A conjecture in relation to Loewner's conjecture

Abstract

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 $d^{n}f:={\sum }_{i=0}^n$ $\left(\begin{array}{c}n\\ i\end{array}\right)$ $\left({\partial }_x^{n-i}{\partial }_y^{i}f\right)d{x}^{n-i}d{y}^i$ and ${\widetilde{𝒟}}_{d^{n}f}$ a finitely many-valued one-dimensional distribution obtained from $d^{n}f$: for example, ${\widetilde{𝒟}}_{d^{1}f}$ is the one-dimensional distribution defined by the gradient vector field of $f$; ${\widetilde{𝒟}}_{d^{2}f}$ 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 ${\widetilde{𝒟}}_{d^{n}f}$ 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 ${\widetilde{𝒟}}_{d^{n}f}$ is not more than one.