A frame energy for immersed tori and applications to regular homotopy classes

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1 A frame enery for immersed tori and applications to reular homotopy classes Andrea Mondino, Tristan Rivière Abstract :The paper is devoted to study the Dirichelet enery of movin frames on -dimensional tori immersed in the euclidean 3 m-dimensional space This functional, called Frame enery, is naturally linked to the Willmore enery of the immersion and on the conformal structure of the abstract underlyin surface As first result, a Willmore-conjecture type lower bound is established : namely for every torus immersed in R m, m 3, and any movin frame on it, the frame enery is at least π and equalty holds if and only if m 4, the immersion is the standard Clifford torus up to rotations and dilations), and the frame is the flat one Smootheness of the critical points of the frame enery is proved after the discovery of hidden conservation laws and, as application, the minimization of the Frame enery in reular homotopy classes of immersed tori in R 3 is performed Math Class 3C7, 58E15, 58E3, 49Q1, 53A3, 35R1, 35J35, 35J48, 35J5 I Introduction The purpose of this paper is to study the Dirichlet enery of movin frames associated to tori immersed in R m, m 3 Movin frames have been played a key role in the modern theory of immersed surfaces startin from the pioneerin works of Darboux 9], Goursat 14], Cartan 5], Chern 7]-8], etc note also that in the book of Willmore 4], the theory of surfaces is presented from Cartan s point of view of movin frames, and the recent book of Hélein 16] is devoted to the role of movin frames in modern analysis of submanifolds; see also the recent introductory book of Ivey and Landsber 18]) Indeed, due to the stron link between movin frames on an immersed surface and the conformal structure of the underlyin abstract surface see later in the introduction for more explanations), the importance of selectin a best movin frame in surface theory is comparable to fixin an optimal aue in physical problems for instance for the study of Einstein s equations of eneral relativity it is natural to work in the aue of the so called harmonic coordinates, for the analysis of Yan-Mills equation it is convenient the so called Coulomb aue, etc) Before oin to the description of the main results of the present paper, the objects of the investiation of the this work must be defined Let T be the abstract -torus seen as -dimensional smooth manifold) and let Φ : T R m, m 3, be a smooth immersion let us start with smooth immersions, then we will move to weak immersions) One denotes with T ΦT ) the tanent bundle to ΦT ), a pair e := e 1, e ) ΓT ΦT )) ΓT ΦT )) is said a movin frame on Φ if, for every x T, the couple e 1 x), e x)) is a positive orthonormal basis for T xφt ) with positive we mean that we fix a priori an orientation of ΦT ) and that the movin frame arees with it) Given Φ and e as above we define the frame enery as the Dirichelet enery of the frame, ie F Φ, e) := 1 d e dvol, I1) 4 T where d is the exterior differential alon Φ, dvol is the area form iven by the immersion Φ this can be seen equivalently as the restriction to ΦT ) of the -dimensional Hausdorff measure on R m, or as the volume form associated to the pullback metric := Φ R m) where R m is the euclidean metric on Department of Mathematics, ETH Zentrum, CH-893 Zürich, andreamondino@mathethzch A M is supported by the ETH fellowship Department of Mathematics, ETH Zentrum, CH-893 Zürich, Switzerland 1

2 R m ), and d e is the lenth of the exterior differential of the frame which is iven in local coordinates by d e = k=1 d e k = i,j,k=1 ij xi e k xj e k ; in the paper u v or u, v) denotes the scalar product of vectors in R m Let n be the unit simple m -multivector ivin the normal space to Φ in terms of the Hode duality operator in R m x1φ xφ n := Rm x1φ x, Φ I) In R 3, for instance, it can be written in terms of vector product n := x 1 Φ x Φ x1 Φ x Φ I3) Let π T : R m T ΦT ) and π n : R m N ΦT ) the orthonormal projections on the tanent and on the normal space respectively Recall that the second fundamental form I of the immersion Φ is defined by Iij := π n x ix j Φ) I4) and the mean curvature H is iven by half of its trace H := 1 ij Iij I5) Notice that, writin d e i = π T d e i ) + π n d e i ) = d e i, e i+1 ) e i+1 + π n d e i )-where Z -indeces are used-, the frame enery decomposes as F Φ, e) = 1 e 1 d e dvol + 1 T 4 I dvol I6) T The tanential part F T Φ, e) := 1 T e 1 d e dvol, is equal to the L -norm of the covariant derivative of the frame with respect to the Levi-Civita connection, whereas the normal part corresponds to the Willmore enery after havin applied Gauss Bonnet theorem: W Φ) := H dvol = 1 T 4 I dvol, I8) T where the above defined W is the so-called Willmore functional, we et I7) F Φ, e) = F T Φ, e) + W Φ) I9) Let us observe that the frame enery F is invariant under scalin and under conformal transformations of the metric, but not under conformal transformations of R m to this purpose note that, by definition, the movin frame has to be orthonormal with respect to the extrinsic metric, ie R m, but the norm of the derivative as well as the volume form is computed with respect to the intrinsic metric ) Therefore, even if natural on its own, F can be seen as a more coercive Willmore enery where the extra term F T prevents the deenerations caused by the action of the Moebius roup of R m and the deeneration of the confomal class of the abstract torus More precisely we have the the followin proposition Proposition I1 For every C >, the metrics induced by the framed immersions in F 1, C]) are contained in a compact subset of the moduli space of the torus Let us mention that the proof of Proposition I1 is remarkably elementary and makes use just of the Fenchel lower bound 13] on the total curvature of a closed curve in R m Combinin Proposition I1 with the celebrated results of Li-Yau 1] and Montiel-Ros 5] on the Willmore conjecture, we manae to prove the followin sharp lower bound with riidity) on the frame enery

