smooth manifolds with Ricci curvature bounded below by K - ε.
It is the measured Gromov-Hausdorff limit of weighted manifolds with Bakry-Emery ⌈N⌉-Ricci at least K - ε.
The CD condition on the glued space holds only when the stated boundary conditions are satisfied, generalizing Kosovskiĭ's theorem for sectional curvature.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This construction yields new examples of CD spaces by gluing known ones that satisfy the boundary matching.
The conditions ensure the synthetic Ricci bound does not drop at the interface.
Analogous gluing arguments may extend to non-smooth spaces such as RCD(K,N) spaces.

Load-bearing premise

The boundary conditions that the sum of the second fundamental forms is non-negative and the sum of the normal derivatives of the weights is bounded by the trace of that sum.

What would settle it

A counterexample where the second fundamental forms sum to a negative tensor yet the glued space still satisfies CD(K,N), or where the conditions hold but CD fails.

read the original abstract

We consider Riemannian manifolds $M_i$, ${i=0,1}$, with boundary and $\Phi_i\in C^{\infty}(M_i)$ non-negative such that the pair $(M_i, \Phi_i)$ 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 $\Phi_0=\Phi_1$ on $Y_0\simeq Y_1$. We assume that $\Pi_0+\Pi_1=\Pi \geq 0$ (*), and $d\Phi_0(\nu_0)+ d\Phi_1(\nu_1)\leq \mbox{tr}\Pi$ on $Y_0\simeq Y_1$ (**) where $\Pi_i$ is the second fundamental form and $\nu_i$ 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 $\Phi d\mathcal H^n$ satisfies the curvature-dimension condition $CD(K,\lceil N \rceil)$ where $\Phi: M\rightarrow [0,\infty)$ arises tautologically from $\Phi_1$ and $\Phi_2$. Moreover, $(M, \Phi d\mathcal H^n)$ is the collapsed Gromov-Hausdorff limit of smooth, $\lceil N \rceil$-dimensional Riemannian manifolds with Ricci curvature bounded from below by $K- \epsilon$ 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 $K-\epsilon$. 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 Kosovski\u{\i} 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends Kosovskii's sectional gluing theorem to Bakry-Emery N-Ricci with explicit boundary conditions, a converse, and limit statements.

read the letter

The core contribution is a gluing theorem: if two manifolds with Bakry-Emery N-Ricci bounded below by K have isometric boundaries where the second fundamental forms sum to something non-negative and the weights satisfy a derivative inequality, then the glued space with the combined measure satisfies CD(K, ceil(N)). They also prove the converse, that the CD condition on the glued space forces those boundary conditions, and note that the glued object arises as a collapsed GH limit of smooth manifolds with almost the same curvature bound. This is a direct extension of the unweighted sectional case, and the converse plus the limit claims are the parts that add value beyond a routine application. The boundary conditions are stated explicitly rather than hidden, which helps for constructing examples. The ceiling on N and the switch between weighted and unweighted statements are handled in the abstract without obvious contradictions. The argument appears to rest on standard comparison techniques rather than circular fitting. Soft spots are minor: the full details of how the ceiling interacts with the measure in the limit arguments would need checking in the body, and the low-dimensional or collapsed cases might require extra care, but nothing in the stated claims looks load-bearing or inconsistent. This is for readers already working in synthetic Ricci curvature or Bakry-Emery geometry who need tools for building spaces or verifying limits. It is a substantive but narrow extension, so it belongs in a specialized journal rather than a broad one. I would send it to peer review; the combination of direct and converse statements plus the limit results makes it worth a referee's time even if revisions are needed on the technical steps.

Referee Report

0 major / 4 minor

Summary. The paper proves a gluing theorem for Bakry-Emery curvature-dimension conditions: given Riemannian manifolds with boundary (M_i, Φ_i) each satisfying BE N-Ricci curvature ≥ K, with isometric boundary components Y_0 ≃ Y_1 where Φ_0 = Φ_1, and satisfying the boundary conditions Π_0 + Π_1 = Π ≥ 0 (*) and dΦ_0(ν_0) + dΦ_1(ν_1) ≤ tr Π (**), the metric glued space M = M_0 ∪_I M_1 equipped with the measure Φ dH^n satisfies CD(K, ⌈N⌉). The glued space is also realized as a collapsed GH limit of smooth ⌈N⌉-dimensional manifolds with Ricci ≥ K-ε and as an mGH limit of weighted manifolds with BE ⌈N⌉-Ricci ≥ K-ε. Conversely, any such glued space satisfies CD(K, N) only if (*) and (**) hold, generalizing Kosovskii's theorem (including the unweighted Ricci case).

