Measure rigidity of Ricci curvature lower bounds

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• 作者：Fabio Cavalletti^a ; ^{fabio.cavalletti@unipv.it" class="auth_mail" title="E-mail the corresponding author} ; Andrea Mondino^b ; ^{andrea.mondino@math.uzh.ch" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Metric measure spaces ; Ricci curvature ; Measure contraction property ; Disintegration theorem
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：2 January 2016
• 年：2016
• 卷：286
• 期：Complete
• 页码：430-480
• 全文大小：764 K

文摘

The measure contraction property, an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si1.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=62a42330066212ef30bb510e8d5d9961" title="Click to view the MathML source">MCPan>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">athvariant="sans-serif">MCPath>an>an>an> for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker modifications) implies on the measure, on its support and on the geodesics of the space.

We start our investigation from the Euclidean case by proving that if a positive Radon measure an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si2.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=6cd1e448b33bb7877d63aa046b095cee" title="Click to view the MathML source">man>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">athvariant="sans-serif">math>an>an>an> over an id="mmlsi167" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si167.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=73abb465ee98cfaf4975d3b908ece883" title="Click to view the MathML source">R^dan>an class="mathContainer hidden">an class="mathCode">ath altimg="si167.gif" overflow="scroll">athvariant="double-struck">Rdath>an>an>an> is such that an id="mmlsi4" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si4.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=0187c37dfa055d83fca249694870d3e7" title="Click to view the MathML source">(R^d,|⋅|,m)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">alse">(athvariant="double-struck">Rd,alse">|⋅alse">|,athvariant="sans-serif">malse">)ath>an>an>an> verifies a weaker variant of an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si1.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=62a42330066212ef30bb510e8d5d9961" title="Click to view the MathML source">MCPan>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">athvariant="sans-serif">MCPath>an>an>an>, then its support an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si5.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=7e309b9b5a33c81d0ed1725680e18ca8" title="Click to view the MathML source">spt(m)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">athvariant="normal">sptalse">(athvariant="sans-serif">malse">)ath>an>an>an> must be convex and an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si2.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=6cd1e448b33bb7877d63aa046b095cee" title="Click to view the MathML source">man>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">athvariant="sans-serif">math>an>an>an> has to be absolutely continuous with respect to the relevant Hausdorff measure of an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si5.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=7e309b9b5a33c81d0ed1725680e18ca8" title="Click to view the MathML source">spt(m)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">athvariant="normal">sptalse">(athvariant="sans-serif">malse">)ath>an>an>an>. This result is then used as a starting point to investigate the rigidity of an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si1.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=62a42330066212ef30bb510e8d5d9961" title="Click to view the MathML source">MCPan>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">athvariant="sans-serif">MCPath>an>an>an> in the metric framework.

We introduce the new notion of reference measure   for a metric space and prove that if an id="mmlsi398" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si398.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=f622c78fb9a660884c9c182ee0282664" title="Click to view the MathML source">(X,d,m)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si398.gif" overflow="scroll">alse">(X,athvariant="sans-serif">d,athvariant="sans-serif">malse">)ath>an>an>an> is essentially non-branching and verifies an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si1.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=62a42330066212ef30bb510e8d5d9961" title="Click to view the MathML source">MCPan>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">athvariant="sans-serif">MCPath>an>an>an>, and μ   is an essentially non-branching an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si1.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=62a42330066212ef30bb510e8d5d9961" title="Click to view the MathML source">MCPan>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">athvariant="sans-serif">MCPath>an>an>an> reference measure for an id="mmlsi7" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si7.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=8349b5a402d3a7819e0bae628452a794" title="Click to view the MathML source">(spt(m),d)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si7.gif" overflow="scroll">alse">(athvariant="normal">sptalse">(athvariant="sans-serif">malse">),athvariant="sans-serif">dalse">)ath>an>an>an>, then an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si2.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=6cd1e448b33bb7877d63aa046b095cee" title="Click to view the MathML source">man>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">athvariant="sans-serif">math>an>an>an> is absolutely continuous with respect to μ  , on the set of points where an inversion plan exists. As a consequence, an essentially non-branching an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003552&_mathId=si1.gif&_user=111111111&_pii=S0001870815003552&_rdoc=1&_issn=00018708&md5=62a42330066212ef30bb510e8d5d9961" title="Click to view the MathML source">MCPan>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">athvariant="sans-serif">MCPath>an>an>an> reference measure enjoys a weak type of uniqueness, up to densities. We also prove a stability property for reference measures under measured Gromov–Hausdorff convergence, provided an additional uniform bound holds.

In the final part we present concrete examples of metric spaces with reference measures, both in smooth and non-smooth setting. The main example will be the Hausdorff measure over an Alexandrov space. Then we prove that the following are reference measures over smooth spaces: the volume measure of a Riemannian manifold, the Hausdorff measure of an Alexandrov space with bounded curvature, and the Haar measure of the subRiemannian Heisenberg group.