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

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Abstract

We consider the following Toda system

where γ i >−1, δ 0 is Dirac measure at 0, and the coefficients a ij form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result:

$$\sum_{j=1}^n a_{ij}\int_{\mathbb{R}^2}e^{u_j} dx = 4\pi(2+\gamma _i+\gamma_{n+1-i}), \quad\forall\;1\leq i \leq n.$$

This generalizes the classification result by Jost and Wang for γ i =0, \(\forall\;1\leq i\leq n\). (ii) We prove that if γ i +γ i+1+⋯+γ j ∉ℤ for all 1≤i≤j≤n, then any solution u i 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.

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Authors and Affiliations

  1. Taida Institute of Mathematical of Science, Center for Advanced study in Theoretical Sciences, National Taiwan University, Taipei, Taiwan

    Chang-Shou Lin

  2. Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong

    Juncheng Wei

  3. Département de Mathématiques, UMR 7122, Université de Metz, Bât. A, Ile de Saulcy, 57045, Metz Cedex 1, France

    Dong Ye

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  1. Chang-Shou Lin
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Correspondence to Chang-Shou Lin.

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Lin, CS., Wei, J. & Ye, D. Classification and nondegeneracy of SU(n+1) Toda system with singular sources. Invent. math. 190, 169–207 (2012). https://doi.org/10.1007/s00222-012-0378-3

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  • DOI: https://doi.org/10.1007/s00222-012-0378-3

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