Sobolev vector fields

Given two compact Riemannian manifolds $({\mathcal M},\mathfrak {g})$ and $({\mathcal N},\mathfrak {h})$ , we apply the above construction to Sobolev maps from ${\mathcal M}$ into vector fields in ${\mathcal N}$ (i.e., to maps $\xi : {\mathcal M}\to T{\mathcal N}$ ). To this end, we need to introduce a Riemannian metric on $T{\mathcal N}$ .

The Riemannian metric $\mathfrak {h}$ on $T{\mathcal N}$ induces in a canonical way a Riemannian metric on $T{\mathcal N}$ . Specifically, there exists a one-to-one correspondence between an affine connection $\nabla $ on $T{\mathcal N}$ and a connector operator, $K: TT{\mathcal N}\to T{\mathcal N}$ , such that for $q\in {\mathcal N}$ and $s\in T_q{\mathcal N}$ ,

$$\begin{align*}K_s : T_s T{\mathcal N}\to T_q{\mathcal N}. \end{align*}$$

The connector induces the covariant derivative via

$$\begin{align*}\nabla_s \eta = K^{\mathfrak{h}} \circ D\eta(s) \qquad \text{for every } \eta\in\Gamma(T{\mathcal N})\ \text{and } s\in T{\mathcal N}. \end{align*}$$

Denote by $x\in {\mathbb R}^{\dim {\mathcal N}}$ coordinates on ${\mathcal N}$ ; by choosing a local frame field in this coordinate patch, we obtain local coordinates $(x,v)\in {\mathbb R}^{2\dim {\mathcal N}}$ of $T{\mathcal N}$ . We denote the associated coordinates on $TT{\mathcal N}$ by $(x,v,w,s)\in {\mathbb R}^{4\dim {\mathcal N}}$ . In these coordinates, the projection $\pi :T{\mathcal N}\to {\mathcal N}$ is given by $\pi (x,v) = x$ , its derivative $D\pi :TT{\mathcal N}\to T{\mathcal N}$ by $D\pi (x,v,w,s) = (x,w)$ , and the connector K is given by

$$\begin{align*}K(x,v,w,s)=(x, s+\Gamma(x)[v,w]), \end{align*}$$

where $\Gamma :{\mathbb R}^{\dim {\mathcal N}}\to \operatorname {Bil}({\mathbb R}^{\dim {\mathcal N}})$ is a smooth map into the space of bilinear maps on ${\mathbb R}^{\dim {\mathcal N}}$ (representing the Christoffel symbols). In the following, we will always take the Levi-Civita connection $\nabla ^{\mathfrak {h}}$ , and its corresponding connector $K^{\mathfrak {h}}$ .

The map $D\pi \times K^{\mathfrak {h}} :TT{\mathcal N}\to T{\mathcal N}\times _{\mathcal N} T{\mathcal N}$ is a linear bundle isomorphism (covering the projection $\pi :T{\mathcal N}\to {\mathcal N}$ ), turned into an isometry by introducing the Sasaki metric $S^{\mathfrak {h}}$ on $TT{\mathcal N}$ [Reference SasakiSas58],

(2.1) $$ \begin{align} \langle V,W \rangle_{S^{\mathfrak{h}}} = \langle D\pi(V),D\pi(W) \rangle_{\mathfrak{h}} + \langle K^{\mathfrak{h}}(V),K^{\mathfrak{h}}(W) \rangle_{\mathfrak{h}}. \end{align} $$

By choosing an isometric embedding $\iota :(T{\mathcal N},S^{\mathfrak {h}}) \to ({\mathbb R}^D,\mathfrak {e})$ , where $\mathfrak {e}$ is the Euclidean metric, that is,

$$\begin{align*}\langle d\iota(V),d\iota(W) \rangle_{\mathfrak{e}} = \langle V,W \rangle_{S^{\mathfrak{h}}} \qquad V,W\in TT{\mathcal N}, \end{align*}$$

we can define the Sobolev space $W^{1,p}({\mathcal M};T{\mathcal N})$ as above.

The zero section $\zeta \in \Gamma (T{\mathcal N})$ is an isometric embedding of $({\mathcal N},\mathfrak {h})$ into $(T{\mathcal N},S^{\mathfrak {h}})$ : indeed, for $q\in {\mathcal N}$ and $v,w\in T_q{\mathcal N}$ ,

$$ \begin{align*} \langle v,w \rangle_{\zeta^\#S^{\mathfrak{h}}} &= \langle D\zeta(v),D\zeta(w) \rangle_{S^{\mathfrak{h}}} \\ &= \langle D\pi\circ D\zeta(v),D\pi \circ D\zeta(v) \rangle_{\mathfrak{h}} + \langle K^{\mathfrak{h}}\circ D\zeta(v),K^{\mathfrak{h}}\circ D\zeta(v) \rangle_{\mathfrak{h}} \\ &= \langle v,w \rangle_{\mathfrak{h}}, \end{align*} $$

where the first equality is the definition of the pullback metric $\zeta ^\# S^{\mathfrak {h}}$ , and in the last passage we used the fact that $D\pi \circ D\zeta = {\operatorname {Id}}_{T{\mathcal N}}$ and $K^{\mathfrak {h}}\circ D\zeta =0$ . Thus, $\jmath : {\mathcal N}\to {\mathbb R}^D$ given by $\jmath = \iota \circ \zeta $ is an isometric embedding of $({\mathcal N},\mathfrak {h})$ into $({\mathbb R}^D,\mathfrak {e})$ (although for our uses, one can choose any other isometric embedding of ${\mathcal N}$ into Euclidean space).

For a map $\xi :{\mathcal M}\to T{\mathcal N}$ , we denote $f_\xi = \pi \circ \xi : {\mathcal M}\to {\mathcal N}$ . Note that by the definition of the Sasaki metric,

(2.2) $$ \begin{align} |D\xi|^2 = |Df_\xi|^2 + |K^{\mathfrak{h}}\circ D\xi|^2, \end{align} $$

or equivalently,

(2.3) $$ \begin{align} |d\iota\circ D\xi|^2 = |d\jmath\circ D f_\xi|^2 + |d\jmath\circ K^{\mathfrak{h}}\circ D\xi|^2. \end{align} $$

(Note that $d\jmath :T{\mathcal N}\to {\mathbb R}^D$ rather than $\iota :T{\mathcal N}\to {\mathbb R}^D$ is the linear isometry.) The following lemma asserts that strong and weak convergence of a sequence $\xi _n$ implies the corresponding convergence of $f_{\xi _n}$ .

Proof. Let $\iota :(T{\mathcal N},S^{\mathfrak {h}})\to ({\mathbb R}^D,\mathfrak {e})$ and $\jmath : ({\mathcal N},\mathfrak {h})\to ({\mathbb R}^D,\mathfrak {e})$ be defined as above. Let $\xi \in W^{1,p}({\mathcal M};T{\mathcal N})$ . Since ${\mathcal N}$ is compact, the fact that $f_\xi \in W^{1,p}({\mathcal M};{\mathcal N})$ follows from (2.2) (or equivalently, (2.3)).

Let $\xi _n \rightharpoonup \xi $ in $W^{1,p}({\mathcal M};T{\mathcal N})$ (i.e., $\iota \circ \xi _n \rightharpoonup \iota \circ \xi $ in $W^{1,p}({\mathcal M};{\mathbb R}^D)$ ). By (2.3) and the compactness of ${\mathcal N}$ , $\jmath \circ f_{\xi _n}$ is bounded in $W^{1,p}({\mathcal M};{\mathbb R}^D)$ (and equi-integrable if $p=1$ ), and thus $\jmath \circ f_{\xi _n}$ weakly converges to some $F\in W^{1,p}({\mathcal M};{\mathbb R}^D)$ (modulo a subsequence). We need to show that $F = \jmath \circ f_\xi $ . Since weak $W^{1,p}$ -convergence implies strong $L^p$ -convergence, we can move to a subsequence and obtain that $\iota \circ \xi _n \to \iota \circ \xi $ almost everywhere, and thus, for that same subsequence, $\jmath \circ f_{\xi _n}\to \jmath \circ f_\xi $ almost everywhere, which by the uniqueness of the limit implies that $F=\jmath \circ f_\xi $ . As we could have started with any subsequence of $\xi _n$ and obtain the result for a sub-subsequence, the claim holds for the whole sequence.

