Abstract
For smooth mappings of the unit disc into the oriented Grassmannian manifold \({\mathbb {G}}_{n,2}\), Hélein (Harmonic Maps Conservation Laws and Moving Frames, Cambridge University Press, Cambridge, 2002) conjectured the global existence of Coulomb frames with bounded conformal factor provided the integral of \(|{{\varvec{A}}} |^2\), the squared-length of the second fundamental form, is less than \(\gamma _n=8\pi \). It has since been shown that the optimal bounds that guarantee this result are: \(\gamma _3 = 8\pi \) and \(\gamma _n = 4\pi \) for \(n \ge 4\). For isothermal immersions in \({\mathbb {R}}^3\), this hypothesis is equivalent to saying the integral of the sum of the squares of the principal curvatures is less than \(\gamma _3\). The goal here is to prove that when \(n=3\) the same conclusion holds under weaker hypotheses. In particular, it holds for isothermal immersions when \(|{{\varvec{A}}} |^2\) is integrable and the integral of \(|K |\), where K is the Gauss curvature, is less than \(4\pi \). Since \(2|K |\le |{{\varvec{A}}} |^2\) this implies the known result for isothermal immersions, but \(|K |\) may be small when \(|{{\varvec{A}}} |^2\) is large. The method, which is purely analytic, is then developed to examine the case \(n=3\) when \(|\varvec{A} |\) is only square-integrable. The possibility of extending that result in the language of Grassmannian manifolds to the case \(n>3\) is outlined in an Appendix.
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1 Introduction
In applications, it is often important to understand the global behaviour of an immersed surface, or even a curve, in \({\mathbb {R}}^n\). Although the theory of curves is non-trivial, much has been done since the work of Schur [20] (see also Chern [2, Sects. 4, 5]) and recently the focus has been on surfaces. For example, a question in conformal geometry is how to decide whether a surface in \({\mathbb {R}}^n\) has bi-Lipschitz isothermal coordinates, in other words is there a locally Lipschitz map with a locally Lipschitz inverse which maps locally onto a flat surface? A related problem is to understand how the existence of Coulomb moving frames, generated by mappings of a disc into the Grassmanian manifold \({\mathbb {G}}_{n,2}\), depends on the gradients of the mappings.
In this context, Hélein conjectured, see (1.12), (1.13) and the text in between, that when \(n=3\) a Coulomb frame generated by mapping the unit disc into the unit sphere \({\mathbb {S}}^2\) exists if the Dirichlet integral of the mapping is strictly less than \(8\pi \). This was proved by Schätzle [18, Proposition 5.1] and that \(8\pi \) is optimal was shown by Kuwert & Li [10, last line p. 331]. For an isothermal immersed surface in \({\mathbb {R}}^3\) this is equivalent to saying that the logarithm of the conformal factor is in \(L^\infty \cap W^{1,2}\) when the integral of the sum of squares of principal curvatures \(|\varvec{A} |^2=\kappa _1^2 + \kappa _2^2\) is strictly less than \(8\pi \).
In Sect. 3, it is shown that in \({\mathbb {R}}^3\) the same conclusion holds when the integral of \(|K|\), where \(K = \kappa _1\kappa _2\) is the Gauss curvature, is less than \(4\pi \). Since \(2|K|\le |\varvec{A}|^2\), and \(|K|\) can be small when \(|\varvec{A}|^2\) large, this result is different from, and stronger than, the Hélein conjecture when \(n=3\).
In Sect. 4, the method of Sect. 3 is extended to investigate the case when \(n=3\) but \(|\varvec{A} |^2\) is merely integrable. Finally, for \(n>3\) an analogue of the result of Sect. 4 is sketched in terms of Grassmannian manifolds in an Appendix.
Results like these for surfaces in \({\mathbb {R}}^3\) are important because of applications that lead to variational problems in which the unknown variables describe surfaces. For example, in biological membrane theory [6, 12] the Helfrich functional involves the Willmore energy (1.11) and questions arise about the regularity of critical points with square-integrable second fundamental forms; in hydroelasticity [15, 16] the profile of steady waves on fluid bounded above by a frictionless elastic sheet is governed by critical points of a Lagrangian involving the square of the surface mean curvature; in general relativity [8], when the universe is modelled as \({\mathbb {R}}^3\) with a metric tensor, the quasi-local Hawking mass energy \(m(\Sigma )\) for a domain \(\Omega \subset {\mathbb {R}}^3\) which is bounded by a closed surface \(\Sigma \) involves the surface integral of \(H^2\), where H is the mean curvature.
To be accessible to mathematicians who need results for applications but are not professional geometers, the paper deals mainly with the case \(n=3\) using analysis and partial differential equation techniques. Complex projective spaces and Grassmannian manifolds arise only briefly, first when citing the literature in Sect. 1.2 and later, in Appendix A which discusses the case \(n>3\). The rest of Sect. 1 is a survey of work on Hélein’s conjecture [5] in the case \(n=3\) which is relevant to the present contribution. The layout of the rest of the paper is set out in Sect. 1.5.
1.1 Isothermal immersions
Let \(D_1\) denote the closed unit disc centred at the origin in \({\mathbb {R}}^2\). A smooth mapping \(\Psi : D_1\rightarrow {\mathbb {R}}^3\) is called an isothermal immersion and \(e^f\) is its conformal factor if, with \( \partial _i={\partial }/{\partial X_i}\),
Then, the coefficients of the first fundamental form [7, Sect. 2.2], [17, Sect. 6.1.1] of the surface \(\Psi (D_1)\) are \(E=G=e^{2f}\) and \(F = 0\). Since \(\partial _{12}\Psi = \partial _{21}\Psi \) and \(\partial _k( {{\textbf{e}}}_i\cdot {\textbf{e}}_j)=0\), \(i,j,k \in \{1,2\}\), it follows that
and hence
Let \({{\textbf{n}}}(X) = \textbf{e}_1(X)\times \textbf{e}_2(X)\). Then, since \(\partial _1\Psi \), \(\partial _2\Psi \) are normal to \({{\textbf{n}}}\), the coefficients of the second fundamental form [7, Sect. 2.2], [17, Sect. 7.1] of \(\Psi (D_1)\) are
and, by (1.1), its Gauss curvature K [7, Sect. 2.2], [17, Cor. 8.1.3] is
Now since \(\textbf{e}_i\cdot {{\textbf{n}}}= 0\), \(\textbf{e}_i\cdot \textbf{e}_j =\delta _{ij}\) and \(\Vert \textbf{e}_j\Vert ^2=1\) on \(D_1\), \(i,j= 1,2\),
Therefore,
Note also that \(\Vert {{\textbf{n}}}\Vert ^2 = 1\) implies that \({{\textbf{n}}}\cdot \partial _i{{\textbf{n}}}=0\), and hence that \(\partial _i {{\textbf{n}}}=-( {{\textbf{n}}}\cdot \partial _i{{\textbf{e}}}_1){{\textbf{e}}}_1 -({{\textbf{n}}}\cdot \partial _i{{\textbf{e}}}_2){{\textbf{e}}}_2.\) Therefore,
and it follows that
Thus,
where K is the Gauss curvature of the surface \(\Psi (D_1)\). To estimate the \(L^2\)-norm of \(|\nabla f |\) it suffices to show that the right-hand side of (1.6) is in \( W^{-1,2}\).
Now the isothermal immersion \(\Psi \) has an associated Gauss map, namely the unit vector \({{\textbf{n}}}(X)= \textbf{e}_1(X)\times \textbf{e}_2(X),\,X \in D_1\), which is normal to the immersed surface. Since \(\Vert {{\textbf{n}}}\Vert ^2 =1\) on \(D_1\), it follows from (1.3) that
and hence
where
and H, the mean curvature of \(\Psi (D_1)\) [7, Sect. 2.2], [17, Corollary 8.1.3], is
Since \(2H = \kappa _1 + \kappa _2\) and \(K = \kappa _1\kappa _2\), where \(\kappa _1,\kappa _2\) are the principal curvatures of the surface \(\Psi (D_1)\), it follows that \(|\varvec{A} |^2 = \kappa _1^2 +\kappa _2^2\). Here \(|\varvec{A} |^2\), which is referred to as the squared-length of the second fundamental form of the surface, is independent of the parametrization \(\Psi \) of \(\Psi (D_1)\). (For a general surface, \(|\varvec{A} |^2 \) is the square of the Hilbert–Schmidt norm [3, Ch. XI] of its Weingarten map [17, Sects. 7.2 and 8.1] on the tangent space.) Moreover,
This shows for isothermal imbedding that
where \(\mu _g = e^{2f} dX\), and
where \(\kappa _1,\kappa _2\) are the principal curvatures of \(\Psi (D_1)\). In general,
and equality holds for zero-mean curvature (minimal) surfaces. For completeness, note that in this notation the Willmore energy of a surface is given by
1.2 Hélein’s conjecture
Motivated by Toro’s work [22, 23], Müller and Sv̆erák [13] investigated properties of immersions of the plane \({\mathbb {R}}^2\) into Euclidean space \({\mathbb {R}}^n\) when the second fundamental form is square-integrable. To do so, they reformulated the problem in terms of the oriented Grassmannian manifold \(\mathbb G_{n,2}\) of two-dimensional oriented subspaces of \({\mathbb {R}}^n\) embedded in complex projective space \(\mathbb C\mathbb P^{n-1}\), and used compensated-compactness methods from the theory of partial differential equations.
In his monograph, Hélein proved a result [5, Lemma 5.1.4] on mappings from the unit disc \(D_1\) in \({\mathbb {R}}^2\) into the Grassmannian manifold \({\mathbb {G}}_{n,2}\), which has been widely used when analysing variational problems in the theory of surfaces with bounded Willmore energy. In doing so, he did not assume that the mapping from the disc to \({\mathbb {G}}_{n,2}\) corresponds to assigning oriented tangent spaces to a surface.
Theorem
(Hélein [5, Lemma 5.1.4]) For a mapping \({{\textbf{n}}}: D_1 \rightarrow {\mathbb {G}}_{n,2}\) with
there exists \( ({{\textbf{e}}}_1(X), {{\textbf{e}}}_2(X))\in {{\textbf{n}}}(X),~X \in D_1\), such that
and the frame \({{\textbf{e}}}_i\) forms a so-called Coulomb frame.
He further conjectured [5, Conjecture 5.2.3] that the same conclusion should holds when \(8\pi /3\) in (1.12) is replaced by \(8\pi \). For \(n=3\), \({\mathbb {G}}_{3,2}\) can be identified with the sphere \({\mathbb {S}}^2\) of unit vectors in \({\mathbb {R}}^3\), and \({{\textbf{n}}}\) in (1.12) can be thought of as a mapping \({{\textbf{n}}}:D_1 \rightarrow {\mathbb {S}}^2\). In that case Hélein’s conjecture has the following form in which it is not assumed that \({{\textbf{n}}}\) is a field of normals to a surface in \({\mathbb {R}}^3\).
