The authors analyze the weak limit of sequences of regular axisymmetric maps having an equibounded neo-Hookean energy of the type: \(E(u)=\int_{\Omega }\left\vert Du\right\vert ^{2}+H(\det Du)dx\), where \(u:\Omega \rightarrow \mathbb{R}^{3}\) is the deformation field, \(\Omega \subset \mathbb{R}^{3}\) the reference configuration, and \(H:(0,+\infty )\rightarrow \lbrack 0,+\infty )\) a convex function satisfying \(\lim_{t\rightarrow +\infty }H(t)/t=\lim_{s\rightarrow 0}H(s)=+\infty \) and \(\int_{B(0,3)}(H(\det Du)dx<\infty \), under the assumption that these maps have finite surface energy. The set \(\Omega \subset \mathbb{R}^{3}\) is axisymmetric if \(\Omega =\cup _{x\in \Omega }(\partial B_{\mathbb{R}^{2}}((0,0),\left\vert (x_{1},x_{2})\right\vert )\times \{x_{3}\})\). A function \(u\in C(U,\mathbb{R} ^{3})\) satisfies the INV property in a bounded open set \(U\) in \(\mathbb{R} ^{3}\) if for every point \(x_{0}\in U\) and a.e. \(r\in (0,\mathrm{dist}(x_{0},\partial U))\), \(u(x)\in \mathrm{im}_{T}(u,B(x_{0},r))\) for a.e. \(x\in B(x_{0},r)\) and \( u(x)\notin \mathrm{im}_{T}(u,B(x_{0},r))\) for a.e. \(x\in \Omega \setminus B(x_{0},r)\), where \(\mathrm{im}_{T}(u,B(x_{0},r))=\{y\in \mathbb{R}^{3}\setminus u(\partial U):\deg (u,U,y)\neq 0\}\). The main result proves that if \(u\) is the \( H^{1}(B(0,3),\mathbb{R}^{3})\) axisymmetric map of S. Conti and C. De Lellis [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 2, No. 3, 521–549 (2003; Zbl 1114.74004)], there exists a sequence of axisymmetric maps \( (u_{n})_{n}\subset H^{1}(B(0,3),\mathbb{R}^{3})\) such that \(u_{n}\) is bi-Lipschitz, for every \(n\in \mathbb{N}\); \(u_{n}\rightarrow u\) in \( H^{1}(B(0,3),\mathbb{R}^{3})\); \(u\) is one-to-one a.e., but it does not satisfy the INV property. Moreover, one has the equality \(\mathrm{Det} Du=(\det Du) \mathcal{L}^{3}+\frac{\pi }{6}(\delta _{(0,0,1)}-\delta _{(0,0,0)})\) and the divergence identities are not satisfied; \[\lim_{n\rightarrow \infty }\int_{B(0,3)}\left\vert Du_{n}\right\vert ^{2}dx=\int_{B(0,3)}\left\vert Du\right\vert ^{2}dx+2\pi ;\] \[\lim_{n\rightarrow \infty }\int_{B(0,3)}H(\det Du_{n})dx=\int_{B(0,3)}H(\det Du)dx;\] \(u_{3}^{-1}\) has SBV regularity, its jump set is the sphere \(\Gamma =\{(y_{1},y_{2},y_{3}):y_{1}^{2}+y_{2}^{2}+(y_{3}-\frac{1}{2})^{2}=\frac{1}{ 2^{2}}\}\), the amplitude of the jump is 1, and \(\left\Vert D^{s}u_{3}^{-1}\right\Vert =\pi \). The authors define \(\mathcal{A}=\{u\in H^{1}(\Omega ,\mathbb{R}^{3}):u=b \text{ in } \Omega \setminus \widetilde{\Omega }, \, u \text{ is one-to-one a.e., } \det Du>0 \text{ a.e., and } E(u)<+\infty \}\), \[\mathcal{A} ^{r}=\{u\in \mathcal{A}: \mathrm{Div}((\mathrm{adj} Du)g\circ u)=(\mathrm{div} g)\circ u\det Du, \, \forall g\in C_{c}^{1}(\mathbb{R}^{3},\mathbb{R} ^{3}\},\] and \(\mathcal{A}_{s}^{r}=\{u\in \mathcal{A}^{r}:u\) is axisymmetric\(\}\). The second main result proves that if \(u\notin \overline{ \mathcal{A}}_{s}^{r}\) is such that \(E(u)<+\infty \), there exists a countable set \(C(u)\) of points such that \[\mathrm{Det} Du=(\det Du)\mathcal{L}^{3}+\sum_{a\in C(u)}\mathrm{Det} Du(\{a\})\delta _{a};\] \(u^{-1}\in SBV(\Omega _{b},\mathbb{R}^{3})\). Let \(J_{u^{-1}}\) be the jump set of the inverse \(u^{-1}\), \((u^{-1})^{\pm }\) its lateral traces, and for \(\xi ,\xi ^{\prime }\in \mathbb{R}^{3}\) \(\Gamma _{\xi }^{\pm }=\{y\in J_{u^{-1}}:(u^{-1})^{\pm }(y)=\xi \}\), \(\Gamma _{\xi }=\Gamma _{\xi }^{+}\cup \Gamma _{\xi }^{-}\), \(\Gamma _{\xi ,\xi ^{\prime }}=\Gamma _{\xi }\cup \Gamma _{\xi ^{\prime }}\), \(\left\Vert D^{s}u^{-1}\right\Vert =\sum_{\xi ,\xi ^{\prime }\in C(u)}\left\vert \xi -\xi ^{\prime }\right\vert \mathcal{H}^{2}(\Gamma _{\xi ,\xi ^{\prime }})\); Let \(x\in \Omega \) and \(r>0\) be such that \(B(x,r)\subset \Omega \) and \(\deg(u,\partial B(x,r),\cdot )\) is well defined, and set \(\Delta _{x-r}=\deg(u,\partial B(x,r),\cdot )-\chi _{\mathrm{im}_{G}(u,B(x,r))}\). Then \(\Delta _{x-r}\in BV(\mathbb{R}^{3})\) and is integer-valued. There exists \(\Delta _{x}\in BV(\mathbb{R}^{3})\) integer-valued such that \(\Delta _{x-r_{n}}\rightarrow \Delta _{x}\) weakly\(^{\ast }\) in \(BV(\mathbb{R}^{3})\) for all sequences \(r_{n}\rightarrow 0\) such that \(\Delta _{x-r_{n}}\) is well defined; \(\Delta _{x}\neq 0\) if and only if \(x\in C(u)\); For \(\xi \in C(u)\), \(\Gamma _{\xi }=\cup _{k\in \mathbb{Z}}\partial ^{\ast }\{y\in \mathbb{R} ^{3}:\Delta _{\xi }(y)=k\}\) \(\mathcal{H}^{2}\)-a.e.; \(\sum_{\xi \in C(u)}\Delta _{\xi }=0\). For the proof, the authors define the surface energy \(\mathcal{E}\) to measure the creation of new surface of a deformation and the geometric and topological images of maps. They build the optimal recovery sequence. They also analyze the properties of maps in \(\overline{ \mathcal{A}}_{s}^{r}\) with finite surface energy.