Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds
We prove that a metric measure space $(X,d,m)$ satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space $W^{1,2}$ is Hilbert is rectifiable. That is, a $RCD^*(K,N)$-space is rectifiable, and in particular for $m$-a.e. point the tangent cone is unique and euclidean of dimens...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Published: |
European Mathematical Society
2019
|
| _version_ | 1826306419999113216 |
|---|---|
| author | Mondino, A Naber, A |
| author_facet | Mondino, A Naber, A |
| author_sort | Mondino, A |
| collection | OXFORD |
| description | We prove that a metric measure space $(X,d,m)$ satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space $W^{1,2}$ is Hilbert is rectifiable. That is, a $RCD^*(K,N)$-space is rectifiable, and in particular for $m$-a.e. point the tangent cone is unique and euclidean of dimension at most $N$. The proof is based on a maximal function argument combined with an original Almost Splitting Theorem via estimates on the gradient of the excess. To this aim we also show a sharp integral Abresh-Gromoll type inequality on the excess function and an Abresh-Gromoll-type inequality on the gradient of the excess. The argument is new even in the smooth setting. |
| first_indexed | 2024-03-07T06:47:42Z |
| format | Journal article |
| id | oxford-uuid:fb6b6f15-c3ca-4656-8b51-2fdb5c4bdbc9 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:47:42Z |
| publishDate | 2019 |
| publisher | European Mathematical Society |
| record_format | dspace |
| spelling | oxford-uuid:fb6b6f15-c3ca-4656-8b51-2fdb5c4bdbc92022-03-27T13:13:41ZStructure Theory of Metric-Measure Spaces with Lower Ricci Curvature BoundsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:fb6b6f15-c3ca-4656-8b51-2fdb5c4bdbc9Symplectic Elements at OxfordEuropean Mathematical Society2019Mondino, ANaber, AWe prove that a metric measure space $(X,d,m)$ satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space $W^{1,2}$ is Hilbert is rectifiable. That is, a $RCD^*(K,N)$-space is rectifiable, and in particular for $m$-a.e. point the tangent cone is unique and euclidean of dimension at most $N$. The proof is based on a maximal function argument combined with an original Almost Splitting Theorem via estimates on the gradient of the excess. To this aim we also show a sharp integral Abresh-Gromoll type inequality on the excess function and an Abresh-Gromoll-type inequality on the gradient of the excess. The argument is new even in the smooth setting. |
| spellingShingle | Mondino, A Naber, A Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds |
| title | Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds |
| title_full | Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds |
| title_fullStr | Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds |
| title_full_unstemmed | Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds |
| title_short | Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds |
| title_sort | structure theory of metric measure spaces with lower ricci curvature bounds |
| work_keys_str_mv | AT mondinoa structuretheoryofmetricmeasurespaceswithlowerriccicurvaturebounds AT nabera structuretheoryofmetricmeasurespaceswithlowerriccicurvaturebounds |