1.3 Hélein’s conjecture, \(n=3\)
Let \({{\textbf{n}}}: D_1\rightarrow {\mathbb {S}}^2\) satisfy
Then, there exist an orthonormal moving frame \(({{\textbf{e}}}_1, {{\textbf{e}}}_2)\) and a function f such that the vectors \({{\textbf{e}}}_i(X)\) are orthogonal to \({{\textbf{n}}}(X)\), the frame \(({{\textbf{e}}}_1(X), {{\textbf{e}}}_2(X), {{\textbf{n}}}(X))\) has positive orientation, \({{\textbf{n}}}\cdot ({{\textbf{e}}}_1\times {{\textbf{e}}}_2)>0\), and
Following the Grassmannian approach [18, Appendix A], Schätzle confirmed the conjecture for all n by proving the following.
Proposition (Schätzle [18, Propositon 5.1]) Let \(\Psi : D_1 \subset {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^n\) be a conformal immersion with induced metric \(g_{ij} = e^{2f}\delta _{ij}\) such that the second fundamental form \(\varvec{A}\) satisfies
for some \(\delta >0\). Then, there exists a smooth solution \(v: D_1 \rightarrow {\mathbb {R}}\) of
satisfying
The proof is based on Kuwert& Schätzle [9], and on the Müller–Sv̆erák estimates of the Kähler form in complex projective space.
Remark 1.1
Suppose \(\Psi : D_1\rightarrow {\mathbb {R}}^3\) is an isothermal immersion with conformal factor \(e^f\). Then, \(v-f\) is harmonic by (1.6) and (1.19). It then follows from (1.7) and (1.18) that Hélein’s conjecture for \(n=3\) is a Corollary of Schätzle’s proposition above, at least for immersions. That the result is optimal is shown by examples using Enneper’s surface following [10, Corollary 5.1].
1.4 Refinement of Hélein’s conjecture, \(\varvec{n=3}\)
Theorem 1.2
Suppose
Then, there exists a function f and an orthonormal moving frame \(({{\textbf{e}}}_1, {{\textbf{e}}}_2)\) such that \({{\textbf{e}}}_i(X)\perp {{\textbf{n}}}(X)\), \(i=1,2\), the frame \(({{\textbf{e}}}_1(X), {{\textbf{e}}}_2(X), {{\textbf{n}}}(X))\) has positive orientation, and (1.14)–(1.17) are satisfied with \(c(\delta )\) replaced by \(\frac{c}{\delta }\Vert \nabla {\textbf{n}}\Vert _{L^2(D_1)}\).
Proof
After elementary analysis that yields preliminary estimates, the proof is completed in Sect. 3.4 using standard techniques from the theory of partial differential equations. \(\square \)
Remark 1.3
Since (1.13) implies (1.20) and that \(\frac{c}{\delta }\Vert \nabla {\textbf{n}}\Vert _{L^2(D_1)}\) is bounded by \(8\pi c/\delta \), Theorem 1.2 implies (1.14)–(1.17). Moreover, since examples of Enneper’s minimal surfaces [10, pp. 331, 332] show that \(8\pi \) is optimal in (1.18) when \(n=3\), it follows from (1.8) and (1.9) that \(4\pi \) is optimal in (1.20), because \(|\varvec{A} |^2 = 2|K |\) for minimal surfaces and equality holds in (1.10), see [14, Sect. 5].
1.5 Organisation of the paper
Section 2 is a brief description of Theorems 2.2 and 2.4 which are the main results on which the rest of the discussion relies.
Section 3 is a self-contained yet elementary proof of Theorem 2.2. The notation is set out in Sect. 3.1, estimates are developed in Sect. 3.2, and the main part of the proof is in Sect. 3.3. In Sect. 3.4, Theorem 1.2, which is an improvement on Hélein’s conjecture for \(n=3\), is proved using Theorem 2.2 and the continuation argument in Hélein’s book [5].
In Sect. 4, the theory of Sect. 3 is extended to the case when hypothesis (2.4) in Theorem 2.2 is not satisfied. The main result is Theorem 2.4, and its proof relies on constructions from Sect. 3, a theory of regular points set out in Sect. 4.2, and the co-area formula [4, Theorem 3.2.22] from geometric measure theory.
Appendix A is a brief discussion of the possibility of extending Theorem 2.4 to immersions of \(D_1\) into \({\mathbb {R}}^n,~n>3\), in the language of Grassmannian manifolds.
2 Notation and main results
2.1 Notation
Let
Definition
A map \({{\textbf{u}}} \in W^{1,2}(D_1, {\mathbb {S}}^2)\) which has finite energy
is said to be smooth, written \({{\textbf{u}}} \in C^\infty (D_1,{\mathbb {S}}^2)\), if for some \(r>1\) there is an infinitely differentiable \({{\textbf{v}}}: D^\circ _r\rightarrow {\mathbb {S}}^2\) with \({{\textbf{v}}} (X) = {{\textbf{u}}}(X)\) almost everywhere on \(D_1\). Let \( C^\infty _0(D_1,{\mathbb {S}}^2) = \{{{\textbf{u}}} \in C^\infty (D_1,{\mathbb {S}}^2): {{\textbf{u}}} \text { has compact support in } D_1^\circ \}\) and let \(W^{1,2}_0(D_1,{\mathbb {S}}^2)\) be the completion of \(C^\infty _0(D_1,{\mathbb {S}}^2)\) in \(W^{1,2}(D_1,{\mathbb {S}}^2)\).
Lemma 2.1
\({C^\infty (D_1,{\mathbb {S}}^2)}\) is dense in \(W^{1,2}(D_1, {\mathbb {S}}^2)\).
Proof
For \({{\textbf{u}}} \in W^{1,2}(D_1, {\mathbb {S}}^2)\) and \(r>1\), let \({{\textbf{u}}}_r(X) = {{\textbf{u}}}(X/r),\, X \in D_r.\) Then, \({{\textbf{u}}}_r \in W^{1,2}(D_r, {\mathbb {S}}^2)\), its restriction, \(\hat{{{\textbf{u}}}}_r\), to \(D_1\) is in \(W^{1,2}(D_1, {\mathbb {S}}^2)\), and \(\hat{{{\textbf{u}}}}_r \rightarrow {{\textbf{u}}}\) in \(W^{1,2}(D_1, {\mathbb {S}}^2)\) as \(r \rightarrow 1\). Now from [19, Sect. 4] (see also [1]) it follows that there is a smooth function
Hence, \(\Vert \hat{{{\textbf{v}}}}_r-{{{\textbf{u}}}}\Vert _{W^{1,2}(D_1, \mathbb S^2)} \rightarrow 0 \text { as } r \rightarrow 1\) and, since \(\hat{{{\textbf{v}}}}_r \in C^\infty (D_1,{\mathbb {S}}^2)\), the proof is complete. \(\square \)
Now when \({{\textbf{n}}}\in W^{1,2}(D_1,{\mathbb {S}}^2)\) put
and note that \(\Phi \in L^1(D_1,{\mathbb {S}}^2)\) and
Since \(\partial _i{{\textbf{n}}},\,i=1,2\), are orthogonal to the unit vector \({{\textbf{n}}}\) and the vector field \(\partial _1\textbf{n}\times \partial _2{{\textbf{n}}}\) is parallel to \({{\textbf{n}}}\), it follows that
Thus, the area on \({\mathbb {S}}^2\) of the image under \({{\textbf{n}}}\) of an area element dX of \(D_1\) is
2.2 Main results
A corollary (see Theorem 1.2) of Theorem 2.2, is that (1.14)–(1.17) in Hélein’s conjecture hold when hypothesis (1.13) is replaced by (2.4).
Theorem 2.2
Suppose \({{\textbf{n}}} \in W^{1,2}(D_1, {\mathbb {S}}^2)\) satisfies
Then, there exist \(\Omega _i\in L^2(D_1)\) with \(\Vert \Omega _i\Vert _{L^2(D_1)}\le (8\pi /\delta )\Vert \nabla \textbf{n}\Vert _{L^2(D_1)}\), \(i = 1,2\), and for every \(\zeta \in C^\infty _0(D_1)\),
In particular, for \(\delta \) in (2.4) and an absolute constant c,
Remark 2.3
Since \( |\Phi |\le |\partial _1{{\textbf{n}}} ||\partial _2{{\textbf{n}}} |\le \frac{1}{2}|\nabla {{\textbf{n}}} |^2, \) condition (2.4) is satisfied if (1.13), the hypothesis of Hélein’s conjecture, holds, but not vice versa.
To investigate what can be said when there is no restriction on the energy of \({{\textbf{n}}}\) except that it is finite, let \(\mathcal A\subset {\mathbb {S}}^2\) and \( {\mathcal {F}}:={{\textbf{n}}}^{-1}({\mathcal {A}}) = \big \{X\in D_1:\, {{\textbf{n}}}(X)\in {\mathcal {A}}\big \}.\)
Theorem 2.4
If \({{\textbf{n}}} \in W^{1,2}(D_1, {\mathbb {S}}^2)\) and \({\mathcal {A}}\subset {\mathbb {S}}^2\) is Borel with positive measure \(\mu \), there exist \(\Omega _i\in L^2(D_1),\,i=1,2\), such that for all \(\zeta \in C^\infty _0(D_1)\),
and \(\Vert \Omega _i\Vert _{L^2(D_1)}\le {c}\,\mu ^{-1/2}\,\Vert \nabla {{\textbf{n}}}\Vert _{L^2(D_1)}\), where c is an absolute constant. Thus,
Remark 2.5
To see that Theorem 2.4 implies Theorem 2.2, let \({\mathcal {A}} = {\mathbb {S}}^2\setminus {{\textbf{n}}}(D_1)\) so that \(\chi _{_{{\mathcal {F}}}} = 0\) on \(D_1\) and \(\mu > 0\) when (2.4) holds.
Theorem 2.4 implies that the pre-image of every subset \({\mathcal {A}} \subset {\mathbb {S}}^2\) of positive measure, however small, contains significant information about the singularity of \(\Phi \).