Next, assume that $\xi _n \to \xi $ strongly in $W^{1,p}({\mathcal M};T{\mathcal N})$ . Since, a fortiori, $\xi _n \rightharpoonup \xi $ in $W^{1,p}({\mathcal M};T{\mathcal N})$ , it follows from the previous clause that $f_{\xi _n} \rightharpoonup f_\xi $ in $W^{1,p}({\mathcal M};{\mathcal N})$ , or equivalently, that $\jmath \circ f_{\xi _n}\rightharpoonup \jmath \circ f_\xi $ in $W^{1,p}({\mathcal M};{\mathbb R}^D)$ . We need to show that $d(\jmath \circ f_{\xi _n})\to d(\jmath \circ f_\xi )$ in $L^p({\mathcal M};T^*{\mathcal M}\otimes {\mathbb R}^D)$ . It is sufficient to prove this for a subsequence, and thus we can assume, without loss of generality, that $D\iota \circ D\xi _n \to D\iota \circ D\xi $ almost everywhere. Since $D\iota $ is an embedding, it follows that $D\xi _n \to D\xi $ almost everywhere (as maps from ${\mathcal M}\to T^*{\mathcal M}\otimes TT{\mathcal N}$ ). In particular, $D\pi \circ D\xi _n\to D\pi \circ D\xi $ almost everywhere (as maps ${\mathcal M} \to T^*{\mathcal M}\otimes T{\mathcal N}$ ), i.e., $D f_{\xi _n} \to D f_{\xi _n}$ almost everywhere, and thus also $d(\jmath \circ f_{\xi _n})\to d(\jmath \circ f_\xi )$ . To complete the proof, it remains to show that

(2.4) $$ \begin{align} \lim_{n\to\infty} \int_{\mathcal M} |Df_{\xi_n}|^p\, \text{d}\text{Vol}_{\mathfrak{g}} = \int_{\mathcal M} |Df_\xi|^p\, \text{d}\text{Vol}_{\mathfrak{g}}. \end{align} $$

Since $D f_{\xi _n} \to D f_{\xi _n}$ almost everywhere, we also have that $|Df_{\xi _n}|^p \to |Df_\xi |^p$ almost everywhere, and since

$$\begin{align*}|Df_{\xi_n}|^p \le |D\xi_n|^p \qquad\text{ and }\qquad |Df_\xi|^p \le |D\xi|^p, \end{align*}$$

and since it follows from the strong $W^{1,p}$ -convergence of $\xi _n$ that

$$\begin{align*}\lim_{n\to\infty} \int_{\mathcal M} |D\xi_n|^p\, \text{d}\text{Vol}_{\mathfrak{g}} = \int_{\mathcal M} |D\xi|^p\, \text{d}\text{Vol}_{\mathfrak{g}}, \end{align*}$$

the limit (2.4) follows from a refinement of the dominated convergence theorem (e.g., [Reference Evans and GariepyEG15, Theorem 1.20]).

The following lemma, which will be needed in the sequel, addresses a situation where $\xi _n\rightharpoonup \xi $ in $W^{1,p}$ along with $K^{\mathfrak {h}} \circ D\xi _n$ converging strongly in $L^p$ . The fact that the limit is, as one would expect, $K^{\mathfrak {h}} \circ D\xi $ is somewhat nontrivial, since the two converging sequences have to be interpreted with respect to two different maps into ${\mathbb R}^D$ . Since this suffices in the application below, we assume that p is large enough.

Proof. Since $p>\dim {\mathcal M}$ , $\xi _n \to \xi $ uniformly; hence, we can work with local coordinates. Let $X:\Omega \subset {\mathbb R}^{\dim {\mathcal M}} \to {\mathcal M}$ and $Y: \tilde \Omega \subset {\mathbb R}^{\dim {\mathcal N}} \to {\mathcal N}$ be local coordinate systems, and denote $(x_n , v_n) = (DY)^{-1} \circ \xi _n \circ X$ , and similarly $(x,v) = (DY)^{-1} \circ \xi \circ X$ , both maps $\Omega \to \tilde \Omega \times {\mathbb R}^{\dim {\mathcal N}}$ . The uniform convergence $\xi _n\to \xi $ implies that $(x_n,v_n) \to (x,v)$ uniformly. The boundedness of $\xi _n$ in $W^{1,p}({\mathcal M};T{\mathcal N})$ implies the boundedness of the coordinate expressions $(x_n,v_n)$ in $W^{1,p}(\Omega , {\mathbb R}^{2\dim {\mathcal N}})$ ; hence, $(x_n,v_n) \rightharpoonup (x,v)$ in $W^{1,p}(\Omega , {\mathbb R}^{2\dim {\mathcal N}})$ to $(x,v)$ . In these coordinates,

$$\begin{align*}D\xi_n = (x_n,v_n, Dx_n[\cdot], Dv_n[\cdot]): \Omega \to (\tilde\Omega\times{\mathbb R}^{\dim{\mathcal M}}) \times \operatorname{Hom}({\mathbb R}^{\dim{\mathcal N}}, {\mathbb R}^{2\dim {\mathcal N}}). \end{align*}$$

Hence,

$$\begin{align*}K^{\mathfrak{h}} \circ D\xi_n = (x_n,Dv_n[\cdot] + \Gamma(x_n)[v_n,Dx_n[\cdot]]). \end{align*}$$

Since $Dv_n \rightharpoonup Dv$ and $Dx_n \rightharpoonup Dx$ in $L^p$ , $\Gamma $ is smooth and $(x_n,v_n)\to (x,v)$ uniformly, we obtain that

$$\begin{align*}(x_n,Dv_n[\cdot] + \Gamma(x_n)[v_n,Dx_n[\cdot]]) \rightharpoonup (x,Dv[\cdot] + \Gamma(x)[v,Dx[\cdot]]) \end{align*}$$

in $L^p(\Omega ;\tilde \Omega \times \operatorname {Hom}({\mathbb R}^{\dim {\mathcal M}},{\mathbb R}^{\dim {\mathcal N}}))$ . We have thus proved that $K^{\mathfrak {h}} \circ D\xi _n \rightharpoonup K^{\mathfrak {h}} \circ D\xi $ weakly in $L^p$ in every coordinate patch.

Add now the fact that $dj \circ K^{\mathfrak {h}}\circ D\xi _n \to dj \circ T$ strongly in $L^p$ . By moving to subsequences, we can assume that $dj \circ K^{\mathfrak {h}}\circ D\xi _n \to dj \circ T$ a.e. Since $Dj:{\mathcal N}\to {\mathbb R}^{2K}$ is an embedding, it follows that $K\circ D\xi _n \to T$ a.e. Therefore, in coordinates, for almost every $p\in \Omega $ ,

$$\begin{align*}(x_n(p),D_pv_n[\cdot] + \Gamma(x_n(p))[v_n(p),D_px_n[\cdot]]) \to T(p)[\cdot]. \end{align*}$$

Thus, $T = K \circ D\xi $ a.e. in every coordinate patch; hence, a.e.

The following is an immediate application of [Reference Alpern, Kupferman and MaorAKM22, Prop. D.1]:

Proposition 2.3. Let $p>\dim {\mathcal M}$ . Then, $\xi _n\to \xi $ in $W^{1,p}({\mathcal M};T{\mathcal N})$ if and only if $f_{\xi _n}\to f_\xi $ in $W^{1,p}({\mathcal M};{\mathcal N})$ , and for some isometric embedding $\jmath :{\mathcal N}\to {\mathbb R}^D$ ,

$$\begin{align*}d\jmath\circ \xi_n \to d\jmath\circ \xi \qquad {\mathrm{in}\ L^p({\mathcal M};{\mathbb R}^D)}, \end{align*}$$

and

$$\begin{align*}d\jmath\circ K^{\mathfrak{h}}\circ D\xi_n \to d\jmath\circ K^{\mathfrak{h}} \circ D\xi \qquad {\mathrm{in}\ L^p(T^*{\mathcal M};{\mathbb R}^D)}. \end{align*}$$

Sobolev immersions

Let $\dim {\mathcal M}=d$ and $\dim {\mathcal N}=d+1$ , and let $f\in W^{1,p}({\mathcal M};{\mathcal N})$ satisfy $\operatorname {rank} df=d$ a.e. We may define a.e. a map $\mathfrak {n}_f\in L^\infty ({\mathcal M};T{\mathcal N}$ ) covering f, such that $\mathfrak {n}_f(q)\in T_{f(q)}{\mathcal N}$ is a unit vector, normal to the image of $Df_q$ , and for every oriented basis $(u_1,\dots ,u_d)\subset T_q{\mathcal M}$ ,

$$\begin{align*}(df_q(u_1),\dots,df_q(u_d),\mathfrak{n}_f(q))\subset T_{f(q)}{\mathcal N} \end{align*}$$

is an oriented basis. If $\mathfrak {n}_f \in W^{1,p}({\mathcal M};T{\mathcal N})$ , we say that f belongs to the set of p -Sobolev immersions (this is the definition in (1.3)). Since there is a one-to-one correspondence between f and $\mathfrak {n}_f$ , $\operatorname {Imm}_p({\mathcal M};{\mathcal N})$ can be identified with a subset of $W^{1,p}({\mathcal M};T{\mathcal N})$ , and specifically a subset of $W^{1,p}({\mathcal M};S{\mathcal N})$ , where $S{\mathcal N}\subset T{\mathcal N}$ is the sphere bundle of ${\mathcal N}$ . It is, however, not a closed subset, as the rank condition is not closed under $W^{1,p}$ -convergence.

For $f\in \operatorname {Imm}_p({\mathcal M};{\mathcal N})$ , the total energy (1.2) is more rigorously written as

(2.5) $$ \begin{align} {\mathcal E}_p(f) = \int_{\mathcal M} \left(\operatorname{dist}^p(Df,\operatorname{O}(\mathfrak{g},\mathfrak{h})) + |Df\circ S + K^{\mathfrak{h}}\circ D\mathfrak{n}_f|^p\right)\,\text{d}\text{Vol}_{\mathfrak{g}}, \end{align} $$

as this notation does not rely on non-smooth pullback bundles and their corresponding pullback connections.

