Existence of solutions for the general noncompact Toda system (1.2)

Determine the existence of solutions on compact Riemann surfaces Σ to the general noncompact affine Toda system associated to τ-primitive maps into G/T, namely the PDE system ΔΣ w_j = Σ_{k∈I+} Cˆ_{jk} e^{w_k} − Σ_{k∈I−} Cˆ_{jk} e^{w_k} for j ∈ I with the linear relation Σ_{j=0}^r m_j w_j = 0, where Cˆ is the affine Cartan matrix, the index partition I = I+ ∪ I− is induced by a compatible Coxeter automorphism τ and Cartan involution σ, and m_j are the affine coefficients. Establish existence beyond the scalar a_2^{(2)} case and clarify any necessary conditions on the holomorphic data to control terms with positive signs.

Background

The paper develops the geometric Toda equations for noncompact symmetric spaces, classifies compatible Coxeter automorphisms, and proves existence and uniqueness for the totally noncompact case via stability (Theorem 1.3), including the cyclic and simple non-affine settings. These results cover systems with all simple affine roots noncompact, yielding equations with favorable sign structure (equation (1.5)).

The authors note that outside these cases, the general noncompact system (equation (1.2)) involves terms with positive signs that likely require additional bounds on the holomorphic sections to control the analysis. Beyond the special scalar a_2^{(2)} case, they indicate the existence theory has not been studied, highlighting a substantive unresolved problem in extending the current methods to the full range of noncompact Toda systems.