Remark 2.6
If \(\{{\mathcal {A}}_j:1\le j \le N\}\) is a family of mutually disjoint subsets of \({\mathbb {S}}^2\), each with measure \( \mu _j\), the corresponding family \(\{{\mathcal {F}}_j:1\le j \le N\}\) of their inverse images under \({{\textbf{n}}}\) are mutually disjoint in \(D_1\) and, by (2.7), for each j there exist \(\Omega ^j_i\in L^2(D_1),\,i=1,2\), such that, for all \(\zeta \in C^\infty _0(D_1)\),
Now let \(\mu = \sum _{j=1}^N \mu _j\), \({\mathcal {A}} = \cup _{j=1}^N {\mathcal {A}}_j\), \({\mathcal {F}} = \cup _{j=1}^N {\mathcal {F}}_j\), and sum over j to obtain
where
Thus, \({{\widetilde{\Omega }}}_i\) satisfies (2.7) when \({\mathcal {A}} = \cup _{j=1}^N {\mathcal {A}}_j\) and \({\mathcal {A}}_j \cap {\mathcal {A}}_k = \emptyset \), \(j \ne k\). For example if \(\cup _{j=1}^N {\mathcal {A}}_j = {\mathbb {S}}^2\),
since \(\mu = 4\pi \) when \({\mathcal {A}} = {\mathbb {S}}^2\) and, by (2.7),
3 Refining Hélein’s conjecture when \(\varvec{n = 3}\)
The idea underlying the proof of Theorem 2.2 is, for a given \({{\textbf{n}}}\in W^{1,2}(D_1, {\mathbb {S}}^2)\), to write \(\Phi \) in weak divergence form,
and establish appropriate estimates. Since such a representation on the whole disc \(D_1\) is not always possible, even when the corresponding \({{\textbf{n}}}\) is smooth, because the resulting \(\omega _i\) may have strong singularities, the task is limited to showing that an appropriate representation is possible under the hypotheses of Theorem 2.2. Since, by Lemma 2.1, every \({{\textbf{n}}} \in W^{1,2}(D_1,{\mathbb {S}}^2)\) can be approximated in \(W^{1,2}(D_1,{\mathbb {S}}^2)\) by a sequence \(\{{{\textbf{n}}}_k\}\) of vector fields in \(C^\infty (D_1,{\mathbb {S}}^2)\), it suffices to prove the theorem for smooth \({{\textbf{n}}}\) satisfying (2.4).
3.1 Construction of \(\varvec{\omega }_i\) and \(\Gamma \)
The representations of a vector field \({{\textbf{n}}}:D_1 \rightarrow {\mathbb {S}}^2\) by Cartesian coordinates and by spherical coordinates are related as follows:
where \({{\textbf{n}}}= (n_1,n_2,n_3)\), \(\vartheta \) are \(\varphi \) are functions of X with \(\vartheta :D_1 \rightarrow [0,\pi )\) and \(\varphi :D_1 \rightarrow (0,2\pi ]\). Then, formal partial differentiation yields that
whence
Then, since
and
the following formula apparently gives \(\Phi \) in the divergence form (3.1)
However, since \(s_1^2 +s_2^2 + s_3^2 = 1\) for \(\varvec{s} \in {\mathbb {S}}^2\), the mapping
is real-analytic on \({\mathbb {S}}^2\) except at \({{{\varvec{k}}}} = (0,0,1)\). It follows that when \({{\textbf{n}}}: D_1 \rightarrow {\mathbb {S}}^2\) is smooth, \(\omega _i,i=1,2,\) in (3.2) is smooth where \({{\textbf{n}}}(X)\ne {{\varvec{k}}}\), but there may be singularities for X elsewhere. Thus, (3.2) may not hold, even in the sense of distributions on \(D_1\), if \({{\textbf{n}}}(X) ={{\varvec{k}}}, X \in D_1\). The following remarks are key to overcoming this difficulty.
Remark 3.1
-
(i)
The mappings from \({\mathbb {S}}^2\) to \({\mathbb {R}}\) defined for \( s_3 \ne 1\) in (3.3),
$$\begin{aligned} (s_1,s_2,s_3) \mapsto \left|\frac{(s_3+1)s_i}{ s_1^2+s_2^2}\right|\le \frac{2|s_i |}{{ s_1^2+s_2^2}} \le \frac{2}{\sqrt{s_1^2+s_2^2}}, \quad i=1,2, \end{aligned}$$are in \(L^p({\mathbb {S}}^2)\) for \(p<2\), and the singularities in (3.2) are correspondingly weak.
-
(ii)
Since rotation of the Cartesian coordinate system in which \({\mathbb {S}}^2\) is parameterized changes the location of the poles of \({\mathbb {S}}^2\), Müller & Sv̆erák [13] showed that singularities in (3.2) can be dealt with by integrating over a set of rotated coordinates.
-
(iii)
In polar coordinates \( {{\varvec{s}}} =(\cos \phi \sin \theta , \sin \phi \sin \theta , \cos \theta ) \in {\mathbb {S}}^2\) and \({{\varvec{k}}} = (0,0,1)\),
$$\begin{aligned} \int _{{{\varvec{s}}} \in {\mathbb {S}}^2}\frac{ dS_{{{\varvec{s}}}}}{|{{\varvec{s}}} - {{\varvec{k}}} |} = \int _0^\pi \frac{2\pi \sin \theta }{2\sin (\theta /2)} ~d\theta = 2\pi \int _0^\pi \cos (\theta /2) \,d\theta = 4\pi . \end{aligned}$$(3.4)
Here the approach is similar to Müller and Sv̆erák [13], except that in what follows the field \({{\textbf{n}}}\) is rotated instead of the coordinate system. To see that this is possible without changing \(\Phi \) in (2.1), let \({{\textbf{U}}}\) be a rotation matrix (a \(3\times 3\) orthogonal matrix with determinant 1), the transpose of the columns of which form an orthonormal basis \(U_i,\, i= 1,2,3\), for \({\mathbb {R}}^3\) with
Then, for a given \({{\textbf{n}}}:D_1\rightarrow {\mathbb {S}}^2\), let \(({{\textbf{U}}} {\textbf{n}})(X) = {{\textbf{U}}} ({{\textbf{n}}}(X))\in {\mathbb {S}}^2,\,X \in D_1\), and put
It follows that
and therefore, since \({{\textbf{U}}}\) is orthogonal,
Therefore, replacing \({{\textbf{n}}}\) with \({{\textbf{m}}}\) in formula (3.2) yields
where \(W_i:D_1 \rightarrow {\mathbb {R}}\) depends on \({{\textbf{n}}}:D_1 \rightarrow {\mathbb {S}}^2\) and the rotation matrix \({{\textbf{U}}}\) as
Clearly, as with (3.3), the singularities of \(W_i\) occur at \(X \in D_1\) where \({{\textbf{m}}}(X)= {{\varvec{k}}}\), equivalently where \({{\textbf{n}}}(X)= {{\textbf{U}}}^{-1}{{\varvec{k}}}\). Therefore, (3.5) holds pointwise at \(X \in D_1\) if \({{\textbf{m}}}(X)\ne {{\varvec{k}}}\).
The next step is to parameterize a suitable family of rotation matrices. So for fixed \({{\varvec{n}}}' \in {\mathbb {S}}^2_0 = {\mathbb {S}}^2 \setminus \{{{\varvec{k}}},-{\varvec{k}}\}\) (note that \({{\varvec{n}}}'\) is not a function of X), let
where \(\cos \theta '\ne \pm 1\), and let \(\textbf{U}({{\varvec{n}}}')\) denote the rotation matrix the transpose (equivalently the inverse) of which is
when \(\lambda ':= {n'_1}^2+{n'_2}^2 = 1-{n_3'}^2\ne 0\). Thus, \({{\varvec{n}}}' \mapsto {{\textbf{U}}} ({{\varvec{n}}}')\) in (3.8) depends real analytically on \((n'_1,n'_2,n'_3) ={{{\varvec{n}}}'}\in {\mathbb {S}}^2_0\).
Since \({{\textbf{U}}}({{\varvec{n}}}')^{-1}({{\varvec{k}}})={{\varvec{n}}}'\) by (3.8), it follows that for any \({{{\varvec{n}}}'} \in {\mathbb {S}}_0^2\) and \({{{\varvec{n}}}\in {\mathbb {S}}^2}\)
which means \({{\varvec{n}}}' = {{\varvec{n}}}\) if and only if \({{\varvec{m}}} = {{\varvec{k}}}\) and, since \(\varvec{m}=(m_1,m_2,m_3) \in {\mathbb {S}}^2\),
Definition 3.2
Now for \(({{{\varvec{n}}}}, {{\varvec{n}}}') \in \Sigma \) let
and for \(({{{\varvec{n}}}}, {{\varvec{n}}}') \in \Sigma \) and \(\varvec{\xi }\in {\mathbb {R}}^3\) let
where \({{\textbf{U}}}({{\varvec{n}}}')\varvec{\xi }= \big (({{\textbf{U}}}({{\varvec{n}}}')\varvec{\xi })_1, ({{\textbf{U}}}({{\varvec{n}}}')\varvec{\xi })_2, ({{\textbf{U}}}({{\varvec{n}}}')\varvec{\xi })_3\big )\) and \(\Gamma ({{\varvec{n}}}, {{\varvec{n}}}', \cdot ):{\mathbb {R}}^3 \rightarrow {\mathbb {R}}\) is linear for fixed \(({{{\varvec{n}}}, {{\varvec{n}}}'}) \in \Sigma \).
Then, for fixed \({{\varvec{n}}}' \in {\mathbb {S}}^2_0\) and smooth \({{\textbf{n}}}: D_1 \rightarrow {\mathbb {S}}^2\) with \(({{\textbf{n}}}(X), {{\varvec{n}}}') \in \Sigma \), \(X \in D_1\), put \({{\textbf{m}}}(X) = {{\textbf{U}}}({{\varvec{n}}}')({{\textbf{n}}}(X))\) and note that \(\partial _i {{\textbf{m}}}(X) = \partial _i ({{\textbf{U}}}({{\varvec{n}}}'){{\textbf{n}}})(X) = {\textbf{U}}({{\varvec{n}}}')(\partial _i{{\textbf{n}}}(X))\). Then, \({{\textbf{m}}} (X) \ne \varvec{k}\) and by (3.6) and (3.11),
The proof of Theorem 2.2 depends on estimates of \(\Gamma \) and \(\omega _i\) in terms of \({{\varvec{n}}}'\in {\mathbb {S}}^2_0\) and \({{\textbf{n}}}:D_1 \rightarrow {\mathbb {S}}^2\) in the next section.