Trivializations

Let $\xi _n\in W^{1,p}({\mathcal M};T{\mathcal N})$ be a sequence converging to $\xi \in W^{1,p}({\mathcal M};T{\mathcal N})$ uniformly (hence, in particular, $f_{\xi _n} \to f_\xi $ uniformly). The compactness of ${\mathcal M}$ and ${\mathcal N}$ implies the existence of finite open covers ${\mathcal U} = \{U_i\}$ and ${\mathcal V} = \{V_i\}$ of ${\mathcal M}$ and ${\mathcal N}$ such that $T{\mathcal N}|_{V_i}$ is a trivial bundle for every i, and such that $f_{\xi _n}(U_i)\subset V_i$ for n large enough. For every such U and V (to simplify notations, we omit the index i), let $R:V\to \operatorname {SO}(\mathfrak {h},{\mathbb R}^{\dim {\mathcal N}})$ be a smooth orthonormal frame (which may or may not be a coordinate frame); namely, for every $q\in V$ and $\eta ,\xi \in T_q{\mathcal N}$ ,

$$\begin{align*}\langle \eta,\xi \rangle_{\mathfrak{h}} = \langle R \circ \eta,R \circ \xi \rangle_{\mathfrak{e}}. \end{align*}$$

Such a frame exists by the triviality of $T{\mathcal N}|_V$ . Then, restricting to U,

$$\begin{align*}R \circ Df_{\xi_n}, R \circ Df_\xi \in L^p\Omega^1(U;{\mathcal N}\times {\mathbb R}^{\dim{\mathcal N}}), \end{align*}$$

where R acts on the vector part $df_{\xi _n}$ of $Df_{\xi _n}$ . Note that $R\circ Df_{\xi _n} - R\circ Df_\xi \ne R\circ (Df_{\xi _n} - Df_\xi )$ ; in fact, the right-hand side is not well defined, as the images of $\xi _n$ and $\xi $ do not belong to the same fiber of $T{\mathcal N}$ .

Proposition 2.4. Assume that $p>\dim {\mathcal M}$ . Then,

$$\begin{align*}f_{\xi_n} \to f_\xi \qquad{\mathrm{in}\ W^{1,p}({\mathcal M};{\mathcal N})} \end{align*}$$

if and only if

$$\begin{align*}R\circ Df_{\xi_n} \to R \circ Df_\xi \qquad{\mathrm{in}\ L^p\Omega^1(U;{\mathcal N}\times{\mathbb R}^{\dim{\mathcal N}})} \end{align*}$$

for every $U\in {\mathcal U}$ . Likewise, if

$$\begin{align*}f_{\xi_n} \rightharpoonup f_\xi \qquad{\mathrm{in}\ W^{1,p}({\mathcal M};{\mathcal N})}, \end{align*}$$

then

$$\begin{align*}R\circ Df_{\xi_n} \rightharpoonup R \circ Df_\xi \qquad{\mathrm{in}\ L^p\Omega^1(U;{\mathcal N}\times{\mathbb R}^{\dim{\mathcal N}})} \end{align*}$$

for every $U\in {\mathcal U}$ .

2.1 Coordinate representation of the energy

In this section, we present some of the geometric constructs and the energy (1.2) in coordinates for the benefit of readers who are more used to this notation. For simplicity, assume that both ${\mathcal M}$ and ${\mathcal N}$ are covered by single coordinate charts,

$$\begin{align*}x_{\mathcal M}:{\mathcal M}\to{\mathbb R}^d \qquad\text{ and }\qquad x_{\mathcal N}:{\mathcal N}\to{\mathbb R}^{d+1}. \end{align*}$$

A map $f:{\mathcal M}\to {\mathcal N}$ is represented by a map ${\mathbb R}^d\supset x_{\mathcal M}({\mathcal M}) \to {\mathbb R}^{d+1}$ ,

$$\begin{align*}x \mapsto x_{\mathcal N}\circ f \circ x_{\mathcal M}^{-1}, \end{align*}$$

with components $f^\alpha $ , where for the sake of clarity, we use Latin indexes for the body manifold ${\mathcal M}$ and Greek indexes for the space manifold ${\mathcal N}$ .

Throughout this section, we will denote the projection of a vector bundle onto its base by $\pi $ , where the type (e.g., $T{\mathcal M}\to {\mathcal M}$ or $TT{\mathcal N}\to T{\mathcal N}$ ) should be clear from the context. The tangent bundle $T{\mathcal M}$ has coordinates $(x_{{\mathcal M}},\dot {x}_{{\mathcal M}}):T{\mathcal M}\to {\mathbb R}^d\times {\mathbb R}^d$ , where, with a slight abuse of notations which will be used repeatedly, $x_{{\mathcal M}} = x_{\mathcal M}\circ \pi $ and $\dot {x}_{{\mathcal M}} = dx_{\mathcal M}$ ; the metric $\mathfrak {g}$ is represented by a matrix-valued function $x_{\mathcal M}({\mathcal M})\to {\mathbb R}^d\otimes {\mathbb R}^d$ , with entries $\mathfrak {g}_{ij}$ , such that for $V \in T{\mathcal M}$ ,

$$\begin{align*}|V|^2 = \mathfrak{g}_{ij}(x_{{\mathcal M}}(V))\, \dot{x}_{{\mathcal M}}^i(V) \dot{x}_{{\mathcal M}}^j(V), \end{align*}$$

with the Einstein summation convention over repeated indexes. Similarly, the tangent bundle $T{\mathcal N}$ has coordinates $(x_{{\mathcal N}},\dot {x}_{{\mathcal N}}):T{\mathcal N}\to {\mathbb R}^{d+1}\times {\mathbb R}^{d+1}$ ; the metric $\mathfrak {h}$ is represented by a matrix-valued function $x_{\mathcal N}({\mathcal N})\to {\mathbb R}^{d+1}\otimes {\mathbb R}^{d+1}$ , with entries $\mathfrak {h}_{\alpha \beta }$ , so that for a vector $W\in T{\mathcal N}$ ,

$$\begin{align*}|W|^2 = \mathfrak{h}_{\alpha\beta}(x_{{\mathcal N}}(W))\, \dot{x}_{{\mathcal N}}^\alpha(W) \dot{x}_{{\mathcal N}}^\beta(W). \end{align*}$$

For $f:{\mathcal M}\to {\mathcal N}$ , the fiber of the pullback bundle $f^*T{\mathcal N}$ at a point $p\in {\mathcal M}$ is canonically identified with the fiber $T_{f(p)}{\mathcal N}$ ; its coordinates are $(x_{{\mathcal M}},\dot {x}_{{\mathcal N}}):f^*T{\mathcal N}\to {\mathbb R}^d\times {\mathbb R}^{d+1}$ , where $x_{{\mathcal M}} = x_{{\mathcal M}}\circ \pi $ and $\dot {x}_{{\mathcal N}}$ is identified with $\dot {x}_{\mathcal N}: T{\mathcal N}\to {\mathbb R}^{d+1}$ , via the canonical identification of fibers of $f^*T{\mathcal N}$ with fibers of $T{\mathcal N}$ . The coordinate representation of $df\in \Gamma (f^*T{\mathcal N})$ is the map $x_{\mathcal M}({\mathcal M})\to \operatorname {Hom}({\mathbb R}^d,{\mathbb R}^{d+1})$ given by $\partial _if^\alpha $ , whereas the coordinate representation of $Df:T{\mathcal M}\to T{\mathcal N}$ is the map $x_{\mathcal M}({\mathcal M})\to {\mathbb R}^{d+1}\times \operatorname {Hom}({\mathbb R}^d,{\mathbb R}^{d+1})$ , given by $(f^\alpha ,\partial _if^\alpha )$ .

We denote by $\mathfrak {g}^{1/2}: x_{\mathcal M}({\mathcal M})\to \operatorname {Hom}({\mathbb R}^d,{\mathbb R}^d)$ and $\mathfrak {h}^{1/2}:x_{\mathcal N}({\mathcal N})\to \operatorname {Hom}({\mathbb R}^{d+1},{\mathbb R}^{d+1})$ the symmetric positive-definite square roots of the matrices $\mathfrak {g}$ and $\mathfrak {h}$ ; that is,

$$\begin{align*}\mathfrak{g}_{ij} = \delta_{kl} (\mathfrak{g}^{1/2})_i^k (\mathfrak{g}^{1/2})_j^\ell \qquad\text{ and }\qquad \mathfrak{h}_{\alpha\beta} = \delta_{\gamma\eta} (\mathfrak{h}^{1/2})_\alpha^\gamma (\mathfrak{h}^{1/2})_\beta^\eta. \end{align*}$$

The stretching energy of $f:{\mathcal M}\to {\mathcal N}$ takes the form

$$\begin{align*}{\mathcal E}_p^S(f) = \int_{x_{\mathcal M}({\mathcal M})} \operatorname{dist}^p(Q(x),O(d)) \det(\mathfrak{g}^{1/2}(x))\, dx, \end{align*}$$

where

$$\begin{align*}Q^\alpha_i(x) = (\mathfrak{h}^{1/2}(f(x)))_\beta^\alpha \, \partial_j f^\beta(x) \,(\mathfrak{g}^{-1/2}(x))_i^j, \end{align*}$$

or, in matrix form,

$$\begin{align*}Q(x) = h^{1/2}(f(x)) \circ df(x) \circ \mathfrak{g}^{-1/2}(x), \end{align*}$$

and the distance here is with respect to the Euclidean Frobenius norm.

We proceed with the double tangent $TT{\mathcal N}$ , which has coordinates

$$\begin{align*}(x_{{\mathcal N}},\dot{x}_{{\mathcal N}},v_{{\mathcal N}},\dot{v}_{{\mathcal N}}):TT{\mathcal N}\to{\mathbb R}^{d+1}\times{\mathbb R}^{d+1}\times{\mathbb R}^{d+1}\times{\mathbb R}^{d+1}, \end{align*}$$

where $x_{{\mathcal N}} = x_{{\mathcal N}}\circ \pi $ , $\dot {x}_{{\mathcal N}} = \dot {x}_{{\mathcal N}}\circ \pi $ , $v_{{\mathcal N}} = \dot {x}_{{\mathcal N}}\circ D\pi $ and $\dot {v}_{{\mathcal N}} = d\dot {x}_{{\mathcal N}}$ . That is, a vector in $T_{(x_{\mathcal N},\dot {x}_{\mathcal N})}T{\mathcal N}$ represents a direction of change, in $T{\mathcal N}$ , from the point $(x_{\mathcal N},\dot {x}_{\mathcal N})$ ; the coordinate $v_{{\mathcal N}}$ represents the direction of change in the coordinate $x_{\mathcal N}$ , and $\dot {v}_{{\mathcal N}}$ the direction of change in the coordinate $\dot {x}_{{\mathcal N}}$ .

