Posted by: matheuscmss | November 5, 2017

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 {3}-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 {M^d} of low dimensions {3\leq d\leq 7} 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 {M^d} be a closed manifold of dimension {3\leq d\leq 7}. Then, a generic Riemannian metric {g} on {M^d} has a lot of minimal hypersurfaces: the union of all of its closed (smooth) embedded minimal hypersurfaces is a dense subset of {M}.

Remark 1 The hypothesis {3\leq d\leq 7} of low dimensionality is related to the fact that area-minimizing minimal hypersurfaces in dimensions {>7} 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 {M^d} be a closed manifold and {\mathcal{M}} be the space of {C^{\infty}} Riemannian metrics on {M^d}.

Given an open subset {U\subset M^d}, let {\mathcal{M}_U} be the subset of Riemannian metrics {g\in\mathcal{M}} possessing a non-degenerateclosed (smooth) embedded minimal hypersurface passing through {U}. (Here, non-degenerate means that all Jacobi fields are trivial.)

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

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

Proposition 2 Let {\Sigma} be a closed (smooth) embedded minimal hypersurface in the Riemannian manifold {(M^{n+1},g)}. Then, we can perform conformal perturbations to find a sequence of metrics {(g_i)_{i\in\mathbb{N}}} converging to {g} (in {C^{\infty}}-topology) such that {\Sigma} is a non-degenerate minimal hypersurface of {(M^{n+1}, g_i)} for all sufficiently large {i\in\mathbb{N}}.

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

Fix a bump function {h:M\rightarrow \mathbb{R}} supported in a small neighborhood of {\Sigma} and coinciding with the square {d_g(x,\Sigma)^2} of the distance function {d_g(x,\Sigma)} nearby {\Sigma}.

The metrics {g_i=\exp(2h/i)g} are conformal to {g}, and they converge to {g} in the {C^{\infty}}-topology. Furthermore, the features of the distance function imply that {\Sigma} is a minimal hypersurface of {g_i} such that the Jacobi operator {L_{\Sigma, g_i}} acting on normal vector fields verify

\displaystyle L_{\Sigma, g_i}(X) = L_{\Sigma, g}(X) - \frac{2n}{i}X

for all {i\in\mathbb{N}}. In particular, the spectrum of {L_{\Sigma, g_i}} is derived from the spectrum of {L_{\Sigma, g}} by translation by {2n/i}. Therefore, {0} doesn’t belong to the spectrum of {L_{\Sigma, g_i}} for all {i} surfficiently large, and, hence, {\Sigma} is a non-degenerate minimal hypersurface of {(M,g_i)} for all large {i}. \Box

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

Theorem 3 Let {M^d} be a closed manifold of dimension {3\leq d\leq 7}. Then, for any open subset {U\subset M}, one has that {\mathcal{M}_U} is dense in {\mathcal{M}}.

In fact, assuming Theorem 3, we can easily deduce Theorem 1: if {(U_i)_{i\in\mathbb{N}}} is a countable basis of open subsets of {M^d}, then Theorem 3 ensures that {\mathcal{G}=\bigcap\limits_{i\in\mathbb{N}} \mathcal{M}_{U_i}} is a countable intersection of open and dense subsets of the Baire space {\mathcal{M}}; in other terms, {\mathcal{G}} is a residual / generic subset of {\mathcal{M}} such that any {g\in\mathcal{G}} satisfies the conclusions of Theorem 1 (thanks to the definition of {\mathcal{M}_U} and our choice of {(U_i)_{i\in\mathbb{N}}}).

Remark 2 Note that, since {\mathcal{M}} is a Baire space, it follows from Baire category theorem that {\mathcal{G}} is a dense subset of {\mathcal{M}}.