3.2 Estimates \(\varvec{\Gamma }\) and \(\varvec{\omega _i}\)
Lemma 3.3
The function \(\Gamma \) in (3.11b) is real-analytic on \(\Sigma \times {\mathbb {R}}^3\) and
Proof
Since \( {{\varvec{n}}}'\mapsto {{\textbf{U}}}({{\varvec{n}}}') \) and \(({{\varvec{n}}}, {{\varvec{n}}}')\mapsto {{\textbf{U}}}({{\varvec{n}}}'){{\varvec{n}}}= {{\varvec{m}}} = (m_1,m_2,m_3)\in \mathbb S^2\) are real-analytic on \({\mathbb {S}}^2\times {\mathbb {S}}^2_0\times {\mathbb {R}}^3\), \(\Gamma \) in (3.11) will be real-analytic on \(\Sigma \times {\mathbb {R}}^3\) if \(\displaystyle ({{\varvec{n}}}, {{\varvec{n}}}') \mapsto \frac{1}{1-m_3}\) is real-analytic on \(\Sigma \). So it suffices to observe from (3.9) that \(m_3 \ne 1\) when \(({{\varvec{n}}},{{\varvec{n}}}') \in \Sigma \). Now, when \(({{\varvec{n}}}, {{\varvec{n}}}') \in \Sigma \) and \({{\varvec{m}}} \ne {{\varvec{k}}}\), it follows from (3.11)(b),
by (3.10). Thus, (3.13) holds and the proof is complete. \(\square \)
Definition 3.4
For \({{\varvec{n}}}' \in {\mathbb {S}}^2_0\) and smooth \({{\textbf{n}}}:D_1 \rightarrow {\mathbb {S}}^2\), let
Lemma 3.5
For fixed \({{\varvec{n}}}' \in {\mathbb {S}}^2_0\) and smooth \({{\textbf{n}}}:D_1 \rightarrow {\mathbb {S}}^2\), the functions \(\omega _i({{\textbf{n}}}, {{\varvec{n}}}'),\,i=1,2\), in (3.12) are infinitely differentiable at \(X \in \Pi ({{\textbf{n}}},{{\varvec{n}}}')\). Moreover,
Proof
For fixed \({{\varvec{n}}}'\in {\mathbb {S}}^2_0\), the mapping \(X\rightarrow {{\textbf{n}}}(X)\) takes \(\Pi ({{\textbf{n}}},{{\varvec{n}}}')\) to \({\mathbb {S}}^2\setminus \{{{\varvec{n}}}'\}\) and since \( ({\mathbb {S}}^2\setminus \{{{\varvec{n}}}'\})\times \{{{\varvec{n}}}'\}\subset \Sigma , \) it follows from Lemma 3.3 that \(\Gamma ({{\varvec{n}}}, {{\varvec{n}}}', \varvec{\xi })\) is real-analytic with respect to \(({{\varvec{n}}},\varvec{\xi })\) at \(({{\textbf{n}}}(X), \varvec{\xi }_0)\), when \(X \in \Pi ({{\textbf{n}}}, {{\varvec{n}}}')\) and \(\varvec{\xi }_0\in {\mathbb {R}}^3\) is arbitrary. Therefore, since \({{\textbf{n}}}\) is infinitely differentiable on \(D_1\), by (3.12) the functions \(\omega _i({{{\textbf{n}}}}, {{\varvec{n}}}')\), \(i=1,2\), are infinitely differentiable on \(\Pi ({{\textbf{n}}},{{\varvec{n}}}')\). Now by (3.6), \(\omega _i({{\textbf{n}}}, {{\varvec{n}}}')(X) = W_i({{\textbf{n}}}, {{\textbf{U}}}({{\varvec{n}}}'))(X)\), \(X \in D_1\), where in the definition of \(W_i\), \( {\textbf{m}} (X)= {{\textbf{U}}}({{\varvec{n}}}'){{\textbf{n}}}(X)\ne {{\varvec{k}}}\) since \({\textbf{n}}(X)\ne {{\varvec{n}}}'\). It follows that
The equality in (3.14) follows from (3.5) and (3.6), since \({{\textbf{n}}}(X) \ne {{\varvec{n}}}'\) when \(X \in \Pi ({{\textbf{n}}},\varvec{n}')\), and the inequality (3.14) is then immediate from (3.13). This completes the proof. \(\square \)
Lemma 3.5 concerns smoothness of \(\omega _i\) at \(X \in \Pi ({{\textbf{n}}}, \varvec{n}')\) for fixed \({{\varvec{n}}}'\). The next lemma deals with their joint smoothness with respect to X and \({{\varvec{n}}}'\).
Lemma 3.6
Suppose \({{\textbf{n}}}:D_1 \rightarrow {\mathbb {S}}^2\) is smooth, \( K\subset {\mathbb {S}}^2_0\) and \( E\subset \Sigma \) are compact, and \(G \subset D_1\) is such that \(({{\textbf{n}}}(X), {{\varvec{n}}}')\in E\text { for all } X\in G \text { and } {{\varvec{n}}}'\in K\). Then, there is a neighbourhood \( O_ K\) of K such that the functions \((X,{{\varvec{n}}}')\mapsto \omega _i({{\textbf{n}}}(X), {{\varvec{n}}}')\), \(i=1,2,\) are infinitely differentiable on \( G\times O_K\).
Proof
Since \(D_1\) is compact and \({{\textbf{n}}}:D_1 \rightarrow {\mathbb {S}}^2\) is smooth, \(|\partial _i{{\textbf{n}}} |\le b\) on \( D_1\) for some \(b \in {\mathbb {R}}\). Let \( B=\{\varvec{\xi }: |\varvec{\xi }|\le b,\,\varvec{\xi }\in {\mathbb {R}}^3\}\) and \( V= E\times B\). Since V is compact in \( \Sigma \times {\mathbb {R}}^3\), by Lemma 3.3, the function \(\Gamma \) is real-analytic on a neighborhood \( O_ V\) of V. Moreover, the mapping
is infinitely differentiable on \(D_1\times {\mathbb {S}}^2_0\) and maps \( G\times K\) into V. The smoothness of \(\omega _i\) on \( G\times O_{ K}\) follows from (3.12). \(\square \)
3.3 Proof of Theorem 2.2
As noted at the beginning of Sect. 3, it suffices to consider smooth maps \({{\textbf{n}}} \) which takes the disc \(D_1\) into the unit sphere \({\mathbb {S}}^2\) and which, by hypothesis (2.4), satisfy \(\text {meas}\, \big ( \textbf{n}(D_1)\big )<4\pi -\delta \) (see (2.3)).
Let \( A= {{\textbf{n}}}(D_1)\cup \{\pm {{\varvec{k}}}\}\subset {\mathbb {S}}^2\). Then, \(\text {meas}\,({\mathbb {S}}^2\setminus A)>\delta \), since \(\text {meas}\,({\mathbb {S}}^2)= 4\pi \) and \(\text {meas}\,( A)<4\pi -\delta \). Hence, there is a compact \(K \subset \mathbb S^2\setminus A\) with \(\text {meas}\,( K)\ge \delta \). Since \(D_1\) is compact and \({{\textbf{n}}}\) is continuous on \(D_1\), \({{\textbf{n}}}(D_1)\) is compact and, for some \(\sigma >0\) independent of \({{\varvec{n}}}'\in K\) and of \(X\in D_1\),
By Definition 3.4, \(\Pi ({{\textbf{n}}}, {{\varvec{n}}}')= D_1\) for all \({{\varvec{n}}}'\in K\) and by Lemma 3.6, with \( G=D_1\) and \(E = D_1\times K\), there is a neighborhood \( O_ K\) of K such that \(\omega _i({{\textbf{n}}}, {{\varvec{n}}}'),\,i=1,2\), are infinitely differentiable on \(D_1\times O_ K\). Therefore, by Lemma 3.5
and
where \(|{{\textbf{n}}}(X)-{{\varvec{n}}}' |\ge \sigma >0\), \((X, {{\varvec{n}}}') \in D_1\times K\). Now for \(X \in D_1\) let
where the integration over K is with respect to the measure on \({\mathbb {S}}^2\). Then, (3.14) yields the estimate
which gives
Now multiplying (3.16) by \(\zeta \in C^\infty _0(D_1)\) and integrating with respect to \({{\varvec{n}}}'\) over \(K \subset {\mathbb {S}}^2\) yields (2.5). This completes the proof.
3.4 Proof of Theorem 1.2
In this section, Theorem 1.2, which improves Hélein’s conjecture when \(n=3\), is deduced from Theorem 2.2 by a continuation argument similar to that in [5].
3.4.1 Step 1. A priori bounds
The first observations are similar to those of Sect. 1.1.
Lemma 3.7
Suppose, for smooth \({{\textbf{n}}}: D_1\rightarrow {\mathbb {S}}^2\), there exist an orthonormal frame \(({{\textbf{e}}}_1, {{\textbf{e}}}_2)\) were the \({\textbf{e}}_i\) are smooth on \(D_1\) and orthogonal to \({{\textbf{n}}}\), and \(({{\textbf{e}}}_1, {{\textbf{e}}}_2, {{\textbf{n}}})\) has positive orientation. Suppose also that f is smooth and satisfy (1.15) and (1.16). Then,
and
Proof
Since \({{\textbf{n}}}\cdot \partial _i{{\textbf{n}}}=\partial _i({{\textbf{n}}}\cdot {{\textbf{e}}}_j)=0\) for all i, j, it follows that
and hence that
Moreover,
since
and similarly
Substituting these observations into (3.19) gives
and (3.17) follows from (1.16). Next,
and (1.15) imply (3.18). \(\square \)
Lemma 3.8
For a smooth \({{\textbf{n}}}:D_1 \rightarrow {\mathbb {S}}^2\) which satisfies (2.4), let \(({{\textbf{e}}}_1, {{\textbf{e}}}_2, {{\textbf{n}}})\) be an orthonormal moving frame with positive orientation which, together with a smooth function \(f:D_1 \rightarrow {\mathbb {R}}\), satisfies (1.15) and (1.16). Then,
Here, the constant C depends only on k and \({{\textbf{n}}}\), c is a constant and \(\delta \) is given by (2.4).
Proof
By (3.17) of the preceding Lemma,
and it follows from standard estimates of solutions of the Dirichlet problem for Poisson’s equation that
In particular,
from which it follows by (3.18) and induction that
This with (3.23) implies (3.20). By (2.4), \({{\textbf{n}}}\) satisfies the hypotheses of Theorem 2.2, and hence by (2.6), for an absolute constant c,
This and (3.22) yield that
Now from (3.22) and the Wente–Topping inequality [24, 21, Theorem 1], it follows that \( \Vert f\Vert _{L^\infty (D_1)}\le (c/\delta )\Vert \nabla {{\textbf{n}}}\Vert _{L^2(D_1)}\). Hence, (3.21) holds and the proof is complete. \(\square \)
3.4.2 Step 2. A parameterized family of normal vector fields
For any smooth \({{\textbf{n}}}:D_1 \rightarrow {\mathbb {S}}^2\) satisfying (2.4), consider the family of vector fields
Note that \({{\textbf{n}}}_0\) is a constant vector field and that, for all \(\lambda \in [0,1]\), \({{\textbf{n}}}_\lambda \) satisfies
Corollary 3.9
Let \(f_\lambda , {{\textbf{e}}}_{\lambda ,i}\in C^\infty (D_1)\) be solutions to equations (1.15)–(1.16) with \({{\textbf{n}}}\) replaced by \({{\textbf{n}}}_\lambda \). Then,
where \(\delta \) is given in (2.4) and c and \(C(k,{{\textbf{n}}})\) are constant independent of \(\lambda \in [0,1]\).
