Complete minimal submanifolds with nullity in Euclidean spheres

Transcription

1 Complete minimal submanifolds wit nullity in Euclidean speres M. Dajczer, T. Kasioumis, A. Savas-Halilaj and T. Vlacos Abstract In tis paper we investigate m-dimensional complete minimal submanifolds in Euclidean speres wit index of relative nullity at least m 2 at any point. Tese are austere submanifolds in te sense of Harvey and Lawson [19] and were initially studied by Bryant [3]. For any dimension and codimension tere is an abundance of non-complete examples fully described by Dajczer and Florit [7] in terms of a class of surfaces, called elliptic, for wic te ellipse of curvature of a certain order is a circle at any point. Under te assumption of completeness, it turns out tat any submanifold is eiter totally geodesic or as dimension tree. In te latter case tere are plenty of examples, even compact ones. Under te mild assumption tat te Omori-Yau maximum principle olds on te manifold, a trivial condition in te compact case, we provide a complete local parametric description of te submanifolds in terms of 1-isotropic surfaces in Euclidean space. Tese are te minimal surfaces for wic te standard ellipse of curvature is a circle at any point. For tese surfaces, tere exists a Weierstrass type representation tat generates all simply connected ones. Let M m be a complete m-dimensional Riemannian manifold. In [10] we considered te case of minimal isometric immersions into Euclidean space f : M m R n, m 3, satisfying tat te index of relative nullity is at least m 2 at any point. Under te mild assumption tat te Omori-Yau maximum principle olds on M m, we concluded tat any f must be trivial, namely, just a cylinder over a complete minimal surface. Tis result is global in nature since for any dimension tere are plenty of non-complete examples oter tan open subsets of cylinders. It is natural to expect rater different type of conclusions wen considering a similar global problem for minimal isometric immersions into nonflat space forms. For instance, for submanifolds in te yperbolic space one would guess tat under te same condition on te relative nullity index tere exist many non-trivial examples, and tat a kind of triviality conclusion will only old under a strong additional assumption. Te tird autor would like to acknowledge financial support from te grant DFG SM 78/6-1. 1

2 Tis paper is devoted to te case of minimal submanifolds in round speres. Tus, in te sequel f : M m S n, m 3, will be a minimal isometric immersion into Euclidean spere wit index of relative nullity at least m 2 at any point. As in te Euclidean case, it is known tat are plenty of non-complete examples of any dimension. On one and, under te assumption of completeness of M m we ave tat f as to be totally geodesic unless m = 3. On te oter and, for dimension m = 3 we will see tat tere are plenty of non trivial examples, even compact ones. Notice tat te class of submanifolds studied in tis paper are precisely te minimal δ(2)-ideal submanifolds considered by Cen; see [5]. From te results in [7] it follows tat any example for dimension m = 3 can locally be constructed as follows: Let g : L 2 R n+1, n 4, be an elliptic surface wose first curvature ellipse is always a circle. Ten, te map ψ g : T 1 L S n defined on te unit tangent bundle of L 2 and given by ψ g (x, w) = g w (1) parametrizes (outside singular points) a minimal immersion f : M 3 S n wit index of relative nullity at least one at any point. Minimal surfaces are elliptic, but te latter class of surfaces is muc larger. In fact, tat a surface g : L 2 R n is elliptic means tat given a (ence any) basis X, Y of te tangent plane T x L at any x L 2 te second fundamental form α g : T L T L N g L of g satisfies aα g (X, X) + 2bα g (X, Y ) + cα g (Y, Y ) = 0 were a, b, c R verify ac b 2 > 0. Equivalently, in any local system of coordinates (u, v) of L 2 any coordinate function of g is a solution of te (same) elliptic PDE of type a b u2 u v + c 2 v + d 2 u + e v = 0 were a, b, c, d, e are smoot functions suc tat ac b 2 > 0. Our main result sows tat for complete submanifolds te condition of ellipticity of te generating surface as to be restricted to minimality. For minimal surfaces, te first curvature ellipse as an elliptic surface coincides wit te standard ellipse of curvature, namely, te image in te normal space of te second fundamental form restricted to te unit circle in te tangent plane. Teorem 1. Let f : M m S n, m 3, be a minimal isometric immersion wit index of relative nullity at least m 2 at any point. If M m is complete ten f is totally geodesic unless m = 3. Moreover, if te Omori-Yau maximum principle olds on M 3 ten along an open dense subset f is locally parametrized by (1) were g : L 2 R n+1 is a minimal surface wose first curvature ellipse is always a circle. 2

