On Lawson-Osserman Constructions

On Lawson-Osserman Constructions

Abstract.

Lawson-Osserman [31] constructed three types of non-parametric minimal cones of high codimensions based on Hopf maps between spheres, which correspond to Lipschitz but non- solutions to the minimal surface equations, thereby making sharp contrast to the regularity theorem for minimal graphs of codimension 1. In this paper, we develop the constructions in a more general scheme. Once a mapping between unit spheres is composited of a harmonic Riemannian submersion and a homothetic (i.e., up to a constant factor, isometric) minimal immersion, certain twisted graph of can yield a non-parametric minimal cone. Because the choices of the second component usually form a huge moduli space, our constructions produce a constellation of uncountably many examples. For each such cone, there exists an entire minimal graph whose tangent cone at infinity is just the given one. Moreover, new phenomena on the existence, non-uniqueness and non-minimizing of solutions to the related Dirichlet problem are discovered.

2010 Mathematics Subject Classification:
58E20, 53A10, 53C42.

1. Introduction

The research on minimal graphs in Euclidean spaces has a long and fertile history. Among others, the Dirichlet problem (cf. [24, 4, 14, 35, 31]) is a central topic in this subject:

Let be a bounded and strictly convex domain with boundary of class for . It asks, for a given function of class with , what kind of and how many functions exist so that each such satisfies the minimal surface equations in the weak sense (or equivalently, the graph of is minimal in the sense of [1]) and .

When , we have a fairly profound understanding.

  • Given arbitrary boundary data of class , by the works of J. Douglas [15], T. Radó [38, 39], Jenkins-Serrin [24] and Bombieri-de Giorgi-Maranda [4], there exists a unique Lipschitz solution to the Dirichlet problem.

  • Furthermore, due to the works of E. de Giorgi [14] and J. Moser [35], this solution turns out to be analytic.

  • Each solution gives an absolutely area-minimizing graph by virtue of the convexity of and §5.4.18 of [19]. As a consequence, it is stable.

Utterly unlike the above, the situation for becomes much more complicated. Even when (the unit Euclidean disk), H. B. Lawson and R. Osserman [31] discovered astonishing phenomena that reveal essential differences.

  • For , some real analytic boundary data can be constructed so that there exist at least three different analytic solutions to the Dirichlet problem. Moreover, one of them corresponds to an unstable minimal surface.

  • For and , the Dirichlet problem is generally not solvable. In fact, for each that is not homotopic to zero, there exists a positive constant depending only on , such that the problem is unsolvable for the boundary data , where is a constant no less than .

  • For certain boundary data, there exists a Lipschitz solution to the Dirichlet problem which is not .

As shown in [31], the nonexistence and irregularity of the Dirichlet problem are intimately related as follows. Given that represents a non-trivial element of , the Dirichlet problem for is solvable when is small (due to the implicit function theorem) but unsolvable for large . This leads Lawson-Osserman to suspect there exists a critical value which supports some sort of singular solution. In particular, for the Hopf map with or ,

(1.1)

with

(1.2)

is the principal orbit of maximal volume under certain group action, and hence presents a minimal sphere (cf. W.Y. Hsiang [23]). Then the minimal cone over is the graph of

(1.3)

Hence, its restriction over gives a Lipschitz solution to the Dirichlet problem for boundary data .

To develop constructions akin to Lawson-Osserman’s in a more general framework, we introduce the following concepts.

Definition 1.1.

For a smooth map , if there exists an acute angle , such that

(1.4)

is a minimal submanifold of , then we call a Lawson-Osserman map (LOM), the associated Lawson-Osserman sphere (LOS), and the cone over the corresponding Lawson-Osserman cone (LOC).

Similarly, for an LOM , the associated is the graph of

(1.5)

Thus the restriction over provides a Lipschitz solution to the Dirichlet problem for the boundary data with .

Assume is an LOM that is not a totally geodesic isometric embedding. Then is called an LOMSE if the nonzero singular values of are equal for each . As varies, these values give a continuous function . One can deduce that equals a constant and that has constant rank (see Theorem 2.5 (ii)). Moreover, all components of this vector-valued function , i.e. are harmonic spherical functions of degree (see Theorem 2.8). Accordingly, we call such an LOMSE of (n,p,k)-type. It is worth noting that, the Hopf map from onto is an LOMSE of -type, for . Hence the LOMSEs and corresponding LOSs, LOCs are natural generalizations of Lawson-Osserman’s original constructions.

In this paper, we shall study LOMSEs systematically from several viewpoints.