Proof
In the light of (3.24), the proof is the same as for \(\lambda = 1\) in Lemma 3.8. \(\square \)
3.4.3 Step 3. Parameter continuation
Denote by \({\mathcal {L}}\) the set of \(\lambda \in [0,1]\) for which the system (1.15), (1.16) has an infinitely differentiable solution \(\{f_\lambda , {{\textbf{e}}}_{\lambda , i}:\,i=1,2\}\), and note that \(0\in {\mathcal {L}}\). Indeed, since \({{\textbf{n}}}_0=\text {const.}\), the function \(f_0=0\) and an arbitrary pair of constant vectors \({{\textbf{e}}}_{0,i}\) with
satisfy (1.15) and (1.16) with \({{\textbf{n}}} = {\textbf{n}}_0\).
To show that \({\mathcal {L}}\) is closed let \(\lambda _n \in {\mathcal {L}}\) and \(\lambda _n \rightarrow \lambda \) as \(n\rightarrow \infty \). Then, by Corollary 3.9 there is a sequence \(\{n_\ell \} \subset {\mathbb {N}}\) such that the solutions, \({{\textbf{e}}}_{\lambda _{n_{\ell },i}},\,i=1,2\), and \(f_{\lambda _{n_\ell }}\) to problem (1.15)–(1.16) with \({{\textbf{n}}}\) replaced by \({{\textbf{n}}}_{\lambda _{n_\ell }}\), converge in \(C^k(D_1)\) for all k, to functions denoted by \({{\textbf{e}}}_{\lambda , i}\) and \(f_\lambda \). Obviously these functions are infinitely differentiable and satisfy equations (1.15)–(1.16) with \({{\textbf{n}}}\) replaced by \(\textbf{n}_\lambda \). Hence, \({\mathcal {L}}\) is closed in [0, 1].
Now following Hélein [5], to show \({\mathcal {L}}\) is open let \(\lambda _0\in {\mathcal {L}}\) and \(I_0\subset [0,1]\) be a segment with endpoints \(\lambda _0\) and \(\lambda _0+t_0\), \(0<|t_0 |<1\). The goal is to prove that, for sufficiently small \(t_0\) and all \(t\in I_0\), equations (1.15) and (1.16) with \({{\textbf{n}}}={\textbf{n}}_{\lambda _0+t}\) have a smooth solution. To simplify notation, let \({{\textbf{n}}}^0\) and \({{\textbf{n}}}^t\) denote \({{\textbf{n}}}_{\lambda _0}\) and \({{\textbf{n}}}_{\lambda _0+t}\), and denote solutions of (1.15) and (1.16) with \({{\textbf{n}}}={{\textbf{n}}}^t\) by \({{\textbf{e}}}^t_i\) and \(f^t\).
Then, for \(t \in I_0\) and \(X\in D_1\), define a family of orthogonal projections \({\mathbb {P}}^t(X):{\mathbb {R}}^3 \rightarrow \{{{\textbf{n}}}^t(X)\}^\perp \subset {\mathbb {R}}^3 \) by
Since \(\lambda _0\in {\mathcal {L}}\), there exist \(C^\infty \) vector fields \({{\textbf{e}}}_i^0\), \(i=1,2\), which satisfy (1.15) and (1.16) with \({{\textbf{n}}}={{\textbf{n}}}^0\) and, since \({\textbf{n}}^0\in C^\infty ({\mathbb {R}}^2)\), there exists \(t_0>0\) such that
Then, for \(t \in I_0\), let \( \overline{{{\textbf{e}}}}^{(t)}_i(X)= {\mathbb {P}}^t(X)\,{{\textbf{e}}}_i^0(X), ~ i=1,2\), and note, since \({\mathbb {P}}^0{{\textbf{e}}}_i^0={{\textbf{e}}}_i^0\), that for \(X\in D_1\), and \(|t |\le |t_0 |\),
Therefore, with their dependence on \(t \in I_0\) suppressed for convenience of notation, orthonormal vector fields \(\textbf{e}^*_i\), \(i=1,2\), are well defined on \(D_1\) by
Since \({{\textbf{n}}}^t\in C^\infty (D_1)\), it follows that \(\textbf{e}_i^*\in C^\infty (D_1)\), and obviously the orthonormal triplets \(({{\textbf{e}}}_1^*, {{\textbf{e}}}_2^*, {{\textbf{n}}}^t)\) have positive orientation, \(t \in I_0\). The aim now is to find a solution of (1.15) and (1.16) with \({{\textbf{e}}}^t_i\), \(i=1,2\), given by
for some smooth function \(\vartheta :D_1 \rightarrow {\mathbb {R}}\) yet to be determined. Since \({{\textbf{e}}}^*_i \cdot {{\textbf{e}}}^*_j = \delta _{ij}\), by (3.25), \(({{\textbf{e}}}_1^t, {{\textbf{e}}}_2^t, {{\textbf{n}}}^t)\) is an orthonormal triple with positive orientation. Now define vector fields which are infinitely differentiable on \(D_1\) by
and note that
Next, as in [5] note that the variational problem
has a unique, infinitely differentiable minimiser \(\vartheta \in {\mathcal {V}}\) which satisfies
where \(\varvec{\nu }\) is the unit normal to \(\partial D_1\). By (3.27), this can be rewritten
For \(t \in I_0\), let \({{\textbf{e}}}_1^t \,d{{\textbf{e}}}_2^t (X)= (h^t_1(X), h^t_2(X)),~X \in D_1\), and put
Then, \(f^t\in C^\infty (D_1)\) and, since \(\text {div} (h^t_1,h^t_2)=0\) by (3.28),
Hence, by (3.26), \(f^t\) and \({{\textbf{e}}}_i^t,\, i=1,2\), satisfy (1.15). Also, by the second part of (3.28), \(f^t\) is constant on \(\partial D_1\), and the constant c can be chosen so that \(f^t=0\) on \(\partial D_1\). Therefore, (1.16) follows because
Since \(({{\textbf{e}}}_1^t, {{\textbf{e}}}_2^t)\) and \(f^t\) satisfy equations (1.15) and (1.16) with \({{\textbf{n}}}={{{\textbf{n}}}}^t\) for all \(|t |\le |t_0 |\), \({\mathcal {L}}\) is open in [0, 1]. Since \(\mathcal L\ne \emptyset \) is also closed and [0, 1] is connected, \({\mathcal {L}} =[0,1]\). When \(t=1\) this shows \({{\textbf{e}}}_1^1, {{\textbf{e}}}_2^1, f^1\) satisfy (1.15)–(1.17) holds by Lemma 3.8. Thus, Hélein’s conjecture for \(n=3\) with hypothesis (2.4) is established, and the proof is complete.
4 Generalising beyond \(\varvec{8\pi }\) when \(\varvec{n=3}\)
By developing the work of previous sections and using some classical results from integral geometry, this section is devoted a proof of Theorem 2.4 when \({{\textbf{n}}}\in W^{1,2}(D_1,{\mathbb {S}}^2)\) without further restrictions. Appendix A formulates, without proof, an analogous result for \(n>3\).
4.1 Preliminaries on geometric integration
Let \(\text { card}(E)\) denote the number of points in a finite set E and let \(\textrm{card}(E)=\infty \) if E is infinite. Then, for \(E \subset {\mathbb {R}}^m\), \(\textrm{card}(E)\) is finite when its 0-Hausdorff measure \({\mathcal {H}}^0(E)\) is finite, and if \(E\subset \mathbb R^m\) is finite, every function g defined on E is \({\mathcal {H}}^0\)-measurable.
For fixed \({{\textbf{n}}}\in C^\infty (D_1,{\mathbb {S}}^2)\) and \({{\varvec{n}}}' \in {\mathbb {S}}^2\), let
Theorem 4.1
For \({{\textbf{n}}}\in C^\infty (D_1,{\mathbb {S}}^2)\) and \(g\in L^1(D_1)\),
where \(\Phi ={{\textbf{n}}}\cdot (\partial _1{{\textbf{n}}} \times \partial _2{{\textbf{n}}})\).
Proof
In the co-area formula [4, Theorem 3.2.22] let \(\nu = 3\), \(n=m=\mu =2\) and
Then, \(\text { ap}\,J_\mu Df(w)=|\Phi |\), two-dimensional Hausdorff measure coincide with Lebesgue measure on \(D_1\) and \({\mathbb {S}}^2\), and with \({\mathcal {H}}^0\) denoting zero-dimensional Hausdorff measure,
Therefore, (4.1) is a special case of identity (3) in Theorem 3.2.22 of [4]. \(\square \)
Corollary 4.2
Under the hypotheses of Theorem 4.1, for all Borel sets \({\mathcal {A}}\subset {\mathbb {S}}^2\),
Proof
If \(\chi _{_{{\mathcal {F}}}}\) and \(\chi _{_{{\mathcal {A}}}}\) are the characteristic functions of \({\mathcal {F}}\) and \({\mathcal {A}}\), then \(\chi _{_{{\mathcal {F}}}}(X)= \chi _{_{{\mathcal {A}}}}({{\textbf{n}}}(X))\), and in particular \(\chi _{_{{\mathcal {F}}}}(X)=\chi _{_{{\mathcal {A}}}}({{\varvec{n}}}')\) for all \( X\in Y({{\textbf{n}}},{{\varvec{n}}}')\). It follows from (4.1) with g replaced by \(\chi _{_{{\mathcal {F}}}}\, g\) that
which proves the assertion. \(\square \)
4.2 Regular points and their properties
The following lemma shows that, for fixed \({{\textbf{n}}}\in C^\infty (D_1, {\mathbb {S}}^2)\), the set \(Y({{\textbf{n}}},{{\varvec{n}}}')\) is well behaved for most \({{\varvec{n}}}'\in {\mathbb {S}}^2\). Recall from Sect. 2.1 that \(D_1^\circ \) is the interior of \(D_1\).
Lemma 4.3
For \(N>1\), \(N \in {\mathbb {N}}\), there is a compact set \(\mathcal Q_N\subset {\mathbb {S}}^2\) such that:
-
(a)
\({{\varvec{n}}}'\in {\mathcal {Q}}_N\) implies that \(Y({{\textbf{n}}}, {{\varvec{n}}}') \subset D_1^\circ \), \(\textrm{card}(Y({{\textbf{n}}},{{\varvec{n}}}')) \le N\),
$$\begin{aligned} \int _{D_1}\frac{dX}{|{{\textbf{n}}}(X)-{{\varvec{n}}}' |}\le N \text { and }~ |{{\varvec{n}}}'\pm {{\varvec{k}}} |\ge \frac{1}{N}. \end{aligned}$$(4.2) -
(b)
Each \(A\in Y({{\textbf{n}}},{{\varvec{n}}}')\), \({{\varvec{n}}}'\in {\mathcal {Q}}_N\), is non-degenerate, meaning \(\Phi (A)\ne 0\) and \(\partial _i{{\textbf{n}}}(A),i=1,2\), are linearly independent.