Let us now explain the proof of Theorem 3. Given a neighborhood {\mathcal{V}} of a smooth Riemannian metric {g\in\mathcal{M}} on a closed manifold {M^d}, and an open subset {U\subset M}, our goal is to show that

\displaystyle \mathcal{V}\cap\mathcal{M}_U\neq\emptyset

For this sake, we apply White’s bumpy metric theorem asserting that we can find {g'\in\mathcal{V}} such that all closed (smooth) immersed minimal hypersurfaces in {(M,g')} are non-degenerate.

If {g'\in\mathcal{M}_U}, then we are done. If {g'\notin\mathcal{M}_U}, then all closed (smooth) embedded minimal hypersurfaces in {(M,g')} avoid {U}. In this case, we can naively describe the idea of Irie–Marques–Neves to perturb {g'} to get {g''\in\mathcal{V}\cap\mathcal{M}_U} as follows:

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

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 {g'\in\mathcal{V}\setminus\mathcal{M}_U} as above. Take {h} a non-negative smooth bump function supported in {U} such that {h(x_0)=1} for some {x_0\in U}.

Consider the family {g'(t)=(1+th)g'} of conformal deformations of {g'}. Note that {\textrm{vol}(M,g'(t))>\textrm{vol}(M,g')} for all {t>0}.

From now on, we fix once and for all {t_0>0} such that {g'(t)\in\mathcal{V}} for all {t\in[0,t_0]}.

3. Weyl law for the volume spectrum

Roughly speaking, the {k}width {\omega_k(M,g)} of a Riemannian manifold {(M^{n+1},g)} is the following min-max quantity. We consider the space {\mathcal{Z}_n} of closed hypersurfaces of {M^{n+1}}, and {k}sweepouts {\Phi:\mathbb{RP}^k\rightarrow\mathcal{Z}_n} of {M}, i.e., {\Phi} is a continuous map from the {k}-dimensional real projective space {\mathbb{RP}^k} to {\mathcal{Z}_n} which is “homologically non-trivial” and, a fortiori, {\Phi} is not a constant map.

Remark 3 Intuitively, a {k}-sweepout {\Phi} is a non-trivial way of filling {M} with {k}-parameter family of hypersurfaces (which is “similar” to the way the {1}-parameter family of curves {\{z=\textrm{constant}\}\cap S^2} fills the round {2}-sphere {S^2=\{(x,y,z)\in\mathbb{R}^3: x^2+y^2+z^2=1\}}).

If we denote by {\mathcal{P}_k} the set of {k}-sweepouts {\Phi} such that no concentration of mass occur (i.e., {\lim\limits_{r\rightarrow 0}\sup\{n\textrm{-dimensional volume of }\Phi(x)\cap B_r(p): x\in \mathbb{RP}^k, p\in M\} = 0}), then the {k}-width is morally given by

\displaystyle \omega_k(M,g) = \inf\limits_{\Phi\in\mathcal{P}_k} \sup\{n\textrm{-dimensional volume of }\Phi(x): x\in\mathbb{RP}^k\}

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

The {k}-width {\omega_k(M,g)} varies continuously with {g} (cf. Lemma 2.1 in Irie–Marques–Neves’ paper). Moreover, it “counts” minimal hypersurfaces (cf. Proposition 2.2 in Irie–Marques–Neves’ paper): if {M^{n+1}} has dimension {3\leq n+1\leq 7}, then, for each {k\in\mathbb{N}}, there is a finite collection {\{\Sigma_1,\dots,\Sigma_N\}} of mutually disjoint, closed, smooth, embedded, minimal hypersurfaces in {(M,g)} with (stability) indices {\sum\limits_{i=1}^N\textrm{index}(\Sigma_i)\leq k} such that

\displaystyle \omega_k(M,g)=\sum\limits_{i=1}^N m_i\textrm{vol}_g(\Sigma_i) \ \ \ \ \ (1)

for some integers {m_1,\dots, m_N\in\mathbb{N}}. (Here, the stability index is the quantity of negative eigenvalues of the Jacobi operator.)

