Matzoh ball soup in spaces of constant curvature

In this paper, we generalize Magnanini-Sakaguchi's result [MS3] from Euclidean space to spaces of constant curvature. More precisely, we show that if a conductor satisfying the exterior geodesic sphere condition in the space of constant curvature has initial temperature 0 and its boundary is kept at temperature 1 (at all times), if the thermal conductivity of the conductor is inverse of its metric, and if the conductor contains a proper sub-domain, satisfying the interior geodesic cone condition and having constant boundary temperature at each given time, then the conductor must be a geodesic ball. Moreover, we show similar result for the wave equations and the Schr\"{o}dinger equations in spaces of constant curvature.