3 Theorem I1 Let Φ : T R m be a smooth immersion of the -dimensional torus into the Euclidean 3 m-dimensional space and let e = e 1, e ) be any movin frame alon Φ Then the followin lower bound holds: F Φ, e) := 1 d e dvol π I1) 4 T Moreover, if in I1) equality holds then it must be m 4, ΦT ) R m must be, up to isometries and dilations in R m, the Clifford torus T Cl := S 1 S 1 R 4 R m, I11) and e must be, up to a constant rotation on T ΦT )), the movin frame iven by θ, ϕ ), where of course θ, ϕ) are natural flat the coordinates on S 1 S 1 Remark I1 Let us mention that, thanks to I9), in codimension one, the lower bound I1) follows by the recent proof of the Willmore conjecture by Marques and Neves ] usin min-max principle; the approach here is a more direct enery based consideratton Indeed from their result non just the frame enery, but the Willmore functional W Φ) is bounded below by π for any smooth immersed torus, and W Φ) = π if and only if Φ is a conformal transformation of the Clifford torus Curiously, our lower bound seems to work better in codimension at least two, where it becomes sharp and riid; clearly, in codimension one it is not sharp because of the nonexistence of flat immersions of the torus in R 3 and because of the Marques-Neves proof of the Willmore conjecture Let us also mention that Toppin 37, Theorem 6], usin aruments of interal eometry very far from our proof), obtained an analoous lower bound on an analoous frame enery for immersed tori in S 3 under the assumption that the underlyin conformal class of the immersion is a rectanular flat torus For variational matters the framework of smooth immersions has to be relaxed to a weaker notion of immersion introduced by the second author in 3], that we recall below Given any smooth reference metric on T the definition below is independent of the choice of a smooth ), the map Φ : T R m is called weak immersion if the followin properties hold 1 Φ W 1, T, R m ) and the pullback metric Φ := Φ R m is equivalent to, ie there exists a constant C Φ > 1 such that C 1 Φ Φ C Φ Φ as quadratic forms, denoted by n L T, Λ m R m ) the normal space defined ae by I) or more simply in R 3 by I3)), it holds n W 1, T ); or, equivalently, the second fundamental form I defined ae in I4) is L interable over T The space of weak immersions Φ from T into R m is denoted by ET, R m ) Recall also that iven a weak immersion, up to a local bilipschitz diffeomorphism, we can assume it is locally conformal so it induces a smooth conformal structure on the torus this result is a consequence of a combination of works of Toro 38]-39], Müller-Sverak 6], Hélein 16] and the second author 3]; for a comprehensive discussion see 3]) Let us remark that Proposition I1 and Theorem I1 holds for weak immersions as well for more details see Section V) In order to perform the calculus of variations of the frame enery, in Section III1 we establish that the Frame enery is differentiable in ET, R m ) and we compute the first variation of the tanential frame enery F T which, combined with the first variation of the Willmore functional 8] and with I9), ives the first variation of the frame enery F As for the Willmore enery as well as for many important eometric problems as Harmonic maps, CMC surfaces, Yan Mills, Yamabe, etc) the equation we obtain is critical It is therefore challenin to prove the reularity of critical points of the frame enery 3

4 Inspired by the work of Hélein 16] on CMC surfaces and of the second author on Willmore surfaces 8] see also 4] for the manifold case and 3] for a comprehensive discussion), in order to study the reularity of the critical points of the frame enery we discover some new hidden conservation laws: in Subsection III we find some new identities for eneral weak conformal immersions, and then in Subsection III3 we use these identities in order to deduce a system of conservation laws satisfied by the critical points of the frame enery In particular, this system of conservation laws yields to an elliptic system involvin Jacobian nonlinearities which can be studied usin interability by compensation theory for a comprehensive treatment see 9]) Thanks to this special form, we are able to show reularity of the solutions of this critical system, namely we prove the followin result Theorem I Let Φ be a weak immersion of the disc D into R 3 and let e = e 1, e ) be a movin frame on Φ such that Φ, e) is a critical point of the frame enery F Then, up to a bilipschitz reparametrization we have locally that Φ is conformal and e is the coordinate movin frame associated to Φ, ie e 1, e ) = x1 Φ x1 Φ, x ) Φ x Φ Moreover, there exist ρ, 1) such that Φ Bρ) is a C immersion Let us observe that for sake of simplicity of the presentation, this work is more focused in the codimension one case; the hiher codimensional case will be the object of a forthcomin paper, many of the aruments carry on in a similar way Now let us discuss an application of the tools developed in this paper to the study of reular homotopy classes of immersions Let us first recall some classical facts about reular homotopies, startin from the definition: iven a smooth closed surface Σ, two smooth immersions f, : Σ R m are said reularly homotopic if there exists a smooth map H : Σ, 1] R m, called reular homotopy between f and, such that H, ) = f ), H, 1) = ) and H t ) := H, t) : Σ R m is an immersion for every t, 1], everythin up to diffeomorphisms of Σ In his celebrated paper 35] of 1958, Smale proved that any couple of smooth immersions of the - sphere into R 3 are reularly homotopic, ie homotopic via a one parameter family of immersions see also 36] for the hiher dimensional results) The same is not true for immersions of the -sphere in R 4 where indeed there are countably many reular homotopy classes An year later, Hirsch 17] eneralized the ideas of Smale to arbitrary submanifolds and in particular he proved that the reular homotopy classes of immersions of any fixed smooth closed surface in a Euclidean space of codimension hiher than two trivialize, ie every two immersions of a fixed surface are reularly homotopic this follows from the fact that the second homotopy roup of the Stiefel manifold V R m ) is null for m 5 ) Remarkably, the case of tori immersed in R 3 differs from the one of the spheres Indeed, as proved by Pinkall in ], there are exactly two reular homotopy classes of immersed tori in R 3 : the standard one the one of a classical rotational torus, say ) and the nonstandard one a knotted torus, for an explicit example we refer to 7]) One could address the question of a canonical rapresentant for each of the two classes As an application of the tools developed in this paper, we prove the existence of a smooth minimizer of the frame enery within each of the two reular homotopy classes; such a minimizer can be seen as a canonical raprensentant of its reular homotopy class It is proven below that the notion of reular homotopy class extend to the eneral framework of weak immersions see Proposition V7) The followin is the last main result of the present paper Theorem I3 Fix σ a reular homotopy class of immersions of the -torus T into R 3 Then there exists a smooth conformal immersion Φ : T R 3, with Φ x1 Φ σ, such that, called e := e 1, e ) := x1 Φ, x ) Φ x Φ the coordinate movin frame, the couple Φ, e) minimizes the frame enery F amon all weak immersions of T into R 3 lyin in σ and all W 1, movin frames on ΦT ): F Φ, { e) = min F Φ, e) : Φ ET, R 3 ), } Φ σ, e W 1, T ) I1) 4

