  We derive the conformal Ward identities for the correlation functions of the
Stress--Energy tensor in probabilistic Liouville Conformal Field Theory on
compact Riemann surfaces by varying the correlation functions with respect to
the background metric. The conformal Ward identities show that the correlation
functions of the Stress--Energy tensor can be expressed as a differential
operator with meromorphic coefficient acting on the correlation functions of
the primary fields of Liouville Conformal Field Theory.
  Variations of the metric come in three different forms: reparametrizations,
conformal scalings and deformations of the conformal structure. Conformal
symmetry makes it easy to treat variations of the metric that do not deform the
conformal structure. Variations that deform the conformal structure have to be
treated separately, and this part of the computation relies on regularity and
integrability properties of the correlation functions of Liouville Conformal
Field Theory.
