# PDE Seminar

## Classification and nondegeneracy of $SU(n+1)$ Toda systems

### Dong Ye 教授 (University of Lorraine-Metz, France)

#### 2012年7月12日(周四)下午 4:30-5:30 闵行校区行政楼12楼PDE中心1202报告厅

Abstract. We consider the following Toda system

\begin{align*}

\Delta u_i + \sum_{j = 1}^n a_{ij}e^{u_j} = 4\pi\gamma_{i}\delta_{0} \;\; \text{in }\mathbb R^2, \quad \int_{\mathbb R^2}e^{u_i} dx < \infty,\;\; \forall\; 1\leq i \leq n,

\end{align*}

where $\gamma_{i} > -1$, $\delta _0$ is Dirac measure at $0$, and the coefficients $a_{ij}$ form the standard tridiagonal Cartan matrix. We completely classify the solutions and obtain the quantization result:

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

According to the values of $\gamma_i$, the solution manifolds $\cal M$ have dimensions ranging from $n$ to $n(n+2)$. In particular, if $\gamma_i+\gamma_{i+1}+\ldots+\gamma_j \notin \mathbb Z$ for all $1\leq i\leq j\leq n$, then any solution $u_i$ is radially symmetric w.r.t. $0$. These results generalize the classification result by Jost-Wang for $\gamma_i=0$, $\forall \;1\leq i\leq n$, and the result by Prajapat-Tarantello for $n = 1$.

Moreover, we prove that any solution is non-degenerate, which means that the bounded solution set to the linearized equation at any solution $u$ is just the tangent space $T_u\cal M$. These are fundamental results in order to understand the blow-up behavior of the Toda system, and we show some applications to understand holomorphic curves from $S^2$ into $\mathbb{CP}^n$.

This is a joint work with C. Lin and J. Wei.