Heat kernel approach to geometric analysis on metric measure spaces with Ricci curvature bounded below
Heat kernel approach to geometric analysis on metric measure spaces with Ricci curvature bounded below
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Let $(X, d, m)$ be a compact metric measure space with Ricci听curvature bounded below in a synthetic sense, so-called an RCD space.听The heat kernel allows us to embed the space into $L^2$ for any time听$t>0$, and the pull-back $g_t$ defines a geometric flow on the space.听The geometric flow $g_t$ has various applications to metric measure听geometry, including a resolution of a conjecture raised by De听Philippis-Gigli. In this talk we discuss Sobolev maps between RCD
spaces via $g_t$, instead of using Nash's embedding in the smooth听setting. In particular, we discuss a compatibility with Korevaar-Schoen听theory, and a nonlinear analogue of Cheeger's differentiability听theorem for Sobolev functions.
This talk is based on a joint work with听Yannick Sire.