New formulas for the Laplacian of distance functions and applications
The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially non-bra...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Mathematical Sciences Publishers
2020
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| _version_ | 1797075083863261184 |
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| author | Cavalletti, F Mondino, A |
| author_facet | Cavalletti, F Mondino, A |
| author_sort | Cavalletti, F |
| collection | OXFORD |
| description | The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially non-branching MCP(K,N)-spaces). Such a representation formula makes apparent the classical upper bounds and also some new lower bounds, together with a precise description of the singular part. The exact representation formula for the Laplacian of 1-Lipschitz functions (in particular for distance functions) holds also (and seems new) in a general complete Riemannian manifold. We apply these results to prove the equivalence of CD(K,N) and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic Splitting Theorem for infinitesimally Hilbertian essentially non-branching spaces verifying MCP(0,N). |
| first_indexed | 2024-03-06T23:45:16Z |
| format | Journal article |
| id | oxford-uuid:70b04fc2-969f-4be9-9732-30514080fc09 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T23:45:16Z |
| publishDate | 2020 |
| publisher | Mathematical Sciences Publishers |
| record_format | dspace |
| spelling | oxford-uuid:70b04fc2-969f-4be9-9732-30514080fc092022-03-26T19:38:47ZNew formulas for the Laplacian of distance functions and applicationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:70b04fc2-969f-4be9-9732-30514080fc09EnglishSymplectic Elements at OxfordMathematical Sciences Publishers2020Cavalletti, FMondino, AThe goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially non-branching MCP(K,N)-spaces). Such a representation formula makes apparent the classical upper bounds and also some new lower bounds, together with a precise description of the singular part. The exact representation formula for the Laplacian of 1-Lipschitz functions (in particular for distance functions) holds also (and seems new) in a general complete Riemannian manifold. We apply these results to prove the equivalence of CD(K,N) and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic Splitting Theorem for infinitesimally Hilbertian essentially non-branching spaces verifying MCP(0,N). |
| spellingShingle | Cavalletti, F Mondino, A New formulas for the Laplacian of distance functions and applications |
| title | New formulas for the Laplacian of distance functions and applications |
| title_full | New formulas for the Laplacian of distance functions and applications |
| title_fullStr | New formulas for the Laplacian of distance functions and applications |
| title_full_unstemmed | New formulas for the Laplacian of distance functions and applications |
| title_short | New formulas for the Laplacian of distance functions and applications |
| title_sort | new formulas for the laplacian of distance functions and applications |
| work_keys_str_mv | AT cavallettif newformulasforthelaplacianofdistancefunctionsandapplications AT mondinoa newformulasforthelaplacianofdistancefunctionsandapplications |