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Talks by Mariel Sáez

Short-time existence for the network flow

Pontificia Universidad Católica de Chile

The network flow is a system of parabolic differential equations that describes the motion of a family of curves in which each of them evolves under curve-shortening flow. This problem arises naturally in physical phenomena and its solutions present a rich variety of behaviors. The goal of this talk is to describe some properties of this geometric flow and to discuss an alternative proof of short-time existence for non-regular initial conditions. The methods of our proof are based on techniques of geometric microlocal analysis that have been used to understand parabolic problems on spaces with conic singularities. This is joint work with Jorge Lira, Rafe Mazzeo, and Alessandra Pluda.

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On the Evolution by Fractional Mean Curvature

Pontificia Universidad Católica de Chile

In this work we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric objects that in turn yield the preservation of certain quantities, such as the positivity of the fractional mean curvature.

Mean curvature flow without singularities

Pontificia Universidad Católica de Chile

We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow with one additional dimension.

Seminario 1ª Planta

Mariel Sáez

Pontificia Universidad Católica de Chile (Chile)

Number of talks
3
Number of visits
2
Last visit
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