We further denote by $\Gamma : x_{\mathcal N}({\mathcal N})\to \operatorname {Hom}({\mathbb R}^{d+1}\otimes {\mathbb R}^{d+1},{\mathbb R}^{d+1})$ the Christoffel symbols of the Levi-Civita connection, with coordinates $\Gamma ^\alpha _{\beta \gamma }$ ; the connection map $K^{\mathfrak {h}}:TT{\mathcal N}\to T{\mathcal N}$ is defined by

$$\begin{align*}x_{{\mathcal N}}\circ K^{\mathfrak{h}} = x_{{\mathcal N}} \qquad\text{ and }\qquad \dot{x}_{{\mathcal N}}\circ K^{\mathfrak{h}} = \dot{v}_{{\mathcal N}} + \Gamma(x_{{\mathcal N}})[\dot{x}_{{\mathcal N}},v_{{\mathcal N}}]. \end{align*}$$

Let $\xi ,Y\in \mathfrak {X}({\mathcal N})$ be vector fields having coordinate representation $x_{\mathcal N}({\mathcal N})\to {\mathbb R}^{d+1}$ ,

$$\begin{align*}\xi^\alpha = \dot{x}_{\mathcal N}^\alpha\circ \xi \circ x_{{\mathcal N}}^{-1} \qquad\text{ and }\qquad Y^\alpha = \dot{x}_{\mathcal N}^\alpha\circ Y \circ x_{{\mathcal N}}^{-1}. \end{align*}$$

The coordinate representation of $D\xi (Y):{\mathcal N}\to TT{\mathcal N}$ is the map $x_{\mathcal N}({\mathcal N}) \to {\mathbb R}^{d+1}\times {\mathbb R}^{d+1}\times {\mathbb R}^{d+1}$ , given by

$$\begin{align*}(Y^\alpha,\xi^\alpha, Y^\beta\, \partial_\beta\xi^\alpha). \end{align*}$$

Thus, the coordinate representation of $K^{\mathfrak {h}}\circ D\xi (Y):{\mathcal N}\to T{\mathcal N}$ is the map $x_{\mathcal N}({\mathcal N})\to {\mathbb R}^{d+1}$ , given by

$$\begin{align*}Y^\beta\, \partial_\beta\xi^\alpha + \Gamma^\alpha_{\beta\gamma}(x) Y^\beta \xi^\gamma, \end{align*}$$

thus showing that indeed $K^{\mathfrak {h}}\circ D\xi (Y) = \nabla _Y\xi $ .

We proceed to write the bending energy in explicit form. The reference shape operator S is represented by a map $x_{\mathcal M}({\mathcal M})\to \operatorname {Hom}({\mathbb R}^d,{\mathbb R}^d)$ with indexes $S_i^j$ . The unit normal $\mathfrak {n}_f$ is represented by a function $x_{\mathcal M}({\mathcal M})\to {\mathbb R}^{d+1}$ such that

$$\begin{align*}(\partial_1 f^\alpha(x),\dots,\partial_df^\alpha(x),\mathfrak{n}_f^\alpha(x)) \end{align*}$$

is an oriented basis for ${\mathbb R}^{d+1}$ , and $\mathfrak {n}_f(x)$ is a unit vector normal to range of $\nabla f(x)$ with respect to the inner-product $\mathfrak {h}_{\alpha \beta }(f(x))$ . With this, we have

$$\begin{align*}{\mathcal E}_p^B(f) = \int_{x_{\mathcal M}({\mathcal M})} \left(\mathfrak{g}^{ij}(x) \mathfrak{h}_{\alpha\beta}(f(x)) A_i^\alpha(x) A_j^\beta(x)\right)^{p/2} \,\det(\mathfrak{g}^{1/2}(x))\, dx, \end{align*}$$

where

$$\begin{align*}A_i^\alpha(x) = \partial_i \mathfrak{n}_f^\alpha(x) + \Gamma^\alpha_{\beta\gamma}(f(x)) \partial_i f^\beta(x) \mathfrak{n}_f^\gamma(x) + \partial_i f^\gamma(x) S_\gamma^\alpha(x) \end{align*}$$

represents the discrepancy between the target shape operator S and the shape operator of f as defined in (1.4).

3.1 The relaxed functional

Denote as above $d = \dim {\mathcal M}$ . Consider the vector bundle of rank $d+1$ , $T{\mathcal M}\oplus {\mathbb R}\to {\mathcal M}$ endowed with the product metric

$$\begin{align*}\langle (v,s),(w,t) \rangle_G = \langle v,w \rangle_{\mathfrak{g}} + st, \qquad\text{for } q\in{\mathcal M}, v,w\in T_q{\mathcal M}\ \text{and } s,t\in{\mathbb R}. \end{align*}$$

The orientation of $T{\mathcal M}\oplus {\mathbb R}$ is induced by the orientation of $T{\mathcal M}$ : if $(u_1,\dots ,u_d)$ is an oriented basis for $T_q{\mathcal M}$ , then, $(u_1,\dots ,u_d,1)$ is an oriented basis for $T_q{\mathcal M}\oplus {\mathbb R}$ .

For $\xi \in W^{1,p}({\mathcal M};T{\mathcal N})$ , we introduce the map

$$\begin{align*}Df_\xi\oplus \xi \in L^p({\mathcal M};(T{\mathcal M} \oplus {\mathbb R})^*\otimes T{\mathcal N}), \end{align*}$$

which is a linear bundle morphism covering $f_\xi $ defined a.e. by

$$\begin{align*}(Df_\xi\oplus \xi)_q(v,t) = d_qf_\xi(v) + t\, \xi(q) \in T_{f_\xi(q)}{\mathcal N}, \end{align*}$$

where $q\in {\mathcal M}$ , $v\in T_q{\mathcal M}$ and $t\in {\mathbb R}$ . That is, on the left-hand side we identify $\xi (q)\in T_{f_\xi (q)}{\mathcal N}$ with $\xi (q)\in \operatorname {Hom}({\mathbb R},T_{f_\xi (q)}{\mathcal N})$ .

Definition 3.1. The relaxed energy functional $\tilde {{\mathcal E}}_p : W^{1,p}({\mathcal M};T{\mathcal N})\to [0,\infty )$ is defined by

$$ \begin{align*} \tilde{{\mathcal E}}_p(\xi) &= \int_{\mathcal M} \operatorname{dist}^p(Df_\xi\oplus \xi,\operatorname{SO}(G,\mathfrak{h}))\,\text{d}\text{Vol}_{\mathfrak{g}} + \int_{\mathcal M} |Df_\xi\circ S + K^{\mathfrak{h}}\circ D\xi|^p\,\text{d}\text{Vol}_{\mathfrak{g}} \\ &\equiv \tilde{{\mathcal E}}_p^S(\xi) + \tilde{{\mathcal E}}_p^B(\xi). \end{align*} $$

Proposition 3.2. The functional $\tilde {{\mathcal E}}_p$ is a relaxation of ${\mathcal E}_p$ , in the sense that

$$\begin{align*}{\mathcal E}_p(f) = \tilde{{\mathcal E}}_p(\mathfrak{n}_f) \qquad \text{for all } f\in \operatorname{Imm}_p({\mathcal M},{\mathcal N}). \end{align*}$$

Furthermore, $\tilde {{\mathcal E}}_p^S(\xi ) = 0$ if and only if $f_\xi \in \operatorname {Imm}_p({\mathcal M};{\mathcal N})$ , $\xi = \mathfrak {n}_{f_\xi }$ and $Df\in \operatorname {O}(\mathfrak {g},\mathfrak {h})$ a.e.

Proof. We first prove the second claim. Let $\tilde {{\mathcal E}}_p^S(\xi ) = 0$ ; that is, $Df_\xi \oplus \xi \in \operatorname {SO}(G,\mathfrak {h})$ a.e. Let $v\in T{\mathcal M}$ . Then,

$$\begin{align*}|Df_\xi(v)|_{\mathfrak{h}} = |(Df_\xi\oplus\xi)(v,0)|_{\mathfrak{h}} = |(v,0)|_G = |v|_{\mathfrak{g}}, \end{align*}$$

thus proving that $Df_\xi \in \operatorname {O}(\mathfrak {g},\mathfrak {h})$ . In particular, $Df_\xi $ has full rank a.e. Moreover, a.e., and for every $v\in T{\mathcal M}$ ,

$$\begin{align*}\langle Df_\xi(v),\xi \rangle_{\mathfrak{h}} = \langle (Df_\xi\oplus\xi)(v,0),(Df_\xi\oplus\xi)(0,1) \rangle_{\mathfrak{h}} = \langle (v,0),(0,1) \rangle_G = 0, \end{align*}$$

whereas

$$\begin{align*}\langle \xi,\xi \rangle_{\mathfrak{h}} = \langle (Df_\xi\oplus\xi)(0,1),(Df_\xi\oplus\xi)(0,1) \rangle_{\mathfrak{h}} = \langle (0,1),(0,1) \rangle_G = 1, \end{align*}$$

(i.e., $\xi = \mathfrak {n}_{f_\xi }$ ) and since $\xi \in W^{1,p}({\mathcal M};T{\mathcal N})$ , it follows that $f_\xi \in \operatorname {Imm}_p({\mathcal M};{\mathcal N})$ . The other direction is immediate.

