Abstract: This talk explores the "inward flow," a contracting inhomogeneous curvature flow of closed immersed hypersurfaces in Euclidean space. The normal speed of this curvature flow is given by a positive increasing function which is asymptotic to the mean curvature. This inhomogeneity allows for novel behaviours not seen in standard mean curvature flow, including the loss of embeddedness without curvature blow-up and the appearance of higher multiplicity tangent flows.

We will outline the core analytical results governing this flow, primarily a Huisken–Sinestrari-type convexity estimate which ensures asymptotic weak convexity near singularities. Using these tools, we establish a compactness theorem for blow-up sequences, showing they converge to convex ancient solutions to mean curvature flow. Next, we examine the formation of collapsing singularities and present a noncollapsing result for surfaces of revolution. Finally, we conclude with a construction of unique, weak level-set solutions.