5 Let us conclude the introduction with some comment and open problems As already observed in Remark I1, from Theorem I1 it follows that the lobal minimizer of the frame enery for weak) immersions of T into R m, for m 4, is the Clifford torus; instead it is still an open problem to identify who is the minimizer for immersions into R 3 We expect it to be the Clifford torus as well We also expect the minimizer of the nonstandard reular homotopy class of immersed tori into R 3 to be the diaonal double cover of the Clifford torus proposed by Kusner in the framework of the Willmore problem, pae 333] Both of these are open problems, as well as the existence of a minimizer of the frame enery amon reular homotopy classes of tori immersed into R 4 ; indeed, in codimension two, there is an extra difficulty iven by the possibility of havin loss of homotopic complexity in the concentration points of the frame enery to exclude this, in our arument we use Lemma VI9, which is not true in codimension two) Notice finally that in codimension reater or equal to three, by the aforementioned result of Hirsch, there is just one reular homotopy class of immersed tori, and by Theorem I1 the lobal minimizer is the Clifford torus, up to isometries and rescalins, with riidity The paper is oranized as follows Section II is devoted to the proofs of Proposition I1 and Theorem I1, namely the bound on the conformal class and the lower bound with riidity) on the frame enery In Section III it is established the system of conservation laws satisfied by the critical points of the Frame enery; more precisely, in Subsection III1 we establish the Frechet differentiability of F in the space of weak immersion and compute the first variation formula, in Subsection III we discover some eneral conservation laws associated to a eneral weak conformal immersion, and in Subsection III3 these conservation laws are used to obtain a system of conservation laws involvin jacobian quadratic non linearities satisfied by the critical points of the Frame enery In Section IV the peculiar form of the aforementioned system is exploited in order to deduce the reularity of the critical points of the Frame enery via the theory of interability by compensation, namely Theorem I is proved In Section V the above tools of the calculus of variations are applied to prove the existence of a minimizer of the Frame enery in reular homotopy classes, namely Theorem I3 Finally in the Appendices we recall some classical eometric computations in conformal coordinates used in Section III, a Lemma of functional analysis used in the proof of the reularity theorem, and a Lemma of differential topoloy used in the proof of Theorem I3 II A lower bound -with riidity- for the frame enery in R m, the analoue of the Willmore conjecture II1 Reduction to conformal immersions of flat tori and coordinate movin frames Let Φ : T R m be a smooth immersion of the torus into the euclidean 3 m-dimensional space The oal of this section is to prove Lemma II1: namely to reduce the problem of calculatin the infimum of the frame enery amon all smooth immersions of T into R m and all movin frames, to the case of coordinate movin frames associated to smooth conformal immersions of tori lyin in the moduli space of conformal structures We will proceed with consecutive reductions Reduction 1: e satisfies the Coulomb condition Since in this section we are interested in ivin a lower bound on the frame enery F, we can assume that the frame e minimizes its tanential part F T := 1 e T, d e 1 dvol ; this is equivalent to say that e is a Coulomb frame, ie it satisfies the Coulomb condition d e 1, d e ) =, II13) which reads in local isothermal coordinates as div e 1, e ) = for more details about Coulomb frames and the Chern method see 16] or 3]) Reduction : e is a coordinate movin frame Recall that, by usin the Chern movin frame method and the fact that e is Coulomb, we can can cover the torus T by finitely many balls {B k } k=1,,n such that for every ball there exists a diffeomorphism f k : B k B k such that Φ f k is a smooth conformal 5

6 immersion of B k into R m and e j = x j Φ f k ) xj Φ f k ), II14) ie the movin frame e is the coordinate movin frame associated to the smooth conformal immersion Φ Reduction 3: the reference torus is flat Now the local conformal coordinates on T define a smooth conformal structure on T therefore, by the Uniformization Theorem, there exists a diffeomorphism ψ from a flat torus Σ ie Σ is the quotient of R modulo a Z lattice) into our T such that f 1 k ψ is a conformal diffeormorphism; it follows that Φ ψ = Φ f k f 1 k ψ is a smooth conformal immersion of Σ into R m Moreover, recallin that the property of bein Coulomb for a movin frame is invariant under conformal chanes of metric this property is a direct consequence of equation II13) and of the invariance of the Hode operator under conformal chanes of metric), we et that e ψ is a Coulomb movin frame on Σ Now observe that we have another natural Coulomb frame on the flat torus Σ iven by the conformal immersion Φ ψ, namely f j = y j Φ ψ) yj Φ ψ), satisfyin f 1 d f = dλ, II15) where y 1, y are standard coordinates on Σ and λ = lo yi Φ ψ) ) is the conformal factor Clearly, we can the two movin frames e and f via a rotation in the tanent space, ie e 1 + i e = e iθ f 1 + i f ) for some smooth function θ : Σ S 1 Observe that, usin II15) and interatin by parts we et F T Φ, e) = e 1 d e dy = f 1 df dθ dy = f 1 df + dθ < f 1 df, dθ > dy Σ Σ Σ = f 1 df + dθ < dλ, dθ > dy = f 1 df + dθ dy Σ F T Φ, f), II16) with equality if and only of e is a constant rotation of f this will be useful to prove the part of the riidity statement involvin the frame) Reduction 4: the flat torus lies in the moduli space of conformal structures As last reduction, we want to reduce the problem to the case when Σ is a flat torus in the canonical moduli space M of conformal structures of tori composed by the parallelorams in R whose edes are 1, ) and τ = τ 1, τ ) M where M is the strip M := {τ = τ 1, τ ) R : τ >, 1 < τ 1 < 1, τ 1 and τ 1 if τ = 1} II17) Indeed a classical result of Riemann surfaces see for instance 19, Section 7]) says that up to composition with a linear transformation which preserves the orientation more precisely up to composition with a projective unimodular trasformation in P SL, Z)), the conformal structure of our flat torus Σ is isomorphic to the one of a flat torus described by the paralleloram iven by 1, ) and τ 1, τ ) M, where M was defined in II17) We can finally summarize the discussion in the followin lemma Lemma II1 Let T be the abstract torus ie the unique smooth orientable -dimensional manifold of enus one) Call β1 m the infimum of the frame enery F Φ, e) amon all smooth immersions Φ of T into R m and all the movin frames e = e 1, e ) alon Φ Denote also β m the infimum of the frame enery F Φ, f) amon all smooth conformal immersions Φ of any flat torus Σ described by any lattice in R of the form 1, ), τ 1, τ ))-where τ 1, τ ) M is defined in II17); here f is the coordinate movin frame associated to Φ, ie f j := xj Φ/ xj Φ, j = 1,, x 1, x ) bein flat coordinates on Σ Then β1 m = β m In other words, in order to compute the infimum of F amon all smooth immersions of tori and all movin frames, it is enouh to restrict to coordinate movin frames associated to smooth conformal immersions of flat tori lyin in M, the moduli space of conformal structures of tori Σ 6

