The splitting theorem in non-smooth context

Abstract

We prove that an infinitesimally Hilbertian CD(0,N) space containing a line splits as the product of $R$ and an infinitesimally Hilbertian CD(0,N-1) space. By `infinitesimally Hilbertian' we mean that the Sobolev space $W^{1,2}(X,d,m)$, which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence.

Final version as appeared in Memoirs of the AMS

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• Isoperimetric sets in spaces with lower bounds on the Ricci curvature
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• On Interpolation and Curvature via Wasserstein Geodesics
• Sobolev spaces on warped products
• Measure rigidity of Ricci curvature lower bounds
• The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds
• Maximal volume entropy rigidity for ${\mathsf{RCD}}^{\ast }(-(N-1),N)$ spaces
• Local spectral convergence in $RC{D}^{\ast }(K,N)$ spaces
• Second order differentiation formula on compact $RC{D}^{\ast }(K,N)$ spaces
• Weyl's law on $RC{D}^{\ast }(K,N)$ metric measure spaces
• Rigidity for the spectral gap on $RCD(K,\infty )$-spaces
• An overview on the proof of the splitting theorem in non-smooth context
• An upper bound on the revised first Betti number and a torus stability result for RCD spaces
• A new notion of angle between three points in a metric space
• A sufficient condition to a regular set of positive measure on RCD spaces
• Recognizing the flat torus among $RC{D}^{\ast }(0,N)$ spaces via the study of the first cohomology group
• Mixed curvature almost flat manifolds
• Sobolev-to-Lipschitz Property on QCD-spaces and Applications
• From volume cone to metric cone in the nonsmooth setting
• Weakly non-collapsed RCD spaces are strongly non-collapsed
• Ricci curvature lower bounds on Sasakian manifolds
• Cheeger-Colding-Tian theory for conic Kahler-Einstein metrics
• The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces
• Collapsed Ricci limit spaces as non-collapsed $RCD$ spaces
• The isometry group of an ${\mathsf{RCD}}^{\ast }$-space is Lie
• Rigidity of manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound
• A Bishop type inequality on metric measure spaces with Ricci curvature bounded below
• Monotonicity formulas for harmonic functions in $\mathrm{RCD}(0,N)$ spaces
• Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces
• The large scale structure of complete $4$-manifolds with nonnegative Ricci curvature and Euclidean volume growth
• Noncompact RCD(0,N) spaces with linear volume growth
• A characterization of non-collapsed $RCD(K,N)$ spaces via Einstein tensors
• Sobolev mappings between RCD spaces and applications to harmonic maps: a heat kernel approach
• Characterization of equality in Zhong-Yang type (sharp) spectral gap estimates for metric measure spaces
• Nonlinear geometric analysis on Finsler manifolds
• On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: asymptotic volume ratio, volume entropy and rigidity
• Characterization of Low Dimensional $RC{D}^{\ast }(K,N)$ spaces
• Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature
• Certain min-max values related to the $p$-energy and packing radii of Riemannian manifolds and metric measure spaces
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• The Measure Preserving Isometry Groups of Metric Measure Spaces
• Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry
• Failure of topological rigidity results for the measure contraction property
• Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth Have Finitely Generated Fundamental Groups
• An almost rigidity Theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth
• Isometric immersions of RCD spaces