A characterization of LOMSEs will be established in Theorem 2.5, which asserts that each of them can be written as the composition of a Riemannian submersion from with connected fibers and a homothetic minimal immersion into . In fact, the submersion, which determines , has to be a Hopf fibration over a complex projective space, a quaterninonic projective space or the octonionic projective line, according to the wonderful result in [46]; while the choices of the second component for each even integer usually form a moduli space of large dimension (see [9, 36, 44, 42, 43]), yielding a huge number of LOMSEs as well as the associated LOSs and LOCs. Note that except for the three original Lawson-Osserman cones, we always have . Therefore, ’ is not homotopic to zero’ is not a requisite to span a non-parametric minimal cone.

Although there exist uncountably many LOMSEs, for each of them both the nonzero singular value and the acute angle for the associated LOS are constants depending on in a discrete manner (see Theorem 2.8). Consequently, we gain interesting gap phenomena for certain geometric quantities of LOSs or LOCs associated to LOMSEs, e.g. angles between normal planes and a fixed reference plane, volumes, Jordan angles and slope functions, see Corollary 2.9. We remark that rigidity properties for these quantities of compact minimal submanifolds in spheres or entire minimal graphs in Euclidean spaces have drawn attention in many literatures [2, 20, 26, 27, 28, 29, 12, 37, 30].

Motivated by the argument of Lawson-Osserman [31], we seek for analytic solutions to Dirichlet problem for the boundary data as well. A good candidate (compared with (1.5)) turns out to be

(1.6)

Here is a smooth positive function on for some , satisfying . If

(1.7)

is a minimal submanifold and , then Morrey’s regularity theorem [34] ensures an analytic solution to the minimal surface equations through the origin. Since the minimality is invariant under rescaling, for and produce a series of minimal graphs. Therefore, in the -plane, every intersection point of the graph of and the straight line corresponds to an analytic solution to the Dirichlet problem for .

In particular, when is an LOMSE, the minimal surface equations can be reduced to (3.18), a nonlinear ordinary differential equation of second order, equivalent to an autonomous system (3.25) in the -plane for , and . With the aid of suitable barrier functions, we obtain a long-time existing bounded solution, whose orbit in the phase space emits from the origin - a saddle critical point and limits to - a stable critical point (see Propositions 3.3-3.4).

Quite subtly, there are two dramatically different types of asymptotic behaviors aroud depending on the values of :

  1. is a stable center when or ;

  2. is a stable spiral point when , or , .

As a consequence, the graphs of the solutions to (3.18) are illustrated below, respectively for LOMSEs of Type (I) and Type (II).

Much interesting information can be read off from the above pictures:

  1. For each LOMSE , there exists an entire analytic minimal graph whose tangent cone at infinity is exactly the LOC associated to (see Theorem 3.5).

  2. For an LOMSE of Type (II), there exist infinitely many analytic solutions to the Dirichlet problem for ; meanwhile, it also has a singular Lipschitz solution which corresponds to the truncated LOC (see Theorem 3.6).

  3. For Type (II), although a Lipschitz solution arises for the boundary data , there exists an such that the Dirichlet problem still has analytic solutions for whenenver .

  4. By the monotonicity of density for minimal submanifolds (currents) in Euclidean spaces (see [19, 13]), LOCs associated to LOMSEs of Type (II) are all non-minimizing (see Theorem 3.7).

To the knowledge of the authors of the present paper, it seems to be the first time to have phenomena (B)-(C) observed, and hard to foresee the occurrence from the classical theory of partial differential equations.

By the machinery of calibrations, the LOC associated to the Hopf map from onto (i.e. the LOMSE of -type) was shown area-minimizing by Harvey-Lawson [22]. It would be interesting to consider whether the associated LOC is area-minimizing for an LOMSE of -type. In Theorem 3.7, we establish a partial negative answer to the question. On the other hand, in a subsequent paper [50], we explore this subject from a different point of view and confirm that all LOCs associated to LOMSEs of -type are area-minimizing.


2. Lawson-Osserman maps

2.1. Preliminaries on harmonic maps

Let and be Riemannian manifolds and be a smooth mapping from to . The energy desity of at is defined to be

(2.1)

Here is an orthonormal basis of . The total energy is the integral of over .

Let and be Levi-Civita connections w.r.t. and respectively. Then the second fundamental form of is given by

(2.2)

whose trace under is the tensor field of

(2.3)

If vanishes indentically, then is called a harmonic map. When , is called totally geodesic. The first variation formula asserts that is harmonic if and only if it is a critical point of functional .

For a smooth function , one can see that where is the Laplace-Beltrami operator for . Hence is harmonic if it is a harmonic function in the usual sense.