7 II Proof of Theorem I1: lower bound and riidity for the frame enery From now on we will work with a torus as in the reduced case: Σ = R /Z τz) is the flat quotient of R by the lattice enerated by the two vectors 1, ), τ 1, τ )-where τ 1, τ ) M is defined in II17); we will denote π θ Σ := arccos τ 1 3, π ), II18) 3 where the interval π 3, ) π 3 comes directly from the definition of M as in II17) One of the key technical results of this paper is the control of the conformal class in terms of the frame enery, namely Proposition I1; this is implied by the followin lower bound Proposition II Let Σ = R /Z τz), with τ M as above, be a flat torus Let Φ : Σ R m, m 3, be a smooth conformal immersion and let e be the coordinate frame associated to Φ: e j := xj Φ/ xj Φ, j = 1,, x 1, x ) bein flat coordinates on Σ Then the followin lower bound holds true ) ] e 4λ 1 + cos4 θ I Σ sin 11 + sin θ I + 4 cos θ I cot θ ) d e1 e 1, e ) + d e e, e 1 ) ] dvol θ 4π τ + 1 ), τ where θ := θ Σ := arccos τ 1 π 3, π 3 ) II19), dvol is the area form on Σ induced by the pullback metric = Φ R m, and λ = lo xi Φ ) is the conformal factor In particular II19) implies the followin lower bound on the frame enery of Φ, e): F Φ, e) := 1 4 Σ d e dvol π τ + 1 ) sin ) θ τ sin θ + cos 4 θ II) Proof of Proposition II First of all recall that by the classical Fenchel Theorem the oriinal proof of Fenchel 13], see also 1], was for closed curves immersed in R 3 The result was eneralized to immersions in R m, m 3, by Borsuk 4] with a different proof), iven a smooth closed curve γ : S 1 R m one has k ds π, II1) γ where k := d ds γs) is curvature of γ-here s is the arclenht parameter The stratey is to apply Fenchel Theorem to the curves γ x := Φγ x )), γ y := Φγ y )) where γ x ) :, τ ] Σ and γ y ) :, 1] Σ are iven by γ x t) := x + t cot θ, t), γ y t) := y cot θ + t, y), II) for every x, 1] and y, τ ] Notice that γ x and γ y are nothin but the parallel curves of the vectors eneratin the lattice of Σ Now applyin Fenchel theorem to γ y, recallin that Φ is conformal with λ := lo x Φ = lo y Φ so that x Φ = e λ e 1, we have π k ds = γ y L γy) d ds γ L γ y) y ds = d e1 e 1 ds = 1 d e1 e 1 e λ dx, where L γ y ) is of course the lenth of the curve γ y ) Squarin the above inequality, usin Cauchy-Schartz and interatin with respect to y, τ ] ives 4π τ τ 1 d e1 e 1 e λ dxdy = Σ d e1 e 1 dvol II3) Analoously, called e θ := cos θ e 1, sin θ e ), observin that γ x / γ x = e θ we et L γx) π k ds = d γ x ds γ L γ x) x ds = d e θ e θ ds = 1 τ d sin θ e θ e θ e λ dy 7

8 Aain, squarin the above inequality, usin Cauchy-Schartz and interatin with respect to x, 1] ives 4π 1 1 τ τ sin d e θ θ e θ e λ dy dx = 1 sin d e θ θ e θ dvol II4) A straihtforward computation usin the definition of e θ ives d e θ e θ = π n d e θ e θ ) ) ) + d e e θ θ, e 1 + d e θ e θ, e II5) = e 4λ cos 4 θ I 11 + sin 4 θ I + 4 sin θ cos θ ] I 1 + cos θ d e1 e 1, e ) + sin θ d e e, e 1 ) Σ Combinin II4) and II5) we obtain cos e 4λ 4 ] θ I sin 11 + sin θ I + 4 cos θ I 1 θ Σ + cot θ d e1 e 1, e ) + d e e, e 1 ) ] dvol 4π τ II6) Observin that d e1 e 1 = e 4λ I 11 + d e1 e 1, e ) and puttin toether II4) with II6) ives the first claim II19) In order to obtain II), let us recall that d e := d ei e j = e 4λ I 11 + I + ] I 1 + d e1 e 1, e ) + d e e, e 1 ) II7) i,j=1 Observe also that by the definition of M as in II17), we have θ π/3, π/3], therefore we et that 4 cos θ 1 and 1 + cot θ 4 < Now, combinin the last trionometric estimates with II) and II7), we conclude that II) holds true Now we can prove the lower bound and the riidity statement) for the frame enery of immersed tori in arbitrary codimension, namely Theorem I1 Notice the analoy with the Willmore conjecture proved by Marques-Neves in codimension one but still open in arbitrary codimension) Proof of Theorem I1 First of all, thanks to Lemma II1 we can assume that the reference torus is flat so, followin the notations above, it will be denoted with Σ) and is iven by the quotient of R via the Z lattice enerated by the vectors 1, ), τ 1, τ ) with τ 1, τ ) M defined in II17), the immersion Φ : Σ R m is conformal, e is the coordinate movin frame associated to Φ: e i = x i Φ xi Φ Once this reduction is perfomed, we proved in Proposition II that the lower bound II) holds, namely F Φ, e) := 1 d e dvol π τ + 1 ) sin ) θ 4 τ sin, II8) θ + cos 4 θ Σ where θ = θ Σ = arccos τ 1 Notice that the expression above is symmetric with respect to the map τ 1 τ 1, so it is enouh to consider τ 1, τ ) M +, where M + := M {τ 1 } From now on we denote with f : M + R the function ) fτ, θ) := τ + 1τ sin ) θ sin II9) θ + cos 4 θ A first attempt would be to prove that f is bounded below by on the whole M + However, by an elementary computation, it is easy to check that fτ = 1, θ = π/) = and fτ = sin θ, θ) < for θ π/3, π/) ; or, in other words, except for τ =, 1), on the arc of circle S 1 M + we always have f < 8

9 In order to overcome this difficulty let us recall that, by equation I9), we can write F Φ, e) = F T Φ, e) + W Φ) II3) where W Φ) = Σ H Φ dvol is the Willmore functional of Φ and F Φ T -defined in I7)-is a non neative functional Let us denote { Ω LY MR := τ 1, τ ) : τ 1 1 ) } + τ 1) 1 M +, 4 and recall that if τ Ω LY MR the Willmore conjecture holds true see 5, Corollary 7]; this remarkable result of Montiel and Ros extend a previous celebrated result of Li and Yau 1]), namely one has W Φ) π for every smooth conformal immersion Φ : R /Z τz) with τ Ω LY MR II31) A direct computation shows that f ΩLY MR with equality if and only if τ = 1 and θ = π, II3) { where, of course, Ω LY MR = τ 1, τ ) : } τ 1 ) 1 + τ 1) = 1 4 M + Observin that the function τ fτ, θ) is monotone strictly increasin for τ 1, the lower bound II3) implies that f M + \Ω LY MR with equality if and only if τ = 1 and θ = π II33) The claimed lower bound for the frame enery I1) follows then by combinin on one hand II8), II9) II33) and on the other hand II3) with II31) Now let us discuss the riidity statement From the work of Montiel and Ros in particular combinin Corollary 6 and the estimate 111) in 5]), we already know that W Φ) > π for every smooth conformal immersion Φ : R /Z τz) with τ Ω LY MR, II34) where Ω LY MR is the interior of the reion Ω LY MR as subset of M + Therefore, combinin on one hand II8), II9) II33) and on the other hand II3) with II34) we et that if F Φ, e) = π then Φ is a smooth conformal embeddin recall that if Φ has self intersection then by 1] one has W Φ) 8π) of the flat square torus-ie τ = 1, θ = π - into Rm At this point the riidity statement would follow by the work of Li-Yau 1], where they prove that the Clifford torus is the unique minimizer of the Willmore enery in its conformal class In any case, below, we wish to ive an elementary proof of the riidity Observin that the flat square torus lies in Ω LY MR, we have aain by II31) that π = F Φ, e) = W Φ) + F T Φ, e) π + F T Φ, e), and since F T is non-neative we obtain F T Φ, e) = 1 e 1 d e dvol,1] Φ e 1 d e on, 1] II35) A simple computation shows that e 1 d e = dλ which, in our settin writes more easily as e 1 e = λ), where λ = lo x1 Φ) is the conformal factor Therefore the conformal factor of the immersion Φ is constant and, up to a scalin in R m, Φ is actually an isometric embeddin of the square torus into R m At this point, repeatin the proof of Proposition II we observe that now θ = so the curves γ x and γ y are the coordinate curves Moreover equality must hold in Fenchel Theorem II1); it follows that γ x, γ y are planar convex curves with curvatures respectively k x ), k y ) Since also in the Schwarz inequality brinin respectively to II3) and II4) there must be equality, it follows that the curvatures k x ), k y ) are constant, so γ x and γ y are two planar circles of constant radius one whose plane may depend on x 9

