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- W4311031889 abstract "Abstract We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in full generality. One of the auxiliary results of independent interest that we obtain is about the link between the properties <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>tr</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mi>Hess</m:mi> <m:mo></m:mo> <m:mi>f</m:mi> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi mathvariant=normal>Δ</m:mi> <m:mo></m:mo> <m:mi>f</m:mi> </m:mrow> </m:mrow> </m:math> operatorname{tr}(operatorname{Hess}f)=Delta f on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>U</m:mi> <m:mo>⊆</m:mo> <m:mi mathvariant=sans-serif>X</m:mi> </m:mrow> </m:math> Usubseteq{mathsf{X}} for every 𝑓 sufficiently regular, <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi mathvariant=fraktur>m</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>c</m:mi> <m:mo></m:mo> <m:msup> <m:mi mathvariant=script>H</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> mathfrak{m}=cmathscr{H}^{n} on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>U</m:mi> <m:mo>⊆</m:mo> <m:mi mathvariant=sans-serif>X</m:mi> </m:mrow> </m:math> Usubseteq{mathsf{X}} for some <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>c</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> c>0 , where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>U</m:mi> <m:mo>⊆</m:mo> <m:mi mathvariant=sans-serif>X</m:mi> </m:mrow> </m:math> Usubseteq{mathsf{X}} is open and 𝖷 is a – possibly collapsed – RCD space of essential dimension 𝑛." @default.
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- W4311031889 date "2022-12-01" @default.
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- W4311031889 title "Weakly non-collapsed RCD spaces are strongly non-collapsed" @default.
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