Cones over metric measure spaces and the maximal diameter theorem

By C. Ketterer. 2013

153 citations recorded by the source corpus.

The main result of this article states that the (K;N)-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD^*(KN;N+1) if and only if the underlying space satisfies RCD^*(N-1;N). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner's inequality that was established for metric measure spaces by Erbar, Kuwada and Sturm and independently by Ambrosio, Mondino and Savar\'e. As corollary of our result and the Cheeger-Gigli-Gromoll splitting theorem we obtain a maximal diameter theorem for metric measure spaces that satisfy a reduced Riemannian curvature-dimension condition.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/2ab67bafa4f99a9f9996dfeda16cca930499d07e

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In Applied Mathematics

Citations in the corpus

Needle decompositions and isoperimetric inequalities in Finsler geometry (2015, methodology): “pts in such a direction). 39 (E) Rigidity. In [CM1], some rigidity results for the isoperimetric inequality in positively curved spaces were obtained with the help of the maximal diameter rigidity in [Ke] for metric measure spaces satisfying the Riemannian curvature-dimension condition. In the Finsl”
Obata’s Rigidity Theorem for Metric Measure Spaces (2014, methodology): “u;PsXφ) ≥ 0 if t,s→ 0 (ﬁrst tthen s), and we obtain ΓX,V 2 (u;φ) ≥ 0 for u∈ D(ΓX,V 2 ) and φ∈ D∞(LX,V ). For instance, this works precisely like in paragraph 3. and 4. of the proof of Theorem 3.23 in [Ket]. Hence, ChV,X satisﬁes BE(κ+ K,∞) and by the equivalence result from [AGS] (X,d X,e−Vdm X) sat”
Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, background): “Note that (Xn,dn,mn) is a RCD∗(N+n−2, N+n−1)-space (see [36]) and that (Xn,dn) are isometric to a hemisphere of the n-dimensional unit sphere Sn(1) as metric spaces.”
Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds (2015, methodology): “But then by the Maximal Diameter Theorem [43] we get that (X∞, d∞,m∞) is isomorphic to a spherical suspension [0, π]×N−1sin Y for some RCD∗(N − 2, N − 1) space (Y, dY ,mY ) with mY (Y ) = 1.”
Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature (2020, background): “Moreover, according to Ketterer [32] the same holds if we consider M to be a CD∗(n − 1, n) space.”
On quotients of spaces with Ricci curvature bounded below (2017, background): “Moreover, examples of m.m. spaces that satisfy the MCPcondition in which a finite group of measure-preserving isometries acts but do not have a principal orbit type have been constructed in [46] by Ketterer and Rajala, and generalized in [68] by the fourth author.”
Second order differentiation formula on RCD*$(K,N)$ spaces (2018, background): “1 Introduction In the last ten years there has been a great interest in the study of metric measure spaces with Ricci curvature bounded from below, see for instance [44], [55], [56], [29], [5], [6], [27], [7], [48], [49], [30], [24], [26], [38], [9], [47], [13], [12].”
Local spectral convergence in RCD∗(K,N) spaces (2017, unknown): “kej i(j) − ˆei(j)kH1,2(BR(x i(j))) → 0. Proposition 4.15 (Local convergence for metric cones). Let us consider the metric cone (C(Z),dC(Z),r 2m Z) over a compact RCD∗(N − 2,N − 1)-space (Z,dZ,mZ) (by [K15] (C(Z),dC(Z),r2mZ) is an RCD∗(0,N)-space). Then H1,2 0 (BR(p),d C(Z),r 2m Z) = Hˆ1,2 0 (BR(p),d”
From volume cone to metric cone in the nonsmooth setting (2015, methodology): “We remark that such information about the structure of the sphere will come after we proved the cone structure of the original space via a direct use of Ketterer’s results [20] about lower Ricci bounds for cones.”
New stability results for sequences of metric measure spaces with uniform Ricci bounds from below (2016, result): “ application is related to almost spherical suspension theorems of positive Ricci curvature. For simplicity we discuss here only the case when N≥ 2 is an integer, but our results (as those in [St06], [K15a], [K15b], [CM15b]) cover also the case N∈ (1,∞). In [CM15b] Cavalletti-Mondino proved that for”
New formulas for the Laplacian of distance functions and applications (2018, background): “The stronger rigidity statement under the stronger RCD(K,N) assumption is a direct consequence of the Maximal Diameter Theorem proved by Ketterer [37] in the RCD(K,N)-setting.”
Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below (2014, methodology): “i) The constructions made here are compatible with all the analytic tools developed so far for the study of RCD spaces, which in turn already produced non-trivial geometric consequences like the Abresch-Gromoll inequality [35], the splitting theorem [31], the maximal diameter theorem [38] and rectif”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “The following corollary is a consequence of the Maximal Diameter Theorem [43], and of the compactness/stability of the class of RCD(K,N) spaces, for some fixed K > 0 and N > 1, with respect to the measured Gromov-Hausdorff convergence.”
On the topology and the boundary of N–dimensional RCD(K,N) spaces (2019, background): “. Thanks to the stability of (resp. non-collapsed) RCD(K,N) spaces, it follows that the (resp. non-collapsed) pmGH limits of such objects are (resp. non-collapsed) RCD(K,N) as well. (2) By [Pet11] or [Ket15a] or [GGKMS18], a cone (resp. a spherical suspension) over RP2 is an example of a non-collaps”