Complete Solutions of Toda Equations and Cyclic Higgs Bundles Over Non-compact Surfaces.

The article focuses on the existence, uniqueness, and properties of complete solutions to the Toda equation associated with holomorphic |$r$|-differentials on general non-compact Riemann surfaces. It establishes a natural correspondence between solutions of the Toda system and |$G_r$|-invariant harmonic metrics on cyclic Higgs bundles constructed from |$r$|-differentials. The main result proves that for any non-compact Riemann surface |$X$| and holomorphic |$r$|-differential |$q$| (nonzero unless |$X$| is hyperbolic), there exists a unique complete and real solution to the Toda equation, inducing complete metrics on |$X$|. The paper further provides quantitative curvature and metric estimates for these solutions, discusses conditions for uniqueness or non-uniqueness depending on the completeness of the metric induced by |$|q|^{2/r}$|, and develops existence proofs via the method of super-subsolutions, including detailed constructions on the unit disk, hyperbolic surfaces, the complex plane, and parabolic surfaces. The work also includes a classification of tame harmonic bundles on parabolic Riemann surfaces and relates the Toda solutions to geometric structures such as harmonic maps and maximal surfaces in pseudo-Riemannian spaces.