Local curvature-dimension condition implies measure-contraction property

By Fabio Cavalletti, Karl‐Theodor Sturm. 2011

69 citations recorded by the source corpus.

Abstract We prove that for non-branching metric measure spaces the local curvature condition CD loc ( K , N ) implies the global version of MCP ( K , N ) . The curvature condition CD ( K , N ) introduced by the second author and also studied by Lott and Villani is the generalization to metric measure space of lower bounds on Ricci curvature together with upper bounds on the dimension. This paper is the following step of Bacher and Sturm (2010) [1] where it is shown that CD loc ( K , N ) is equivalent to a global condition CD ⁎ ( K , N ) , slightly weaker than the usual CD ( K , N ) . It is worth pointing out that our result implies sharp Bishop–Gromov volume growth inequality and sharp Poincare inequality.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/21d01dd33fdd95c477e85945a5b4040e1fde6bd6

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “While a great deal of progress has been made in this latter general framework from both the analytic, geometric and structural points of view, see for instance [2, 3, 4, 5, 6, 7, 9, 14, 15, 18, 27, 33, 35, 37, 36, 43, 50, 59, 63, 66], the isoperimetric problem has remained elusive.”
An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature (2014, result): “Notice that our argument for such dimension reduction is in fact the same used by Cavalletti-Sturm in [12].”
Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below (2013, background): “For arbitrary K we have that any CD(K,N)-space is also CD(K,N), but the converse implication is currently not perfectly understood (it is known that CD(K,N) implies CD(K, N), where K = N−1 N K ≤ K but the equivalence is open; for some recent progress see [10] and [9]).”
Optimal Maps and Exponentiation on Finite-Dimensional Spaces with Ricci Curvature Bounded from Below (2013, unknown): “We turn to the formulation of the CD∗(K,N) condition, coming from [6], to which we also refer for a detailed discussion of its relation with the CD(K,N) condition (see also [9] and [8]).”
Cones over metric measure spaces and the maximal diameter theorem (2013, unknown): “Bacher and Sturm observed that this situation can be avoided if and only if the diameter of the underlying space is smaller than π.”
Monge problem in metric measure spaces with Riemannian curvature-dimension condition (2013, unknown): “Since RCD(K, N) implies MCP(K, N), see [18] and [11], and the L(2) optimal plans are unique, the estimate (6.”
Structure theory of metric measure spaces with lower Ricci curvature bounds (2014, background): “While a great deal of progress has be en made in this latter general framework, see for instance [3, 5, 6, 7, 8, 9, 10, 14, 15, 27, 28, 29, 30, 32, 33, 34, 40, 41, 42, 46], the structure theory on such metric-measure spaces is still much less developed tha n in the case of smooth limits.”
On the differential structure of metric measure spaces and applications (2012, background): “The drawback of the CD(K,N) condition is that the geometric and functional inequalities which are its consequence sometime occur with non-sharp constants (although some positive recent development in this direction can be found in [12]).”
Characterization of Low Dimensional RCD*(K, N) Spaces (2015, background): “The precise de nition and the proof can be found in [10].”
The globalization theorem for the Curvature-Dimension condition (2016, methodology): “In [30], this strategy was implicitly employed by Cavalletti and Sturm to show that CDloc(K,N) implies the measure-contraction property MCP(K,N), which in a sense is a particular case of CD(K,N) when one end of theW2geodesic is a Dirac delta at a point o ∈ X (see [72, 56]).”
Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds (2015, methodology): “[69] T. Rajala and K.T. Sturm, Non-branching geodesics and optimal maps in strong CD(K,∞)-spaces, Calc.”
On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces (2013, background): “[8] K. Bacher and K.-Th. Sturm.”
The Splitting Theorem in Non-Smooth Context (2013, unknown): “Actually, our situation is simpler than that of these papers because we have at disposal a product structure which is absent in [18], [17].”
Li–Yau and Harnack type inequalities in RCD∗(K,N) metric measure spaces (2013, background): “We refer to this source also for a detailed discussion of the relation of the CD∗(K,N) with the CD(K,N) condition previously introduced by Lott-Villani [29] and Sturm [38] (for recent development about the relations between CD(K,N) and CD∗(K,N) see also [15] and [14]).”