-
(c)
For an absolute constant c,
$$\begin{aligned} \text { meas }({\mathbb {S}}^2\setminus {\mathcal {Q}}_N)\,\le \, \frac{c}{N}\Big (\int _{D_1}|\nabla {{\textbf{n}}} |^2\, dX+1\Big ). \end{aligned}$$
Thus, the set \({\mathcal {Q}}_\infty = \cup _{_{N \in \mathbb N}}{\mathcal {Q}}_N\) of regular points has full measure in \({\mathbb {S}}^2\).
Proof
The set \(R_0=\{{{\varvec{n}}}' \in {\mathbb {S}}^2: Y({{\textbf{n}}},{{\varvec{n}}}') \cap \partial D_1\ne \emptyset \}\subset {{\textbf{n}}}(\partial D_1)\) has zero measure in \({\mathbb {S}}^2\), because \(\partial D_1\) has zero measure and \({{\textbf{n}}}:D_1 \rightarrow {\mathbb {S}}^2\) is smooth. From (4.1) with \(g\equiv 1\) and (2.2),
whence
Since, by Fubini’s theorem,
where c is an absolute constant, it follows that
Note also from (3.4) that
Finally, it follows from (4.3)–(4.5) that
Hence, there is an open set \(O_N\supset R^*\) such that, for an absolute constant c,
and \({\mathcal {Q}}_N = {\mathbb {S}}^2 \setminus O_N\) satisfies parts (a) and (c).
To prove (b), suppose that some \(A \in Y({{\textbf{n}}},{{\varvec{n}}}')\) is degenerate, i.e. \({{\textbf{n}}}(A) = {{\varvec{n}}}'\) and \(\partial _i{{\textbf{n}}}(A)\), \(i=1,2\), are linearly dependent. Since the mapping \({{\textbf{n}}}(X)\) is infinitely differentiable,
for some constant \(\alpha \). In the first case (the second is similar) for \(X \in D_1\),
where \(c>0\) is some constant. Hence,
which contradict (4.2). This completes the proof of Lemma 4.3. \(\square \)
If \(Y({{\textbf{n}}},{{\varvec{n}}}') \ne \emptyset \) for \({{\textbf{n}}}\in C^\infty (D_1, {\mathbb {S}}^2)\) and \( {{\varvec{n}}}' \in {\mathcal {Q}}_N\), let
Lemma 4.4
There exists \(r_N>0\) such that, for \(\varvec{n}'\in {\mathcal {Q}}_N\),
Proof
Suppose no \(r_N>0\) satisfies the first inequality (4.6). Then, since \({\mathcal {Q}}_N\) and \(D_1\) are compact, there exist \(i\in \{1,\cdots , N\}\) and \(\{{{\varvec{n}}}_k'\}\subset {\mathcal {Q}}_N\) with \({{\varvec{n}}}'_{k} \rightarrow {{\varvec{n}}}' \in {\mathcal {Q}}_N\) and \(A_{i}({{\varvec{n}}}_k') \rightarrow A\) where \(|A |= 1\). Since \({{\textbf{n}}}(A_{i}({{\varvec{n}}}_k')) = {{\varvec{n}}}_k'\), it follows that \({{\textbf{n}}}(A) = {{\varvec{n}}}'\). But \(A \in \partial D_1\) and \({{\varvec{n}}}' \in {\mathcal {Q}}_N\) is false, by Lemma 4.3.
If the second inequality (4.6) is false, there exist \(\{i_{_k}\}, \{j_{_k}\}\), and \(\{{{\varvec{n}}}_k'\}\) such that
Taking subsequence if necessary, assume \(A_{i_{_k}}({{\varvec{n}}}_k')\rightarrow A_i\) and \({{\varvec{n}}}_k'\rightarrow {{\varvec{n}}}'\) as \(k\rightarrow \infty \). Then, \({{\textbf{n}}}(A_i)={{\varvec{n}}}'\), since \({{\textbf{n}}}\) is continuous, and \({{\varvec{n}}}'\in {\mathcal {Q}}_N\), since \({\mathcal {Q}}_N\) is compact.
Since the mapping \(X \mapsto {{\textbf{n}}}(X)\) is non-degenerate at \(A_i \in D_1\), its derivation \(D{{\textbf{n}}}(A_i)\) maps \(\mathbb R^2\) onto \( T_{{{\varvec{n}}}'} ({\mathbb {S}}^2)\), the tangent space to \({\mathbb {S}}^2\) at \({{\textbf{n}}}'\), and its bounded inverse has \(\Vert D{\textbf{n}}(A_i)^{-1}\Vert \ne 0\). Now, since \({{\textbf{n}}}:X \rightarrow {\mathbb {S}}^2\) is smooth, there exists \(\rho >0\) such that
So, for \(X',X''\in D_\rho (A_i)\), the disc of radius \(\rho \) about \(A_i\),
Since \(X'+t(X''-X')\in D_\rho (A_i)\), it follows from (4.8) that
Therefore, since \({{\textbf{n}}}(A_{i_{_k}})={{\textbf{n}}}(A_{j_{_k}})={{\varvec{n}}}_k'\) for all k and, by (4.7), for k sufficiently large \(A_{i_{_k}}({{\varvec{n}}}'), A_{j_{_k}}({{\varvec{n}}}') \in D_\rho (A_i)\),
But \(A_{i_{_k}}({{\varvec{n}}}')\ne A_{j_{_k}}({{\varvec{n}}}')\) for all k. This contradiction completes the proof. \(\square \)
Lemma 4.5
For \({{\varvec{n}}}_0'\in {\mathcal {Q}}_N\) and \(\delta >0\), let \( {\mathcal {W}}_\delta ({{\varvec{n}}}_0')=\{ {{\varvec{n}}}\in {\mathbb {S}}^2: \, |{{\varvec{n}}}_0'-{{\varvec{n}}}' |<\delta \}. \) Then, there exists \(\delta >0\) such that \(\mathrm {card\,}(Y({{\textbf{n}}},{{\varvec{n}}}'))=\mathrm { card\,}(Y({{\textbf{n}}},{{\varvec{n}}}_0'))\) for all \({{\varvec{n}}}'\in {\mathcal {W}}_\delta ({{\textbf{n}}}_0')\). Moreover, \(Y({{\textbf{n}}},{{\varvec{n}}}')\) can be labelled \(A_\ell ({{\varvec{n}}}'),\,1 \le \ell \le n\le N\), where
Proof
Let \(n = \mathrm {card\,}(Y({{\textbf{n}}},{{\varvec{n}}}_0'))\) where \(Y({{\textbf{n}}},{{\varvec{n}}}_0') = \{A_\ell ({{\varvec{n}}}_0'): 1 \le i\le n\}\), and choose an arbitrary \(\ell \in \{1,\cdots , n\}\). Since the mapping \(X \mapsto {{\textbf{n}}}(X)\) is non-degenerate at \(X =A_\ell ({{\varvec{n}}}'_0)\), its derivative \(D{{\textbf{n}}}(A_\ell ({{\varvec{n}}}'_0)):{\mathbb {R}}^2\rightarrow T_{{{\varvec{n}}}_0'} (\mathbb S^2)\), has bounded inverse. Hence, by the inverse function theorem, there exist \(\delta _\ell >0\) and \(\rho _\ell \in (0, r_N)\) (see Lemma 4.4) such that the equation \({{\textbf{n}}}(X)={{\varvec{n}}}'\) has a unique solution \(X=A_\ell ({{\varvec{n}}}')\) in the disc \(D_{\rho _\ell }(A_\ell ({{\varvec{n}}}'_0))\) for every \({{\varvec{n}}}'\in \mathcal W_{\delta _\ell }({{\varvec{n}}}_0')\). Moreover, the mappings \({{\varvec{n}}}' \mapsto A_\ell ({{\varvec{n}}}')\) are infinitely differentiable on \(\mathcal W_{\delta }({{\varvec{n}}}'_0)\), where \(\delta =\min \{\delta _\ell : 1 \le \ell \le n\}\), and \(A_\ell ({{\varvec{n}}}')\rightarrow A_\ell \) as \({{\varvec{n}}}'\rightarrow {{\varvec{n}}}_0'\) for all \(1 \le \ell \le n\).
It remains to show that for sufficiently small \(\delta \), all the solutions \(X \in D_1\) of the equation \({{\textbf{n}}}(X)={{\varvec{n}}}',\,{{\varvec{n}}}'\in {\mathcal {W}}_\delta ({{\varvec{n}}}'_0)\), are given by \(\{A_\ell ({{\varvec{n}}}'), \,1\le i\le n\}\). If no such \(\delta >0\) exists, then there are sequences
such that for all \(\ell \),
Since \(\rho _\ell >0\) is independent of k, after passing to a subsequence if necessary, \(B_k\rightarrow B\) as \(k\rightarrow \infty \) and hence \({{\textbf{n}}}(B)={{\varvec{n}}}_0'\) and \(|B-A_\ell ({{\varvec{n}}}_0') |\ge \rho _\ell ,\,1\le \ell \le n_0 \). This is false since \(A_\ell ({{\varvec{n}}}_0'),\,1\le \ell \le n_0\), are the only solutions of \({{\textbf{n}}}(X)={{\varvec{n}}}_0'\). \(\square \)
4.3 Properties of \(\varvec{\omega _i(X,{{\varvec{n}}}')}\)
Henceforth, for fixed \({{\textbf{n}}}\in C^\infty (D_1, {\mathbb {S}}^2)\) and any \({{\varvec{n}}}' \in {\mathbb {S}}^2_0\), let
where the rotation matrix \({{\textbf{U}}}({{\varvec{n}}}')\) is given by (3.8). The next two lemmas concern the regularity with respect to X and \({{\varvec{n}}}'\in {\mathcal {Q}}_N\) of functions which, since \({{\textbf{n}}}\) is fixed, will be written \(\omega _i(X, {{\varvec{n}}}')\) instead of \(\omega _i({{\textbf{n}}}, {{\varvec{n}}}')(X)\) in (3.12). Therefore, for \(i=1,2\),
are well defined since \({{\varvec{n}}}' \ne \pm \varvec{k}\) because \({{\varvec{n}}}' \in {\mathcal {Q}}_N\), but are singular when \(m_3(X)=1\), equivalently when \({{\textbf{n}}}(X) = {{\varvec{n}}}'\) by (3.9). The notation \({\mathcal {Q}}_N\subset {\mathbb {S}}^2\), \(r_N>0\) and \(Y({{\textbf{n}}},{{\varvec{n}}}_0')\) are as in Lemmas 4.4 and 4.5.