3 A minimal surface g : L 2 R n wose first curvature ellipse is a circle at any point is called a 1-isotropic surface. Te above result sould be complemented by te fact tat tere is te Weierstrass type representation from [9] tat generates all simply connected 1-isotropic surfaces. It goes as follows: start wit any nonzero olomorpic map in a simply connected domain α 0 : U C C n 4 and let α 1 : U C n 2 be given by α 1 = β 1 ( 1 φ 2 0, i(1 + φ 2 0), 2φ 0 ) were φ 0 = z α 0 dz and β 1 0 is any olomorpic function. Now let α 2 : U C n be given by α 2 = β 2 ( 1 φ 2 1, i(1 + φ 2 1), 2φ 1 ) were φ 1 = z α 1 dz and β 2 0 is any olomorpic function. Ten g = Re{α 2 } is a 1-isotropic surface in R n. Te Omori-Yau maximum principle olds on M m if for any function ϕ C 2 (M) bounded from above tere exists a sequence of points {x j } j N suc tat ϕ(x j ) > sup ϕ 1/j, ϕ(x j ) 1/j and ϕ(x j ) 1/j M for any j N. Tere are fairly general assumptions tat imply te validity of te Omori-Yau maximum principle on a Riemannian manifold; see [1]. For instance, it is applicable on complete Riemannian manifolds wose Ricci curvature does not decay fast to. We see next tat tere are plenty of complete (even compact) examples of treedimensional minimal submanifolds in speres wit index of relative nullity at least one at any point. Hopf lifts: If g : L 2 CP n, n 2, is a substantial olomorpic curve, ten te Hopf fibration H: S 2n+1 CP n induces a circle bundle M 3 over L 2. Tis lifting induces an immersion f : M 3 S 2n+1 suc tat g π = H f, were π : M 3 L 2 is te projection map. Suc submanifolds are minimal wit index of relative nullity at least one if n = 2 (see [13]) or n = 3 (see [21]). Moreover, if L 2 is compact, ten M 3 is also compact. Tubes over minimal 2-speres: Due to te work of Calabi, Cern, Barbosa and oters, it is known tat minimal 2-speres in speres are pseudoolomorpic (isotropic) in substantial even codimension. Calabi [4] proved tat any suc surface in S 2n is nicely curved if its area is 2πn(n + 1), and Barbosa sowed [2] tat te space of tese surfaces is diffeomorpic to SO(2n + 1, C)/SO(2n + 1, R). According to Proposition 5 below suc surfaces produce examples of compact tree-dimensional minimal submanifolds in S 2n wit index of relative nullity one. Among te second family of examples given above, tere are te submanifolds produced from pseudoolomorpic surfaces g : S 2 S 6 wit area 24π wic are olomorpic wit respect to te nearly Kaeler structure in S 6. For instance, tis is te situation 3

4 of te Veronese surface in S 6. In tis case, te compact submanifolds M 3 are Lagrangian (also called totally real) in S 6 ; see [13]. Corollary 2. Let f : M 3 S 6 be an isometric immersion wit index of relative nullity at least one at any point. Assume tat f is Lagrangian wit respect to te nearly Kaeler structure in S 6. If M 3 is complete and te Omori-Yau maximum principle olds, ten f is locally parametrized by (1) along an open dense subset of M 3 were g is a 2-isotropic surface in R 6 (respectively, R 7 ) and f is substantial in S 5 (respectively, S 6 ). Tat te surface g is 2-isotropic means tat it is 1-isotropic and tat te second ellipse of curvature is also a circle at any point. Hence, in te case of R 6 we ave tat g is congruent to a olomorpic curve in C 3 R 6. It follows from te results in [11] tat te universal cover of any of te complete treedimensional submanifolds considered in Teorem 1 admits a one-parameter associated family of isometric immersions of te same type. Moreover, tat family is trivial if and only if te (local) generating minimal surface is congruent to a olomorpic curve. We refer to Lotay [21] for a discussion about te existence of suc an associated family in te case of yet anoter family of examples. 1 Te relative nullity foliation In tis section, we recall basic facts on te relative nullity foliation for submanifolds in space forms to be used in te sequel witout furter reference. Let f : M m Q n c denote an isometric immersion of an m-dimensional Riemannian manifold M m into te Euclidean space R n (c = 0) or te unit spere S n (c = 1). Te relative nullity subspace of f at x M m is te tangent subspace given by D(x) = {X T x M : α f (X, Y ) = 0 for all Y T x M} were α f : T M T M N f M denotes te second fundamental form of f. Te dimension ν(x) of D(x) is called te index of relative nullity of f at x M m. For simplicity, in te sequel we call k(x) = m ν(x) te rank of f at x M m. Notice tat k(x) is te rank of te Gauss map of f at x M m. If f as constant rank on an open subset U M m, it is an elementary fact tat te relative nullity distribution D along U is integrable and tat te leaves of te relative nullity foliation are totally geodesic submanifolds of M m and under f of Q n c. Let U M m be an open subset were te index of relative nullity ν = s > 0 is constant. Te following is a well known fundamental result in te teory of isometric immersions (cf. [6]). 4

