A counterexample to gluing theorems for MCP metric measure spaces

Abstract

Perelman's doubling theorem asserts that the metric space obtained by gluing along their boundaries two copies of an Alexandrov space with curvature κ is an Alexandrov space with the same dimension and satisfying the same curvature lower bound. We show that this result cannot be extended to metric measure spaces satisfying synthetic Ricci curvature bounds in the MCP sense. The counterexample is given by the Grushin half-plane, which satisfies the MCP(0, N) if and only if N 4, while its double satisfies the MCP(0, N) if and only if N 5.

Sub-Riemannian interpolation inequalities

“…Finally, by taking A = y ∈ G 2 , and using the fact that the Grushin plane admits a one-parameter group of metric dilations, we obtain the following result. On the other hand, the Grushin half -planes satisfy the MCP(K, N ) if and only if N ≥ 4 and K ≤ 0, see [Riz17]. 7.4.…”

“…On the other hand, the Grushin half -planes satisfy the MCP(K, N ) if and only if N ≥ 4 and K ≤ 0, see [Riz17]. 7.4.…”

Distortion Coefficients of the $$\alpha $$-Grushin Plane

“…It seems that the Grushin structures behave in such a way that points q 0 and q lying on the same horizontal line (with x 0 = 0) provide the sharpest N where β t (q 0 , q) ≥ t N holds for all t ∈ [0, 1]. This is also what happens when α = 1 (see [4,Proposition 62.] and [19,Theorem 8. ] for Grushin half-planes).…”

Heat flow with Dirichlet boundary conditions via optimal transport and gluing of metric measure spaces

“…In [14], metric measure spaces supporting Dirichlet forms are glued together. There is also a very recent preprint by Rizzi which shows that gluing does not preserve the dimension in the measure-contraction property [19]. Apart from curvature bounds, the doubling of manifolds with boundary has also been applied by other communities to produce a related manifold without boundary, see for instance [2].…”