Properties of gradient maps associated with action of real reductive group
Let \(M\) be a Kähler manifold with Kähler form \(\omega\) and \(\mathbb U^{\mathbb C}\) a group of holomorphic transformations of \(M\), which is the complexification of a connected, compact Lie subgroup \(\mathbb U \subset \mathbb U^{\mathbb C}\), whose action on \(M\) is \(\omega\)-symplectic and admits a moment map \(\mu: M \to \mathfrak u = \operatorname{Lie}(\mathbb U)\). Let also \(G \subset \mathbb U^{\mathbb C}\) be a closed Lie subgroup admitting a Cartan decomposition of the form \(G=K{\cdot}\operatorname{exp}(\mathfrak p)\), with \(K = G \cap \mathbb U\) and \(\mathfrak p = \operatorname{Lie}(G) \cap (i \mathfrak u)\). Denoting by \(\langle\cdot, \cdot\rangle\) an \(\operatorname{Ad}_{\mathbb U}\)-invariant scalar product on \(\operatorname{Lie}(\mathbb U)\), we may consider the application \(\mu^{(\mathfrak p)}: M \to \mathfrak p\), called gradient map, defined by \[\langle \mu^{(\mathfrak p)}(x), X\rangle := -\langle \mu(x), i X\rangle,\] and its associated squared norm function \(f^{(\mathfrak p)} := \| \mu^{(\mathfrak p))}\|^2\). \N\NThe authors determine several properties of the restrictions \(f^{(\mathfrak p)}|_N\) of such a squared norm to \(G\)-invariant compact and connected real submanifolds \(N \subset M\). For instance, they prove: \N\N(a) Without any assumption of real analyticity, the limit of the negative gradient flow of \(f^{(\mathfrak p)}|_N\) always exists and is unique; \N\N(b) Any \(G\)-orbit of \(N\) collapses to an orbit of the maximal compact subgroup \(K = G \cap \mathbb U\); \N\N(c) If two critical points of \(f^{(\mathfrak p)}|_N\) are in the same \(G\)-orbit, then they are also in the same \(K\)-orbit; \N\N(d) If the \(G\)-action on \(N\) has just two orbits, then \(f^{(\mathfrak p)}|_N\) is Morse-Bott. \N\NThe authors also present convexity results for the images of the gradient map \(\mu^{(\mathfrak p)}\) in the cases that \(\mathfrak p\) is abelian.
scientific article; zbMATH DE number 7885854
scientific article