A characterization of non-collapsed RCD(K,N) spaces via Einstein tensors
[Une caractérisation des espaces RCD(K,N) non-collapsed via les tenseurs d’Einstein]
Honda, Shouhei1 ; Zhu, Xingyu2
1 Tohoku University, Sendai (Japan)
2 Institut für Agewandte Mathematik, Universität Bonn, Bonn (Germany)
Annales de l'Institut Fourier, Online first, 46 p.

Nous étudions le deuxième terme principal dans le développement des métriques c(n)t (n+2)/2 g t induites par le plongement via le noyau de la chaleur dans L 2 sur un espace RCD(K,N) compact. Nous montrons que la propriété de ce terme d’avoir une divergence nulle est vérifiée au sens faible et asymptotique si, et seulement si, l’espace est non-collapsed, quitte à multiplier la mesure de référence par un scalaire. Cela semble nouveau même pour les variétés riemanniennes pondérées, c’est-à-dire munies d’une mesure de référence. De plus, un exemple nous indique que le résultat ne peut pas être généralisé au cas non compact. En ce sens, notre résultat est optimal.

We investigate the second principal term in the expansion of the metrics c(n)t (n+2)/2 g t induced by the heat kernel embedding into L 2 on a compact RCD(K,N) space. We prove that the divergence free property of this term holds in the weak, asymptotic sense if and only if the space is non-collapsed up to multiplying the reference measure by a constant. This seems new even for weighted Riemannian manifolds. Moreover an example tells us that the result cannot be generalized to the noncompact case. In this sense, our result is sharp.

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DOI : 10.5802/aif.3652
Classification : 53C21, 53C23
Keywords: Einstein tensor, Heat kernel, Spectrum, Ricci curvature.
Mot clés : Tenseur d’Einstein, noyau de la chaleur, spectre, courbure de Ricci.
Honda, Shouhei 1 ; Zhu, Xingyu 2

1 Tohoku University, Sendai (Japan)
2 Institut für Agewandte Mathematik, Universität Bonn, Bonn (Germany) 
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     title = {A characterization of non-collapsed $\mathrm{RCD}(K,N)$ spaces via {Einstein} tensors},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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     language = {en},
     note = {Online first},
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Honda, Shouhei; Zhu, Xingyu. A characterization of non-collapsed $\mathrm{RCD}(K,N)$ spaces via Einstein tensors. Annales de l'Institut Fourier, Online first, 46 p.

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