The first claim follows from the next lemma.

Lemma 3.3. For $f\in \operatorname {Imm}_p({\mathcal M},{\mathcal N})$ , the following identity holds:

$$\begin{align*}\operatorname{dist}(Df\oplus \mathfrak{n}_f,\operatorname{SO}(G,\mathfrak{h})) = \operatorname{dist}(Df,\operatorname{O}(\mathfrak{g},\mathfrak{h})). \end{align*}$$

Proof. This is a linear-algebraic statement. Let $(V_1,\mathfrak {g}_1)$ , $(V_2,\mathfrak {g}_2)$ and $(W,\mathfrak {h})$ be oriented d-, k- and $(d+k)$ -dimensional inner-product spaces. Let $A\in \operatorname {Hom}(V_1,W)$ and $B\in \operatorname {Hom}(V_2,W)$ satisfy

1. (a) $A\oplus B \in \operatorname {Hom}(V_1\oplus V_2,W)$ is orientation-preserving (with respect with the natural orientation of $V_1\oplus V_2$ ).

2. (b) ${\operatorname {Image}}(A) = ({\operatorname {Image}}(B))^\perp $ .

3. (c) $B\in \operatorname {O}(\mathfrak {g}_2,\mathfrak {h})$ .

Then,

$$\begin{align*}\operatorname{dist}(A\oplus B,\operatorname{SO}(\mathfrak{g}_1\oplus \mathfrak{g}_2,\mathfrak{h})) = \operatorname{dist}(A,\operatorname{O}(\mathfrak{g}_1,\mathfrak{h})). \end{align*}$$

In the present case, $k=1$ , $A = Df$ and $B = \mathfrak {n}_f$ . Indeed, by definition, $Df\oplus \mathfrak {n}_f$ is orientation-preserving, $\mathfrak {n}_f$ is a unit vector, and G is a product metric.

The fact that ${\mathcal E}_p(f) = \tilde {{\mathcal E}}_p(\mathfrak {n}_f)$ implies that Theorem 1.1 is proved if we prove the following relaxed version:

Theorem 3.4. Suppose that there exists a sequence $\xi _n \in W^{1,p}({\mathcal M};T{\mathcal N})$ satisfying

$$\begin{align*}\lim_{n\to\infty} \tilde{{\mathcal E}}_p(\xi_n) \to 0. \end{align*}$$

Then there exists a subsequence of $\xi _n$ converging strongly in $W^{1,p}({\mathcal M};T{\mathcal N})$ to a smooth limit $\xi $ . Furthermore, $\tilde {{\mathcal E}}_p(\xi )=0$ , implying that $f_\xi \in \operatorname {Imm}_p({\mathcal M};{\mathcal N})$ is a smooth isometric immersion, satisfying $K^{\mathfrak {h}}\circ D\mathfrak {n}_{f_\xi } = - Df_\xi \circ S$ (i.e., S is the shape operator of the limit).

3.2 Basic compactness considerations

Proof. Let $\tilde {{\mathcal E}}_p(\xi _n) \to 0$ . The uniform boundedness of the stretching energy $\tilde {{\mathcal E}}_p^S(\xi _n)$ and the compactness of ${\mathcal N}$ imply that $\xi _n$ is a bounded sequence in $L^p({\mathcal M},T{\mathcal N})$ and that $Df_{\xi _n}$ is a bounded sequence in $L^p({\mathcal M};T^*{\mathcal M}\otimes T{\mathcal N})$ . Indeed,

$$\begin{align*}|Df_\xi\oplus \xi| \ge \max\{|Df_\xi|, |\xi|\}, \end{align*}$$

and

$$\begin{align*}\operatorname{dist}(Df_\xi\oplus \xi,\operatorname{SO}(G,\mathfrak{h})) \ge |Df_\xi\oplus \xi| - \sqrt{d+1}, \end{align*}$$

since every element of $\operatorname {SO}(G,\mathfrak {h})$ is of norm $\sqrt {d+1}$ . The boundedness of the bending energy $\tilde {{\mathcal E}}_p^B(\xi _n)$ implies in turn that $K^{\mathfrak {h}}\circ D\xi _n$ is a bounded sequence in $L^p({\mathcal M};T^*{\mathcal M}\otimes T{\mathcal N})$ . By the definition (2.1) of the Sasaki metric, this implies that $D\xi _n$ is a bounded sequence in $L^p({\mathcal M};T^*{\mathcal M}\otimes TT{\mathcal N})$ (i.e., $\xi _n$ is bounded in $W^{1,p}({\mathcal M};T{\mathcal N})$ ). The claim on $\xi _n$ follows immediately, and the claim on $f_{\xi _n}$ follows from Lemma 2.1.

3.3 The case $p>d$

We first prove Theorem 3.4 for $p>d$ ; this restriction is relaxed further below. The analysis for $p>d$ can also be applied directly to the functional ${\mathcal E}_p:\operatorname {Imm}_p({\mathcal M};{\mathcal N})\to [0,\infty )$ ; the extension to $p\le d$ in the next part, however, cannot.

Analysis of the limiting map

Let ${\mathcal U}$ , ${\mathcal V}$ and R define trivializations of $T{\mathcal N}$ as in Section 2, and let $U\in {\mathcal U}$ ; note that $TU = T{\mathcal M}|_U$ is a vector bundle over U. Then,

$$\begin{align*}\alpha_n = R\circ (Df_{\xi_n}\oplus \xi_n) \in L^p\Gamma((TU\oplus{\mathbb R})^*\otimes{\mathbb R}^{d+1}), \end{align*}$$

which is a section of a vector bundle over U, is a trivialization of $Df_{\xi _n}\oplus \xi _n$ satisfying

$$\begin{align*}\operatorname{dist}(Df_{\xi_n}\oplus \xi_n,\operatorname{SO}(G,\mathfrak{h})) = \operatorname{dist}(\alpha_n,\operatorname{SO}(G,\mathfrak{e})). \end{align*}$$

The fact that $\tilde {{\mathcal E}}_p(\xi _n) \to 0$ implies in particular that

(3.1) $$ \begin{align} \lim_{n\to\infty} \int_U \operatorname{dist}^p(\alpha_n,\operatorname{SO}(G,\mathfrak{e}))\,\text{d}\text{Vol}_{\mathfrak{g}} = 0. \end{align} $$

It follows from (3.1) that $\alpha _n$ is an $L^p$ -bounded sequence of sections of $(TU\oplus {\mathbb R})^*\otimes {\mathbb R}^{d+1}$ . By the fundamental theorem of Young measures (see [Reference Kupferman, Maor and ShacharKMS19, Section 3.1] for a version for vector bundles), there exists an $L^p$ -Young measure $\nu \in Y^p(U,(TU\oplus {\mathbb R})^*\otimes {\mathbb R}^{d+1})$ such that

$$\begin{align*}\alpha_n \xrightarrow{Y} \nu. \end{align*}$$

That is, for every vector bundle $E\to U$ and every (generally nonlinear) bundle map $W:(TU\oplus {\mathbb R})^*\otimes {\mathbb R}^{d+1}\to E$ for which $W(\alpha _n):U\to E$ is $L^1$ -weakly precompact,

$$\begin{align*}W(\alpha_n) \rightharpoonup \left\{q \to \int_{(TU\oplus{\mathbb R})^*\otimes{\mathbb R}^{d+1}} W(A) \, d\nu_q(A)\right\} \qquad { \mathrm{in}\ L^1\Gamma(E)}. \end{align*}$$

Take $W(A)=\operatorname {dist}(A,SO(G,\mathfrak {e}))$ . Since $W(\alpha _n)$ is bounded in $L^p(U)$ for $p>1$ , it is $L^1$ -weakly precompact, from which follows that

$$ \begin{align*} 0 &= \lim_{n\to\infty} \int_U \operatorname{dist}(\alpha_n,\operatorname{SO}(G,\mathfrak{e}))\,\text{d}\text{Vol}_{\mathfrak{g}} \\\ &= \int_U \int_{(TU\oplus{\mathbb R})^*\otimes{\mathbb R}^{d+1}} \operatorname{dist}(A,\operatorname{SO}(G,\mathfrak{e}))\, d\nu(A)\, \text{d}\text{Vol}_{\mathfrak{g}} \end{align*} $$

(i.e., $\nu _q$ is supported on $\operatorname {SO}(G,\mathfrak {e})$ for a.e. $q\in U$ ).

