The \(n\)-dimensional analogue of a variational problem of Euler
type: Item
instance of: scholarly article
title: The \(n\)-dimensional analogue of a variational problem of Euler
zbMATH Open document ID: 1547.5301
author: Ulrich Dierkes
author: Gerhard Huisken
published in: Mathematische Annalen
publication date: 2024-08-06T00:00:00Z
review text: Let \(M\subset \mathbb{R}^{n+1}\) be a smooth hypersurface, and \(f\) a \(C^2\) function from \(\mathbb{R}^{n+1}\) into \(\mathbb{R}\). The authors consider the weighted area-functional \[\mathcal{E}_f(M):=\int_M f(x) d\mathcal{H}_n(x)\]\Nand derive first and second variation formulae for the integral \(\mathcal{E}_f(M)\). Note that \(M\) is stationary for \(f\) if the relation \N\[\Nf H =\sum_{i=1}^{n+1} f_{x_i} \nu_i\N\]\Nholds, where \(\nu_i\) are the components of the normal vector and \(H\) is the mean curvature. A stationary hypersurface is said to be stable if the second derivative of the functional \(\mathcal{E}_f(M)\) is non-negative. \N\NThe authors study the case \(f(x)=|x]^\alpha\) and prove that a stationary surface\N\N-- cannot be compact if \(\alpha > -n\), \N\N-- is necessarily a sphere centered at the origin if \(\alpha=-n\) (if we suppose \(M\) to be compact),\N\N-- and in the case \(\alpha < -n\) the closure of \(M\) contains the origin.\N\NThey also study the case of minimal cones over products of spheres with vertex at the origin: they are stationary for \(\mathcal{E}_f\) and the authors investigate their stability. In addition, complementing non-existence results for nontrivial stable cones are formulated and proved.\N\NThe authors conclude by proving minimizing properties of some symmetric cones under suitable conditions on \(\alpha\) and the dimension \(n\). Also planes through the origin and spheres centered at the origin are minimizers in certain cases. All results are proved by using some kind of calibration theory.
reviewed by: Antoine Henrot
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 7892680
zbMATH Keywords: weighted area functional
zbMATH Keywords: minimal cones
MaRDI profile type: Publication
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DOI: 10.1007/S00208-023-02726-3
description: scientific article; zbMATH DE number 7892680