Quoting p. 142 of the trusty [Ashcroft-Mermin](https://books.google.com/books?id=FRZRAAAAMAAJ) (who write $\mathcal E_F$ for your $\mu$):

> For each partially filled band there will be a surface in $k$-space separating the occupied from the unoccupied levels. The set of all such surfaces is known as the *Fermi surface*, and is the generalization to Bloch electrons of the free electron Fermi sphere. The parts of the Fermi surface arising from individual partially filled bands are known as the *branches* of the Fermi surface. We shall see (Chapter 12) that a solid has metallic properties provided that a Fermi surface exists.

> Analytically, the branch of the Fermi surface in the $n$th band is that surface in $k$-space (if there is one) determined by $$\mathcal E_n(\mathbf k)=\mathcal E_F.$$

As to "what $\mathcal E_n(\mathbf k)$ is": it arises from looking for eigenfunctions of a "single-electron" hamiltonian
$$
\left(-\frac{\hbar^2}{2m}\nabla^2+U(\mathbf r)\right)\psi(\mathbf r)=E\psi(\mathbf r)
$$
in the "Bloch" form: $\psi(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u(\mathbf r)$ with (Bravais lattice) periodic boundary conditions on $u$, i.e.
$$
\left(\frac{\hbar^2}{2m}\left(\frac1i\nabla + \mathbf k\right)^2+U(\mathbf r)\right)u_{\mathbf k}(\mathbf r)=\mathcal E(\mathbf k)u_{\mathbf k}(\mathbf r).
$$
> Because of the periodic boundary condition we can regard [this] as a Hermitian eigenvalue problem restricted to a single primitive cell of the crystal. Because the eigenvalue problem is set in a fixed finite volume, we expect on general grounds to find an infinite family of solutions with *discretely* spaced eigenvalues, which we label with the band index $n$.