Interface evolution driven by curvature and related problems

Many natural and technological processes involve shapes that change over time. Examples include how soap films move, how crystals grow or shrink, how different materials separate into distinct regions, or how images are smoothed in computer graphics. In all these situations, the boundary of a shape moves with a speed that depends on its curvature -- a quantity that describes how much the shape bends. Understanding such motions is important both for describing physical phenomena and for developing reliable mathematical models. This project aims to deepen the mathematical understanding of curvature-driven evolution in several challenging directions. 1. Shapes made of several regions. Many materials or physical systems consist of several phases, each occupying part of space. Their boundaries form networks that meet at junctions, and these networks may have angles or singular points. We study how such multiphase crystalline networks evolve over time, how singularities form or disappear, and how special self-similar shapes (called self-shrinkers) behave. We also aim to understand which partitions of the plane minimize perimeter when the geometry is influenced by a preferred direction, such as in square or hexagonal crystals. 2. Flows that smooth shapes. Another major theme is the study of the gradient flow of the total variation, a model that describes how irregular shapes or signals become smoother in time. This flow is deeply connected to mean curvature flow. We investigate questions of existence, the regularity of the evolution in time, how the energy changes, and how the jump set -- the parts where a function has sharp edges -- moves during the evolution. 3. Links with phase-field models. In physics and numerical simulation, curved interfaces are often approximated by so-called phase- field models. These are easier to compute but harder to analyze. We study how such approximations relate to the true geometric motion and address conjectures of De Giorgi on when and how these diffuse models capture the correct mean-curvature evolution. The project develops new ideas and explores open problems that are central in modern geometric analysis. Our approach combines tools from geometric measure theory, the calculus of variations, nonlinear partial differential equations, and the theory of evolving networks. Early results already show strong promise. The project includes collaborations with leading international experts Prof. Giovanni Bellettini (U. Udine and ICTP of Trieste, Italy) and Prof. Matteo Novaga (U. Pisa, Italy) and will involve the training of a PhD student, helping to strengthen the research environment at the University of Vienna.