Asymptotic behaviour of certain families of harmonic bundles on Riemann surface


Takuro Mochizuki

13:00:00 - 13:50:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

Let $(E,overline{partial}_E,theta)$ be a stable Higgs bundle of degree $0$ on a compact connected Riemann surface. Once we fix a flat metric $h_{det(E)}$ on the determinant of $E$, we have the harmonic metrics $h_t$ $(t>0)$ for the stable Higgs bundles $(E, overline{partial}_E,ttheta)$ such that $det(h_t)=h_{det(E)}$. In this talk, we will discuss two results on the behaviour of $h_t$ when $t$ goes to $infty$. First, we show that the Hitchin equation is asymptotically decoupled under some asumption for the Higgs field. We apply it to the study of the so called Hitchin WKB-problem. Second, we discuss the convergence of the sequence $(E,overline{partial}_E,theta, h_t)$ in the case where the rank of $E$ is $2$. We explain a rule to determine the parabolic weights of a ``limiting configuration'', and we show the convergence of the sequence to the limiting configuration in an appropriate sense.