10 and y respectively Finally, we claim that the plane is indenpedent of x and y Indeed, since θ = and e 1 d e = e d e 1 = by II35), the estimate II19) reduces to π 1 I I] dvol II36),1] e 4λ But, on the other hand, usin I6), II35) and the enery assumption, we have that π = F Φ, e) = 1 4 I dvol = 1 e 4λ I,1] I + I1] dvol,1] II37) Combinin II36) and II37) yields I 1 which, combined with II35), implies that d dx γ x y) = x 1x Φ = xx d 1Φ = dy γ y x) on, 1] We conclude that the plane where γ x resp γ y ) lies does not depend on x resp y) and then, up to rotations, Φ, 1] ) = S 1 S 1 R 4 R m Let us conclude by discussin the riidity of the frame From the discussion of Subsection II1 it should be clear that if Φ, e) attains the minimal value π, then the frame e must be a Coulomb frame this is because in particular it minimizes the tanential frame enery F T ); moreover, called f ) := θ, ϕ the flat coordinate movin frame on S 1 S 1, the estimate II16) ives that F T Φ, e) F T Φ, f) with equality if and only if e is a constant rotation of f This was exactly our claim III Geometric systems of conservation laws associated to the Frame enery III1 First variation formula for the frame enery For simplicity of presentation, and since our applications are in in codimension one, here we present the formulas of the first variation to the frame enery for weak conformal immersions in the euclidean three space R 3 ; the hiher codimensional computations are similar but notationally more involved and can be performed alon the same lines as in 8] Since by equation I9) the frame enery is the sum of the tanential frame enery F T and of the Willmore enery W, and since the first variation formula for W is well known see for instance 4] for the classical form of the equation, and 8]-3] for the diverence form in R m, and 4] for the diverence form in Riemannian manifolds), here we compute the first variation of the tanential frame enery F T This is the content of the next proposition Before statin it let us introduce some notations Let Φ ED, R 3 ) be a weak conformal immersion, λ = lo x1φ ) = lo xφ ) the conformal factor and e := e 1, e ) = e λ x1φ, xφ) the associated orthonormal frame For any smooth vector field w Cc R 3, R 3 ), we call Φ t x) := Φx) + t w Φx)) ED, R 3 ) the perturbed weak immersion and we consider the followin orthonormal frame associated to Φ t : e 1,t := e n t = e 1 + t e λ x1 w, n) n + ot), III38) where, in the second equality, we used VI115); the second vector of the frame is therefore e,t = e 1,t n t = e + t e λ x w, n) + ot) III39) Proposition III3 Let Φ ED, R 3 ) be a weak conformal immersion, λ = lo x1 Φ ) = lo x Φ ) the conformal factor and e := e 1, e ) = e λ x1 Φ, x Φ) the associated orthonormal frame Then, for any smooth perturbation w C c R 3, R 3 ) with w Φ D ) =, called Φ t x) := Φx) + t w Φx)) ED, R 3 ) the perturbed weak immersion and e t := e 1,t, e,t ) the associated movin frame 1

11 defined in III38)-III39), it holds d dt F T Φ t, e t )) := d ] 1 e,t d e 1,t dt t dvol t ) D ] ] = w, d I e, d e 1 ) + d e, d e 1 ) e, d e 1 ) 1 e d e 1 D dφ ]) = w, div I D e, e 1 ) e, e 1 ) e, e 1 ), Φ ]) + 1 e, e 1 ) Φ III4) where in the last formula we use the followin notation: div is the diverence in R with euclidean metric, = x, x1 ), u, v) or u v denotes the euclidean scalar product in R 3 and < u, v > denotes the scalar product in D, ), where = Φ R 3 is the pullback on D of the euclidean metric in R 3 In the second formula, denotes the Hode duality with respect to, d is the Cartan differential, and is the restriction of forms with respect to Proof Usin the expression of e t iven in III38)-III39) we compute d dt e,t, d e 1,t ) = d ij t e,t, xi e 1,t ) e,t, xj e 1,t ) dt ij, We have in one hand e,t, xi e 1,t ) = e, xi e 1 ) + t e λ x w, n) xi e 1, n) + t e λ x1 w e, xi n) Hence usin VI118) we obtain = e, xi e 1 ) + t x w, n) I i1 e λ t x1 w, n) I i e λ + ot) III41) d dt e,t, d e 1,t ) t ) = e 4λ ij ] xiφ, xj w) + xj Φ, xi w) e, xi e 1 ) e, xj e 1 ) +e 4λ e, xi e 1 ) x w, n) I 1i x1 w, n) I i ] i = e, d e 1 ) e, d e 1 )] dφ, d w I e, e 1 ), d w III4) Combinin now the variation of the volume form VI119)-computed in the appendix-and III4), we obtain d dt F T Φ t, e t )) = e, d e 1 ) e, d e 1 ) 1 ] ] D e, d e1 dφ, d w I e, e 1 ), d w dvol The thesis follows with an interation by parts recallin that by assumption w Φ D ) = III Some eneral conservation laws for conformal immersions The oal of the present section is to prove the followin Proposition Proposition III4 Let Φ ED, R 3 ) be a weak conformal immersion, λ = lo x1φ ) = lo xφ ) the conformal factor and e := e 1, e ) = e λ x1φ, xφ) the associated coordinate orthonormal frame Then the followin identities hold dφ) e I e d e 1 ) + d e 1 e d e 1 1 e d e 1 ] dφ )] = III43) and dφ) e I e d e 1 ) + d e 1 e d e 1 1 e d e 1 ] dφ )] = < dλ, dd > III44) where dd = I dφ n III45) 11

