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In this paper we establish the best constant $\widetilde A_{opt}(\bar{M})$for the Trace Nash inequality on a $n-$dimensional compact Riemannian manifoldin the presence of symmetries, which is an improvement over the classical casedue to the symmetries which arise and reflect the geometry of manifold. This isparticularly true when the data of the problem is invariant under the action ofan arbitrary compact subgroup $G$ of the isometry group $Is(M,g)$, where allthe orbits have infinite cardinal.

Sharp Nash inequalities on manifolds with boundary in the presence of symmetries