On the failure of the Poincaré Lemma for \(\overline\partial_M\). II. (English) Zbl 1092.32503

Summary: The purpose of this paper is to repair some inaccuracies in the formulation of the main result of part I of this paper [the authors, Math. Ann. 324, No. 2, 213–224 (2002; Zbl 1014.32027)]. As were written, the main theorems 4.1 and 4.2 of part I are in fact in contradiction to earlier results of one of the authors [M. Nacinovich, Ann. Pol. Math. 46, 213–235 (1985; Zbl 0606.58046)]. In the process of writing an erratum, we actually discovered some new phenomena. As we found these quite interesting, it has led us to incorporate the needed corrections into this self contained article.
An unfortunate misprint, in which \(R^{-1}\) got replaced by \(R\), led the authors to misinterpret what their proof in part I actually demonstrated. Upon closer scrutiny, we realized that there are two distinct ways to proceed.
One is that we may still obtain the original conclusions of our main theorems, provided we slightly strengthen our original hypothesis (cf. Theorems 5.1 and 5.2). This however entails a much more complicated argument, involving the CR structure of the characteristic bundle, which is of considerable independent interest (cf. Theorems 4.3 and 4.6).
The other is that if we stick to our original hypothesis, then the conclusions we obtain are slightly weaker than originally claimed, but in our opinion still interesting. In fact a new invariant comes into play, which measures the rate of shrinking, even in the situation where the local Poincaré lemma is valid.
Recall that here we make an important distinction between the vanishing of the cohomology and the validity of the Poincaré lemma: Consider the inhomogeneous problem \(\overline\partial_Mu=f\), to be solved for \(u\), with given data \(f\) satisfying \(\overline\partial_Mf=0\) in some domain \(U\) containing a point \(x_0\). The vanishing of the cohomology in \(U\) refers to the situation in which, no matter how \(f\) is prescribed in \(U\), there is always a solution \(u\) in \(U\) (i.e., no shrinking). The validity of the Poincaré lemma at \(x_0\) requires only that a solution \(u\) exist in a smaller domain \(V_f\) with \(x_0\in V_f\subset U\) (i.e., there is some shrinking which might, in principle, depend on \(f)\). Our new invariant measures the relative rate of shrinking of \(V_f\) with respect to radius \((U)\), as \(U\) shrinks to the point \(x_0\). Under our original hypothesis, we are able to show that the cohomology of small convex neighborhoods of \(x_0\) is always infinite dimensional, with respect to any choice of the Riemannian metric (cf. Theorems 7.2 and 7.3). This means that special shapes are needed, if one is to have the vanishing of the cohomology for small sets.
In Section 6 we have listed a number of natural examples satisfying the slightly strengthened hypothesis. They illustrate how common it is for the Poincaré lemma to fail. At the end of Section 7 we give a very simple example satisfying our original hypothesis. It illustrates how, even if the Poincaré lemma were to be valid, a shrinking must occur which goes like radius \((V_f)\simeq Cr^{3/2}\), where radius \((U)\simeq r\), as \(r\to 0\).


32V05 CR structures, CR operators, and generalizations
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
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