Significance. If the claims hold, the result supplies an explicit gluing construction for CD(K, N) spaces in the weighted setting, together with approximation by smooth manifolds, extending classical comparison geometry techniques to the Bakry-Emery and synthetic CD framework. The necessity direction strengthens the characterization of when gluing preserves lower curvature bounds.

minor comments (4)

Abstract, line beginning 'where Φ: M→[0,∞) arises tautologically from Φ1 and Φ2': this is a typographical error; the indices should read Φ0 and Φ1.
Abstract: the notation 'M=M0∪_I M1' and 'Y0≃Y1' is introduced without prior definition; a brief sentence clarifying the gluing map I and the identification would improve readability.
Abstract: the phrase 'the pair (M_i, Φ_i) admits Bakry-Emery N-Ricci curvature bounded from below by K' is used without an explicit reference or short definition of the Bakry-Emery tensor; adding a parenthetical or citation to the standard definition would aid readers.
Abstract, final sentence: 'dim_{M_i}≤N' appears without explanation of the subscript notation; consistent use of 'dim M_i ≤ N' or a clarifying clause is recommended.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of our gluing theorem for Bakry-Emery CD(K,N) spaces and for the positive assessment of its significance. The recommendation for minor revision is noted. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes an if-and-only-if theorem: the glued space satisfies CD(K,⌈N⌉) precisely when the stated boundary conditions (*) and (**) hold on the interface. Sufficiency is shown via gluing arguments that reduce to standard Bakry-Émery comparison geometry on the pieces; necessity generalizes an external result of Kosovskiĭ (not a self-citation). The collapsed and measured GH-limit statements follow directly from the construction of the glued measure Φ dH^n without any parameter fitting or redefinition of the target curvature bound. No step equates the conclusion to its inputs by construction, and the central claim retains independent geometric content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard definition of Bakry-Emery N-Ricci curvature, the synthetic CD(K,N) condition, and the usual notions of second fundamental form and inner normal on a boundary; no new free parameters or invented entities are introduced.

axioms (2)

standard math Bakry-Emery N-Ricci curvature is defined via the standard formula involving the Hessian of log Phi and the Ricci tensor.
Invoked in the opening sentence of the abstract when stating the hypothesis on (Mi, Phi_i).
standard math The curvature-dimension condition CD(K,N) is the synthetic notion introduced by Lott-Villani and Sturm.
The conclusion of the theorem is stated directly in terms of CD(K, ceil(N)).

pith-pipeline@v0.9.0 · 5957 in / 1450 out tokens · 21754 ms · 2026-05-24T07:12:19.574286+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages · 1 internal anchor

[1]

Alexander and Richard L

Stephanie B. Alexander and Richard L. Bishop, Curvature bounds for warped products of metric spaces, Geom. Funct. Anal. 14 (2004), no. 6, 1143--1181. 2135163 (2007d:53061)

work page 2004
[2]

33, American Mathematical Society, Providence, RI, 2001

Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. 1835418 (2002e:53053)

work page 2001
[3]

Bakry and Michel \' E mery, Diffusions hypercontractives, Lecture Notes in Math., vol

D. Bakry and Michel \' E mery, Diffusions hypercontractives, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985. 889476

work page 1985
[4]

Christian B\" a r and Bernhard Hanke, Boundary conditions for scalar curvature, Perspectives in scalar curvature. V ol. 2, World Sci. Publ., Hackensack, NJ, [2023] 2023, pp. 325--377. 4577919

work page 2023
[5]

McCann, and Eric Woolgar, Inscribed radius bounds for lower Ricci bounded metric measure spaces with mean convex boundary , SIGMA, Symmetry Integrability Geom

Annegret Burtscher, Christian Ketterer, Robert J. McCann, and Eric Woolgar, Inscribed radius bounds for lower Ricci bounded metric measure spaces with mean convex boundary , SIGMA, Symmetry Integrability Geom. Methods Appl. 16 (2020), paper 131, 29 (English)

work page 2020
[6]

Elia Bru \`e , Aaron Naber , and Daniele Semola , Fundamental Groups and the Milnor Conjecture , arXiv e-prints (2023), arXiv:2303.15347

work page arXiv 2023
[7]

