Gabriella Tarantello, Università di Roma "Tor Vergata"¶
...since we've been talking about bubbling, I'm going to show you how bubbling also occurs in the theory of vortices.
This morning, we focused on planar Chern-Simons vortices (topological type). This amounted to proving the following...
Theorem: The problem:
$$ -\Delta u = \frac{4}{k^2} e^u (1 - e^u) - 4 \pi \sum_{j=1}^N \delta_{p_j}$$
with boundary condition $1 -e^u \in L^1 (\mathbb{R}^2) \iff u \rightarrow 0$ as $|x| \rightarrow +\infty$
admits a unique solution for fixed $\{ p_1, \dots, p_N \}$ repeated with multiplicity $\implies$ vortex configuration given as a minimizer of the energy with (Higgs field) $\phi$ with fixed degree = $N$.
We also have in this case that the total energy $\frac{4}{k^2} \int_{\mathbb{R}^2} e^u (1 - e^u) = 4 \pi N$, so quantized.
with the (different) boundary condition $e^u \in L^1 (\mathbb{R}^2) \implies u \rightarrow - \infty \implies e^u \rightarrow 0$ which in turn tells me that $|\phi|$ admits a zero at $\infty$.
$\implies$ total energy $ = \frac{4}{k^2} \int_{\mathbb{R}^2} e^u (1 - e^u)$ is no longer quantized by the parameter $N$. We typically call this integral $\beta$ and we investigate for which $beta$ we can perform an analysis. We have $^u \rightarrow 0$ but the convergence to zero is like $|x|^{2N - \frac{\beta}{2\pi}}$. The decay of $u$ is connected to the value of $\beta$.
We have a nice ansatz: $u = v + \sum_{j=1}^N \ln |x - p_j|^2.$ We can determine that $\beta > 4 \pi (N+1)$. In fact, this is not sharp and the right size is $\beta > 8 \pi (N+1)$. The range of $\beta$ is big. For each such beta, we know there is a solution.
Tomorrow will focus on the nontopological vortex case. Today, we will focus our efforts on the topological case.
in $(M,g)$ where $M$ is a closed surface. There is again an analogous situation with two different boundary conditions leading to "toplogical" and "nontopological" solutions.
...moving fast..discussing constraint equations...I'm not keeping up with the typing...reverse Hölder inequality....leads to a necessary condition for existence: $$ \frac{4 \pi N k^2}{|M|} \leq 1.$$
Theorem: There exists a critical $k_c >0$ (depending upon various parameter) such that the Chern-Simons Vortex equation admits a solution on $M \iff 0 < k \leq k_c$. Furthermore, $0<k\leq k_c$ the problem admits two solutions $u_{\pm}$ such that
...I got tired of typing...