For general oriented inner-product spaces $(V,\mathfrak {g})$ and $(W,\mathfrak {h})$ of equal dimension s, and a linear map $A\in \operatorname {Hom}(V,W)$ , the determinant and the cofactor of A, $\det A \in \operatorname {Hom}({\mathbb R};{\mathbb R}) \simeq {\mathbb R}$ and $\operatorname {cof} A \in \operatorname {Hom}(V,W)$ can be defined in a basis-independent manner by

$$\begin{align*}\det A = \star_{\mathfrak{h}} \circ (\wedge^s A)\circ \star_{\mathfrak{g}} \qquad\text{ and }\qquad \operatorname{cof} A = (-1)^{s+1} \star_{\mathfrak{h}} \circ (\wedge^{s-1} A)\circ \star_{\mathfrak{g}}, \end{align*}$$

where $\star _{\mathfrak {g}}:\Lambda ^k V\to \Lambda ^{s-k}V$ and $\star _{\mathfrak {h}}:\Lambda ^k W\to \Lambda ^{s-k}W$ are the hodge-dual isomorphisms of the respective spaces, and $\wedge ^k A$ is the k-minor of A, defined by

$$\begin{align*}\wedge^k A (v_1,\dots,v_k) = A(v_1) \wedge \dots\wedge A(v_k). \end{align*}$$

These definitions can be equivalently obtained by choosing positive orthonormal bases for V and W and evaluating the determinant and cofactor of their matrix representations in these bases. It is well known that

$$\begin{align*}A\in \operatorname{SO}(\mathfrak{g},\mathfrak{h}) \qquad\text{if and only if}\qquad \operatorname{cof} A = A \quad\text{ and }\quad \det A = 1. \end{align*}$$

With that, consider next the test functions $W(A) = A$ , $W(A) = \operatorname {cof} A$ and $W(A)=\det A$ . By definition, they are compositions of minors of A of ranks 1, d and $(d+1)$ , respectively, and hodge-dual isometries. Since $\mathfrak {h}$ and $\mathfrak {h}^{-1}$ are bounded uniformly (in coordinates) and $Df_{\xi _n}$ is bounded in $L^p$ , both the cofactor and the determinant of $\alpha _n$ are uniformly bounded in $L^{p/d}$ , and therefore, $W(\alpha _n)$ are $L^1$ -weakly precompact for all above choices of W. It follows from the definition of the Young measure limit that

$$ \begin{align*} \alpha_n &\rightharpoonup \left\{q \to \int_{(TU\oplus{\mathbb R})^*\otimes{\mathbb R}^{d+1}} A \, d\nu_q(A)\right\} &{ \mathrm{in}\ L^1\Gamma((TU\oplus{\mathbb R})^*\otimes{\mathbb R}^{d+1})} \\ \operatorname{cof}(\alpha_n) &\rightharpoonup \left\{q \to \int_{(TU\oplus{\mathbb R})^*\otimes{\mathbb R}^{d+1}} \operatorname{cof} A \, d\nu_q(A)\right\} & { \mathrm{in}\ L^1\Gamma((TU\oplus{\mathbb R})^*\otimes{\mathbb R}^{d+1})} \\ \det(\alpha_n) &\rightharpoonup \left\{q \to \int_{(TU\oplus{\mathbb R})^*\otimes{\mathbb R}^{d+1}} \det A \, d\nu_q(A)\right\} & { \mathrm{in}\ L^1(U)}. \end{align*} $$

Since $\nu _q$ is supported on $\operatorname {SO}(G,\mathfrak {e})$ for a.e. $q\in U$ , and on that set $\operatorname {cof} A=A$ and $\det A=1$ , it follows that

(3.2) $$ \begin{align} \alpha_n &\rightharpoonup \left\{q \to \int_{\operatorname{SO}(G,\mathfrak{e})} A \, d\nu_q(A)\right\} & { \mathrm{in}\ L^1\Gamma((TU\oplus{\mathbb R})^*\otimes{\mathbb R}^{d+1})} \nonumber \\ \operatorname{cof}(\alpha_n) &\rightharpoonup \left\{q \to \int_{\operatorname{SO}(G,\mathfrak{e})} A \, d\nu_q(A)\right\} & { \mathrm{in}\ L^1\Gamma((TU\oplus{\mathbb R})^*\otimes{\mathbb R}^{d+1})} \\ \det(\alpha_n)& \rightharpoonup 1 & { \mathrm{in}\ L^1(U)}. \nonumber \end{align} $$

Recall that $\xi _n \rightharpoonup \xi $ (hence also $f_{\xi _n} \rightharpoonup f_\xi $ ) in $W^{1,p}$ . It follows from the second clause of Proposition 2.4 that

(3.3) $$ \begin{align} \alpha_n \rightharpoonup \alpha = R\circ(Df_\xi\oplus\xi) \qquad{\mathrm{in}\ L^p\Gamma((TU\oplus{\mathbb R})^*\otimes{\mathbb R}^{d+1})}. \end{align} $$

Lemma 3.6. The weak $L^p$ -convergence $\alpha _n\rightharpoonup \alpha $ implies that

(3.4) $$ \begin{align} \nonumber \operatorname{cof}(\alpha_n) &\rightharpoonup \operatorname{cof}(\alpha) \qquad { \mathrm{in}\ L^{p/d}\Gamma((TU\oplus{\mathbb R})^*\otimes{\mathbb R}^{d+1})} \\ \det(\alpha_n) &\rightharpoonup \det(\alpha) \qquad { \mathrm{in}\ L^{p/d}(U)}. \end{align} $$

Proof. Since the hodge-dual operator is a linear isometry and $\xi _n\to \xi $ uniformly, it suffices to show that

$$\begin{align*}\wedge^k \alpha_n \rightharpoonup \wedge^k \alpha \qquad { \mathrm{in}\ L^{p/d}\Gamma(\Lambda^k(TU\oplus{\mathbb R})^*\otimes \Lambda^k {\mathbb R}^{d+1})} \end{align*}$$

for $k=d,d+1$ . We show more generally that for every $k=1,\dots ,d+1$ ,

$$\begin{align*}\wedge^k \alpha_n \rightharpoonup \wedge^k \alpha \qquad {\mathrm{in}\ L^{p/\min(k,d)}\Gamma(\Lambda^k(TU\oplus{\mathbb R})^*\otimes\Lambda^k {\mathbb R}^{d+1})}. \end{align*}$$

We start by noting that for $v_1,\ldots ,v_k \in T_qU$ and $t_1,\dots ,t_k\in {\mathbb R}$ ,

$$ \begin{align*} \wedge^k\alpha_n((v_1,t_1),\dots,(v_k,t_k)) &= \wedge_{j=1}^k (R_{f_{\xi_n}(q)}\circ D_qf_{\xi_n}(v_j) + t_j R_{f_{\xi_n}(q)}\circ\xi_n(q)) \\ &= \wedge_{j=1}^k R_{f_{\xi_n}(q)}\circ D_qf_{\xi_n}(v_j) \\ &\qquad + \sum_{\ell=1}^k (-1)^\ell t_\ell \wedge_{j\ne \ell} \left(R_{f_{\xi_n}(q)}\circ D_qf_{\xi_n}(v_j)\right)\wedge (R_{f_{\xi_n}(q)}\circ \xi_n(q)). \end{align*} $$

That is,

$$\begin{align*}\wedge^k\alpha_n = \wedge^k (R\circ Df_{\xi_n}\oplus 0) + \wedge^{k-1} (R\circ Df_{\xi_n}\oplus 0) \wedge (0\oplus R\circ \xi_n). \end{align*}$$

Since $\xi _n\to \xi $ uniformly, and since the product of a weakly $L^p$ -convergent sequence and a uniformly convergence sequence is weakly $L^p$ -convergent, it suffices to show that

$$\begin{align*}\wedge^k (R\circ Df_n) \rightharpoonup \wedge^k (R\circ Df) \qquad {\mathrm{in}\ L^{p/\min(k,d)}\Omega^k(U;\Lambda^k {\mathbb R}^{d+1})}. \end{align*}$$

Since ${\mathcal M}$ and ${\mathcal N}$ are compact, the orthonormal frame R can be replaced by any other smooth frame – for example, $R = dh$ , where $h:V\to {\mathbb R}^{d+1}$ is a local coordinate system. The claim reduces then to the standard weak-continuity of minors, proved inductively on k (see, for example, [Reference RindlerRin18, Lemma 5.10]).

Since $\operatorname {cof}(AB) = \operatorname {cof}(A)\,\operatorname {cof}(B)$ and $\det (AB) = \det (A)\,\det (B)$ , and since R is an orientation-preserving isometry,

$$ \begin{align*} \operatorname{cof}(R\circ A) &= R\circ \operatorname{cof}(A)\\ \det(R\circ A) &= \det(A), \end{align*} $$

we obtain by combining (3.4) with (3.2) that

$$\begin{align*}\operatorname{cof}(Df_\xi\oplus \xi) = Df_\xi\oplus \xi \qquad\text{ and }\qquad \det(Df_\xi\oplus \xi) = 1 \qquad\text{a.e.,} \end{align*}$$

which implies that

$$\begin{align*}Df_\xi\oplus \xi \in \operatorname{SO}(G,\mathfrak{h}) \qquad\text{a.e.} \end{align*}$$

Thus, $\tilde {{\mathcal E}}_p^S(\xi ) = 0$ . By Proposition 3.2, $f_\xi \in \operatorname {Imm}_p({\mathcal M};{\mathcal N})$ with $Df_\xi \in \operatorname {O}(\mathfrak {g},\mathfrak {h})$ , and $\xi = \mathfrak {n}_{f_\xi }$ .

Thus, we have obtained at this stage that $\xi _n \rightharpoonup \xi $ in $W^{1,p}$ where, $\xi = \mathfrak {n}_{f_\xi }$ . In order to complete the proof for the case $p>d$ , we need to show that the convergence is in fact strong and that the shape operator of $\mathfrak {n}_{f_\xi }$ is S, and deduce from Theorem 1.2 that the limit is smooth.