5 Proposition 3. Let γ : [0, b] M m be a geodesic curve suc tat γ([0, b)) is contained in a leaf of relative nullity contained in U. Ten also ν(γ(b)) = s. In te sequel, we decompose any tangent vector field X on M m as X = X v + X according to te ortogonal splitting T M = D D. Te splitting tensor C : D D D is given by C(T, X) = XT = ( X T ) for any T D and X D. We also regard C as a map C : Γ(D) Γ(End(D )). Te following differential equations for te endomorpism C T = C(T, ) are a well known easy consequence of te Codazzi equation: were I is te identity map, and S C T = C T C S + C S T + c T, S I (2) ( XC T )Y ( Y C T )X = C v X T Y C v Y T X (3) for any S, T Γ(D) and X, Y Γ(D ). For a proof we refer to [6] or [8]. 2 Elliptic submanifolds In tis section, we recall from [7] te notion of elliptic submanifold of a space form as well as several of teir basic properties. Let f : M m Q n c be an isometric immersion. Te l t -normal space N f l (x) of f at x M m for l 1 is defined as N f l (x) = span{ α l+1 f (X 1,..., X l+1 ) : X 1,..., X l+1 T x M }. Here α 2 f = α f and for s 3 te so called s t -fundamental form is te symmetric tensor α s f : T M T M N fm defined inductively by α s f(x 1,..., X s ) = π s 1 ( X s X 3 α f (X 2, X 1 ) ) were π k stands for te projection onto (N f 1 N f k 1 ). An isometric immersion f : M m Q n c of rank 2 is called elliptic if tere exists a (necessary unique up to a sign) almost complex structure J : D D suc tat te second fundamental form satisfies α f (X, X) + α f (JX, JX) = 0 5

6 for all X D. Notice tat J is ortogonal if and only f is minimal. Let f : M m Q n c be substantial and elliptic. Te former means tat te codimension cannot be reduced. Assume also tat f is nicely curved wic means tat for any l 1 all subspaces N f l (x) ave constant dimension and tus form subbundles of te normal bundle. Notice tat any f is nicely curved along connected components of an open dense subset of M m. Ten, along tat subset te normal bundle splits ortogonally and smootly as N f M = N f 1 Nτ f f (4) were all N f l s ave rank 2, except possibly te last one tat as rank 1 in case te codimension is odd. Tus, te induced bundle f T Q n c splits as were N f 0 = f D. Setting f T Q n c = f D N f 0 N f 1 N f τ f τ o f = { τ f if n m is even τ f 1 if n m is odd it turns out tat te almost complex structure J on D induces an almost complex structure J l on eac N f l, 0 l τ f o, defined by J l α l+1 f (X 1,..., X l, X l+1 ) = α l+1 f (X 1,..., X l, JX l+1 ) were α 1 f = f. Te l t -order curvature ellipse E f l (x) N f l (x) of f at x M m for 0 l τ o f is E f l (x) = { α l+1 f (Z θ,..., Z θ ) : Z θ = cos θz + sin θjz and θ [0, π) } were Z D (x) as unit lengt and satisfies Z, JZ = 0. From ellipticity suc a Z always exists and E f l (x) is indeed an ellipse. We say tat te curvature ellipse E f l of an elliptic submanifold f is a circle for some 0 l τf o if all ellipses E f l (x) are circles. Tat te curvature ellipse E f l is a circle is equivalent to te almost complex structure J l being ortogonal. Notice tat E f 0 is a circle if and only if f is minimal. An elliptic submanifold f is called l-isotropic if all ellipses of curvature up to order l are circles. Ten f is called isotropic if te ellipses of curvature of any order are circles. Substantial isotropic surfaces in R 2n are olomorpic curves in C n R 2n. Isotropic surfaces in speres are also referred to as pseudoolomorpic surfaces. For tis class of surfaces a Weierstrass type representation was given in [12]. 6

