On asymptotic behavior of the second Chern forms on degenerating K\"ahler-Einstein surfaces

We study an asymptotic behavior of the second Chern forms of canonical metrics on a degenerating family of K\"ahler surfaces with the central fibre having ADE-singularities. We investigate a function on the unit disc defined by fiber integrals of the forms with a smooth test function on the family. We show a lower bound of the H\"older exponent of the function at the origin. Our main results consists of two cases: one is a bound of H\"older exponent along a line for cscK-metrics, using Biquard-Rollin's a priori estimates for cscK-metrics, and the other is a bound of H\"older exponent at the origin for Ricci-flat metrics.