Measurement of singly Cabibbo-suppressed decays $D^{0}\to\pi^{0}\pi^{0}\pi^{0}$, $\pi^{0}\pi^{0}\eta$, $\pi^{0}\eta\eta$ and $\eta\eta\eta$

Using a data sample of $e^+e^-$ collision data corresponding to an integrated luminosity of 2.93 $fb^{-1}$ collected with the BESIII detector at a center-of-mass energy of $\sqrt{s}= 3.773~GeV$,we search for the singly Cabibbo-suppressed decays $D^{0}\to\ pi^{0}\ pi^{0}\ pi^{0}$, $\ pi^{0}\ pi^{0}\eta$, $\ pi^{0}\eta\eta$ and $\eta\eta\eta$ using the double tag method. The absolute branching fractionsare measured to be $\mathcal{B}(D^{0}\to\ pi^{0}\ pi^{0}\ pi^{0}) = (2.0 \pm 0.4 \pm 0.3)\times 10^{-4}$, $\mathcal{B}(D^{0}\to\ pi^{0}\ pi^{0}\eta) = (3.8 \pm 1.1 \pm 0.7)\times 10^{-4}$ and $\mathcal{B}(D^{0}\to\ pi^{0}\eta\eta) = (7.3 \pm 1.6 \pm 1.5)\times 10^{-4}$ with the statistical significances of $4.8\sigma$, $3.8\sigma$ and $5.5\sigma$, respectively, where the first uncertainties are statistical and the second ones systematic. No significant signal of $D^{0}\to\eta\eta\eta$ is found, and the upper limit on its decay branching fractionis set to be $\mathcal{B}(D^{0}\to\eta\eta\eta) < 1.3 \times 10^{-4}$ at the $90\%$ confidence level.