Bradley Lewis Burdick, Metrics of P ositive R icci C urvature on C onnected S ums: P rojective S paces, P roducts, and P lumbings , ProQuest LLC, Ann Arbor, MI, 2019, Thesis (Ph.D.)--University of Oregon. 4035402

work page 2019
[8]

, Ricci-positive metrics on connected sums of projective spaces, Differential Geom. Appl. 62 (2019), 212--233. 3881650

work page 2019
[9]

Walsh, and David J

Boris Botvinnik, Mark G. Walsh, and David J. Wraith, Homotopy groups of the observer moduli space of R icci positive metrics , Geom. Topol. 23 (2019), no. 6, 3003--3040. 4039184

work page 2019
[10]

99 (2014), 136--151

Fabio Cavalletti, Monge problem in metric measure spaces with R iemannian curvature-dimension condition , Nonlinear Anal. 99 (2014), 136--151. 3160530

work page 2014
[11]

McCann, and Michael Schmuckenschl \"a ger, A R iemannian interpolation inequality \`a la B orell, B rascamp and L ieb , Invent

Dario Cordero-Erausquin, Robert J. McCann, and Michael Schmuckenschl \"a ger, A R iemannian interpolation inequality \`a la B orell, B rascamp and L ieb , Invent. Math. 146 (2001), no. 2, 219--257. 1865396 (2002k:58038)

work page 2001
[12]

Fabio Cavalletti and Andrea Mondino, Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower R icci curvature bounds , Invent. Math. 208 (2017), no. 3, 803--849. 3648975

work page 2017
[13]

PDE 13 (2020), no

, New formulas for the L aplacian of distance functions and applications , Anal. PDE 13 (2020), no. 7, 2091--2147. 4175820

work page 2020
[14]

Fabio Cavalletti and Emanuel Milman, The globalization theorem for the curvature-dimension condition, Invent. Math. 226 (2021), no. 1, 1--137. 4309491

work page 2021
[15]

Guido De Philippis and Nicola Gigli, Non-collapsed spaces with R icci curvature bounded from below , J. \' E c. polytech. Math. 5 (2018), 613--650. 3852263

work page 2018
[16]

Blaine Lawson, Jr., Spin and scalar curvature in the presence of a fundamental group

Mikhael Gromov and H. Blaine Lawson, Jr., Spin and scalar curvature in the presence of a fundamental group. I , Ann. of Math. (2) 111 (1980), no. 2, 209--230. 569070

work page 1980
[17]

Nicola Gigli, Andrea Mondino, and Giuseppe Savar \'e , Convergence of pointed non-compact metric measure spaces and stability of R icci curvature bounds and heat flows , Proc. Lond. Math. Soc. (3) 111 (2015), no. 5, 1071--1129. 3477230

work page 2015
[18]

Bang-Xian Han, Measure rigidity of synthetic lower R icci curvature bound on R iemannian manifolds , Adv. Math. 373 (2020), 107327, 31. 4129481

work page 2020
[19]

Hirsch, Differential topology, Grad

Morris W. Hirsch, Differential topology, Grad. Texts Math., vol. 33, Springer, Cham, 1976 (English)

work page 1976
[20]

Christian Ketterer, Ricci curvature bounds for warped products, J. Funct. Anal. 265 (2013), no. 2, 266--299. 3056704

work page 2013
[21]

, Ricci curvature bounds for warped products, J. Funct. Anal. 265 (2013), no. 2, 266--299. 3056704

work page 2013
[22]

, The Heintze - Karcher inequality for metric measure spaces , Proc. Am. Math. Soc. 148 (2020), no. 9, 4041--4056 (English)

work page 2020
[23]

, Stability of metric measure spaces with integral R icci curvature bounds , J. Funct. Anal. 281 (2021), no. 8, Paper No. 109142, 48. 4271791

work page 2021
[24]

Vitali Kapovitch, Martin Kell, and Christian Ketterer, On the structure of RCD spaces with upper curvature bounds , Math. Z. 301 (2022), no. 4, 3469--3502. 4449717

work page 2022
[25]

Vitali Kapovitch , Christian Ketterer , and Karl-Theodor Sturm , On gluing Alexandrov spaces with lower Ricci curvature bounds , accepted in Com. Anal. Geom. (2020), arXiv:2003.06242

work page arXiv 2020
[26]

