Open Access Published by De Gruyter Open Access

Volume 5 Issue 1

January 2020

Issue of Geometric Flows

Contents
Journal Overview

Open Access February 17, 2020

On the Surface Diffusion Flow with Triple Junctions in Higher Space Dimensions

H. Garcke, M. Gößwein

Page range: 1-39

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Abstract

We show short time existence for the evolution of triple junction clusters driven by the surface diffusion flow. On the triple line we use the boundary conditions derived by Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard equation with degenerated mobility. These conditions are concurrency of the triple junction, angle conditions between the hypersurfaces, continuity of the chemical potentials and a flux-balance. For the existence analysis we first write the geometric problem over a fixed reference surface and then use for the resulting analytic problem an approach in a parabolic Hölder setting.

Open Access March 25, 2020

On the Convergence of the Elastic Flow in the Hyperbolic Plane

Marius Müller, Adrian Spener

Page range: 40-77

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Abstract

We examine the L 2 -gradient flow of Euler’s elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by constructing a class of curves whose lengths blow up in infinite time. The convergence results follow from a constrained sharp Reilly-type inequality.

Open Access October 31, 2020

A dynamical approach to the variational inequality on modified elastic graphs

Shinya Okabe, Kensuke Yoshizawa

Page range: 78-101

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Abstract

We consider the variational inequality on modified elastic graphs. Since the variational inequality is derived from the minimization problem for the modified elastic energy defined on graphs with the unilateral constraint, a solution to the variational inequality can be constructed by the direct method of calculus of variations. In this paper we prove the existence of solutions to the variational inequality via a dynamical approach. More precisely, we construct an L 2 -type gradient flow corresponding to the variational inequality and prove the existence of solutions to the variational inequality via the study on the limit of the flow.

Open Access December 31, 2020

Simply connected translating solitons contained in slabs

Francesco Chini

Page range: 102-120

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Abstract

In this work we show that 2-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of ℝ 3 with entropy strictly less than 3 must be mean convex and thus, thanks to a result by Spruck and Xiao are convex. Recently, such 2-dimensional convex translating solitons have been completely classified, up to an ambient isometry, as vertical planes, (tilted) grim reaper cylinders, Δ -wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples by Ho man, Martín and White show that the bound on the entropy is necessary.

About this journal

THE JOURNAL IS CLOSED FOR SUBMISSIONS

Geometric Flows was an open access journal which exposes readers to the research frontier concerning geometric flow theory. It featured papers presenting the state-of-the-art of a rapidly developing field lying at the interface between geometric analysis, applied mathematics, theoretical and mathematical physics.

The journal was published in Open Access model and is archived in Portico.

All authors retain copyright, unless – due to their local circumstances – their work is not copyrighted. The copyrights are governed by the Creative-Commons license.

The journal was edited by: Mauro Carfora (University of Pavia, Italy) and Carlo Mantegazza (Università degli Studi di Napoli "Federico II", Italy)