Energy Concentration and A Priori Estimates forB₂andG₂Types of Toda Systems

Abstract

For Toda systems with Cartan matrix either B 2 or G 2 , we prove that the local mass of blowup solutions at its blowup points converges to a finite set. Further more this finite set can be completely determined for B 2 Toda systems, while for G 2 systems we need one additional assumption. As an application of the local mass classification we establish a priori estimates for corresponding Toda systems defined on Riemann surfaces.Recently equation (1.2) is found to have deep connection with the classical Lame equation and also the Painleve VI equation. For example for the Painleve VI equation with certain parameters, some non-existence theorem of (1.2) plays a key part in the proof of the smoothness of the solutions with unitary monodromy group. The interested readers may read into [9] and [12] for more in-depth discussions.

Degree counting for Toda system with simple singularity: One point blow up

“…This generalizes an earlier result by Lin and Zhang [39]. We notice that for (1.14) with singular sources, the number of the possibility of the local mass relies heavily on the coefficients α p,i of the singular source, as the coefficient becomes larger, the number of possibility gets bigger.…”

B 2 and G2 Toda systems on compact surfaces: A variational approach

“…, K 1 we only have estimates for f 1,u on Ω k , so we apply the scalar Moser-Trudinger inequality, that is Theorem 2.8. By (22) we get again the integral of Q B2 , hence we can argue as before:…”

“…To apply such arguments we will need some compactness conditions on the solutions of (1), (2), which were recently proved in [22]. This result, combined with a standard monotonicity trick from [23,28], allows to apply such min-max methods for the problem (1) as long as neither ρ 1 nor ρ 2 are integer multiples of 4π; the same holds true for (2) under assuming an extra upper bound on both parameters.…”

“…This is because, in the general case, the compactness assumptions which are needed to use this approach, and in particular quantization of local blow-up limits, are not known to hold. This seems to be quite a difficult open problem in general, as it has been solved just for few specials 2-dimensional systems (see [21,22]). Such a compactness result would imply, for given A and ρ, existence of solutions.…”

“…After the submission of this manuscript, a new result [22] concerning compactness of solutions of (1) and (2) was published. It basically improves Theorem 2.2 and Corollary 2.3 and extends them to the case of singular B 2 , G 2 systems, namely…”