The minimal and next minimal volumes of normal KSBA stable surfaces with $p_g\ge 2$

In this paper we investigate the minimal and the next minimal volumes of normal KSBA stable surfaces with $p_g\ge 2$. We show that in case of $\mathrm{\mid }{K}_X\mathrm{\mid }$ not composed with a pencil, the minimal and next minimal volumes are $2p_g-4$ and $2p_g-4+\frac{1}{3}$. In case of $\mathrm{\mid }{K}_X\mathrm{\mid }$ composed with a pencil, the minimal and next minimal volumes are $\frac{p_g-1}{p_g+1}(p_g-1)$ and $\mathrm{m}\mathrm{i}\mathrm{n}\{\frac{2p_g-2}{2p_g+1}(p_g-1),\frac{(3p_g-2)p_g-4}{3(p_g+2)}\}$. We also characterize the surfaces achieving the minimal volumes.