Given compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$ and $p \in (1, \infty)$, the question of traces for Sobolev mappings consists in characterising the mappings from $\partial \mathcal{M}$ to $\mathcal{N}$ that can arises of maps in the first-order Sobolev space $\smash{\dot{W}}^{1, p} (\mathcal{M}, \mathcal{N})$.
A direct application of Gagliardo's characterisation of traces for the linear spaces $\smash{\dot{W}}^{1, p} (\mathcal{M}, \mathbb{R})$ shows that traces of maps in $\smash{\dot{W}}^{1, p} (\mathcal{M}, \mathcal{N})$ should belong to the fractional Sobolev-Slobodeckij space $\smash{\dot{W}}^{1 - 1/p, p} (\partial \mathcal{M}, \mathcal{N})$. There is however no reason for Gagliardo's linear extension to satisfy the nonlinear constraint imposed by $\mathcal{N}$ on the target.
In the case $p > \dim \mathcal{M}$, Sobolev mappings are continuous and thus traces of Sobolev maps are the mappings of $\smash{\dot{W}}^{1 - 1/p, p} (\partial \mathcal{M}, \mathcal{N})$ that are also restrictions of continuous functions (F. Bethuel, F. Demengel, Extensions for Sobolev mappings between manifolds (1995)).
The critical case $p = \dim \mathcal{M}$ can be treated similarly thanks to their vanishing mean oscillation property (F. Bethuel, F. Demengel, Extensions for Sobolev mappings between manifolds (1995); H. Brezis, L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries (1995); R. Schoen, K. Uhlenbeck, A regularity theory for harmonic maps (1982)).
The case $1 < p < \dim \mathcal{M}$ is more delicate.
It was first proved that when the first homotopy $\pi_{1} (\mathcal{N}), \dotsc, \smash{\pi_{\lfloor p - 1 \rfloor}} (\mathcal{N})$ are trivial, then the trace operator from $\smash{\dot{W}}^{1, p} (\mathcal{M}, \mathcal{N})$ to $\smash{\dot{W}}^{1 - 1/p, p} (\partial \mathcal{M}, \mathcal{N})$ is surjective (R. Hardt, Lin F., Mappings minimizing the $L^p$ norm of the gradient (1987)).
On the other hand, several conditions for the surjectivity have been known: topological obstructions require $\smash{\pi_{\lfloor p - 1 \rfloor}}(\mathcal{N})$ to be trivial whereas analytical obstructions arise unless the groups $\pi_{1} (\mathcal{N}), \dotsc, \pi_{\lfloor p - 1\rfloor}(\mathcal{N})$ are finite (F. Bethuel, A new obstruction to the extension problem for Sobolev maps between manifolds (2014)) and, when $p \ge 2$ is an integer, $\smash{\pi_{p - 1}} (\mathcal{N})$ is trivial (Trace theory for Sobolev mappings into a manifold (2021)).
In a recent work, I have completed the characterisation of the cases where the trace is surjective, proving that the known necessary conditions turn out to be sufficient (J. Van Schaftingen, The extension of traces for Sobolev mappings between manifolds).
I extend the traces thanks to a new construction which works on the domain rather than in the image. When $p \ge \dim \mathcal{M}$ the same construction also provides a Sobolev extension with linear estimates for maps that have a continuous extension, provided that there are no known analytical obstructions to such a control.