Classification and nondegeneracy of SU(n+1) Toda system with singular sources

Abstract

We consider the following Toda system ∆ui + n j=1where γi > −1, δ0 is Dirac measure at 0, and the coefficients aij form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result: This generalizes the classification result by Jost and Wang for γi∈ Z for all 1 ≤ i ≤ j ≤ n, then any solution ui is radially symmetric w.r.t. 0. (iii) We prove that the linearized equation at any solution is non-degenerate. These are fundamental results in order to understand the bubbling behavior of the Toda system.

On a general SU(3) Toda system

“…This result was extended when in (1.8) appear singular sources (see [12]). The purpose of this paper is to investigate more general cases than those of the Cartan matrix and to understand for which matrices γ i j can be expected results of existence of solutions.…”

The sphere covering inequality and its applications

“…arise in many important problems in mathematics, mathematical physics and biology. Such equations have been extensively studied in the context of Moser-Trudinger inequalities, Chern-Simons self-dual vortices, Toda systems, conformal geometry, statistical mechanics of two-dimensional turbulence, self-gravitating cosmic strings, theory of elliptic functions and hyperelliptic curves and free boundary models of cell motility, see [BFR,BL,BL2,BLT,BCLT,BT,BT2,Be,CLS,CaY,CLMP,CLMP2,CK,CFL,CY,CY2,CY3,DJLW,GL,L,L2,LM,LM2,LW,LW2,LWY,Y] and the references cited therein. The sphere covering inequality was recently introduced in [GM], and has been applied to solve various problems about symmetry and uniqueness of solutions of elliptic equations with exponential nonlinearity in dimension n = 2.…”

Toda systems and hypergeometric equations

“…This fact was proved in [LWY12] and [Nie16], and we will recall it in (2.3). The Winvariants W j are polynomials in the ∂ k Ui ∂z k for k ≥ 1 and are meromorphic functions in C ∪ {∞} with poles only at 0, 1 and ∞ (see Proposition 2.5).…”