Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds

This article shows that for generic choice of Riemannian metric on a smooth manifold $M$ of dimension at least three, all prime compact parametrized minimal surfaces within $M$ have transversal self-intersections, and when the dimension of $M$ is four, the tangent planes at any self-intersection are in general position. This implies via geometric measure theory that when $M$ is compact and oriented of dimension four, $H_2(M;{\mathbb Z})$ is generated by homology classes that are represented by imbedded minimal surfaces.

be regarded as a critical point for the energy function In this formula, |df | and dA are calculated with respect to some Riemannian metric on Σ which lies within the conformal class ω ∈ T . The bumpy metric theorem of [4] states that for generic choice of Riemannian metric on a manifold M of dimension at least four, all prime compact oriented parametrized minimal surfaces f : Σ → M are free of branch points and are as nondegenerate (in the sense of Morse theory) as allowed by the group G of conformal automorphisms of Σ. If G is discrete, they are Morse nondegenerate in the usual sense, while if G has positive dimension, they lie on nondegenerate critical submanifolds which have the same dimension as G. (By a nondegenerate critical submanifold for F : M → R, where M is a Banach manifold, we mean a submanfold S ⊂ M consisting entirely of critical points for F such that the tangent space to S at a given critical point is the space of Jacobi fields for F .) A corresponding bumpy metric theorem also holds for nonorientable surfaces; it is proven by use of oriented double covers, as described in §11 of [4]. We consider the subset of the s-fold cartesian product Σ s for s a positive integer, as well as the multidiagonal in the s-fold cartesian product M s , In accordance with [2] , Chapter III, §3, we then say that an immersion f : Σ → M has transversal crossings if for every s > 1, the restriction of . if M has dimension four, then at any self-intersection point, the tangent planes are in general position with respect to the metric, that is, they are not simultaneously complex for any orthogonal complex structure on the tangent space.
We emphasize that the minimal surfaces considered in Theorem 1 are not required to be area-minimizing or even stable.
Recall that according to a well-known theorems of Sacks and Uhlenbeck [7], if M is a compact smooth Riemannian manifold of dimension at least three, a set of generators for π 2 (M ) as a Z[π 1 (M )]-module can be represented by area minimizing minimal two-spheres. Theorem 1 shows that when the metric on M is generic, these generators can be taken to be imbedded minimal two-spheres when M has dimension at least five, and to be immersions with transverse double points when M has dimension four. Moreover, when M has dimension four, the generic condition on the tangent planes enables us to use a result of Frank Morgan [6] to show that if f : Σ → M is a surface of genus g which minimizes area in some homology class, and f has points of self-intersection, then one of the self-intersections can be removed by surgery, producing a surface of larger genus and smaller area in the same homology class. Assuming Theorem 1, we can prove Theorem 2 as follows. Results of Almgren and Chang [1] (see the Main Regularity Result on page 72 of [1]) imply that any homology class is represented by an area minimizing integral current which arises from a smooth submanifold except for possible branch points and self-intersections. This can be represented by a finite collection of parametrized minimal surfaces, each of which is either prime or a branched cover of a prime minimal surface. Let f i : Σ i → M for 1 ≤ i ≤ k be the underlying prime minimal surfaces, where each Σ i is connected. When the metric is generic, it follows from the Main Theorem of [4] that each such f i is free of branch points, while when the dimension of M is at least five, it follows from Theorem 1 that there are no self-intersections, or intersections between different components. When the dimension of M is four, Theorem 1 states that at the self-intersections the two tangent planes of f i cannot be simultaneously complex for any orthogonal complex structure at the point of intersection. It therefore follows from Theorem 2 of [6] that if any f i has nontrivial self-intersections, one of the self-intersections could be eliminated with a decrease in area, thereby contradicting the fact that the current is area minimizing. Thus the f i 's must be imbeddings. Similarly, the area could be decreased if the images of different f i 's were not mutually disjoint, again contradicting area minimization. This proves Theorem 2.