7 Let f : M m Q n c c, (c = 0, 1), be a substantial nicely curved elliptic submanifold. Assume tat M m is te saturation of a fixed simply connected cross section L 2 M m to te relative nullity foliation. Te subbundles in te ortogonal splitting (4) are parallel in te normal connection (and tus in Qc n c ) along D. Hence eac N f l can be seen as a vector bundle along te surface L 2. A polar surface to f is an immersion of L 2 defined as follows: (a) If n c m is odd, ten te polar surface : L 2 S n 1 is te sperical image of te unit normal field spanning N f τ f. (b) If n c m is even, ten te polar surface : L 2 R n is any surface suc tat T x L = N f τ f (x) up to parallel identification in R n. Polar surfaces always exist since in case (b) any elliptic submanifold admits locally many polar surfaces. Te almost complex structure J on D induces an almost complex structure J on T L defined by P J = JP were P : T L D is te ortogonal projection. It turns out tat a polar surface of an elliptic submanifold is necessarily elliptic. Moreover, if te elliptic submanifold as a circular ellipse of curvature ten its polar surface as te same property at te corresponding normal bundle. As a matter of fact, up to parallel identification it olds tat N s = N f τ o f s and J s = ( J f τ o f s ) t, 0 s τ o f. (5) In particular, te polar surface is nicely curved. Notice tat te last l + 1 ellipses of curvature of te polar surface to an l-isotropic submanifold are circles. Note tat in tis case te polar surface is not necessarily minimal. A bipolar surface to f is any polar surface to a polar surface to f. In particular, if we are in case f : M 3 S n 1, ten a bipolar surface to f is a nicely curved elliptic immersion g : L 2 R n. 3 Te local case We discuss next two alternative ways to parametrically describe, at least locally, all sperical tree-dimensional minimal submanifolds of rank two in speres. Tis follows from te results in [7] bearing in mind tat a submanifold is minimal in a spere if and only if te cone saped over it is minimal in te Euclidean space. Let g : L 2 R n+1, n 4, be an elliptic surface and let T 1 L denote its unit tangent bundle. 7

8 Proposition 4. If E g 1 is a circle, ten te map ψ g : T 1 L S n given by ψ g (x, w) = g w is a minimal immersion wit index of relative nullity ν 1 outside te subset of singular points, wic correspond to points were dim N g 1 = 0. Moreover, a regular point (x, w) T 1 L is totally geodesic for ψ g if and only if dim N g 2 (x) = 0. Conversely, any tree-dimensional minimal submanifold in te spere wit ν = 1 at any point can be at least locally parametrized in tis way. Te above parametrization (used for Teorem 1) is called te bipolar parametrization in [7] because g is a bipolar surface to ψ g. Te parametrization in te sequel (used for te examples discussed above) was called in [7] te polar parametrization. Let g : L 2 Q 2n+2c 1 c (c = 0, 1), n 2, be a nicely curved elliptic surface and let M 3 = UNτ g g stand for te unit subbundle of Nτ g g. Proposition 5. If E g τ g 1 is a circle, ten φ g : M 3 S 2n+c given by φ g (x, w) = w is a minimal immersion of rank two and polar surface g. Conversely, any minimal submanifold M 3 in S 2n+c of rank two can locally be parametrized in tis way. 4 Te complete case We first observe tat for complete submanifolds of rank at most two te interesting case is te tree-dimensional one. Te remaining of te paper is devoted to te study of te latter case. Proposition 6. Let f : M m S n, m 3, be a minimal isometric immersion wit index of relative nullity ν m 2 at any point. If M m is complete, ten f is totally geodesic unless m = 3. Te above is an immediate consequence of te following result due to Ferus [17] (see [6, Lemma 6.16] were te proof olds regardless te codimension) since due to minimality we cannot ave points wit index of relative nullity m 1. Lemma 7. Let f : M m S n be an isometric immersion and let U M m be an open subset were te index of relative nullity is constant eiter ν = m 1 or ν = m 2. Ten no leaf of relative nullity contained in U is complete if m 4. In te sequel, let f : M 3 S n be a minimal isometric immersion of a complete Riemannian manifold wit index of relative nullity ν(x) 1 at any x M 3. Let U M 3 be an open subset were ν = 1 suc tat D is a line bundle on U. If tat line bundle is trivial, ten tere is a unique, up to a sign, ortogonal almost complex structure J : D U D U. In tat case set C = C e were e is a unit section of D U. 8

