Some remarks on Huisken's monotonicity formula for mean curvature flow

We discuss a monotone quantity related to Huisken's monotonicity formula and some technical consequences for mean curvature flow.Comment: in "Singularities in nonlinear evolution phenomena and applications", 157-169, CRM Series, 9, Ed. Sc. Norm. Pisa, 200

A note on non lower semicontinuous perimeter functionals on partitions

We consider isotropic non lower semicontinuous weighted perimeter functionals defined on partitions of domains in $\mathbb{R}^n$. Besides identifying a condition on the structure of the domain which ensures the existence of minimizing configurations, we describe the structure of such minima, as well as their regularity

Flow by mean curvature inside a moving ambient space

We show some computations related to the motion by mean curvature flow of a submanifold inside an ambient Riemannian manifold evolving by Ricci or backward Ricci flow. Special emphasis is given to the possible generalization of Huisken's monotonicity formula and its connection with the validity of some Li--Yau--Hamilton differential Harnack--type inequalities in a moving Riemannian manifold.Comment: Revised versio

Motion by curvature of planar networks, II

We prove that the curvature flow of an embedded planar network of three curves connected through a triple junction, with fixed endpoints on the boundary of a given strictly convex domain, exists smooth as long as the lengths of the three curves stay far from zero. If this is the case for all times, then the evolution exists for all times and the network converges to the Steiner minimal connection between the three endpoints