Furthermore, the asymptotic behavior of {\omega_k(M,g)} is described by Weyl’s law for the volume spectrum (conjectured by Gromov and confirmed by Liokumovich–Marques–Neves): for some universal constant {a(n)>0}, one has

\displaystyle \lim\limits_{k\rightarrow\infty} \omega_k(M,g)k^{-\frac{1}{n+1}} = a(n)\textrm{vol}(M,g)^{\frac{n}{n+1}}

In particular, coming back to the context of Section 2, Weyl’s law for the volume spectrum and the fact that {g'(t_0)} has volume strictly larger than {g'} mean that we can select {k_0\in\mathbb{N}} such that the {k_0}-width {\omega_{k_0}(M,g'(t_0))} is strictly larger than the {k_0}-width of {\omega_{k_0}(M,g')}, i.e.,

\displaystyle \omega_{k_0}(M,g'(t_0)) > \omega_{k_0}(M,g') \ \ \ \ \ (2)

4. New minimal hypersurfaces intersecting {U}

We affirm that there exists {0<t_1\leq t_0} such that {g'(t_1)} possesses a closed (smooth) embedded minimal hypersurface passing through {U}.

Otherwise, for each {0\leq t\leq t_0}, all closed (smooth) embedded minimal hypersurface in {(M,g'(t))} would avoid {U}. Since {g'(t)} coincides with {g'} outside {U} (by construction), the “counting property of {k}-widths” in equation (1) would imply that

\displaystyle \omega_{k_0}(M,g'(t))\in\mathcal{C} = \left\{\sum\limits_{j=1}^N m_j\textrm{vol}_{g'}(\Sigma_j): m_j\subset\mathbb{N} \textrm{ and } \Sigma_j \textrm{ minimal in }(M,g')\right\}

for all {t\in[0,t_0]}.

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

Since the {k}-width depends continuously on the metric, the countability of {\mathcal{C}} implies that the function {t\mapsto \omega_{k_0}(M,g'(t))} is constant on {[0,t_0]}. In particular, we would have

\displaystyle \omega_{k_0}(M,g'(t_0)) = \omega_{k_0}(M,g'(0)) = \omega_{k_0}(M,g'),

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

At this point, the argument is basically complete: the metric {g'(t_1)\in\mathcal{V}} has a closed (smooth) embedded minimal hypersurface {\Sigma} passing through {U}; by Proposition 2, we can perturb {g'(t_1)} (if necessary) in order to get a metric {g''\in\mathcal{V}} such that {\Sigma} is a non-degenerate closed (smooth) embedded minimal hypersurface in {(M,g'')}, that is, {g''\in\mathcal{V}\cap\mathcal{M}_U}, as desired.

This proves Theorem 3 and, consequently, Theorem 1.

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Responses

  1. Hi Carlos,

    The restriction dim<8 appears in many of their theorems. How do those examples with singular sets appear? Is there a chance that for generic R. metrics minimal hypersurfaces are actually nice?

    • Hi Andrey,

      Classically, the singular set appears when we try to find area-minimizing hypersurfaces looking like catenoids in Euclidean spaces: if we start in dimension 2 with the four points of coordinates (\pm 1,\pm 1), the area-minimizing “surface” with these points as boundaries is the union of two vertical segments; if we go to three dimensions, the area-minimizing surface passing through the two circles \{(x,y,\pm1): x^2+y^2=1\} is a catenoid rather than a cylinder; in other words, by increasing the dimension, the area-minimizing hypersurface tended to go closer to the origin; if we keep increasing dimensions, then the area-minimizing hypersurface in eight dimension becomes the so-called Simons cone, so that a singularity at the origin pops up.

      As it turns out, a vast “regularity theory” was developed to explain that this phenomenon of non-trivial singular sets doesn’t occur in dimensions < 8 (and also that singular sets usually have codimension at least 8): see e.g. Ambrosio’s Bourbaki talk for a nice introduction to the subject.

