# 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… Expand

#### 19 Citations

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… Expand

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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∘,… Expand

Mean Curvature Flow with Triple Junctions in Higher Space Dimensions

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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°… Expand

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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… Expand

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

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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… Expand

On the Classification of Networks Self–Similarly Moving by Curvature

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Abstract We give an overview of the classification of networks in the plane with at most two triple junctions with the property that under the motion by curvature they are self-similarly shrinking.… Expand

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