Zero sets of Lie algebras of analytic vector fields on real and complex 2-dimensional manifolds, II

TL;DR

This paper proves that under certain tracking conditions, a finite-dimensional Lie algebra of analytic vector fields on 2-manifolds must have zeros within specific compact sets, extending fixed point theorems.

Contribution

It establishes new fixed point results for Lie algebras of analytic vector fields on 2-manifolds under tracking conditions, including complex cases with restrictions on Lie algebra quotients.

Findings

Lie algebra elements tracking a vector field imply the existence of zeros.

Fixed points are guaranteed within certain compact sets under specified conditions.

Results extend classical fixed point theorems to Lie algebra actions on 2-manifolds.

Abstract

On a real ($\mathbb F=\mathbb R$) or complex ($\mathbb F=\mathbb C$) analytic connected 2-manifold $M$ with empty boundary consider two vector fields $X,Y$. We say that $Y$ {\it tracks} $X$ if $[Y,X]=fX$ for some continuous function $f\colon M\rightarrow\mathbb F$. Let $K$ be a compact subset of the zero set ${\mathsf Z}(X)$ such that ${\mathsf Z}(X)-K$ is closed, with nonzero Poincar\'e-Hopf index (for example $K={\mathsf Z}(X)$ when $M$ is compact and $\chi(M)\neq 0$) and let $\mathcal G$ be a finite-dimensional Lie algebra of analytic vector fields on $M$. \smallskip {\bf Theorem.} Let $X$ be analytic and nontrivial. If every element of $\mathcal G$ tracks $X$ and, in the complex case when ${\mathsf i}_K (X)$ is positive and even no quotient of $\mathcal G$ is isomorphic to ${\mathfrak {s}}{\mathfrak {l}} (2,\mathbb C)$, then $\mathcal G$ has some zero in $K$. \smallskip {\bf Corollary.} If $Y$ tracks a nontrivial vector field $X$, both of them analytic, then $Y$ vanishes somewhere in $K$. \smallskip Besides fixed point theorems for certain types of transformation groups are proved. Several illustrative examples are given.