9 Lemma 8. If D U is a trivial line bundle tere are armonic functions u, v C (U) suc tat C = vi uj. (6) Proof: Te proof follows similarly as in [10]. Witout loss of generality we assume tat f is substantial. Denote by A ξ te sape operator of f wit respect to ξ N f M. Te Codazzi equation gives e A ξ D = A ξ D C + A e ξ D for any vector field ξ N f M. In particular, Moreover, te minimality condition is equivalent to A ξ D C = C t A ξ D. (7) A ξ D J = J t A ξ D. (8) First we consider te case n = 4. Let e 1, e 2, e 3 = e be a local ortonormal frame field tat diagonalizes A ξ wit respect to a unit normal vector filed ξ suc tat Je 1 = e 2. Set From te Codazzi equations u = e2 e 1, e 3 and v = e1 e 1, e 3. ( ei A ξ )e 3 = ( e 3 A ξ )e i 1 i 2, we obtain tat e2 e 2, e 3 = v and from te Codazzi equation ( e1 A ξ )e 2, e 3 = ( e2 A ξ )e 1, e 3 tat e1 e 2, e 3 = u, and now (6) follows. Assume now tat f does not reduce codimension to one. Due to te minimality assumption, we ave tat dim N f 1 2. If dim N f 1 = 1 on an open subset V M 3, a simple argument using te Codazzi equation gives tat N f 1 is parallel in te normal bundle, and ence f V reduces codimension to one. Due to real analyticity te same would old globally, and tat as been excluded. Hence, tere is an open dense subset W M 3 were dim N f 1 = 2. From (7) and (8) we ave tat C span{i, J} on U W, and (6) follows easily. We now sow tat u and v are armonic functions. From (2) and (3) we ave ec = C 2 + I (9) and ( XC ) Y = ( Y C ) X (10) 9

10 for any X, Y D. Let e 1, e 2, e 3 be a local ortonormal frame wit Je 1 = e 2 and e 3 D. From (6) we find tat and Equation (9) is equivalent to wereas (10) is equivalent to Te Laplacian of v is given by v = v = e1 e 1, e 3 = e2 e 2, e 3 (11) u = e1 e 2, e 3 = e2 e 1, e 3. (12) e 3 (v) = v 2 u and e 3 (u) = 2uv (13) e 1 (u) = e 2 (v) and e 2 (u) = e 1 (v). (14) 3 e i e i (v) ω 12 (e 1 )e 2 (v) ω 13 (e 1 )e 3 (v) + ω 12 (e 2 )e 1 (v) ω 23 (e 2 )e 3 (v) i=1 were ω ij (e k ) = ek e i, e j for any i, j, k {1, 2, 3}. Using (14) we ave e 1 e 1 (v) + e 2 e 2 (v) = e 1 e 2 (u) + e 2 e 1 (u) = [e 1, e 2 ](u) = e1 e 2 (u) + e2 e 1 (u) = ω 12 (e 1 )e 1 (u) ω 23 (e 1 )e 3 (u) + ω 12 (e 2 )e 2 (u) + ω 13 (e 2 )e 3 (u) = ω 12 (e 1 )e 2 (v) ω 12 (e 2 )e 1 (v) + 2ue 3 (u). Inserting tis equality into te previous equation and making use of (11) and (13) yields v = 0. In a similar form it follows tat also u is armonic. Let A denote te set of totally geodesic points of f. By Proposition 3 te relative nullity distribution D is a line bundle on M 3 A. Being f real analytic, te square of te norm of te second fundamental form is a real analytic function and ence A is a real analytic set. According to Lojasewicz s structure teorem [20, Teorem 6.3.3] te set A locally decomposes as A = V 0 V 1 V 2 V 3 were eac V d, 0 d 3, is eiter empty or a disjoint finite union of d-dimensional real analytic subvarieties. A point x 0 A is called a regular point of dimension d if tere is a neigborood Ω of x 0 suc tat Ω A is a d-dimensional real analytic submanifold of Ω. Oterwise x 0 is said to be a singular point. Te set of singular points is locally a finite union of submanifolds. We want to sow tat A = V 1 unless f is just a totally geodesic tree-spere in S n. After excluding te latter case, we ave from te real analyticity of f tat V 3 is empty. We will proceed now following ideas developed in [10]. In fact, we only sketc te proof of te following fact, wic is similar to te proof of Lemma 2 in [10]. 10

