{"title":"Comments on the regularity of harmonic maps between singular spaces","authors":"Luca Gennaioli, Nicola Gigli, Hui-Chun Zhang, Xi-Ping Zhu","doi":"arxiv-2408.00479","DOIUrl":null,"url":null,"abstract":"In this work we are going to establish H\\\"older continuity of harmonic maps\nfrom an open set $\\Omega$ in an ${\\rm RCD}(K,N)$ space valued into a ${\\rm\nCAT}(\\kappa)$ space, with the constraint that the image of $\\Omega$ via the map\nis contained in a sufficiently small ball in the target. Building on top of\nthis regularity and assuming a local Lipschitz regularity of the map, we\nestablish a weak version of the Bochner-Eells-Sampson inequality in such a\nnon-smooth setting. Finally we study the boundary regularity of such maps.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.00479","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this work we are going to establish H\"older continuity of harmonic maps
from an open set $\Omega$ in an ${\rm RCD}(K,N)$ space valued into a ${\rm
CAT}(\kappa)$ space, with the constraint that the image of $\Omega$ via the map
is contained in a sufficiently small ball in the target. Building on top of
this regularity and assuming a local Lipschitz regularity of the map, we
establish a weak version of the Bochner-Eells-Sampson inequality in such a
non-smooth setting. Finally we study the boundary regularity of such maps.