Christian Ketterer and Andrea Mondino, Sectional and intermediate R icci curvature lower bounds via optimal transport , Adv. Math. 329 (2018), 781--818. 3783428

work page 2018
[27]

Vitali Kapovitch and Andrea Mondino, On the topology and the boundary of N --dimensional RCD ( K , N ) spaces , Geom. Topol. 25 (2021), no. 1, 445--495. 4226234

work page 2021
[28]

N. N. Kosovski , Gluing of R iemannian manifolds of curvature , Algebra i Analiz 14 (2002), no. 3, 140--157. 1921991

work page 2002
[29]

N. N. Kosovskij, Gluing of Riemannian manifolds of curvature at least  . , St. Petersbg. Math. J. 14 (2003), no. 3, 140--157 (English)

work page 2003
[30]

John Lott, Some geometric properties of the B akry-\' E mery- R icci tensor , Comment. Math. Helv. 78 (2003), no. 4, 865--883. 2016700 (2004i:53044)

work page 2003
[31]

John Lott and C \'e dric Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903--991. 2480619 (2010i:53068)

work page 2009
[32]

Pengzi Miao, On a localized Riemannian Penrose inequality , Commun. Math. Phys. 292 (2009), no. 1, 271--284 (English)

work page 2009
[33]

Emanuel Milman, Sharp isoperimetric inequalities and model spaces for the curvature-dimension-diameter condition, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 5, 1041--1078. 3346688

work page 2015
[34]

With applications to relativity , Pure and Applied Mathematics , 103

Barrett O'Neill, Semi- Riemannian geometry. With applications to relativity , Pure and Applied Mathematics , 103. New York - London etc.: Academic Press . xiii, 468 p. (1983)., 1983

work page 1983
[35]

Jiayin Pan, The G rushin hemisphere as a R icci limit space with curvature 1 , Proc. Amer. Math. Soc. Ser. B 10 (2023), 71--75. 4566173

work page 2023
[36]

Perelman, Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers , Comparison geometry, Cambridge: Cambridge University, 1997, pp

G. Perelman, Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers , Comparison geometry, Cambridge: Cambridge University, 1997, pp. 157--163 (English)

work page 1997
[37]

, Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers , 157--163 (English)

work page
[38]

203--219 (English)

Anton Petrunin, Applications of quasigeodesics and gradient curves, Comparison geometry, Cambridge: Cambridge University, 1997, pp. 203--219 (English)

work page 1997
[39]

, Semiconcave functions in A lexandrov's geometry , Surveys in differential geometry. V ol. XI , Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 137--201. 2408266 (2010a:53052)

work page 2007
[40]

, Alexandrov meets L ott- V illani- S turm , M\"unster J. Math. 4 (2011), 53--64. 2869253 (2012m:53087)

work page 2011
[41]

819--898

Conrad Plaut, Metric spaces of curvature k , Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 819--898. 1886682

work page 2002
[42]

Jiayin Pan and Guofang Wei, Examples of R icci limit spaces with non-integer H ausdorff dimension , Geom. Funct. Anal. 32 (2022), no. 3, 676--685. 4431126

work page 2022
[43]

Philipp Reiser , Metrics of Positive Ricci Curvature on Simply-Connected Manifolds of Dimension 6k , arXiv e-prints (2022), arXiv:2210.15610

work page arXiv 2022
[44]

Philipp Reiser, Generalized surgery on R iemannian manifolds of positive R icci curvature , Trans. Amer. Math. Soc. 376 (2023), no. 5, 3397--3418. 4577335

work page 2023
[45]

Luca Rizzi and Giorgio Stefani, Failure of curvature-dimension conditions on sub- R iemannian manifolds via tangent isometries , J. Funct. Anal. 285 (2023), no. 9, Paper No. 110099. 4623954

work page 2023
[46]

Wraith , Intermediate Ricci curvatures and Gromov's Betti number bound , arXiv e-prints (2022), arXiv:2208.13438

Philipp Reiser and David J. Wraith , Intermediate Ricci curvatures and Gromov's Betti number bound , arXiv e-prints (2022), arXiv:2208.13438

work page arXiv 2022
[47]

, A generalization of the Perelman gluing theorem and applications , arXiv e-prints (2023), arXiv:2308.06996

work page arXiv 2023
[48]

