Heat kernel and geometry of metric measure spaces with Ricci curvature lower bounds

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Zhu, Xingyu
Belegradek, Igor
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The thesis is a study of geometric properties of non-collapsed metric measure spaces with Ricci curvature lower bounds. We establish some characterizations of non-collapsed spaces and as a consequence, solve the De Philippis-Gigli conjecture which states that weakly non-collapsed spaces are actually non-collapsed. It is obtained as a corollary of the following equivalence, which holds under mild volume ratio condition: - $tr(Hess_f) = Δf$ on $U ⊆ X$ for every f sufficiently regular, - $m = cH^n$ on $U ⊆ X$ for some $c > 0$, where $U ⊆ X$ is open and $X$ is a - possibly collapsed - RCD space of essential dimension $n$. The method we use is smoothing the canonical Riemannian metric by a family of metrics $g_t$ induced by the heat kernel. We also study the short time expansion of $g_t$, and show that the weakly asymptotically divergence free property of the second term of the expansion is equivalent to the metric measure space being non-collapsed, under the same volume ratio condition as above. The expansion is made explicit for weighted Riemannian manifolds. Finally, we prove an almost everywhere convexity of the regular set $R_n$ which states that for every point in the regular set of essential dimension $R_n$ there is a geodesic lies completely in $R_n$ joining almost every other point in $R_n$. This result can also be interpreted as an almost convexity of the interior of an non-collapsed $RCD(K,N)$ space.
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