Strong convergence and second fundamental form of the limit

From the uniqueness of the limit, combining (3.2) and (3.3),

$$\begin{align*}\int_{\operatorname{SO}(G_x,\mathfrak{e})} A \, d\nu_x(A) = \alpha_x \qquad \text{for a.e. } x\in U. \end{align*}$$

The left-hand side is a convex combination of elements in $\operatorname {SO}(G_x,\mathfrak {e})$ , and the right-hand side is in $\operatorname {SO}(G_x,\mathfrak {e})$ ( $\alpha $ is the composition of an element of $\operatorname {SO}(G,\mathfrak {h})$ and an element of $\operatorname {SO}(\mathfrak {h},\mathfrak {e})$ ). Since $\operatorname {SO}(G_x,\mathfrak {e})$ is strictly convex, this convex combination must be trivial; namely,

$$\begin{align*}\nu_x = \delta_{\alpha_x} \qquad \text{for a.e. } x\in U. \end{align*}$$

As a consequence, for every $W:(TU\otimes {\mathbb R})^*\otimes {\mathbb R}^{d+1}\to {\mathbb R}$ for which $W(\alpha _n)$ is $L^1$ -weakly precompact,

(3.5) $$ \begin{align} \lim_{n\to\infty} \int_U W(\alpha_n)\,\text{d}\text{Vol}_{\mathfrak{g}} = \int_U W(\alpha)\,\text{d}\text{Vol}_{\mathfrak{g}}. \end{align} $$

We now show that $f_{\xi _n}\to f_\xi $ strongly in $W^{1,p}({\mathcal M};{\mathcal N})$ (for the moment, we have only established weak convergence, and hence in particular strong $L^p$ convergence). By the first clause of Proposition 2.4, it suffices to prove that $\alpha _n\to \alpha $ in $L^p\Gamma ((TU\otimes {\mathbb R})^*\otimes {\mathbb R}^{d+1})$ . We would be done if we could use (3.5) for

$$\begin{align*}W(A) = |A - \alpha|^p. \end{align*}$$

However, $W(\alpha _n)$ is only bounded in $L^1(U)$ , which does not guarantee sequential weak precompactness. Instead, let

$$\begin{align*}W(A)=|A- \alpha|^p \,\varphi\left(\frac{|A|}{3\sqrt{d+1}}\right), \end{align*}$$

where $\varphi :[0,\infty )\to {\mathbb R}$ is continuous, nonnegative, compactly supported, and satisfies $\varphi (t)=1$ for $t\le 1$ and $\varphi (t)<1$ for $t>1$ . Clearly, $W(\alpha _n)$ is uniformly bounded, and hence $L^1$ -weakly precompact, which since $W(\alpha )=0$ implies that

$$\begin{align*}\lim_{n\to\infty} \int_U W(\alpha_n) \,\text{d}\text{Vol}_{\mathfrak{g}} = 0. \end{align*}$$

On the set

$$\begin{align*}B_n = \{x\in U ~:~ |\alpha_n|<3\sqrt{d+1}\}, \end{align*}$$

we have $W(\alpha _n) = |\alpha _n - \alpha |^p$ , whereas on its complement $U\setminus B_n$ ,

$$\begin{align*}|\alpha_n - \alpha| \le 2(|\alpha_n| - |\alpha|) \le 2\,\operatorname{dist}(\alpha_n,\operatorname{SO}(G,\mathfrak{e})), \end{align*}$$

where the first inequalities follow from the fact that $|\alpha | = \sqrt {d+1}$ and $|\alpha _n|\ge 3\sqrt {d+1}$ , and the second inequality follows from the reverse triangle inequality. Thus, in $U\setminus B_n$ ,

$$\begin{align*}|\alpha_n - \alpha|^p \le 2^p\,\operatorname{dist}^p(\alpha_n,\operatorname{SO}(G,\mathfrak{e})). \end{align*}$$

Combining the two sets,

$$ \begin{align*} \int_U |\alpha_n- \alpha|^p\,\text{d}\text{Vol}_{\mathfrak{g}} &= \int_{B_n} |\alpha_n- \alpha|^p\,\text{d}\text{Vol}_{\mathfrak{g}} + \int_{U\setminus B_n} |\alpha_n- \alpha|^p\,\text{d}\text{Vol}_{\mathfrak{g}} \\ &\le \int_U W(\alpha_n)\,\text{d}\text{Vol}_{\mathfrak{g}} + 2^p \int_{U} \operatorname{dist}^p(\alpha_n,\operatorname{SO}(G,\mathfrak{e}))\,\text{d}\text{Vol}_{\mathfrak{g}}, \end{align*} $$

and both terms tend to zero as $n\to \infty $ . We conclude that $Df_{\xi _n} \to Df_{\xi }$ in $L^p$ and therefore obtain that $f_{\xi _n} \to f_{\xi }$ in $W^{1,p}({\mathcal M};{\mathcal N})$ . It remains to show that S is the shape operator of $f_\xi $ (i.e., that $K^h\circ D\xi =- Df_{\xi } \circ S$ ) and that $\xi _n\to \xi $ strongly in $W^{1,p}({\mathcal M};T{\mathcal N})$ . We observe that

$$ \begin{align*} d\jmath\circ K^{\mathfrak{h}}\circ D\xi_n &= d\jmath\circ(K^{\mathfrak{h}}\circ D\xi_n + Df_{\xi_n}\circ S) \\ &\quad + (d\jmath\circ Df_\xi - d\jmath\circ Df_{\xi_n})\circ S \\ &\quad - d\jmath\circ Df_\xi \circ S \end{align*} $$

(the composition with $d\jmath $ is necessary in order to be able to add and subtract the various terms). The first term on the right-hand side tends to zero in $L^p$ since the bending energy $\tilde {{\mathcal E}}_p^B(\xi _n)$ tends to zero; the second term tends to zero in $L^p$ since $f_{\xi _n}\to f_\xi $ strongly in $W^{1,p}$ . Thus, the left-hand side converges strongly to $- d\jmath \circ Df \circ S$ .

It follows from Lemma 2.2 that $K^{\mathfrak {h}}\circ D\xi _n \to K^{\mathfrak {h}}\circ D\xi $ in $L^p$ and that $K^h\circ D\xi =- Df \circ S$ , thus proving that S is the shape operator of $f_\xi $ . Since $\xi _n\rightharpoonup \xi $ in $W^{1,p}$ , $f_{\xi _n} \to f_{\xi }$ in $W^{1,p}$ and $K^{\mathfrak {h}}\circ D\xi _n \to K^{\mathfrak {h}}\circ D\xi $ in $L^p$ , we obtain from Proposition 2.3 that $\xi _n\to \xi $ strongly, as needed.

Lastly, $f_\xi $ satisfies the assumptions of Theorem 1.2 (which is proved in Section 4 below), and thus $f_\xi $ is smooth, and by extension so is $\xi =\mathfrak {n}_{f_\xi }$ .

3.4 The case $p\le d$ via Sobolev truncation

If $p \le d$ , we first need to regularize $\xi _n \in W^{1,p}({\mathcal M};T{\mathcal N})$ by replacing them with $\tilde {\xi }_n \in W^{1,\infty }({\mathcal M};T{\mathcal N})$ , which are close to $\xi _n$ in $W^{1,p}$ and also uniformly Lipschitz. This is where the relaxation of the energy comes into play, since even if $\xi _n = \mathfrak {n}_{f_{\xi _n}}$ project onto immersions $f_{\xi _n}\in \operatorname {Imm}_p({\mathcal M};{\mathcal N})$ , we do not know how to approximate them with more regular functions within the space $\operatorname {Imm}_p({\mathcal M};{\mathcal N})$ . This construction follows a similar path as in other manifold-valued elasticity results – namely, [Reference Kupferman, Maor and ShacharKMS19, Reference Krömer and MüllerKM21] – with some technical complications due to the structure of the energy in codimension-1 and the non-compactness of $T{\mathcal N}$ .

First, we need the following pointwise estimate:

Lemma 3.7. Let $\xi \in W^{1,p}({\mathcal M};T{\mathcal N})$ . Denote $M = \|S\|_\infty $ , where S is the shape operator in ${\mathcal E}_p$ . If at a point in ${\mathcal M}$ , $|D\xi | \ge (3+2M)\sqrt {d+1}$ , then at that point,

(3.6) $$ \begin{align} |D\xi| \le (3+2M)\left(\operatorname{dist}(Df_\xi\oplus \xi,\operatorname{SO}(G,\mathfrak{h})) + |Df_\xi\circ S + K^{\mathfrak{h}}\circ D\xi|\right). \end{align} $$

Proof. By the definition of the Sasaki metric,

$$\begin{align*}|D\xi| \le |Df_\xi| + |K^{\mathfrak{h}}\circ D\xi|. \end{align*}$$

Consider first the case where $|Df_\xi | \ge 2\sqrt {d+1}$ . Then, by the triangle inequality and the fact that all the elements in $\operatorname {SO}(G,\mathfrak {h})$ are of norm $\sqrt {d+1}$ ,

(3.7) $$ \begin{align} \operatorname{dist}(Df_\xi\oplus \xi,\operatorname{SO}(G,\mathfrak{h})) \ge |Df_\xi| - \sqrt{d+1} \ge \sqrt{d+1}. \end{align} $$

Thus,

(3.8) $$ \begin{align} \nonumber |D\xi| &\le |Df_\xi| + |K^{\mathfrak{h}}\circ D\xi| \\ \nonumber &\le |Df_\xi| + |Df_\xi\circ S| + |Df_\xi\circ S + K^{\mathfrak{h}}\circ D\xi| \\ &\le (1+M)|Df_\xi| + |Df_\xi\circ S + K^{\mathfrak{h}}\circ D\xi| \\ \nonumber &\le (1+M)\operatorname{dist}(Df_\xi\oplus \xi,\operatorname{SO}(G,\mathfrak{h})) + (1+M)\sqrt{d+1} + |Df_\xi\circ S + K^{\mathfrak{h}}\circ D\xi| \\ \nonumber &\le 2(1+M)\operatorname{dist}(Df_\xi\oplus \xi,\operatorname{SO}(G,\mathfrak{h})) + |Df_\xi\circ S + K^{\mathfrak{h}}\circ D\xi|, \end{align} $$

which implies (3.6).