12 The rest of the section is devoted to the proof of Proposition III4 We start by computin dφ) e I e d e 1 ) + d e 1 e d e 1 1 e d e 1 ] dφ )] e = d e 1 e d e 1 1 e d e 1 ], dφ d Φ III46) = e d e 1 e d e 1 1 e d e 1 ], = where the upper dot means a contraction in the R 3 coordinates; this ives the first part of the thesis, namely III43) The second identity of the thesis is more subtle We compute dφ) e I e d e 1 ) + d e 1 e d e 1 1 e d e 1 ] dφ )] = e 4λ xφ I11 e x1 e 1 ) + ) I 1 e x e 1 ) + e 4λ x1φ I1 e x1 e 1 ) + ) I e x e 1 ) + dφ e d e 1 e d e 1 1 e d e 1 ] dφ ) III47) An elementary computation ives dφ e d e 1 e d e 1 1 e d e 1 ] dφ ) = III48) Hence for the moment the followin formula is not used but it will be useful later in the section) dφ) e I e d e 1 ) + d e 1 e d e 1 1 e d e 1 ] dφ )] = dφ) n, I e d e 1 ) Usin that e d e 1 = dλ = 1 eλ de λ, from III47) we et dφ) e I e d e 1 ) + d e 1 e d e 1 1 e d e 1 ] dφ )] III49) = 1 e λ x Φ n I11 x e λ + I 1 x1 e λ) 1 e λ x1 Φ n I1 x e λ + I x1 e λ) = 1 e λ x Φ n I 11 x e λ + I 1 x1 e λ + H x λ ) 1 e λ x1 Φ n I 1 x e λ I 11 x1 e λ H x1 λ ) So we have established the followin dφ) e I e d e 1 ) + d e 1 e d e 1 1 e d e 1 ] dφ )] = 1 e λ xφ n x I 11 e λ ) + x1 I 1 e λ ) ] 1 e λ x Φ n x I 11 e λ x1 I 1 e λ + H x λ ] III5) III51) 1 e λ x1 Φ n x I 1 e λ ) x1 I 11 e λ ) ] 1 e λ x1 Φ n x I 1 e λ + x1 I 11 e λ H x1 λ ) Usin the expression of H VI14) and Codazzi identity VI16), we obtain dφ) e I e d e 1 ) + d e 1 e d e 1 1 e d e 1 ] dφ )] = 1 e λ x Φ n x H R + x 1 H I] 1 x Φ n x e λ H) III5) 1 e λ x1 Φ n x H I x 1 H R] 1 x1 Φ n x1 e λ H) 1

13 Hence, usin that the first two equalities follow from VI14) and the others from the definintion of I) x1 e λ x1φ) x e λ xφ) = H R, x1 e λ xφ) + x e λ x1φ) = H I, x1φ x1 n = xφ x n = e λ I 1 x1φ xφ, xφ x1 n = e λ I 11 x1φ xφ, x1φ x n = e λ I x1φ xφ, we et H I ] xφ x1 n + x1φ x n + HR x Φ x n HR x 1Φ x1 n = e λ H I I 11 I ) x1 Φ x Φ + e λ H R I 1 x1 Φ x Φ =, III53) since aain HR = 1 e λ I 11 I ) and HI = eλ I 1 We then deduce dφ) e I e d e 1 ) + d e 1 e d e 1 1 e d e 1 ] dφ )] = x1 e λ x1 Φ H H R ] + e λ x Φ H I ] ] x e λ x1φ H I + e λ xφ H + H R ] Thus usin III49) we have proved so far dφ) e I e d e 1 ) + d e 1 e d e 1 1 e d e 1 ] dφ )] = dφ) n, I e d e 1 ) = x1 e 4λ x1 Φ I x Φ I1 ]] x e 4λ x1 Φ I1 + x Φ I11 ]] III54) III55) Now we want to use the second equality in order to write the riht hand side in a more convenient way Observe that on one hand dφ) n, I e d e 1 ) = I 1 x λ x1φ n e 4λ + I x1 λ x1φ n e 4λ I 1 x1 λ x Φ n e 4λ + I 11 x λ x Φ n e 4λ III56) On the other hand it holds x1 e 4λ x1 Φ I x Φ I1 ]] + x e 4λ x1 Φ I1 + x Φ I11 ]] = e λ x1 e λ x1 Φ I x Φ I1 ]] + e λ x e λ x1 Φ I1 + x Φ I11 ]] I x1 λ x1 Φ n e 4λ + I 1 x1 λ x Φ n e 4λ III57) + I 1 x λ x1 Φ n e 4λ I 11 x λ x Φ n e 4λ Combinin III56) and III57) ives x1 e 4λ x1 Φ I x Φ I1 ]] + x e 4λ x1 Φ I1 + x Φ I11 ]] = e λ x1 e λ x1 Φ I x Φ I1 ]] + e λ x e λ x1 Φ I1 + x Φ I11 ]] dφ) n, I e d e 1 ) III58) 13

14 Thus puttin toether this time the second equality of III55) and III58) we have obtained the followin lemma Lemma III For any smooth conformal immersion the followin identity holds e λ x1 e λ x1 Φ I x Φ I1 ]] + e λ x e λ x1 Φ I1 + x Φ I11 ]] =, which also implies x1 e λ x1 Φ I x Φ I1 ]] = x e λ x1 Φ I1 + x Φ I11 ]] III59) III6) Let D be the two vector iven by x1d ] := e λ x1φ I1 + xφ I11 x D := e λ x1 Φ I x Φ I1 ] III61) Usin D, we can rewrite III55) as dφ) e I e d e 1 ) + d e 1 e d e 1 1 e d e 1 ] dφ )] ] ] = x1 e λ xd x e λ x1d III6) We conclude observin that ] x1 e λ xd) x e λ x1d) = e λ x1 λ xd x λ x1d = < dλ, dd > Pluin III63) into III6) we obtain III44) III3 III63) A System of conservation laws involvin Jacobian nonlinearities for the critical points of the Frame enery Observe that the -vector D defined in III45) notice that D is unique up to constants and in the followin we will just be interestes in D), usin the more standard notation in R 3 of vector product instead of the wede product of vectors, can be identified and we will do it) with the vector defined by D = I Φ n III64) Notice that D L D ) Let us start by noticin that if the pair Φ, e), where Φ is a weak immersion of D into R 3 and e is a movin frame on Φ, is a critical point for the frame enery F then, up to a reparametrization, we can assume that e is the coordinate movin frame associated to Φ, ie e = e 1, e ) = Φ x 1 Φ, x Φ) This is the content of the next lemma Let us also explicitely remark that when we say that Φ, e) is a critical point of F we mean with respect to every normal perturbation of Φ and any rotation of the movin frame e Lemma III3 Let Φ be a weak immersion of D into R 3 and e a movin frame on Φ we stress that here e can be any movin frame, ie we do not assume a priori that e = e 1, e ) = Φ x 1 Φ, x Φ)) Assume that Φ, e) is critical for the frame enery F Then there exists a bilipshitz diffeomorphism ψ : D D such that the new weak immersion Φ := Φ ψ is conformal and e is the coordinate movin frame associated to Φ, ie e = e1, e ) = Φ x1 Φ, x Φ) 14