Arthur Schlichting , Gluing Riemannian manifolds with curvature operators at least k , arXiv e-prints (2012), arXiv:1210.2957

work page internal anchor Pith review Pith/arXiv arXiv 2012
[49]

187, Cambridge University Press, Cambridge, 2011, An analytic viewpoint

Barry Simon, Convexity, Cambridge Tracts in Mathematics, vol. 187, Cambridge University Press, Cambridge, 2011, An analytic viewpoint. 2814377 (2012d:46002)

work page 2011
[50]

II , Acta Math

Karl-Theodor Sturm, On the geometry of metric measure spaces. II , Acta Math. 196 (2006), no. 1, 133--177. 2237207 (2007k:53051b)

work page 2006
[51]

von Renesse and Karl-Theodor Sturm, Transport inequalities, gradient estimates, entropy, and R icci curvature , Comm

Max-K. von Renesse and Karl-Theodor Sturm, Transport inequalities, gradient estimates, entropy, and R icci curvature , Comm. Pure Appl. Math. 58 (2005), no. 7, 923--940. 2142879 (2006j:53048)

work page 2005
[52]

Jeremy Wong, An extension procedure for manifolds with boundary, Pacific J. Math. 235 (2008), no. 1, 173--199. 2379775

work page 2008

[1] [1]

Alexander and Richard L

Stephanie B. Alexander and Richard L. Bishop, Curvature bounds for warped products of metric spaces, Geom. Funct. Anal. 14 (2004), no. 6, 1143--1181. 2135163 (2007d:53061)

work page 2004

[2] [2]

33, American Mathematical Society, Providence, RI, 2001

Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. 1835418 (2002e:53053)

work page 2001

[3] [3]

Bakry and Michel \' E mery, Diffusions hypercontractives, Lecture Notes in Math., vol

D. Bakry and Michel \' E mery, Diffusions hypercontractives, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985. 889476

work page 1985

[4] [4]

Christian B\" a r and Bernhard Hanke, Boundary conditions for scalar curvature, Perspectives in scalar curvature. V ol. 2, World Sci. Publ., Hackensack, NJ, [2023] 2023, pp. 325--377. 4577919

work page 2023

[5] [5]

McCann, and Eric Woolgar, Inscribed radius bounds for lower Ricci bounded metric measure spaces with mean convex boundary , SIGMA, Symmetry Integrability Geom

Annegret Burtscher, Christian Ketterer, Robert J. McCann, and Eric Woolgar, Inscribed radius bounds for lower Ricci bounded metric measure spaces with mean convex boundary , SIGMA, Symmetry Integrability Geom. Methods Appl. 16 (2020), paper 131, 29 (English)

work page 2020

[6] [6]

Elia Bru \`e , Aaron Naber , and Daniele Semola , Fundamental Groups and the Milnor Conjecture , arXiv e-prints (2023), arXiv:2303.15347

work page arXiv 2023

[7] [7]

Bradley Lewis Burdick, Metrics of P ositive R icci C urvature on C onnected S ums: P rojective S paces, P roducts, and P lumbings , ProQuest LLC, Ann Arbor, MI, 2019, Thesis (Ph.D.)--University of Oregon. 4035402

work page 2019

[8] [8]

, Ricci-positive metrics on connected sums of projective spaces, Differential Geom. Appl. 62 (2019), 212--233. 3881650

work page 2019

[9] [9]

Walsh, and David J

Boris Botvinnik, Mark G. Walsh, and David J. Wraith, Homotopy groups of the observer moduli space of R icci positive metrics , Geom. Topol. 23 (2019), no. 6, 3003--3040. 4039184

work page 2019

[10] [10]

99 (2014), 136--151

Fabio Cavalletti, Monge problem in metric measure spaces with R iemannian curvature-dimension condition , Nonlinear Anal. 99 (2014), 136--151. 3160530

work page 2014

[11] [11]

McCann, and Michael Schmuckenschl \"a ger, A R iemannian interpolation inequality \`a la B orell, B rascamp and L ieb , Invent

Dario Cordero-Erausquin, Robert J. McCann, and Michael Schmuckenschl \"a ger, A R iemannian interpolation inequality \`a la B orell, B rascamp and L ieb , Invent. Math. 146 (2001), no. 2, 219--257. 1865396 (2002k:58038)

work page 2001

[12] [12]

