Critical Kähler toric metrics for the invariant first eigenvalue (Q2331772)

Let \(\mathcal{M}=(M,\omega,g,\mathbb{T}^n)\) be a toric Kähler manifold. Here \(g\) is a Riemannian metric on \(M\) which is invariant under the torus \(\mathbb{T}^n\) and \(\omega\) is a symplectic manifold with a Hamiltonian \(\mathbb{T}^n\) action. Such a manifold always admits a large family of compatible \(\mathbb{T}^n\) invariant Kähler structures. Let \(\lambda_1^{\mathbb{T}^n}\) be the smallest positive eigenvalue of the Laplacian acting on \(\mathbb{T}^n\) invariant eigenfunctions. One has \(\lambda_1^{\mathbb{T}^n}\) is bounded and one says that \(\lambda_1^{\mathbb{T}^n}\) is critical if it achieves the supremum. The author shows Theorem. Let \(\mathcal{M}\) be a toric Kähler manifold. Then there are no analytic toric Kähler structures compatible with \(\omega\) and in the class \([\omega]\) which are critical for \(\lambda_1^{\mathbb{T}^n}\). Theorem. Let \(k\) be an integer; \(k\) corresponds to an \(S^1\) character. There are no \(\lambda_1^k\) critical \(S^1\) invariant metrics on \(S^2\). Section 2 contains background information on \(\lambda_1\)-critical metrics and on toric Kähler geometry. Section 3 extends results from the Riemannian setting to the context of toric Kähler manifolds; these results are then used in Section 4 to prove the results cited above. In Section 5, a system of PDEs is discussed that is relevant to this setting; the system is not elliptic.