      Concerning your last question (about “generic regularity”), I must confess that it is hard for me to give a quick answer (as I’m not an expert on the Almgren–Pitts theory). I will try to consult Fernando on this matter and I will get back to you as soon as possible.

      Best,

      Matheus

  2. Very interesting about the Simons cone, thanks. The boundaries of the cone are some sort of 7d hyperboloids rather than spheres. I guess in the case of spheres singularities don’t form…

    • Hi Andrey,

      I talked to Fernando and he pointed out to me the following facts.

      First, it is probably a hard problem to develop a theory of generic singularities in higher dimensions: indeed, such a theory would imply a solution for a notorious well-known open problem, namely, establishing the positive mass theorem (https://en.wikipedia.org/wiki/Positive_energy_theorem) in any dimension. Anyhow, so far one of the best results available in dimension eight is the fact (due to Nat Smale) that the singular set consists of isolated singularities.

      Secondly, Fernando believes nevertheless that it could be possible (after *non-trivial* work) to extend the statement of Irie–Marques–Neves theorem by possibly including singular minimal hypersurfaces in the discussion *if* one could generalize White’s bumpy metric theorem for “non-degenerate singular minimal hypersurfaces” (whatever this means…).

      Best,

      Matheus

      • “First, it is probably a hard problem to develop a theory of generic singularities in higher dimensions: indeed, such a theory would imply a solution for a notorious well-known open problem, namely, establishing the positive mass theorem in any dimension”

        I believe the positive mass theorem was proved earlier this year by schoen-yau: https://arxiv.org/abs/1704.05490

  3. Dear Carlos,

    Appreciate for the very nice blog for explaining the result and describe the sketch of proof of it!I am curious with both the statement and proof of this result. Unfortunately, I faced some difficulties on understanding the underly idea clearly.

    1.

    It is easy to see that \mathcal{M}_U is open by the property of non-degenerated, I think the subtle thing that why we need to do a small conformal perbutation to make the hyper surface to be non degenerate maybe is in fact this mean the hypersurfaces \Gamma with metric g could be look as a lots of geodesic segment start form a point x\in \Gamma so there is no topology obstacle because the Jacobi field is trivial, then the openness is just come from a uniformly estimate of a sires of ODE which take charge of the geodesic.

    2.

    So it is suffice to proof dense. Contrasty, we assume there is a metric g and a open neiberghood V of g such that \mathcal{M}_U\cap V=\emptyset. this result is a corollary of the following two fact on k-width of (M,g) which is the number \omega_k(M,g) = \inf\sup \{M(\Phi(x)) : x \in dmn(\Phi)\}:

    \omega_k(W,g) depends continuously on metric g under C^{\infty} topology.

    We have a representation \omega_k(W,g),i.e. \omega_k(W,g) = \sum_{j=1}^N m_j vol_g(\Sigma_j),

    These two argument both have subtle details I do not understand fully, the first one is come from the require $\Phi$ is not mass concentrate, I know this is important to establish continuity. But how could we proof this type of \omega_k(W,g)\neq \emptyset?

    On the other hand, I could not understand what is the exact meaning of k_(W,g) although I know it is come from a definition of maximum-minimum principle so I know look it as some “spectrum”. but how to understand it in a geometric point of view?
    this representation is come from a approximation argument:\omega_k(M,g) \leq L(\Pi_i) \leq \sup{M(\Phi_i(x)) : x \in X(k)}, where L(\Pi_i) =\inf_{\Phi' i\in \Pi_i}\sup_{x\in X(k)}\{M(\Phi' i(x))\}, but I do not understand what is the index of \Sigma_{i,j} and why we still need to use The Compactness Theorem of Sharp to pass to a subsequences?

    I try to understand it but due to I am not very familiar to this area of minimal hyper surface those problem may be very stupid. But I will be very appreciate for an answer.

    best,

    Xiyu

  4. Why is there no article on Wikipedia about this important conjecture? xD I would create it my self, but I am not knowledgeable enough. 😦


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