Relative cyclotomic structures and equivariant complex cobordism

We describe a structure on a commutative ring (pre)cyclotomic spectrum $R$ that gives rise to a (pre)cyclotomic structure on topological Hochschild homology ($THH$) relative to its underlying commutative ring spectrum. This lets us construct $TC$ relative to $R$, denoted $T{C}^R$, and we prove some descent results relating $T{C}^R$ and $TC$. We explore several examples of this structure on familiar $\mathbb{T}$-equivariant commutative ring spectra including the periodic $\mathbb{T}$-equivariant complex cobordism spectrum $MU{P}_{\mathbb{T}}$ and a new (connective) equivariant version of the complex cobordism spectrum $MU$.