Otherwise, if $|Df_\xi | < 2\sqrt {d+1}$ and $|D\xi | \ge (3+2M)\sqrt {d+1}$ , then

$$ \begin{align*} |D\xi| &\le |Df_\xi| + |K^{\mathfrak{h}}\circ D\xi| \\ &\le |Df_\xi| + |Df_\xi\circ S| + |Df_\xi\circ S + K^{\mathfrak{h}}\circ D\xi| \\ &\le (1+M)|Df_\xi| + |Df_\xi\circ S + K^{\mathfrak{h}}\circ D\xi| \\ &\le 2(1+M)\sqrt{d+1} + |Df_\xi\circ S + K^{\mathfrak{h}}\circ D\xi| \\ &\le \frac{2(1+M)}{3 + 2M}|D\xi| + |Df_\xi\circ S + K^{\mathfrak{h}}\circ D\xi|, \end{align*} $$

from which follows that

$$\begin{align*}|D\xi| \le (3 + 2M) |Df_\xi\circ S + K^{\mathfrak{h}}\circ D\xi|, \end{align*}$$

which implies (3.6).

For a given $\rho>2\sqrt {d+1}$ , denote $T^\rho {\mathcal N} := \{v\in T{\mathcal N} ~:~ |v|\le \rho \}$ , which is a compact submanifold of $T{\mathcal N}$ . Denote by $\varphi :[0,\infty )\to [0,1]$ a continuous, nonnegative, compactly supported function satisfying $\varphi (t)=1$ for $t\le 2\sqrt {d+1}$ and $\varphi (t)=0$ for $t\ge \rho $ , and let $\hat {\xi }_n = \varphi (|\xi _n|)\xi _n$ . Note that $\hat {\xi }_n: {\mathcal M} \to T^\rho {\mathcal N}$ is uniformly bounded, and $|D\hat {\xi }_n| \le C|D\xi _n|$ for some C depending only on $\varphi $ . Furthermore,

$$ \begin{align*} \{x\in {\mathcal M} ~:~ \hat{\xi}_n(x) \ne \xi_n(x)\} &\subset \{|\xi_n|>2\sqrt{d+1}\} \\ &\subset \{\operatorname{dist}(Df_{\xi_n}\oplus \xi_n,\operatorname{SO}(G,\mathfrak{h})) > \sqrt{d+1}\}, \end{align*} $$

where the last inclusion follows from the same argument as in (3.7). Since $\tilde {{\mathcal E}}_p(\xi _n)\to 0$ , it follows that

$$\begin{align*}\operatorname{dist}(Df_{\xi_n}\oplus \xi_n,\operatorname{SO}(G,\mathfrak{h})) \to 0 \qquad\text{in measure}. \end{align*}$$

Hence, the volumes of all the sets above tend to zero. On the one hand,

$$ \begin{align*} \int_{\mathcal M} |\xi_n - \hat{\xi}_n|^p\,\text{d}\text{Vol}_{\mathfrak{g}} &= \int_{\{\xi_n \ne \hat{\xi}_n\}} |\xi_n - \hat{\xi}_n|^p\,\text{d}\text{Vol}_{\mathfrak{g}} \\ &\lesssim \int_{\{\xi_n \ne \hat{\xi}_n\}} |\xi_n|^p\,\text{d}\text{Vol}_{\mathfrak{g}} + \int_{\{\xi_n \ne \hat{\xi}_n\}} |\hat{\xi}_n|^p\,\text{d}\text{Vol}_{\mathfrak{g}}. \end{align*} $$

The first term tends to zero as $n\to \infty $ because $|\xi _n|^p$ is equi-integrable (it is bounded in $W^{1,p}$ ); the second term tends to zero because $|\hat {\xi }|$ is uniformly bounded. Moreover,

$$\begin{align*}\int_{\mathcal M} |d\iota \circ D\hat{\xi}_n - d\iota \circ D\xi_n|^p\, \text{d}\text{Vol}_{\mathfrak{g}} \lesssim \int_{\{\operatorname{dist}(Df_{\xi_n}\oplus \xi_n,\operatorname{SO}(G,\mathfrak{h}))> \sqrt{d+1}\}} |D\xi_n|^p\, \text{d}\text{Vol}_{\mathfrak{g}} \lesssim \tilde{{\mathcal E}}_p(\xi_n) \to 0, \end{align*}$$

where the last inequality follows from (3.8). Thus,

$$\begin{align*}\|\iota \circ \hat{\xi}_n - \iota \circ \xi_n\|_{W^{1,p}} \to 0. \end{align*}$$

Denote $u_n = \iota \circ \hat {\xi }_n \in W^{1,p}({\mathcal M};{\mathbb R}^D)$ . By [Reference Friesecke, James and MüllerFJM02, Proposition A.1], there exists a constant $C>0$ , depending on ${\mathcal M},\mathfrak {g},p$ and $\|S\|_\infty $ , and there exists a sequence of Lipschitz maps $\bar {u}_n \in W^{1,\infty }({\mathcal M};{\mathbb R}^D)$ having uniform Lipschitz constant C such that

$$\begin{align*}\text{Vol}_{\mathfrak{g}}\left(\{x\in {\mathcal M} ~:~ \bar{u}(x) \ne u(x)\}\right) \le C\int_{\{|Du_n|>(3+2M)\sqrt{d+1}\}} |Du_n|^p\, \text{d}\text{Vol}_{\mathfrak{g}}, \end{align*}$$

and

$$\begin{align*}\|\bar{u}_n - u_n\|_{W^{1,p}}^p \le C\int_{\{|Du_n|>(3+2M)\sqrt{d+1}\}} |Du_n|^p \text{d}\text{Vol}_{\mathfrak{g}}. \end{align*}$$

Using Lemma 3.7, and the fact that $|Du_n| = |D\hat {\xi }_n|$ (since $\iota $ is an isometric immersion), we obtain that

(3.9) $$ \begin{align} \text{Vol}_{\mathfrak{g}}\left(\{x\in {\mathcal M} ~:~ \bar{u}(x) \ne u(x)\}\right) \le C'\tilde{{\mathcal E}}_p(\hat{\xi}_n), \end{align} $$

and

(3.10) $$ \begin{align} \|\bar{u}_n - u_n\|_{W^{1,p}}^p \le C'\tilde{{\mathcal E}}_p(\hat{\xi}_n), \end{align} $$

for some $C'>0$ . Note that the image of $\bar {u}_n$ is not in $\iota (T{\mathcal N})$ , which means that we cannot identify $\bar {u}_n:{\mathcal M}\to {\mathbb R}^D$ with a function $\bar {\xi }_n:{\mathcal M}\to T{\mathcal N}$ ; to rectify this problem, we resort to a projection. Since $\tilde {{\mathcal E}}_p(\hat {\xi }_n)\to 0$ , it follows from (3.9) that for every $\varepsilon>0$ , and $n\in \mathbb {N}$ large enough (depending on $\varepsilon $ ), every ball of radius $\varepsilon $ in ${\mathcal M}$ contains a point x for which $\bar {u}_n(x) = u_n(x)\in \iota (T^\rho {\mathcal N})$ . Since $\bar {u}_n$ are uniformly Lipschitz, we obtain that

$$\begin{align*}\max_{x\in {\mathcal M}} \operatorname{dist}\left(\bar{u}_n(x), \iota (T^\rho {\mathcal N})\right) \to 0. \end{align*}$$

Thus, for n large enough, $\bar {u}_n(x)$ lies in a tubular neighborhood of $\iota (T{\mathcal N})$ on which the orthogonal projection operator P is well defined and smooth (this is where we use the compactness of $T^\rho {\mathcal N}$ and the reason we had to truncate $\xi _n$ to $\hat {\xi }_n$ – we had to ensure that the image of $\bar {u}_n$ is uniformly close to a compact subset of $\iota (T{\mathcal N})$ ). Define $\tilde {u}_n = P\circ \bar {u}_n$ , and $\tilde {\xi }_n = \iota ^{-1}\circ \tilde {u}_n$ . A direct calculation, as in [Reference Kupferman, Maor and ShacharKMS19, pp. 391–2], shows that $\tilde {u}_n$ and $\tilde {\xi }_n$ are uniformly Lipschitz and satisfy (3.9) and (3.10), and furthermore, that $\tilde {{\mathcal E}}_q(\tilde {\xi }_n)\to 0$ for any $q\in (1,\infty )$ .

We can now apply the previous step for $\tilde {\xi }_n$ , for some $q>d\ge p$ , and obtain that $\tilde {\xi }_n\to \xi =\mathfrak {n}_f$ in $W^{1,q}({\mathcal M};T{\mathcal N})$ for some isometric immersion $f\in \operatorname {Imm}_\infty ({\mathcal M};{\mathcal N})$ whose shape operator is S. Since $\|\tilde {u}_n - u_n\|_{W^{1,p}}\to 0$ , it follows by definition that $\operatorname {dist}_{W^{1,p}}(\tilde {\xi }_n,\hat {\xi }_n) \to 0$ , and thus also $\operatorname {dist}_{W^{1,p}}(\tilde {\xi }_n,\xi _n) \to~0$ . It therefore follows that $\xi _n \to \mathfrak {n}_f$ in $W^{1,p}({\mathcal M};T{\mathcal N})$ .