Fabio Cavalletti and Andrea Mondino, Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower R icci curvature bounds , Invent. Math. 208 (2017), no. 3, 803--849. 3648975

work page 2017

[13] [13]

PDE 13 (2020), no

, New formulas for the L aplacian of distance functions and applications , Anal. PDE 13 (2020), no. 7, 2091--2147. 4175820

work page 2020

[14] [14]

Fabio Cavalletti and Emanuel Milman, The globalization theorem for the curvature-dimension condition, Invent. Math. 226 (2021), no. 1, 1--137. 4309491

work page 2021

[15] [15]

Guido De Philippis and Nicola Gigli, Non-collapsed spaces with R icci curvature bounded from below , J. \' E c. polytech. Math. 5 (2018), 613--650. 3852263

work page 2018

[16] [16]

Blaine Lawson, Jr., Spin and scalar curvature in the presence of a fundamental group

Mikhael Gromov and H. Blaine Lawson, Jr., Spin and scalar curvature in the presence of a fundamental group. I , Ann. of Math. (2) 111 (1980), no. 2, 209--230. 569070

work page 1980

[17] [17]

Nicola Gigli, Andrea Mondino, and Giuseppe Savar \'e , Convergence of pointed non-compact metric measure spaces and stability of R icci curvature bounds and heat flows , Proc. Lond. Math. Soc. (3) 111 (2015), no. 5, 1071--1129. 3477230

work page 2015

[18] [18]

Bang-Xian Han, Measure rigidity of synthetic lower R icci curvature bound on R iemannian manifolds , Adv. Math. 373 (2020), 107327, 31. 4129481

work page 2020

[19] [19]

Hirsch, Differential topology, Grad

Morris W. Hirsch, Differential topology, Grad. Texts Math., vol. 33, Springer, Cham, 1976 (English)

work page 1976

[20] [20]

Christian Ketterer, Ricci curvature bounds for warped products, J. Funct. Anal. 265 (2013), no. 2, 266--299. 3056704

work page 2013

[21] [21]

, Ricci curvature bounds for warped products, J. Funct. Anal. 265 (2013), no. 2, 266--299. 3056704

work page 2013

[22] [22]

, The Heintze - Karcher inequality for metric measure spaces , Proc. Am. Math. Soc. 148 (2020), no. 9, 4041--4056 (English)

work page 2020

[23] [23]

, Stability of metric measure spaces with integral R icci curvature bounds , J. Funct. Anal. 281 (2021), no. 8, Paper No. 109142, 48. 4271791

work page 2021

[24] [24]

Vitali Kapovitch, Martin Kell, and Christian Ketterer, On the structure of RCD spaces with upper curvature bounds , Math. Z. 301 (2022), no. 4, 3469--3502. 4449717

work page 2022

[25] [25]

Vitali Kapovitch , Christian Ketterer , and Karl-Theodor Sturm , On gluing Alexandrov spaces with lower Ricci curvature bounds , accepted in Com. Anal. Geom. (2020), arXiv:2003.06242

work page arXiv 2020

[26] [26]

Christian Ketterer and Andrea Mondino, Sectional and intermediate R icci curvature lower bounds via optimal transport , Adv. Math. 329 (2018), 781--818. 3783428

work page 2018

[27] [27]

Vitali Kapovitch and Andrea Mondino, On the topology and the boundary of N --dimensional RCD ( K , N ) spaces , Geom. Topol. 25 (2021), no. 1, 445--495. 4226234

work page 2021

[28] [28]

N. N. Kosovski , Gluing of R iemannian manifolds of curvature , Algebra i Analiz 14 (2002), no. 3, 140--157. 1921991

work page 2002

[29] [29]

N. N. Kosovskij, Gluing of Riemannian manifolds of curvature at least \( \) . , St. Petersbg. Math. J. 14 (2003), no. 3, 140--157 (English)

work page 2003

[30] [30]

John Lott, Some geometric properties of the B akry-\' E mery- R icci tensor , Comment. Math. Helv. 78 (2003), no. 4, 865--883. 2016700 (2004i:53044)

work page 2003

[31] [31]

John Lott and C \'e dric Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903--991. 2480619 (2010i:53068)

work page 2009

[32] [32]

Pengzi Miao, On a localized Riemannian Penrose inequality , Commun. Math. Phys. 292 (2009), no. 1, 271--284 (English)

work page 2009

[33] [33]

