The null space of the $\overline{\partial }$-Neumann operator

Hörmander, Lars

Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1305-1369.

Soit $\Omega$ une variété complexe de dimension $n$ avec une métrique hermitienne et une frontière $C^{\infty }$, et soit $\sqcap \bigsqcup =\overline{\partial }{\overline{\partial }}^{*}+{\overline{\partial }}^{*}\overline{\partial }$ l'opérateur autoadjoint $\overline{\partial }$-Neumann dans l'espace $L_{0,q}^2\left(\Omega \right)$ des $(0,q)$ formes. Si la forme de Levi a au moins $n-q$ valeurs propres positives ou au moins $q+1$ valeurs propres négatives en chaque point de $\partial \Omega$, il est bien connu que dim $\mathrm{Ker}$ $\sqcap \bigsqcup <\infty$ et que l'image de $\sqcap \bigsqcup$ est l'espace orthogonal. Ici nous démontrons que dim $\mathrm{Ker}$ $\sqcap \bigsqcup =\infty$ si l'image de $\sqcap \bigsqcup$ est fermée et si la signature de la forme de Levi est $n-q-1,q$ en un point de $\partial \Omega$. Le point de départ de la démonstration est une formule explicite pour $\mathrm{Ker}$ $\sqcap \bigsqcup$ quand $\Omega \subset {ℂ}^n$ est borné par deux sphères concentriques et $q=n-1$. Alors $\mathrm{Ker}$ $\sqcap \bigsqcup$ a $n$ multiplicateurs indépendants ; ceci est vrai si et seulement si $\Omega \subset {ℂ}^n$ est borné par deux ellipsoïdes confocaux. Ces modèles conduisent à une asymptotique faible pour le noyau de la projection orthogonale sur $\mathrm{Ker}$ $\sqcap \bigsqcup$ quand l'image de $\sqcap \bigsqcup$ est fermée, aux points de $\partial \Omega$ où la forme de Levi est définie négative $q=n-1$. Des bornes grossières sont aussi données quand la signature est $n-q-1,q$ avec $1\le q<n-1$.

Let $\Omega$ be a complex analytic manifold of dimension $n$ with a hermitian metric and $C^{\infty }$ boundary, and let $\sqcap \bigsqcup =\overline{\partial }{\overline{\partial }}^{*}+{\overline{\partial }}^{*}\overline{\partial }$ be the self-adjoint $\overline{\partial }$-Neumann operator on the space $L_{0,q}^2\left(\Omega \right)$ of forms of type $(0,q)$. If the Levi form of $\partial \Omega$ has everywhere at least $n-q$ positive or at least $q+1$ negative eigenvalues, it is well known that $\mathrm{Ker}$ $\sqcap \bigsqcup$ has finite dimension and that the range of $\sqcap \bigsqcup$ is the orthogonal complement. In this paper it is proved that dim $\mathrm{Ker}$ $\sqcap \bigsqcup =\infty$ if the range of $\sqcap \bigsqcup$ is closed and the Levi form of $\partial \Omega$ has signature $n-q-1,q$ at some boundary point. The starting point for the proof is an explicit determination of $\mathrm{Ker}$ $\sqcap \bigsqcup$ when $\Omega \subset {ℂ}^n$ is a spherical shell and $q=n-1$. Then $\mathrm{Ker}$ $\sqcap \bigsqcup$ has $n$ independent multipliers; this is only true for shells $\Omega \subset {ℂ}^n$ bounded by two confocal ellipsoids. These models lead to asymptotics in a weak sense for the kernel of the orthogonal projection on $\mathrm{Ker}$ $\sqcap \bigsqcup$ when the range of $\sqcap \bigsqcup$ is closed, at points on $\partial \Omega$ where the Levi form is negative definite, $q=n-1$. Crude bounds are also given when the signature is $n-q-1,q$ with $1\le q<n-1$.

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