Higher order Willmore hypersurfaces in Euclidean space
type: Item
instance of: scholarly article
title: Higher order Willmore hypersurfaces in Euclidean space
zbMATH Open document ID: 1179.53011
author: Zhen Guo
publication date: 2009-11-11T00:00:00Z
review text: For \(n>2\), \(x:M^n\to\mathbb{R}^{n+1}\) denotes an \(n\)-dimensional hypersurface; for \(0<i\leq n-1\), \(\sigma_i\) denotes the \(i\)-th mean curvature and \(Q_n=\sum_{i=0}^n(-1)^{i+1} {n \choose i}\sigma_1^{n-i}\sigma_i\). In [Differ. Geom. Appl. 25, No. 5, 543--551 (2007; Zbl 1138.53016)], the author showed that \(W_n(x)=\int_M Q_ndM\) is a conformal invariant, which he called the \textit{Willmore functional of \(x\)}, and he studied a variational problem related to that functional. The author calls an extremal hypersurface for the functional \(W_n\) an \textit{\(n\)-th order Willmore hypersurface}. In the case \(n=2\), \(W_2\) is the well-known Willmore functional \(\int_M(\sigma_1^2 -\sigma_2)^{n/2}\), studied by several authors. The purpose of the paper under review is to construct concrete examples of third-order Willmore hypersurfaces.
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 53A30
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 5629486
zbMATH Keywords: higher-order Willmore functional
zbMATH Keywords: conformal invariant
zbMATH Keywords: Willmore hypersurface
MaRDI profile type: Publication
full work available at URL: s10114-008-6422-y
OpenAlex ID: W2115760749
cites work: Generalized Willmore functionals and related variational problems
cites work: Q4889886
cites work: A conformal differential invariant and the conformal rigidity of hypersurfaces
cites work: Conformal geometry of surfaces in \(S^4\) and quaternions
cites work: A duality theorem for Willmore surfaces
cites work: A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces
cites work: A Global Invariant of Conformal Mappings in Space
cites work: Surfaces with vanishing Möbius form in \(S^n\).
cites work: Q3966739
cites work: An Invariant of Conformal Mappings
cites work: The second variational formula for Willmore submanifolds in \(S^n\).
cites work: Q3764821
cites work: Submanifolds with constant Möbius scalar curvature in \(S^n\)
cites work: Classification of hypersurfaces with parallel Möbius second fundamental form in \(S^{n+1}\)
cites work: Möbius isoparametric hypersurfaces in \(S^{n+1}\) with two distinct principal curvatures
cites work: Willmore submanifolds in a sphere.
cites work: Möbius geometry of hypersurfaces with constant mean curvature and scalar curvature
cites work: Q3816661
cites work: Moebius geometry of submanifolds in \(\mathbb{S}^n\)
DOI: 10.1007/S10114-008-6422-Y
Recommended article: On a generalization of Willmore surfaces for hypersurfaces
Recommended article: Willmore hypersurfaces in a sphere
Recommended article: Willmore inequality on hypersurfaces in hyperbolic space
Recommended article: Hypersurfaces with constant higher order mean curvature in Euclidean space
Recommended article: Q3150742
Recommended article: On the Willmore's theorem for convex hypersurfaces
Recommended article: Hypersurfaces with constant higher order mean curvature in space forms
Recommended article: Hypersurfaces with null higher order mean curvature
Recommended article: Willmore surfaces and \(F\)-Willmore surfaces in space forms
Recommended article: Remarks on hypersurfaces with constant higher order mean curvature in Euclidean space
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5629486