Yau’s conjecture of abundance of minimal hypersurfaces is generically true (in low dimensions)

Last week, my friends Fernando and André uploaded to the arXiv their remarkable paper “Denseness of minimal hypersurfaces for generic metrics” joint with Kei Irie (and, in fact, Fernando sent me a copy of this article about one day before arXiv’s public announcement).

The motivation for the work of Irie–Marques–Neves is a famous conjecture of Yau on the abundance of minimal surfaces.

More precisely, Yau conjectured in 1982 that a closed Riemannian -manifold contains infinitely many (smooth) closed immersed minimal surfaces. Despite all the activity around this conjecture, the existence of infinitely many embedded minimal hypersurfaces in manifolds of positive Ricci curvature of low dimensions was only established very recently by Fernando and André.

In their remarkable paper, Irie–Marques–Neves show that Yau’s conjecture is generically true in low dimensions by establishing the following stronger statement:

Theorem 1 Let be a closed manifold of dimension . Then, a generic Riemannian metric on has a lot of minimal hypersurfaces: the union of all of its closed (smooth) embedded minimal hypersurfaces is a dense subset of .

Remark 1 The hypothesis of low dimensionality is related to the fact that area-minimizing minimal hypersurfaces in dimensions might exhibit non-trivial singular sets (as it was famously proved by Bombieri–De Giorgi–Guisti), but such a phenomenon does not occur in low dimensions for “min-max” minimal hypersurfaces thanks to the regularity theories of Almgren, Pitts and Schoen–Simon.

The remainder of this post is dedicated to the proof of this theorem and, as usual, all eventual errors/mistakes in what follows are my entire responsibility.

1. Description of the key ideas

Let be a closed manifold and be the space of Riemannian metrics on .

Given an open subset , let be the subset of Riemannian metrics possessing a non-degenerateclosed (smooth) embedded minimal hypersurface passing through . (Here, non-degenerate means that all Jacobi fields are trivial.)

It is possible to check that a non-degenerate closed embedded minimal hypersurface in is persistent: more concretely, one can use the definition of non-degeneracy and the inverse function theorem to obtain that any Riemannian metric close to possesses an unique closed embedded minimal hypersurface nearby . In particular, every is open.

Also, let us observe (for later use) that the non-degeneracy condition is not difficult to obtain:

Proposition 2 Let be a closed (smooth) embedded minimal hypersurface in the Riemannian manifold . Then, we can perform conformal perturbations to find a sequence of metrics converging to (in -topology) such that is a non-degenerate minimal hypersurface of for all sufficiently large .

Proof: This statement is Proposition 2.3 in Irie–Marques–Neves paper and its proof goes along the following lines.

Fix a bump function supported in a small neighborhood of and coinciding with the square of the distance function nearby .

The metrics are conformal to , and they converge to in the -topology. Furthermore, the features of the distance function imply that is a minimal hypersurface of such that the Jacobi operator  acting on normal vector fields verify

for all . In particular, the spectrum of is derived from the spectrum of by translation by . Therefore, doesn’t belong to the spectrum of for all surfficiently large, and, hence, is a non-degenerate minimal hypersurface of for all large .

Coming back to Theorem 1, we affirm that our task is reduced to prove the following statement:

Theorem 3 Let be a closed manifold of dimension . Then, for any open subset , one has that is dense in .

In fact, assuming Theorem 3, we can easily deduce Theorem 1: if is a countable basis of open subsets of , then Theorem 3 ensures that is a countable intersection of open and dense subsets of the Baire space ; in other terms, is a residual / generic subset of such that any satisfies the conclusions of Theorem 1 (thanks to the definition of and our choice of ).

Remark 2 Note that, since is a Baire space, it follows from Baire category theorem that is a dense subset of .

Let us now explain the proof of Theorem 3. Given a neighborhood of a smooth Riemannian metric on a closed manifold , and an open subset , our goal is to show that

For this sake, we apply White’s bumpy metric theorem asserting that we can find such that all closed (smooth) immersed minimal hypersurfaces in are non-degenerate.

