In 2012, Gigli and Mantegazza introduced a new geometric flow via heat kernels. They demonstrated that this flow is tangential to the Ricci flow in a suitable weak sense for smooth, compact Riemannian manifolds. A salient feature of this flow is that it can be given meaning for compact RCD metric spaces by interpreting the equation distributionally as a flow of the distance metric. Gigli and Mantegazza further show that the two formulations agree for the smooth, compact manifold case. As a consequence, this flow can be successfully defined for spaces containing certain singularities. An important question is to understand regularity - do singularities disappear along the flow, or are they retained? The quintessential example has been to study manifolds with conical singularities.
In our work, we partially address this regularity question by studying spaces with "geometric singularities", by which we mean a smooth manifold but with a non-smooth metric. When such spaces are also RCD metric spaces with singularities on a closed subset, we obtain a metric tensor on the open non-singular part with regularity corresponding to the regularity of the initial heat kernel. In particular, we demonstrate that a manifold with a finite number of geometric conical singularities remains a smooth manifold away from the cone points for all time along the flow. For "rough" initial metrics, where we expect only continuity of the flow, we demonstrate connections between regularity of the flow and homogeneous Kato square root estimates.