11 Lemma 9. Te set V 2 is empty. Proof: We only ave to sow tat tere are no regular points in V 2. Suppose tat a regular point x 0 V 2 exists. Let Ω M 3 be an open neigborood of x 0 suc tat L 2 = Ω A is an embedded surface. Let e 1, e 2, e 3, ξ 1,..., ξ n 3 be an ortonormal frame adapted to M 3 along Ω near x 0. Te coefficients of te second fundamental form are a ij = α f (e i, e j ), ξ a were 1 i, j, k 3 and 1 a, b n 3. Te Gauss map γ : M 3 Gr(4, n + 1) of f is a map into te Grassmannian of oriented 4-dimensional subspaces in R n+1 defined by γ = f e 1 e 2 e 3 were we regard Gr(4, n + 1) as a submanifold in 4 R n+1 via te map for te Plücker embedding. Ten γ e i = a ijf e ja j,a were e ja is taken by replacing e j wit ξ a in e 1 e 2 e 2. Moreover, it easy to see tat te Gauss map satisfies te partial differential equation γ + α f 2 γ = a ij b ikf e ja,kb i,a b,j k were e ja,kb is obtained by replacing e j wit ξ a and e k wit ξ b in e 1 e 2 e 3. Hence, we may write te latter equation in te form γ(x) + γ (x) 2 γ(x) + G(x, γ ) = 0 were G is real analytic wit G(, 0) = 0. Clearly, we ave tat γ is constant along L 2 and tat γ (n) = 0 on L 2, were n is a unit normal of L 2 M 3. Ten, it follows from te uniqueness part of te Caucy-Kowalewsky teorem (cf. [23]) tat te Gauss map γ must be constant. Tis would imply tat f(m) is a tree-dimensional totally geodesic spere wic contradicts our assumption. Lemma 10. Te set V 0 is empty. Proof: Let Ω be an open neigborood around x 0 V 0 suc tat ν = 1 on Ω {x 0 } and let {x j } j N be a sequence in Ω {x 0 } converging to x 0. Let e j = e(x j ) T xj M be te sequence of unit vectors contained in te relative nullity distribution of f. By passing to a subsequence, if necessary, tere is a unit vector e 0 T x0 M suc tat lim e j = e 0. By continuity, te geodesic tangent to e 0 at x 0 is a leaf of relative nullity outside x 0. But tis is a contradiction in view of Proposition 3. Lemma 11. Te foliation of relative nullity extends analytically over te regular points in te set A. 11

12 Proof: Because A = V 1 its 2-capacity cap 2 (A) must be zero (cf. [16, Teorem 3]). On te oter and, te distribution D extends continuously over te regular points of A. In fact, by te previous lemmas it remains to consider te case wen Ω is an open subset of M 3 suc tat Ω A is a open piece of a great circle in te ambient space. But in tis situation te result follows by an argument of continuity similar tan in te proof of Lemma 10. Let Ω be an open subset of M 3 A and e 1, e 2, e 3 a local frame on Ω as in te proof of Lemma 8. Consider te map F : Ω S n wit values into te unit spere given by F = f e 3. A straigtforward computation using (11), (12) and (14) sows tat its tension field 3 ( ) τ(f ) = F e j F e j F ej e j j=1 vanises, were denotes te Levi-Civita connection of S n. Hence F is a armonic map. Since F is continuous on M 3 and because cap 2 (A) = 0, it follows from a result of Meier [22, Teorem 1] tat F is of class C 2 on M 3. But ten F is real analytic by a result due to Eells and Sampson [14, Proposition p. 117]. Lemma 12. Te set A as no singular points. Proof: Let x 0 A be a singular point. From Lemmas 9 and 10 te set A contains subvarieties of dimension one. It is well known tat te singular points of suc curves are isolated (cf. [20, Teorem 6.3.3]). Moreover, according to Lemma 11 te set of regular points of A contains geodesic curves of te relative nullity foliation. Hence x 0 is an intersection of suc geodesic curves. Let Ω M 3 be an open subset containing x 0 suc tat te restriction of f Ω is injective. Consider a fixed cross section L 2 to D on Ω {x 0 }. Note tat te immersion f can be locally parametrized by te embedding φ: L 2 S 1 S n given by φ(x, t) = exp f(x) ( tf e ) = cos t f(x) + sin t f e(x) were e D L 2. Since x 0 is an intersection point of geodesics in te relative nullity foliation, it follows from te parametrization tat tere are points (x 1, t 1 ), (x 2, t 2 ) L 2 S 1 suc tat φ(x 1, t 1 ) = f(x 0 ) = φ(x 2, t 2 ), wic leads to a contradiction. 5 Te proofs Te proof of our main result relies eavily on te following consequence of te Omori- Yau maximum principle. 12