Emanuel Milman, Sharp isoperimetric inequalities and model spaces for the curvature-dimension-diameter condition, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 5, 1041--1078. 3346688

work page 2015

[34] [34]

With applications to relativity , Pure and Applied Mathematics , 103

Barrett O'Neill, Semi- Riemannian geometry. With applications to relativity , Pure and Applied Mathematics , 103. New York - London etc.: Academic Press . xiii, 468 p. (1983)., 1983

work page 1983

[35] [35]

Jiayin Pan, The G rushin hemisphere as a R icci limit space with curvature 1 , Proc. Amer. Math. Soc. Ser. B 10 (2023), 71--75. 4566173

work page 2023

[36] [36]

Perelman, Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers , Comparison geometry, Cambridge: Cambridge University, 1997, pp

G. Perelman, Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers , Comparison geometry, Cambridge: Cambridge University, 1997, pp. 157--163 (English)

work page 1997

[37] [37]

, Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers , 157--163 (English)

work page

[38] [38]

203--219 (English)

Anton Petrunin, Applications of quasigeodesics and gradient curves, Comparison geometry, Cambridge: Cambridge University, 1997, pp. 203--219 (English)

work page 1997

[39] [39]

, Semiconcave functions in A lexandrov's geometry , Surveys in differential geometry. V ol. XI , Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 137--201. 2408266 (2010a:53052)

work page 2007

[40] [40]

, Alexandrov meets L ott- V illani- S turm , M\"unster J. Math. 4 (2011), 53--64. 2869253 (2012m:53087)

work page 2011

[41] [41]

819--898

Conrad Plaut, Metric spaces of curvature k , Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 819--898. 1886682

work page 2002

[42] [42]

Jiayin Pan and Guofang Wei, Examples of R icci limit spaces with non-integer H ausdorff dimension , Geom. Funct. Anal. 32 (2022), no. 3, 676--685. 4431126

work page 2022

[43] [43]

Philipp Reiser , Metrics of Positive Ricci Curvature on Simply-Connected Manifolds of Dimension 6k , arXiv e-prints (2022), arXiv:2210.15610

work page arXiv 2022

[44] [44]

Philipp Reiser, Generalized surgery on R iemannian manifolds of positive R icci curvature , Trans. Amer. Math. Soc. 376 (2023), no. 5, 3397--3418. 4577335

work page 2023

[45] [45]

Luca Rizzi and Giorgio Stefani, Failure of curvature-dimension conditions on sub- R iemannian manifolds via tangent isometries , J. Funct. Anal. 285 (2023), no. 9, Paper No. 110099. 4623954

work page 2023

[46] [46]

Wraith , Intermediate Ricci curvatures and Gromov's Betti number bound , arXiv e-prints (2022), arXiv:2208.13438

Philipp Reiser and David J. Wraith , Intermediate Ricci curvatures and Gromov's Betti number bound , arXiv e-prints (2022), arXiv:2208.13438

work page arXiv 2022

[47] [47]

, A generalization of the Perelman gluing theorem and applications , arXiv e-prints (2023), arXiv:2308.06996

work page arXiv 2023

[48] [48]

Arthur Schlichting , Gluing Riemannian manifolds with curvature operators at least k , arXiv e-prints (2012), arXiv:1210.2957

work page internal anchor Pith review Pith/arXiv arXiv 2012

[49] [49]

187, Cambridge University Press, Cambridge, 2011, An analytic viewpoint

Barry Simon, Convexity, Cambridge Tracts in Mathematics, vol. 187, Cambridge University Press, Cambridge, 2011, An analytic viewpoint. 2814377 (2012d:46002)

work page 2011

[50] [50]

II , Acta Math

Karl-Theodor Sturm, On the geometry of metric measure spaces. II , Acta Math. 196 (2006), no. 1, 133--177. 2237207 (2007k:53051b)

work page 2006

[51] [51]

von Renesse and Karl-Theodor Sturm, Transport inequalities, gradient estimates, entropy, and R icci curvature , Comm

Max-K. von Renesse and Karl-Theodor Sturm, Transport inequalities, gradient estimates, entropy, and R icci curvature , Comm. Pure Appl. Math. 58 (2005), no. 7, 923--940. 2142879 (2006j:53048)

work page 2005

[52] [52]

Jeremy Wong, An extension procedure for manifolds with boundary, Pacific J. Math. 235 (2008), no. 1, 173--199. 2379775

work page 2008