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We present the third in the series of papers describing Poisson properties of planar directed networks in the disk or in the annulus. In this paper we concentrate on special networks$N$_{$u,v$}in the disk that correspond to the choice of a pair ($u, v$) of Coxeter elements in the symmetric group$S$_{$n$}and the corresponding networks $N_{u,v}^\circ$ in the annulus. Boundary measurements for$N$_{$u,v$}represent elements of the Coxeter double Bruhat cell$G$^{$u,v$}⊂GL_{$n$}. The Cartan subgroup$H$acts on$G$^{$u,v$}by conjugation. The standard Poisson structure on the space of weights of$N$_{$u,v$}induces a Poisson structure on$G$^{$u,v$}, and hence on the quotient$G$^{$u,v$}/$H$, which makes the latter into the phase space for an appropriate Coxeter–Toda lattice. The boundary measurement for $N_{u,v}^\circ$ is a rational function that coincides up to a non-zero factor with the Weyl function for the boundary measurement for$N$_{$u,v$}. The corresponding Poisson bracket on the space of weights of $N_{u,v}^\circ$ induces a Poisson bracket on the certain space $ {\mathcal{R}_n} $ of rational functions, which appeared previously in the context of Toda flows.

Let M be a complete non-compact Riemannian manifold whose sectional curvature is bounded between two constants-k and K. Then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in Euclidean space. The purpose of this paper is to give a justi-fication of such a statement.

We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D∈|-KY| is a smooth divisor with D^2=d , then X=Y\D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y=P^2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type I_d fiber appearing as a singular fiber in a rational elliptic surface π: Y →P^1.