GEOMETRIC BOUNDS ON THE RELATIVE INDEX

Abstract

Let $X$ be a compact surface such that $Y\hookrightarrow X$ as a separating, strictly pseudoconvex, real hypersurface;

$$ X\setminus Y=X_+\sqcup X_-, $$

where $X_{+}$ ($X_-$) is the strictly pseudoconvex (pseudoconcave) component of the complement. Suppose further that $X_-$ contains a positively embedded, compact curve $Z$. Under cohomological hypotheses on $(X_-,Z)$ we show that if $\dbarb'$ is a sufficiently small, embeddable deformation of the CR-structure on $Y$, then

$$ \mathrm{R-ind}(\bar{\partial}_{b},\bar{\partial}_{b}')\geq-[\dim H^{0,2}(X_-)+\dim H^0(Z,\mathcal{O}_Z)]. $$

This implies that the set of small, embeddable deformations of the CR-structure on $Y$ is closed, in the $\mathcal{C}^{\infty}$-topology on the set of all deformations.

AMS 2000 Mathematics subject classification: Primary 32V15; 32V30. Secondary 32G07; 32W10; 32W05