13 Lemma 13. Let M m be a Riemannian manifold for wic te Omori-Yau maximum principle olds. If ϕ C (M) satisfies te partial differential inequality ϕ 2ϕ 2, ten sup ϕ = 0. In particular, if ϕ 0 ten ϕ = 0. Proof: See [1, Teorem 2.8] or [18]. Proof of Teorem 1: By Proposition 6 we only ave to consider te case m = 3. We distinguis two cases. Case A =. At first suppose tat te line bundle D is trivial wit e a unit global section. By Lemma 8 tere are armonic functions u, v C (M) suc tat C = vi uj. We claim tat u is nowere zero. To te contrary suppose tat u(x 0 ) = 0 at x 0 M 3. Let γ : R M 3 te maximal integral curve of e emanating from x 0. Te second equation in (13) gives tat u must vanis along γ. Tus te first equation in (13) reduces to v (s) = v 2 (s) + 1, were v(s) = v(γ(s)) is an entire function. But tis is a contradiction since tis equation as no entire solutions. In te sequel we assume tat u > 0. Using (13) one can easily see tat ( (u 1) 2 + v 2) = 2( u 2 + v 2 ) 2((e(u)) 2 + (e(v)) 2 ) and Lemma 13 implies tat C = J. 2 ( (u 1) 2 + v 2) 2, Let U M 3 be te open dense subset were f is nicely curved. Ten let U U be an open connected subset of U tat is te saturation of a simply connected cross section L 2 U to te relative nullity foliation. Hereafter we work on U were f is nicely curved. Hence polar and bipolar surfaces of f U are well defined. Let be a polar surface to f U. We ave seen tat te almost complex structure J on D induces an almost complex structure J on T L defined by P J = JP, were P : T L D is te ortogonal projection. Moreover, is elliptic wit respect to J and (5) olds. In addition, it follows from Proposition 5 tat Eτ 1 is a circle. We claim tat te last curvature ellipse Eτ of is also a circle. In tat case te bipolar surface g : L 2 R n+1 to f is 1-isotropic, and we are done. Observe tat N τ = span{ξ, η} were ξ = f e L 2 and η = f L 2. Using C = J, we obtain tat ξ = f D J P. (15) Consider vector fields X 1,..., X τ, Y T L. Since N τ 1 = N f 0 = f (D ), we ave α τ (X 1,..., X τ ) = f Z 13

14 for some Z D. Keeping in mind te bundle isometries, we obtain tat α τ +1 (X 1,..., X τ, Y ) = ( Y α τ (X 1,..., X τ ) ) = ( Y f Nτ Z ). N τ Taking into account (15) we see tat α τ +1 (X 1,..., X τ, Y ) = Y f Z, ξ ξ + Y f Z, η η = f Z, ξ Y ξ f Z, η Y η = Z, JP Y ξ Z, P Y η. Recall tat te almost complex structure J τ on N τ is given by Since and J τ α τ +1 (X 1,..., X τ, Y ) = α τ +1 (X 1,..., X τ, JY ). α τ +1 (X 1,..., X τ, Y ) = Z, JP Y ξ Z, P Y η α τ +1 (X 1,..., X τ, JY ) = Z, P Y ξ Z, JP Y η, we ave tat te vectors α τ +1 (X 1,..., X τ, Y ) and α τ +1 (X 1,..., X τ, JY ) are perpendicular of te same lengt. Tus Jτ is ortogonal, and proves te claim. Finally, if te line bundle D is not trivial, it suffices to argue for a 2-fold covering Π: M 3 M 3 suc tat te nullity distribution D of f = f Π is a trivial line bundle and Π D = D. Case A. We ave seen tat te relative nullity distribution D can be extended analytically to a line bundle on M 3, denoted again by D, over te set of totally geodesic points A. Witout loss of generality, we may assume tat tere is a global unit section e D, since oterwise we can pass to te 2-fold covering space M 3 = {(x, w) : x M 3, w D(x) and w = 1} and argue as in te previous case. From Lemma 8, we know tat tere exist armonic functions u, v C (M 3 A) suc tat (6) olds on M 3 A. By previous results te functions u and v can be extended analytically to armonic functions on te entire M 3. Moreover, since u is positive on M 3 A and A consists of geodesic curves, by continuity we get tat u 0 on M 3. Ten C + J 2 is globally well defined and, arguing as in te previous case, we conclude again tat C = J on M 3. Te remaining of te proof now goes as before. Proof of Corollary 2: By a result of Ejiri [15] we ave tat f is minimal. Let e 1, e 2, e 3 be a local ortonormal tangent frame suc tat e 3 D. Since f is Lagrangian, we ave 14

