A conjecture in relation to Loewner's conjecture

ANDO, Naoya

J. Math. Soc. Japan, Tome 57 (2005) no. 4, p. 1-20 / Harvested from Project Euclid

Résumé

Let $f$ be a smooth function of two variables $x$ , $y$ and for each positive integer $n$ , let $d^n f$ be a symmetric tensor field of type $(0, n)$ defined by $d^n f:=\sum^n_{i=0}$ $\left(\begin{array}{c}n\\i\end{array}\right)$ $\left( \partial^{n-i}_x \partial^i_y f\right) dx^{n-i} dy^i$ and $\tilde{\mathscr D}_{d^n f}$ a finitely many-valued one-dimensional distribution obtained from $d^n f$ : for example, $\tilde{\mathscr D}_{d^1 f}$ is the one-dimensional distribution defined by the gradient vector field of $f$ ; $\tilde{\mathscr D}_{d^2 f}$ consists of two one-dimensional distributions obtained from one-dimensional eigenspaces of Hessian of $f$ . In the present paper, we shall study the behavior of $\tilde{\mathscr D}_{d^n f}$ around its isolated singularity in ways which appear in [1]--[4]. In particular, we shall introduce and study a conjecture which asserts that the index of an isolated singularity with respect to $\tilde{\mathscr D}_{d^n f}$ is not more than one.

Détail

Publié le : 2005-01-14
Classification:  Loewner's conjecture,  the index conjecture,  Carathéodory's conjecture,  symmetric tensor field,  critical direction,  umbilical point,  many-valued one-dimensional distribution,  index,  37E35,  53A05,  53B25