Theorem 2 is related to an earlier result of Brian White [10] which treats unoriented surfaces. k−1 completion of the space of smooth Riemannian metrics on M .) These completions are Banach manifolds rather than Fréchet manifolds. However, to keep the notation simple, we will continue to denote the completions by Map(Σ, M ) and Met(M ). It is shown in [4] that f is a prime immersed conformal ω-harmonic map }. (2) is a smooth submanifold. The Main Theorem of [4] implies that if g 0 is generic metric on M , then all prime conformal harmonic maps for g 0 are immersed and hence lie in P ∅ . Moreover, for any such metric, each element of N g0 = π −1 2 (g 0 ) ∩ P ∅ is either a nondegenerate critical point for the energy, or lies in a nondegenerate critical submanifold which has the same dimension as the group G of symmetries for Σ. Here Using this, we can calculate the tangent space to P ∅ , the result being where L is the Jacobi operator of E, D 2 F is the derivative with respect to Met(M ) and π V denotes projection into the vertical tangent space at a zero of F . It follows from this expression and from Lemma 6.1 of [4] that if (f, ω, g 0 ) is any element of N g0 , then the projection on the first factor, (If Σ has a positive-dimensional group G of conformal automorphisms, we also use the fact that the orbits of the G-action generate the tangential Jacobi fields.) Thus all pairs (f ′ , ω ′ ) sufficiently close to (f, ω) lie in the image of P ∅ , and can be realized by parametrized minimal surfaces for metrics which are near g 0 .
To prove the first statement of Theorem 1, we construct a countable cover 1. it is the domain for a submanifold chart for P ∅ , 2. the restriction of π 2 : The second condition can be arranged by Theorem 1.6 of [9] and the last condition follows from (3). It follows from standard transversality theory for finite-dimensional manifolds (see §2 of Chapter 3 of [3] or Proposition 3.2 of Chapter III, §3 of [2]) that The second statement is proven by the same argument, modified to the case where Σ is a compact surface with two components instead of one.
The key assertion of Theorem 1 is the last one. Assuming that M has dimension four, we need to construct a variation of the metric which puts a given intersection into general position, the two intersecting planes not being simultaneously complex for an orthogonal complex structure. The argument for the first statement of the Theorem shows that we need only consider one transversal intersection at a time.
Suppose that p and q are distinct points of Σ and that f (p) = f (q), and let V 1 and V 2 be disjoint open neighborhoods of p and q within Σ. We construct coordinates (u 1 , u 2 , u 3 , u 4 ) on a neighborhood U of f (p) in M so that Let g ij be the components of the metric in these coordinates, so that g ab = λ 2 1 δ ab , g rs = λ 2 2 δ rs .
We assume that at the intersection point, f * (T p Σ) and f * (T q Σ) are simultaneously complex for some orthogonal complex structure on T M . (After reordering u 3 and u 4 if necessary, we can then assume without loss of generality that g 13 = g 24 and g 14 = −g 23 .) If we define the Christoffel symbols in terms of the metric the fact that f is harmonic is expressed by the equations We will construct a variation in the metric (ġ ij ) such thatġ ab = 0 =ġ rs and the equations (4) continue to hold. The resulting variationΓ k,ij in the Christoffel symbols will then satisfy the equationṡ Γ b,aa = 0,Γ r,aa = ∂ġ ra ∂u a ,Γ s,rr = 0,Γ a,rr = ∂ġ ra ∂u r .
Thus we want to arrange that a ∂ġ ra ∂u a = 0 along f (V 1 ), and r ∂ġ ra ∂u r = 0 along f (V 2 ).
If we construct a smooth function h : U → R and then set ġ 13ġ14 g 23ġ24 = , we find that the equations (5) are satisfied. We can choose such a function which has compact support within U , and for which ġ 13ġ14 g 23ġ24 (f (p)) is arbitrary. The resulting metric perturbation will preserve conformality and minimality of f as required, yet can be chosen so that after perturbation f * (T p Σ) and f * (T q Σ) will not be simultaneously complex for some orthogonal complex structure on T f (p) M = T f (q) M . This finishes the proof of Theorem 1.