15 tat Je 1, Je 2, Je 3 is an ortonormal frame in te normal bundle of f. Moreover, it is well known tat te 3-linear tensor given by (e i, e j, e k ) = α f (e i, e j ), Je k, i, j, k {1, 2, 3}, is fully symmetric. Away from te totally geodesic points, it is easy to ceck tat te vector fields α f (e 1, e 1 ) and α f (e 1, e 2 ) are perpendicular to eac oter and ave te same lengt. Hence E f 1 is a circle. Suppose at first tat f is substantial in S 6. Assume tat te submanifold is te saturation of a fixed cross section L 2 to te relative nullity foliation. Let : L 2 S 6 be a polar surface to f. From (5) we obtain tat is 1-isotropic. Proceeding as in te proof of Teorem 1, we deduce tat te second ellipse of is also a circle. Terefore, is pseudoolomorpic and any bipolar surface g to f is 2-isotropic in R 7. Now we consider te case were f is substantial in S 5. Take a fixed cross section L 2 to te relative nullity foliation and let : L 2 R 6 be a polar surface to f. As in te previous case, we obtain tat must be isotropic. Terefore, any bipolar surface g to f is an isotropic surface in R 6. References [1] L. Alias, P. Mastrolia and M. Rigoli, Maximum principles and geometric applications. Springer Monograps in Matematics. Springer, Cam, [2] J. Barbosa, On minimal immersions of S 2 into S 2m. Trans. Amer. Mat. Soc. 210 (1975), [3] R. Bryant, Some remarks on te geometry of austere manifolds. Bol. Soc. Brasil. Mat. 21 (1991), [4] E. Calabi, Minimal immersions of surfaces in Euclidean speres. J. Differential Geom. 1 (1967), [5] B. Y. Cen Pseudo-Riemannian geometry, δ-invariants and applications. World Scientific Publising Co. Hackensack, NJ, [6] M. Dajczer et al., Submanifolds and Isometric Immersions, Mat. Lecture Ser. 13, Publis or Peris Inc. Houston, [7] M. Dajczer and L. Florit, A class of austere submanifolds. Illinois J. Mat. 45 (2001), [8] M. Dajczer and D. Gromoll, Rigidity of complete Euclidean ypersurfaces. J. Differential Geom. 31 (1990),

16 [9] M. Dajczer and D. Gromoll, Te Weierstrass representation for complete minimal real Kaeler submanifolds of codimension two. Invent. Mat. 119 (1995), [10] M. Dajczer, T. Kasioumis, A. Savas-Halilaj and T. Vlacos, Complete minimal submanifolds wit nullity in Euclidean space, to appear in Mat. Z. [11] M. Dajczer and T. Vlacos, Te associated family of an elliptic surface and an application to minimal submanifolds. Geom. Dedicata 178 (2015), [12] M. Dajczer and T. Vlacos, A representation for pseudoolomorpic surfaces in speres. Proc. Amer. Mat. Soc. 144 (2016), [13] F. Dillen and L. Vrancken, Totally real submanifolds in S 6 satisfying Cen s equality. Trans. Amer. Mat. Soc. 348 (1996), [14] J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds. Amer. J. Mat. 86 (1964), [15] N. Ejiri, Totally real submanifolds in a 6-spere. Proc. Amer. Mat. Soc. 83 (1981), [16] L. Evans and R. Gariepy, Measure Teory and Fine Properties of Functions. Studies in Advanced Matematics. CRC Press, Boca Raton (1992). [17] D. Ferus, Te rigidity of complete ypersurfaces, Unpublised. [18] T. Hasanis, A. Savas-Halilaj and T. Vlacos. Complete minimal ypersurfaces in te yperbolic space H 4 wit vanising Gauss-Kronecker curvature. Trans. Amer. Mat. Soc. 359 (2007), [19] R. Harvey and B. Lawson, Calibrated geometries. Acta Mat. 148 (1982), [20] S. Krantz and H. Parks, A primer of real analytic functions. Birkäuser Advanced Texts: Basler Lerbücer, Birkäuser Boston, Inc., Boston, MA, [21] J. Lotay, Associative submanifolds of te 7-spere. Proc. Lond. Mat. Soc. 105 (2012), [22] M. Meier, Removable singularities of armonic maps and an application to minimal submanifolds. Indiana Univ. Mat. J. 35 (1986), [23] M. Taylor, Partial differential equations I. Basic teory. Second edition. Applied Matematical Sciences, 115. Springer, New York,

17 Marcos Dajczer IMPA Estrada Dona Castorina, , Rio de Janeiro Brazil Teodoros Kasioumis University of Ioannina Department of Matematics Ioannina Greece Andreas Savas-Halilaj Leibniz Universität Hannover Institut für Differentialgeometrie Welfengarten Hannover Germany savasa@mat.uni-annover.de Teodoros Vlacos University of Ioannina Department of Matematics Ioannina Greece tvlacos@uoi.gr 17