Lemma 4.6
For \({{\varvec{n}}}'_0 \in {\mathcal {Q}}_N\), the functions \(\omega _i,\,i=1,2\), are infinitely differentiable with respect to \((X,{{\varvec{n}}}') \in \big (D_1\setminus \bigcup _{\ell = 1}^n D_{r_N/3}(A_\ell ({{\varvec{n}}}'_0)\big )\times {\mathcal {W}}_\delta ({{\varvec{n}}}_0')\), for some \(\delta >0\).
Proof
Consider the compact sets
where \(D^\circ _{r}(A)\) is the open disc with centre \(A \in D_1\) and radius r. Since \({{\varvec{n}}}_0'\in {\mathbb {S}}^2\setminus \{\pm \varvec{k}\}\) and \({{\textbf{n}}}(X)\ne {{\varvec{n}}}_0'\), \(X \in F\), it follows that \(E \subset \Sigma \) (Definition 3.2) is compact. An application of Lemma 3.6 with \(K=\{\varvec{n}_0'\}\) completes the proof. \(\square \)
Lemma 4.7
For any \({{\varvec{n}}}_0' \in {\mathcal {Q}}_N\), the functions \(\omega _i(\cdot , {{\varvec{n}}}_0')\), \(i=1,2\), are infinitely differentiable on \(D_1\setminus \bigcup _{\ell =1}^n\{A_\ell ({{\varvec{n}}}_0')\}\), and
where c is an absolute constant. In particular, \(\omega _i(\cdot , {{\varvec{n}}}_0')\) is integrable on \(D_1\) and
Proof
Lemma 3.5 yields (4.9), and then (4.10) follows from (4.2) and (4.9). \(\square \)
For \(\zeta \in C^\infty _0 (D_1)\), it follows from Lemma 4.7 and the divergence theorem that for every \({{\varvec{n}}}'\in {\mathcal {Q}}_N\) and \(r \in (0, r_N)\), \(r_N\) in Lemma 4.4,
where
and \((\nu _1,\nu _2)\) is the outward normal to \(\partial D_r(A_\ell ({{\varvec{n}}}'))\).
4.4 Proof of Theorem 2.4
The proof of Theorem 2.4 depends on the calculation in Lemma 4.9 of the limit of (4.11) as \(r \rightarrow 0\), and on the following technical lemma.
Lemma 4.8
\(I(r,{{\varvec{n}}}')\) and \(J(r,{{\varvec{n}}}')\) are continuous with respect to \({{\varvec{n}}}'\) on \({\mathcal {Q}}_N\).
Proof
Fix \({{\varvec{n}}}_0'\in {\mathcal {Q}}_N\). Then, \(\omega _i,\,i=1,2,\) are infinitely differentiable on \(\big (D_1\setminus \bigcup _\ell ^n D_{r/3}(A_\ell ({{\varvec{n}}}'_0))\big )\times {\mathcal {W}}_{\delta }({{\varvec{n}}}_0')\), by Lemma 4.6. Moreover, by Lemma 4.5, for some \(\delta >0\) the function \(A_\ell \in C^\infty (\mathcal W_\delta ({{\varvec{n}}}_0'))\) and \(A_\ell ({{\varvec{n}}}')\rightarrow A_\ell ({{\varvec{n}}}_0')\) as \({{\varvec{n}}}'\rightarrow {{\varvec{n}}}_0'\) in \({\mathcal {Q}}_N\). Therefore, for \(\delta >0\) sufficiently small, \(|A_\ell ({{\varvec{n}}}')-A_\ell ({{\varvec{n}}}'_0) |<r/3\) for all \({{\varvec{n}}}'\in {\mathcal {W}}_\delta ({{\varvec{n}}}_0')\). It follows that \(\partial D_r(A_\ell ({{\varvec{n}}}'_0))\subset D_1\setminus \cup _{\ell =1}^n D_{r/3}(A_\ell ({{\varvec{n}}}'_0))\) for all such \({{\varvec{n}}}'\). Changing coordinates \(X \mapsto X+A_\ell ({{\varvec{n}}}')-A_\ell ({{\varvec{n}}}'_0)\) yields, for all \(1\le \ell \le n\) and \(j=1,2\), that
Since \(X+A_\ell ({{\varvec{n}}}')-A_\ell ({{\varvec{n}}}'_0)\in D_1\setminus \bigcup _\ell D_{r/3}(A_\ell ({{\varvec{n}}}'_0))\big )\), the functions \((X,{{\varvec{n}}}') \mapsto \omega _i(X+A_\ell ({{\varvec{n}}}')-A_\ell ({{\varvec{n}}}'_0), {{\varvec{n}}}')\) are continuous on \(\partial D_r(A_\ell ({{\varvec{n}}}'_0))\times {\mathcal {W}}_\delta ({{\varvec{n}}}_0')\), whence
Therefore, \({{\varvec{n}}}' \mapsto J(r, {{\varvec{n}}}')\) is continuous at every \({{\varvec{n}}}'_0\in {\mathcal {Q}}_N\).
Next set \(G(r, {{\varvec{n}}}')=D_1\setminus \cup _{\ell =1}^n D_r(A_\ell ({{\varvec{n}}}'))\). Then,
Since, for \(|{{\varvec{n}}}'-{{\varvec{n}}}_0' |\ge r/3\), the closure of \(G(r, {{\varvec{n}}}')\cup G(r, {{\varvec{n}}}_0')\) is a compact subset of \( D_1\setminus \cup _{\ell =1}^n D_{r/3}(A_\ell ({{\varvec{n}}}'_0))\), the functions \(\omega _i(X, {{\varvec{n}}}')\) are uniformly continuous and uniformly bounded on \(G(r, {{\varvec{n}}}')\cup G(r, {{\varvec{n}}}_0')\times {\mathcal {W}}_\delta ({{\varvec{n}}}_0')\). Moreover,
as \({{\varvec{n}}}'\rightarrow {{\varvec{n}}}_0'\). It follows that \(I(r, {{\varvec{n}}}')\) is continuous at every point \({{\varvec{n}}}_0'\in {\mathcal {Q}}_N\). \(\square \)
Lemma 4.9
For \({{\varvec{n}}}' \in {\mathcal {Q}}_N\) and \(Y({{\mathrm{\textbf{n}}}},{{{\textbf {n}}}}')=\{A_\ell ({{{\varvec{n}}}}'): 1 \le \ell \le n\}\),
Proof
To prove (4.12), it suffices to note from (4.2) and Lemma 4.7 that the functions \(\omega _i(\cdot , {{\varvec{n}}}'),\,i=1,2\), are integrable in \(D_1\) for every \({{\varvec{n}}}'\in \mathcal Q_N\).
The proof of (4.13) is more complicated. Fix an arbitrary \({{\varvec{n}}}'\in {\mathcal {Q}}_N\) and recall from (3.12) that
where \({{\textbf{m}}}(X)={{\textbf{U}}}({{\varvec{n}}}') {{\textbf{n}}}(X)\) and the orthogonal matrix \({{\textbf{U}}}({{\varvec{n}}}')\) is defined by (3.8). Therefore, \( {{\textbf{m}}}(A_\ell ({{\varvec{n}}}'))= {{\textbf{U}}}({{\varvec{n}}}'){{\varvec{n}}}'={{\varvec{k}}} =(0,0,1) \) for \(A_\ell ({{\varvec{n}}}')\in Y({{\textbf{n}}},{{\varvec{n}}}').\)
Now fix \(\ell \in \{1,\cdots , n\}\) and to simplify notation change the origin of coordinates in \(D_1\) so that \(A_\ell ({{\varvec{n}}}')=0\), \({{\textbf{m}}}(0)={{\varvec{k}}}\) and, as \(X \rightarrow 0\),
Since \(\varvec{\mu }_i\perp {{\varvec{k}}}\), because \(\textbf{m}(X)\in {\mathbb {S}}^2\), it follows that
With \(A_\ell ({{\varvec{n}}}') = 0\), it follows from Lemma 4.3 (b) that \(\partial _i{{\textbf{n}}}(0)\), and hence \(\varvec{\mu }_i \), \(i=1,2\), are linearly independent. Therefore,
for constants \(c_1,c_2\), and \(r>0\) sufficiently small. It follows that
Since \((1-m_3)(1+m_3)=m_1^2+m_2^2\) and \(m_3(0)=1\),
for all sufficiently small r. Therefore,
as \(r\rightarrow 0\). Hence, similarly, for \(i,j = 1,2\), \(i \ne j\),
Since \(\varvec{\mu }_i\perp {{\varvec{k}}}\) in \({\mathbb {R}}^3\), a linear function \({{\textbf{p}}}:{\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2\) is defined by putting
It follows from (4.14) that
and hence
on \(\partial D_r\) for r sufficiently small. It follows that
Since the integral on the right is independent of r, it can be replaced by the same integral over \(\partial D_1\). Therefore, together (4.15) and (4.16) yield
It remains to calculate the right side of (4.17). Since \(\varvec{\mu }_i,\,i=1,2\), are linearly independent, \(\textbf{p}(\partial D_1)\) is a strictly convex curve with the origin an interior point and the function
is defined and continuous on \(\partial D_1\). Now define polar coordinates on \(\partial D_1\) by \(X_1=\cos (\theta +\theta _0)\), \(X_2=\sin (\theta +\theta _0)\) (\(\theta _0\) to be chosen later) and let \(\Psi (\theta )=\Theta (X(\theta ))\). Then,
Therefore, since \(ds = d\theta \), it follows from (4.17) that
To study \(\Psi \), for \(i=1,2\) let
Since the \(\varvec{\mu }_i\) are linearly independent, so are the \(\varvec{\lambda }_i\) and \(\rho _i>0\). Hence,
Now set \(\theta _0=\beta _1\) to obtain
To calculate \( \int _{[0, 2\pi )} \partial _\theta \Psi (\theta ) d\theta = \lim \nolimits _{\theta {\nearrow 2\pi }}\Psi (\theta )-\Psi (0) \) in (4.18) let
Since \(b(0)= \tan \Psi (0)\) and \(\lim \nolimits _{\theta \nearrow \pi /2}b(\theta )= \pm \infty \) when \(\pm \sin \alpha >0\),
Similarly,
and hence
It follows
Finally,
yields
and
Since \(\Psi (\theta )\) is continuous on \([0,2\pi )\), (4.19)–(4.20) lead to
Since
the signum of \(\sin \alpha \) coincides with the orientation of the triple \(({{\varvec{k}}},\varvec{\mu }_1, \varvec{\mu }_2)\). Therefore, since
and \(\det {{\textbf{U}}}({{\varvec{n}}}')=1\), the orientation of the triplet \(({{\varvec{k}}},\varvec{\mu }_1, \varvec{\mu }_2)\) is the same that of \(({{\textbf{n}}}(0),\partial _1{{\textbf{n}}}(0), \partial _1{\textbf{n}}(0))\) which equals \(\text {sign}\, \big ({{\textbf{n}}}(0)\cdot \partial _1{{\textbf{n}}}(0)\times \partial _2 {{\textbf{n}}}(0)\big )=\text {sign}\, \Phi (A_\ell ({{\varvec{n}}}'))\). It follows that \(\text {sign}~(\sin \alpha )=\text {sign}~\Phi (A_\ell ({{\varvec{n}}}'))\). Combining these result with (4.18) and (4.21) and recalling that \(A_\ell ({{\varvec{n}}}')=0\) leads to the identity
where \(Y({{\textbf{n}}},{{\varvec{n}}}') = \{A_1({{\textbf{n}}},{{\varvec{n}}}'),\cdots A_n({{\textbf{n}}},{{\varvec{n}}}')\}\), from which (4.13) follows. \(\square \)
4.4.1 Proof of Theorem 2.4
Since by Lemma 4.8, \(I(r,{{\varvec{n}}}')\), \(J(r, {{\varvec{n}}}')\) are continuous in \({{\varvec{n}}}'\) on \({\mathcal {Q}}_N\), their limits,
are measurable with respect to \({{\varvec{n}}}'\in {\mathcal {Q}}_N\). By Lemma 4.7, \(|I({0, {{\varvec{n}}}'}) |\) is bounded, and hence integrable on \({\mathcal {Q}}_N\) and, since the measurable function \(\zeta \,\text {sign}~\Phi \) is bounded, it follows from Theorem 4.1 that \(J(0, {{\varvec{n}}}')\) is integrable over \({\mathcal {Q}}_N\).
