  The Lorentzian Engle-Pereira-Rovelli-Livine/Freidel-Krasnov (EPRL/FK)
spinfoam model and the Conrady-Hnybida (CH) timelike-surface extension can be
expressed in the integral form $\int e^S$. This work studies the analytic
continuation of the spinfoam action $S$ to the complexification of the
integration domain. Our work extends our knowledge from the real critical
points well-studied in the spinfoam large-$j$ asymptotics to general complex
critical points of $S$ analytic continued to the complexified domain. The
complex critical points satisfying critical equations of the analytic continued
$S$. In the large-$j$ regime, the complex critical points give subdominant
contributions to the spinfoam amplitude when the real critical points are
present. But the contributions from the complex critical points can become
dominant when the real critical point are absent. Moreover the contributions
from the complex critical points cannot be neglected when the spins $j$ are not
large. In this paper, we classify the complex critical points of the spinfoam
amplitude, and find a subclass of complex critical points that can be
interpreted as 4-dimensional simplicial geometries. In particular, we identify
the complex critical points corresponding to the Riemannian simplicial
geometries although we start with the Lorentzian spinfoam model. The
contribution from these complex critical points of Riemannian geometry to the
spinfoam amplitude give $e^{-S_{Regge}}$ in analogy with the Euclidean path
integral, where $S_{Regge}$ is the Riemannian Regge action on simplicial
complex.