15 Proof For the moment fix Φ From the criticality of the movin frame e for the frame enery F- or equivalently for the tanential frame enery F T -with respect to rotations ie with respect to variations of the type e t = e itθ e for θ, π]), a simple computation shows that the frame e satisfies the Coulomb condition div e 1 e ) = At this point the existence of the reparametrization ψ can be performed usin the so called Chern movin frame methos and it is well known see for instance 3]) From now on, if Φ, e) is a critical point of the frame enery F, we will always assume that e is the coordinate orthonormal frame associated to Φ; this is not restrictive, up to bilipschitz reparametrizations of Φ, thanks to Lemma III3 Therefore when sayin that Φ is a critical point of F we will mean that Φ, e) is critical, where e is the coordinate orthonormal frame associated to Φ We remind that by identity I9), we have F Φ, e) = F T Φ, e) + W Φ); combinin the first variation formula of the tanential frame enery F T computed in III4) with the first variation of the Willmore functional W we obtain the one of F Let us recall that the first variation of the Willmore functional W Φ) := Φ H dvol on a weak conformal immersion Φ of a disk D into R 3 with respect a smooth vector field w Cc R 3, R 3 ) with w Φ D ) = is iven by for more detail see 3]) so that, we obtain d F Φ dt + t w) = t= d W Φ dt + t w) = t= D div 1 So we obtained the followin proposition 1 D div H 3 H n + n ] H w, III65) H 3 H n + n H ) III66) I e e 1 ) e e 1 e e 1, Φ) + 1 e e 1 Φ ] w Proposition III5 Let Φ be a weak conformal immersion of the disk D into R 3 Then Φ is a critical point of the frame enery F if and only if 1 div H 3 H n + n H ) I e e 1 ) e e 1 e e 1, Φ) + 1 ] e e 1 Φ = III67) Lemma III4 Let Φ be a weak conformal immersion of the disc D into R 3 critical for the frame enery F Then there exists a vector field L F L, loc D ) with L F L 1 D ) such that LF = 1 H 3 H n + n H ) I e e 1 ) e e 1 e e 1, Φ) + 1 e e 1 Φ Proof By the first variation formula III66), we know that if Φ is critical for the frame enery then 1 div H 3 H n + n H ) III68) I e e 1 ) e e 1 e e 1, Φ) + 1 e e 1 Φ ] = So by the weak Poincaré Lemma more precisely, one takes successively the convolution of the of the diverence free quantity above with the Poisson kernel π) 1 lo r, then takin the diverence and finally subtractin some harmonic vector field one ets the conclusion; for more details see the beinnin of the proof of 31, Theorem VII14]; in particular, the L, reularity of L F follows by a classical result of Adams 1] on Riesz potentials) there exists L F as desired Lemma III5 Let Φ be a weak conformal immersion of the disc D into R 3 critical for the frame enery F and let L F iven by Lemma III4 Then the followin system of conservation laws holds: div LF, R3 ) Φ = ) III69) div LF Φ + λ D + H Φ = 15

16 Therefore there exists a function S F W 1,, ) loc D, R) and a vector field R F W 1,, ) loc D, R 3 ) satisfyin S F = LF, Φ III7) where λ is the mean value of λ on D R 3 R F = L F Φ + λ λ) D H Φ, III71) Proof Combinin Remark III4, the definition of L F in Lemma III4, and Theorem VII14 in 31] in particular see equations VII187) and VII4)), we have L F, Φ = L F Φ + < λ, D > + < H, Φ > = Since div, the last system is equivalent to the desired system III69) The existence and the reularity of S F and R F is analoous to the existence of L F in Lemma III4 Proposition III6 Let Φ be a weak conformal immersion of the disc D into R 3 critical for the frame enery F and let S F W 1,, ) loc D, R), R F W 1,, ) D, R 3 ) be iven by Lemma III5 Then their radients satisfy the followin system: S F = RF, n + λ λ) D, n R F = n RF + S F n + λ λ) D + λ λ) ) D, e e 1 D, e1 e Therefore S F, R F, D, Φ, λ) satisfy the followin elliptic system S F = RF, n + div λ λ) ] D, n III7) R F = n RF + S F n +div λ λ) D + λ λ) )] D, e e 1 D, e1 e D ) = div I Φ n 1 λ λ)] Φ = R F Φ S F, Φ λ = e 1, e III73) As a consequence, we have that S F W 1, loc D, R) and R F W 1, loc D, R 3 ) Remark III A natural question arisin from Proposition III6 is if actually the system III73) is equivalent to the frame enery equation III67); in analoy with the situation in the Willmore framework see ]-31]-33]) we expect this not to be the case More precisely we expect the system III73) to be equivalent to the conformal-constrained Willmore equation A second observation is that we expect the conservation laws on S F and R F to be the associated, via Noether s Theorem, to dilations and rotations transformations that preserve the frame enery, as observed in the introduction); this remark in the context of Willmore surfaces is due to Yann Bernard We will study these questions in a forthcomin work Proof Recall that we use the notation e 1 e = n, III74) 16