Given an isometric immersion , its second fundamental form can be identified with the second fundamental form of in , and its tensor field can be regarded as the mean curvature vector field . Therefore, is harmonic if and only if it is an isometric minimal immersion. Moreover, is totally geodesic if and only if it is an isometric totally geodesic immersion.

For Riemannian submersions, we have the following characterization.

Proposition 2.1.

(see e.g. Proposition 1.12 of [17]) A Riemannian submersion is harmonic if and only if each fiber of is a minimal submanifold of .

Let be Riemannian manifolds, and , be smooth maps. We have the fundamental composition formula for tension fields (see Proposition 1.14 in [17] or §1.4 of [48]):

(2.4)

In particular, for an isometric immersion ,

(2.5)

where is the second fundamental form of in .

2.2. Necessary and sufficient conditions for LOSs

Let be the -dimensional unit sphere, the canonical metric induced by the inclusion map , and the second fundamental form of in .

Given smooth and an acute angle , let

(2.6)

be the embedding associated to and , and . We shall study when is minimal and thus yields an LOS .

Let , and be the position vectors of in , in and in respectively. Then

(2.7)

Here can be viewed from two different angles. On the one hand, is a vector-valued function on and we have . On the other hand, , and consequently by the composition formula (2.5) we have

(2.8)

Here is an orthonormal basis of , the Euclidean inner product, and the mean curvature field of in . We remark that pointwise.

Similarly, for where is the identity map from to and , we derive

(2.9)

where , and

(2.10)

where .

By (2.9) and (2.10), we obtain

(2.11)

Comparing (2.11) and (2.8) produces

(2.12)

We shall employ this relationship for the characterization of LOS.

Theorem 2.2.

For smooth and , is minimal (i.e., is an LOS in ) if and only if the following conditions hold:

  1. is harmonic.

  2. For each and the singular values of ,

Proof..

Firstly, we claim Condition (b) has another two equivalent statements as follows:

  1. The energy density of is everywhere.

  2. The energy density of is everywhere.

Now we give a proof of (b)(c). Due to the theory of singular value decomposition, there exists an orthonormal basis of , such that

(2.13)

Set

(2.14)

Then we have

(2.15)

This implies that is an orthonormal basis of . Here and in the sequel, we call such and the S-bases of and for (w.r.t. and respectively). Then

Therefore (b) is equivalent to (c). Also note that is equivalent to for an acute angle . So (b) and (d) are equivalent as well.

If is an isometric minimal embedding, i.e. , then (2.12) implies and , hence Conditions (a)-(b) hold.

Conversely, when Conditions (a)-(b) hold, substituting and into (2.12) implies Since ,

for every . Hence

(2.16)

and moreover , i.e., is an LOS in .


2.3. Characterizations of trivial LOMs

For an isometric totally geodesic embedding , it is easy to see that is totally geodesic in for arbitrary . We call such a trivial LOM. The following characterizes trivial LOMs from the aspect of singular values.

Proposition 2.3.

For an LOM , the followings are equivalent:

  1. All singular values of are equal at each .

  2. All singular values of are equal to .

  3. is an isometric immersion.

  4. is an isometric totally geodesic embedding.

  5. For every , is totally geodesic.

  6. There exists , such that is a totally geodesic LOS.

Proof..

(i)(ii) immediately follows from Condition (b) in Theorem 2.2; (iii)(iv) is a direct corollary of Condition (a) in Theorem 2.2 and the Gauss equations; and the proofs of (ii)(iii) and (iv)(v)(vi)(i) are trivial. ∎

Corollary 2.4.

Let be a smooth map. Then

  • If and , then cannot be an LOM.

  • If and , then is an LOM if and only if is a trivial one.

Proof..

We shall study each case according to the the values of and .

Case I. . In this case, has only one singular value. By (i) of Proposition 2.3, is an LOM if and only if is a trivial one.

Case II. , . If there were one LOM , then by Theorem 2.2 is harmonic. So is its lifting map . But the strong maximal principle forces to be constant, and hence the same for , which contradicts (ii) of Proposition 2.3. Thus there are no LOMs in this setting.

Case III. . Suppose is an LOM and is the corresponding isometric minimal embedding. Then by Theorem 2.2 and (2.16) and are both harmonic. It is well known that every harmonic map from a -sphere (equipped with arbitrary metric) is conformal (see §I.5 of [40]). For an orthonormal basis of , we have

and

Hence is an isometric immersion. By (iii) of Proposition 2.3, is a trivial LOM. ∎

Remarks.

