Title:
Heat kernel and geometry of metric measure spaces with Ricci curvature lower bounds
Heat kernel and geometry of metric measure spaces with Ricci curvature lower bounds
Author(s)
Zhu, Xingyu
Advisor(s)
Belegradek, Igor
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Abstract
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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Date Issued
2022-04-28
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Dissertation