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Blow-up Solutions for General Toda Systems on Riemann Surfaces

Zhengni Hu, Miaomiao Zhu

TL;DR

This work analyzes the $N$-component Toda system with homogeneous Neumann boundary on a compact Riemann surface under a $k$-symmetry constraint. It develops a singular perturbation framework augmented by Lyapunov–Schmidt reduction, leveraging isothermal coordinates and Green’s functions to pull back planar bubbling profiles and control inter-component interactions through the Cartan matrix. The main result proves the existence of bubbling solutions concentrating at $m$ interior $k$-symmetric centers, with mass parameters satisfying $\rho_i^{\varepsilon} \to 2\alpha_i\pi m$ and the associated measures converging to Dirac masses at the blow-up points; for the $SU(3)$ case this yields asymmetric blow-up configurations. The method, which yields interior blow-up on models like $\mathbb{S}^2$ and $\mathbb{S}^2_+$, advances the bubbling theory for Toda systems on curved manifolds with boundary and identifies the $k$-symmetric centers as the natural blow-up sites; boundary blow-up remains an open question for future work.

Abstract

In this paper, we study general Toda systems with homogeneous Neumann boundary conditions on Riemann surfaces. Assuming the surface satisfies the ``$k$-symmetric'' condition, we construct a family of bubbling solutions using singular perturbation methods, where the concentration rates of different components occur in distinct orders. In particular, we establish the existence of asymmetric blow-up solutions for the $SU(3)$ Toda system. Furthermore, the blow-up points are precisely located at the ``$k$-symmetric'' centers of the surface. Keywords: Toda system, Neumann boundary condition, Blow-up solutions, $k$-symmetry, Finite-dimensional reduction

Blow-up Solutions for General Toda Systems on Riemann Surfaces

TL;DR

This work analyzes the -component Toda system with homogeneous Neumann boundary on a compact Riemann surface under a -symmetry constraint. It develops a singular perturbation framework augmented by Lyapunov–Schmidt reduction, leveraging isothermal coordinates and Green’s functions to pull back planar bubbling profiles and control inter-component interactions through the Cartan matrix. The main result proves the existence of bubbling solutions concentrating at interior -symmetric centers, with mass parameters satisfying and the associated measures converging to Dirac masses at the blow-up points; for the case this yields asymmetric blow-up configurations. The method, which yields interior blow-up on models like and , advances the bubbling theory for Toda systems on curved manifolds with boundary and identifies the -symmetric centers as the natural blow-up sites; boundary blow-up remains an open question for future work.

Abstract

In this paper, we study general Toda systems with homogeneous Neumann boundary conditions on Riemann surfaces. Assuming the surface satisfies the ``-symmetric'' condition, we construct a family of bubbling solutions using singular perturbation methods, where the concentration rates of different components occur in distinct orders. In particular, we establish the existence of asymmetric blow-up solutions for the Toda system. Furthermore, the blow-up points are precisely located at the ``-symmetric'' centers of the surface. Keywords: Toda system, Neumann boundary condition, Blow-up solutions, -symmetry, Finite-dimensional reduction
Paper Structure (8 sections, 13 theorems, 141 equations)

This paper contains 8 sections, 13 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Isothermal coordinates and Green functions
  4. Approximation solutions
  5. The limiting linearized operator
  6. Nonlinear problem
  7. Proof of main results
  8. Technique estimates

Key Result

Theorem 1.3

Let $k >\frac{1}{2} \alpha_N$, where $\alpha_i$ is defined by def:alpha_i for $i=1,\dots, N$, and assume that $\Sigma$ is a $k$-symmetric Riemann surface with smooth boundary. Suppose the potential functions $V_1, \dots, V_N$ are ${\mathfrak R}_k$-invariant, and Then for any $m$ distinct points $\xi^*_1, \dots, \xi^*_m$ in $\Sigma_0$, there exists a family of solutions ${\mathbf u}_\varepsilon =

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • ...and 17 more