Anti-self-dual equation on 4-manifolds with degenerate metric
The author investigates the anti-selfdual (ASD) equation in the following degenerate setting. Let \(D=D_1\cup L\cup D_2\) be a domain in \(\mathbb C\) such that \(L\) is a curve separating \(D_1\) and \(D_2\), and let \(\Sigma\) be a Riemann surface with an \(\roman{SO}(3)\)-bundle \(E\) with \(w_2(E)=\Sigma\). Let \(\chi\colon D\to\mathbb R\) be positive on \(D_1\) and zero on \(D_2\), then \(D\times\Sigma\) is equipped with a warped product metric with warp function \(\chi\). The author considers the ASD equation \[ F_{\mathfrak A}+*F_{\mathfrak A}=0\tag{*} \] for an \(\roman{SO}(3)\)-connection \(\mathfrak A\) on the bundle \(E\) over \(D\times\Sigma\). Over \(D_1\), one gets just the usual ASD-equation. Over \(D_2\), (*) is interpreted as the pseudo-holomorphic curve equation from \(D_2\) to the moduli space of flat connections on \(E\to\Sigma\), which is a Kähler manifold. The first Pontryagin form of \(\mathfrak A\) defines an energy for solutions of (*). The author proves the following removable singularity theorem (Theorem 1.6): Let \(p\in L\), and let \(\mathfrak A\) be a smooth solution of (*) on \((D-\{p\})\times\Sigma\) with finite energy. Then \(\mathfrak A\) is gauge equivalent to a smooth solution on \(D\times\Sigma\). Let \(\mathfrak A_i\) be a sequence of smooth solutions of (*) with uniformly bounded energy. The author shows that there exist finite subsets \(P_1\) of \(D_1\times\Sigma\) and \(P_2\) of \(L\cap D_2\) such that a subsequence of the \(\mathfrak A_i\) converges after gauge transformations on \(((D_1\times\Sigma)-P_1)\cup((L\cup D_2)-P_2)\times\Sigma\) in the \(C^\infty\) topology (Theorem 1.7). Moreover, for each point \(p\in L\cap P_2\), either some energy is lost in the limit near \(p\times\Sigma\), or some subsequence converges on a neighborhood of \(p\times\Sigma\) after gauge transformation (Theorem 1.8). On \(D_1\times\Sigma\), the corresponding results have been established by \textit{K. Uhlenbeck} [Commun. Math. Phys. 83, 31-42 (1982; Zbl 0499.58019)], while over \(D_2\), they follow from work of \textit{M. Gromov} [Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)]. The main difficulty here lies in the behaviour of solutions near \(L\). The author uses that a connection with finite energy is complex gauge equivalent near \(L\times\Sigma\) to a holomorphic family of flat connections on \(\Sigma\).
scientific article; zbMATH DE number 1197115