Finite diffeomorphism theorem for manifolds with lower Ricci curvature and bounded energy

In this paper we prove that the space \(\mathcal {M}(n,\textrm{v},D,\Lambda ):=\{(M^n,g) \text { closed }: ~~\textrm{Ric}\ge -(n-1),~\textrm{Vol}(M)\ge \textrm{v}>0, \text {diam}(M)\le D \text { and } \int _{M}|\textrm{Rm}|^{n/2}\le \Lambda \}\) $\mathcal{M}(n,\text{v},D,\mathrm{\Lambda }):=\{(M^n,g)\text{closed}:\text{Ric}\ge -(n-1),\text{Vol}(M)\ge \text{v}>0,\text{diam}(M)\le D\text{and}{\int }_M|\text{Rm}{|}^{n/2}\le \mathrm{\Lambda }\}$ has at most \(C(n,\textrm{v},D,\Lambda )\) $C(n,\text{v},D,\mathrm{\Lambda })$ many diffeomorphism types. This removes the upper Ricci curvature bound of Anderson-Cheeger’s finite diffeomorphism theorem in Anderson and Cheeger (Geom Funct Anal 1(3):231–252, 1991). Furthermore, if M is Kähler surface, the Riemann curvature \(L^2\) $L^2$ bound could be replaced by the scalar curvature \(L^2\) $L^2$ bound.