  • The Hopf map from onto gives a nontrivial LOM. Thus the restriction on and in Corollary 2.4 is necessary and optimal.

  • For an LOM , the corresponding is flat if and only if is trivial. Hence the second part of the above corollary follows from the rigidity theorems in [11], [2] and [20]. However, it is unknown up to now whether or not there exists a nonflat, non-parametric minimal cone of codimension 2, so the first part of Corollary 2.4 cannot be derived from previous works.


2.4. Nontrivial LOMSEs

It can be observed that three original LOMs, the Hopf maps for and , have singular values and of multiplicities and pointwise. In fact, we can gain the following structure theorem for LOMSEs.

Theorem 2.5.

For smooth , the followings are equivalent:

  1. is a nontrivial LOMSE, namely for each , all the nonzero singular values of are equal.

  2. is an LOM, and has two constant singular values and of multiplicities and respectively everywhere.

  3. There exist a -dimensional Riemannian manifold with , a real number , a map from onto and a map from into , such that , is a harmonic Riemannian submersion with connected fibers, and is an isometric minimal immersion.

Assume satisfies one of the above. Then becomes an LOS exactly when

(2.17)

The proof of the theorem relies on the next two lemmas.

Lemma 2.6.

Let be Riemannian manifolds. Assume is connected and compact, and a smooth map with singular values 0 and 1 of multiplicities and pointwise. Then there exist a Riemannian manifold , a Riemannian submersion whose fibers are all connected, and an isometric immersion , such that .

We save its proof to Appendix §4.1.

Lemma 2.7.

Let be a smooth foliation of -dimensional submanifolds in a manifold , where is an index set. Suppose are Riemannian metrics on , satisfying:

  1. constant , such that for all ;

  2. For every , , and ,
    if and only if .

Then is minimal in if and only if it is minimal in .

Proof..

Let be the Levi-Civita connection for . Then we have (e.g. see §2.3 of [8])

(2.18)

for vector fields on .

With notations for the second fundamental form of in and the mean vector field, we deduce from (2.18) that

Here is a local orthonormal tangent frame field on , such that for every , forms an orthonormal basis of , and in addition, is a vector field on that is orthogonal to leaves.

Similarly, with symbols and for , we have

Since is arbitrary, if and only if . ∎

Proof of Theorem 2.5.

Let be an LOM, the associated LOS in , and the singular values of of multiplicities and respectively. Then Condition (b) of Theorem 2.2 implies

(2.19)

Since varies continuously in , both and have to be constant on . Hence (i)(ii) and (2.17) follows immediately from (2.19).

(ii) means has singular values and of multiplicities and . By Lemma 2.6, there exist a Riemannian manifold , a Riemannian submersion with connected fibers and an isometric immersion , such that . To deduce (iii), it suffices to show both and are harmonic.

By Condition (a) of Theorem 2.2, is harmonic. So is . Moreover, (2.5) leads to

where can be arbitrary orthonormal basis of the tangent plane of at the considered point and the second fundamental form of the immersed in . Observe that and are tangent and normal vectors to respectively. Therefore, is harmonic, and

(2.20)

Assume and . Let and be S-bases of and for accordingly. Then implies that gives an orthonormal basis of and for (i.e., are fiberwise). Hence and by (2.14), is an isometric minimal immersion.

Next, we show is harmonic. By the above, both with and are Riemannian submersions. Since and satisfy Conditions (a)-(b) of Lemma 2.7, together with Proposition 2.1 we gain the harmonicity of from that of w.r.t. . Thus, (ii)(iii).

Finally, the proof of (iii)(i) is quite similar to the idea of showing (ii)(iii), where one instead argues that the minimality of fibers under also guarantees the minimality for based on Lemma 2.7. ∎


2.5. LOMSEs of (n,p,k)-type

Furthermore, in conjunction with Theorem 2.5 and the spectrum theory of Laplacian operators, we show the following properties of LOMSEs.

Theorem 2.8.

Let be an LOMSE with nonzero singular value of multiplicity . Then there exists an integer , such that:

  • For in , each component is a spherical harmonic function of degree .

  • is an LOS associated to if and only if

    (2.21)

We call such an LOMSE of (n,p,k)-type.

Proof..

By Theorem 2.5, there exist a Riemannian manifold , and , such that , is a harmonic Riemannian submersion and is an isometric minimal immersion.

For , by we mean the position vector of in . Then where is the identity map from to . Since is an isometric minimal immersion and a totally geodesic map, we have and thereby via the composition formula (2.5) obtain

(2.22)

Here is an orthonormal basis of . For , (2.22) states precisely

(2.23)