If , then we are done. If , then all closed (smooth) embedded minimal hypersurfaces in avoid . In this case, we can naively describe the idea of Irie–Marques–Neves to perturb to get as follows:

• we perturb  only in to obtain whose volume is strictly larger than the volume ;
• by the so-called Weyl law for the volume spectrum (conjectured by Gromov and recently proved by Liokumovich–Marques–Neves), the –widths of are strictly larger than those of ;
• since -widths “count” the minimal hypersurfaces, the previous item implies that new minimal hypersurfaces in were produced;
• because coincides with outside , the minimal hypersurfaces of avoiding are the exactly same of ; thus, the new minimal hypersurfaces in mentioned above must intersect , i.e., .

In the sequel, we will explain how a slight variant of this scheme completes the proof of Theorem 3.

2. Increasing the volume of Riemannian metrics

Let as above. Take a non-negative smooth bump function supported in such that for some .

Consider the family of conformal deformations of . Note that for all .

From now on, we fix once and for all such that for all .

3. Weyl law for the volume spectrum

Roughly speaking, the –width  of a Riemannian manifold is the following min-max quantity. We consider the space of closed hypersurfaces of , and –sweepouts  of , i.e., is a continuous map from the -dimensional real projective space to which is “homologically non-trivial” and, a fortiori, is not a constant map.

Remark 3 Intuitively, a -sweepout is a non-trivial way of filling with -parameter family of hypersurfaces (which is “similar” to the way the -parameter family of curves fills the round -sphere ).

If we denote by the set of -sweepouts such that no concentration of mass occur (i.e., ), then the -width is morally given by

Remark 4 Formally speaking, the definition of -width involves more general objects than the ones presented above: we construct by replacing hypersurfaces by certain -dimensional flat chains modulo two, we allow arbitrary simplicial complexes in place of , etc.: see Irie–Marques–Neves’ paper for more details and/or references.

The -width varies continuously with (cf. Lemma 2.1 in Irie–Marques–Neves’ paper). Moreover, it “counts” minimal hypersurfaces (cf. Proposition 2.2 in Irie–Marques–Neves’ paper): if has dimension , then, for each , there is a finite collection of mutually disjoint, closed, smooth, embedded, minimal hypersurfaces in with (stability) indices such that

for some integers . (Here, the stability index is the quantity of negative eigenvalues of the Jacobi operator.)

Furthermore, the asymptotic behavior of is described by Weyl’s law for the volume spectrum (conjectured by Gromov and confirmed by Liokumovich–Marques–Neves): for some universal constant , one has

In particular, coming back to the context of Section 2, Weyl’s law for the volume spectrum and the fact that has volume strictly larger than mean that we can select such that the -width is strictly larger than the -width of , i.e.,

4. New minimal hypersurfaces intersecting

We affirm that there exists such that possesses a closed (smooth) embedded minimal hypersurface passing through .

Otherwise, for each , all closed (smooth) embedded minimal hypersurface in would avoid . Since coincides with outside (by construction), the “counting property of -widths” in equation (1) would imply that

for all .

On the other hand, the fact that is bumpy (in the sense of White’s theorem) permits to conclude that the set is countable: indeed, a recent theorem of Sharp implies that the collection of connected, closed, smooth embedded minimal hypersurfaces in with bounded index and volume is finite, so that is countable.

Since the -width depends continuously on the metric, the countability of implies that the function is constant on . In particular, we would have

a contradiction with (2). So, our claim is proved.

At this point, the argument is basically complete: the metric has a closed (smooth) embedded minimal hypersurface passing through ; by Proposition 2, we can perturb (if necessary) in order to get a metric such that is a non-degenerate closed (smooth) embedded minimal hypersurface in , that is, , as desired.

This proves Theorem 3 and, consequently, Theorem 1.

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