RCD gluing conjecture for noncollapsed spaces with boundary

For i=0,1i=0,1, let XiX_i be noncollapsed RCD(K,n)RCD(K,n) spaces with nonempty boundary Xi\partial X_i. Suppose there exists an isometry

I:X0X1.\mathcal I:\partial X_0\rightarrow \partial X_1.

RCD gluing conjecture. The glued metric measure space

(X0IX1,HX0IX1n)(X_0\cup_{\mathcal I} X_1, \mathcal H^n_{X_0\cup_{\mathcal I} X_1})

satisfies the condition RCD(K,n)RCD(K,n). This extends the known gluing result for Alexandrov spaces with curvature bounded below and the corresponding CD(K,N)CD^*(K,N) condition to general noncollapsed RCDRCD spaces; the conjecture is motivated by the expectation that natural notions of boundary for such spaces coincide and that the boundary is closed in the ambient space.

Sources & referencesView supporting material

Primary source

Vitali Kapovitch, Christian Ketterer and Karl-Theodor Sturm, “On gluing Alexandrov spaces with lower Ricci curvature bounds”, arXiv:2003.06242 (2020).

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