# Curvature evolution of nonconvex lens-shaped domains

@inproceedings{Bellettini2009CurvatureEO, title={Curvature evolution of nonconvex lens-shaped domains}, author={Giovanni Bellettini and Matteo Novaga}, year={2009} }

Abstract We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a point, as proved in [Schnürer, Azouani, Georgi, Hell, Jangle, Köller, Marxen, Ritthaler, Sáez, Schulze and Smith, Trans. Amer. Math. Soc.]. Our theorem is the analog of the result of Grayson [J. Diff. Geom. 26: 285–314, 1987] for curvature flow of…

## 21 Citations

### TIME EXISTENCE FOR THE PLANAR NETWORK FLOW

- Mathematics
- 2018

We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions…

### On short time existence for the planar network flow

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We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions…

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- 2014

Signed distance from a smooth boundary.- Mean curvature vector and second fundamental form.- First variations of volume integrals and of the perimeter.- Smooth mean curvature flows.- Huisken's…

### Existence of a lens-shaped cluster of surfaces self-shrinking by mean curvature

- Materials ScienceMathematische Annalen
- 2019

We rigorously show the existence of a rotationally and centrally symmetric “lens-shaped” cluster of three surfaces, meeting at a smooth common circle, forming equal angles of $$120^{\circ }$$120∘,…

### Mean Curvature Flow with Triple Junctions in Higher Space Dimensions

- Mathematics
- 2012

We consider mean curvature flow of n-dimensional surface clusters. At (n−1)-dimensional triple junctions an angle condition is required which in the symmetric case reduces to the well-known 120°…

### Mean Curvature Flow with Triple Junctions in Higher Space Dimensions

- MathematicsArchive for Rational Mechanics and Analysis
- 2013

We consider mean curvature flow of n-dimensional surface clusters. At (n−1)-dimensional triple junctions an angle condition is required which in the symmetric case reduces to the well-known 120°…

### MOTION BY CURVATURE OF PLANAR CURVES WITH TWO FREE END POINTS

- Geology
- 2010

One motivation of our investigation of problem (P) originates from the study of evolution of grain domains in polycrystals. Here by a grain it refers to a periodic lattice structure of composite…

### Motion by curvature of planar curves with end points moving freely on a line

- Mathematics
- 2011

This paper deals with the motion by curvature of planar curves having end points moving freely along a line with fixed contact angles to this line. We first prove the existence and uniqueness of…

### Non–existence of theta–shaped self–similarly shrinking networks moving by curvature

- Mathematics
- 2016

ABSTRACT We prove that there are no networks homeomorphic to the Greek “Theta” letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees, such that under the…

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