Mass conservation for fractional porous-medium flow on noncompact Riemannian manifolds

We prove mass conservation for the spectral fractional porous-medium equation on complete noncompact Riemannian manifolds with Ricci curvature bounded below and a Euclidean-dimensional Faber--Krahn inequality. The result holds for every nonnegative weak dual solution that is spatially integrable at each finite time, for arbitrary nonnegative integrable initial data, all exponents \(m>1\), and all fractional orders \(0<s<1\). Thus the prescribed weak solution class cannot lose mass at spatial infinity. The proof first obtains a mass upper bound from positivity of the integrated Green potential. A positive energy identity after heat regularization then yields global integrability of the nonlinear power on every strictly positive time interval. A conservative linear resolvent converts this integrability into mass equality at every finite time.