17 so, takin the scalar product in R 3 between III71) and n and observin that x1rf, n = e λ LF, e e e 1, n + λ λ) x1d, n = LF, xφ + λ λ) x1d, n xrf, n = e λ LF, e 1 e 1 e, n + λ λ) xd, n = LF, x1φ + λ λ) xd, n we et R F, n = LF, Φ + λ λ) D, n ; III75) recallin III7) and the fact that ) =, the last identity ives S F = RF, n + λ λ) D, n III76) Analoously one computes R F, e i which, combined with III75), ives R F = LF, Φ n LF, n Φ + λ λ) D III77) Takin the vector product of III77) with n ives ) RF n = LF, n Φ + λ λ) D, e e 1 D, e1 e, III78) which, plued in III77) toether with III7), ives R F = n RF + S F n + λ λ) D + λ λ) ) D, e e 1 D, e1 e Applyin the diverence and recallin that div ) we et the first two equations The very definition of D ives the third equation In order to obtain the fourth equation compute the vector product between Φ and RF as in III71): Φ RF = Φ LF Φ) + λ λ) Φ D III79) A short computation ives Φ D ) = Φ I Φ n = I Φ Φ, n n I Φ, Φ = e λ H n = Φ III8) Observin that H Φ Φ = e λ H = Φ, and Φ LF Φ) = L F Φ, Φ Φ Φ, LF = Φ, S F, where in the last equality we recalled III7) and that of of course Φ, Φ = Therefore we can rewrite III79) as 1 λ λ)] Φ = R F Φ S F, Φ The last equation is classical, see for instance 31, VII13)] The improved reularity of λ, S F and R F is quite standard, in any case let us briefly sketch the proof for λ and S F the one for R F is analoous) The fact that λ L D ) follows directly from the Wente-type estimate of Chanillo-Li 6] for more details see also 31, Section VII64]) Let us discuss the reularity of S F : take any ball B D and write S F on B as S F = S + S 1 + S, 17

18 where S C B) is the harmonic extension to B of S F B and S 1, S satisfy the equations S 1 = RF, n on B S 1 = on B, S = div λ λ) ] D, n S = on B on B III81) III8) Since by construction R F W 1,, ) B) and n W 1, B), by a refinement of the Wente inequality due to Bethuel 3] which is based on previous results of Coifman-Lions-Meyer and Semmes), equation III81) implies that S 1 W 1, B) The fact that S W 1, B) follows instead from Stampacchia radient estimates, recallin that λ λ L B, R), D W 1, B, R 3 ) and of course n L B, R 3 ) IV Reularity of critical points for the frame enery: proof of Theorem I The oal of the next Section V is to minimize the frame enery in each reular homotopy class of immersed tori in R 3 and to propose such a minimizer as canonical rapresentant for its own class; to this aim, in the present section we develop the reularity theory for critical points of the frame enery The fundamental startin point is iven by the elliptic system with quadratic jacobian non linearities satisfied by the critical points of the frame enery, namely Proposition III6 The stratey is to show that, for every x D and r > small enouh, the system III73) with Dirichelet boundary condition has at most one solution in a suitable function space for every < ρ r; then, usin a ood slicin arument toether with properties of the trace and harmonic extension, we construct a more reular solution of the system III73) havin the same boundary condition as the initial solution By the uniqueness we infer that the initial solution had to be more reular, namely in a subcritical space, then we conclude with a standard bootstrap arument that the initial solution is actually C Before statin the lemmas let us introduce some notation In the followin, Φ will be a weak conformal immersion of D into R 3 critical for the functional F For any ρ, 1), k R and p 1, ) we will denote E k,p B ρ )) := W k,p B ρ ), R) W k,p B ρ ), R 3 ) W k,p B ρ ), R 3 ) W k+1,p B ρ ), R 3 ), E 1,p B ρ)) := W 1,p B ρ ), R) W 1,p B ρ ), R 3 ) W 1,p B ρ ), R 3 ) W,p W 1,p B ρ ), R 3 )), Let us define LA) := A + B, n div λ λ) ] C, n L B) := B n B A n div λ λ) C + λ λ) )] C, e e 1 C, e1 e L C) := C ) div π n Ψ) Φ n L Ψ) := Ψ 1 + ] B 1 λ λ) Φ + A, Φ, IV83) where Φ, e i, n, λ, λ are the critical weak conformal immmersion, its normalized tanent vectors, its normal vector, its conformal factor and the mean value of the conformal factor on B ρ ); all this terms are seen in system IV83) as coefficients Since λ satisfies the elliptic equation λ = e 1, e, 18

19 from the Wente-type estimate of Chanillo-Li 6] for more details see also 31, Section VII64]) we have λ λ L B ρ)) C e 1 L B ρ)) e L B ρ)) ] C e e 1 dx + n e 1 + n e ) dx B ρ) B ρ) ] C e d e 1 dvol + I dvol, IV84) B ρ) for some C, C, C > independent of Φ Observe that combinin Lemma VI8 and IV84), we et that L is a linear continuous operator from E 1,p B ρ )) to E 1,p B ρ )) for every p 1, ) The key technical lemma for provin the reularity is the followin isomorphism result Lemma IV6 L is an isomorphism between E 1,p and E 1,p ) Let Φ be a weak conformal immersion of D into R 3 critical for the frame enery F Then there exists r > dependin on Φ) such that for every ρ, r] the linear operator L is an isomorphism from E 1,p B ρ)) onto E 1,p B ρ )), for every p 1, ) In particular if E E 1,p B ρ)), for some p 1, ), solves the homoeneous equation L E) =, then E = ae on B ρ ) Proof From the discussion above we already know that L : E 1,p B ρ)) E 1,p B ρ )) is a continuous linear operator Our oal is to prove that L : E 1,p B ρ)) E 1,p B ρ )) is an isomorphism, more precisely we prove that there exists r > such that for every ρ, r) and for every f A, f B, f C, f Ψ ) E 1,p B ρ )), the system LA, B, C, Ψ ) = f A, f B, f C, f Ψ ) has a unique solution A, B, C, Ψ ) E 1,p B ρ)) First of all observe that for every δ > to be fixed later there exists r > such that e d e 1 dvol + I dvol δ IV85) B r) From now on let ρ, r] Observe that, thanks to IV85) and IV84), for any σ > there exists δ > small enouh such that λ λ L B ρ)) σ IV86) As first step we establish a priori estimates on the solutions If A, B, C, Ψ ) E 1,p B ρ)) solve LA, B, C, Ψ ) = f A, f B, f C, f Ψ ) then, usin IV86) and Lemma VI8, and the classical Stampacchia radient estimates for elliptic PDEs, we can estimate all the norms are computed on B ρ )) Ψ L p γ f Ψ L p + A L p + B ] L p C L p γ Ψ L p + B ] L p B L p γ f B W 1,p + ε A L p + B L p + C )] L p A L p γ f A W 1,p + ε A L p + B L p + C )] L p, for some constant γ > Bootstrappin the estimates above we obtain that A, B, C, Ψ ) E 1,p B ρ)) γ ε A, B, C, Ψ ) E 1,p B + f ρ)) A, f B, f C, f ] Ψ ) E 1,p B ρ)) B r) B ρ) Choosin ρ > small enouh such that γ ε 1, the last estimate ives A, B, C, Ψ ) E 1,p B ρ)) γ f A, f B, f C, f Ψ ) E 1,p B ρ)) IV87) Since L is linear, of course, the a priori estimate IV87) ensures uniqueness of the solution to the system LA, B, C, Ψ ) = f A, f B, f C, f Ψ ) in the space E 1,p B ρ)) 19