A conjecture in relation to Loewner's conjecture
type: Item
instance of: scholarly article
title: A conjecture in relation to Loewner's conjecture
zbMATH Open document ID: 1071.53002
author: Naoya Ando
published in: Journal of the Mathematical Society of Japan
publication date: 2005-05-03T00:00:00Z
review text: Let \(f\) be a smooth function of two variables \(x\) and \(y\) on a domain \(D\) of \(\mathbb R^2\). For a positive integer \(n\in\mathbb N\), let \(d^nf\) be a symmetric tensor field of type \((0,n)\) defined by \[ d^nf:=\sum_{i=0}^n\binom ni \frac{\partial^nf}{\partial x^{n-i}\partial y^i}dx^{n-i}dy^i \] and \(\tilde\mathcal D_{d^nf}\) be a finitely many-valued one-dimensional distribution obtained from \(d^nf\) (e.g., \(\tilde{\mathcal D}_{d^1f}\) is the one dimensional distribution defined by the gradient vector field of \(f\); \(\tilde\mathcal D_{d^2f}\) consists of two one-dimensional distributions obtained from one-dimensional eigenspaces of the Hessian of \(f\)). In the paper, the behavior of \(\tilde\mathcal D_{d^nf}\) around an isolated umbilical point of \(d^nf\) is studied. In particular, the index of an isolated umbilical point with respect to \(\tilde\mathcal D_{d^nf}\) is defined and treated. It is shown that the index is a rational number and not always represented as the half of an integer. A conjecture which asserts that the index of an isolated umbilical point with respect to \(\tilde\mathcal D_{d^nf}\) is not greater than one is investigated. It is shown that for \(n\in\{1,2\}\) (respectively, \(n\geq 3\)), this conjecture is equivalent to (respectively, distinct from) Loewner's conjecture. The former conjecture is affirmatively solved in the case where \(f\) is a homogeneous polynomial. In addition, this conjecture is studied in the case where \(f\) is a real-analytic function.
reviewed by: Eugene Ershov
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 37E35
zbMATH DE Number: 2164233
zbMATH Keywords: Loewner's conjecture
zbMATH Keywords: the index conjecture
zbMATH Keywords: Carathéodory's conjecture
zbMATH Keywords: symmetric tensor field
zbMATH Keywords: critical direction
zbMATH Keywords: umbilical point
zbMATH Keywords: many-valued one-dimensional distribution
zbMATH Keywords: index
Wikidata QID: Q123353224
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/1160745810
OpenAlex ID: W2096918595
DOI: 10.2969/JMSJ/1160745810
description: scientific article; zbMATH DE number 2164233