Now for a Borel set \({\mathcal {A}}\subset {\mathbb {S}}^2\) of positive measure, let
and note from Lemma 4.3(c) that \({\mathcal {A}}_N\) has positive measure for all N sufficiently large. Now, for such N, let \(\Omega _{N,i}:D_1 \rightarrow {\mathbb {R}}\), \(i=1,2\), be given by
and note from Lemma 4.7 that
Moreover, (4.22) and Corollary 4.2, with \(g=4\pi \zeta (X)\text {sign}~\Phi (X)\) and \({\mathcal {A}}\) replaced by \({\mathcal {A}}_N\), imply that
and letting \(r\rightarrow 0\) in (4.11) with Lemma 4.9 yields that
Then, integrating both sides over \({\mathcal {A}}_N\) in the light of (4.22)–(4.24) gives
Next, recall from Lemma 4.3 that \({\mathcal {Q}}_N \subset {\mathcal {Q}}_{N+1}\subset {\mathcal {Q}}_\infty \) where \({\mathcal {Q}}_\infty = \bigcup _N {\mathcal {Q}}_N\) and \( \text {meas}\,({\mathcal {E}})=0\) where \( {\mathcal {E}}=\mathbb S^2\setminus {\mathcal {Q}}_\infty . \) To let \(N\rightarrow \infty \) in (4.25), note that \(\mu _N\rightarrow \mu =\text {meas}\,({\mathcal {A}})\),
where
and
where \({\mathcal {F}}_\infty ={{\textbf{n}}}^{-1}({\mathcal {A}}\cap \mathcal Q_\infty ), \quad {\mathcal {F}}\setminus \mathcal F_\infty ={{\textbf{n}}}^{-1}({\mathcal {E}}\cap {\mathcal {A}}).\) To show that
let \(\{G_n\}\subset {\mathbb {S}}^2\) be a decreasing sequence of open sets such that \({\mathcal {E}} \subset G_n\), \(\text {meas}\, (G_n) \rightarrow 0~\text {as}~ n\rightarrow \infty \) and let \(F_n= {{\textbf{n}}}^{-1}(G_n)\). Then, since \({\mathcal {F}}\setminus {\mathcal {F}}_\infty \subset F_n:= {{\textbf{n}}}^{-1}(G_n)\), it suffices to prove that
Now from Theorem 4.1,
and from Corollary 4.2, since \(\text {meas}\,( G_n) \rightarrow 0\),
This yields (4.28) [and hence (4.27)] and it follows that
Hence, (4.25), (4.26) and (4.29) imply that
where \( {\mathcal {F}}={{\textbf{n}}}^{-1}({\mathcal {A}})\), \( \mu =\text {meas}\,({\mathcal {A}}) \) and, by (4.23),
where, with c an absolute constant,
Let \(\Sigma _\rho =\big \{{{\varvec{n}}}'\in {\mathbb {S}}^2: \text {geodesic distance}~ ({{\varvec{n}}}', {{\varvec{k}}})\le \rho \big \}, \text { where } \text {meas}\,(\Sigma _\rho )=\mu . \) Clearly, \(c^{-1}\sqrt{\mu }\le \rho \le c\sqrt{\mu }\) and
where c is an absolute constant. Thus,
This and (4.30) yield (2.7), (2.8), which completes the proof of Theorem 2.4.
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A Sketch of Theorem 2.4 for \(~\varvec{n >3}\)
A Sketch of Theorem 2.4 for \(~\varvec{n >3}\)
For an immersion of the disc \(D_1\) into \({\mathbb {R}}^n\), \(n > 3\), the normal space is multidimensional, the Gauss map \(\varphi \) takes \(D_1\) into the Grassmannian manifold \({\mathbb {G}}_{n,2}\) and, by Schätzle [18, Proposition 5.1], Kuwert and Li [10] and Li et al. [11], the optimal value of the conformal energy in Hélein’s hypothesis is \(4\pi \). This raises the question of whether there is a version of Theorem 2.4 for \(n>3\). This Appendix addresses that possibility and formulates an analogous result, Theorem A.1. However, the approach which is sketched below is necessarily based on the algebro-geometric approach of Müller and Sv̆erák in [13], and is significantly different from the elementary analytic approach developed in the present work for the case \(n=3\).
The main idea of the Müller–Sv̆erák approach is that a Grassmannian manifold \({\mathbb {G}}_{n,2}\) is isomorphic to a quadric embedded in a complex projective space. In this case, the multidimensional analogue of \(\Phi \) in Theorem 2.4 can be identified with the form \(\varphi ^*\omega \), where \(\omega \) is the Kähler form of complex projective space. For a more precise formulation, recall the basic definitions.
The projective space \({\mathbb {C}}{\mathbb {P}}^{n-1}\) can be regarded as the space of all one-dimensional complex subspaces (complex lines containing the origin) in the complex space \({\mathbb {C}}^{n}\). Every such line is completely determined by a director \(\zeta = (\zeta _0, \dots ,\zeta _{n-1})\in {\mathbb {C}}^{n}\), and clearly two directors, \(\zeta \) and \(\lambda \zeta \), \(0\ne \lambda \in {\mathbb {C}}\), define the same line. However, any vector \(\zeta \) defines only one line, which is denoted by the symbol \((\zeta _0:\zeta _1:\cdots :\zeta _{n-1})\), where the \(\zeta _i\) are its homogeneous coordinates in \(\mathbb C{\mathbb {P}}^{n-1}\). The projective space is endowed with the structure of a smooth manifold by the introduction of a system of local charts \(U_\alpha \), \(0\le \alpha \le n-1\),
\(\psi _\alpha :U_\alpha \rightarrow {\mathbb {C}}^{n-1}\) and the corresponding coordinate mappings
In the notation of Müller and Sv̆erák [13], when \(z^\alpha \in \mathbb C^{n-1}\) defines local coordinates in \(U_\alpha \), the Fubini study metric in \({\mathbb {C}}{\mathbb {P}}^{n-1}\) on \(U_0\) is given by the Hermitian form
and the Kähler form in \({\mathbb {C}}{\mathbb {P}}^{n-1}\) is defined by
(In [13], this is acknowledged as an abuse of notation since the standard Kähler form is \(\omega /2\).) Since the Grassmanian manifold \({\mathbb {G}}_{n,2}\) may then be identified with the quadric
every mapping \(\varphi \) from the disc \(D_1\) to the Grassmanian manifold determines a mapping \(\varphi : D_1\rightarrow Q\subset \mathbb C{\mathbb {P}}^{n-1}\). In this framework, the Liouville equation for the conformal factor is
Here, \(*\varphi ^*\omega =\Omega \), where \(\Omega \,dX_1\wedge dX_2\) is the pullback form defined by
To formulate the analogue of Theorem 2.4 for problem (A.1), recall further notation. For every \(a=(a_0,\ldots , a_{n-1})\in {\mathbb {C}}^n\), denote by \(H_a\) the hyperplane in \( \mathbb C{\mathbb {P}}^{n-1}\) defined in homogeneous coordinates by the relation \( \sum _0^{n-1} a_j\,{\overline{\zeta }}_j=0. \) Then, if \(\widehat{{\mathbb {C}}{\mathbb {P}}}^{n-1}\) denotes the set of hyperplanes in \({{\mathbb {C}}{\mathbb {P}}}^{n-1}\), the mapping \(a\rightarrow H_a\) defines the standard isometry between \(\widehat{{\mathbb {C}}{\mathbb {P}}}^{n-1}\) and \({{\mathbb {C}}{\mathbb {P}}}^{n-1}\), and these manifolds can be identified. Then, the following may be regarded as a multidimensional version of Theorem 2.4.
Theorem A.1
For \(\varphi \in W^{1,2}(D_1, {\mathbb {C}}{\mathbb {P}}^{n-1})\) and a Borel set \(A \subset \widehat{{\mathbb {C}}{\mathbb {P}}}^{n-1}\) with \(\text { vol\,} (A)>0\), let \(G\subset D_1\) be the pre-image under \(\varphi \) of A,
Then, a solution to the boundary-value problem
where \(\pi _G\) is the characteristic function of G, satisfies the estimate
Remark
Since the function \(*\varphi ^*\omega \) may be no more than integrable, a solution f of (A.1) may not be in the Sobolev class \(W^{1,2}(D_1)\). More precisely, \(\nabla f\) belongs to the Lorentz space \(L^{2,\infty }(D_1)\), and \(|\nabla f|^2\) is just a measurable non-integrable function. The same conclusion can be drawn for the solution to the boundary-value problem
However, Theorem A.1 asserts that the difference, \(\nabla f^*=\nabla f-\nabla f_G\), is square-integrable on \(D_1\), and hence the pre-image of every set \(A\subset \widehat{{\mathbb {C}}\mathbb P}^{n-1}\) with positive volume, however small, contains information about the strongest singularities of \(\nabla f\).
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Plotnikov, P.I., Toland, J.F. Refinement of Hélein’s conjecture on boundedness of conformal factors when \(n = 3\). Annali di Matematica 202, 1803–1833 (2023). https://doi.org/10.1007/s10231-023-01302-5
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DOI: https://doi.org/10.1007/s10231-023-01302-5