Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates
The Allen–Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang–Wei [WW17]) of the Allen–Cahn equation on a -manifold. Using these, we are able to show for generic metrics on a -manifold, minimal surfaces arising from Allen–Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen–Cahn setting, a strong form of the multiplicity one conjecture and the index lower bound conjecture of Marques–Neves [Mar14, Nev14] in -dimensions regarding min-max constructions of minimal surfaces.
Allen–Cahn min-max constructions were recently carried out by Guaraco [Gua18] and Gaspar–Guaraco [GG16]. Our resolution of the multiplicity one and the index lower bound conjectures shows that these constructions can be applied to give a new proof of Yau’s conjecture on infinitely many minimal surfaces in a -manifold with a generic metric (recently proven by Irie–Marques–Neves [IMN17]) with additional geometric conclusions. Namely, we prove that a -manifold with a generic metric contains, for each positive integer , a two-sided embedded minimal surface with Morse index and area , as conjectured by Marques–Neves.
- 1.1 Notation
1.2 Main results
- 1.2.1 Curvature estimates for stable solutions of (1.1) on -manifolds
- 1.2.2 Strong sheet separation estimates for stable solutions
- 1.2.3 The multiplicity one-conjecture for limits of the Allen–Cahn equation in -manifolds
- 1.2.4 Index lower bounds
- 1.2.5 Applications related to Yau’s conjecture on infinitely many minimal surfaces
- 1.3 One-dimensional heteroclinic solution,
- 1.4 Organization of the paper
- 1.5 Acknowledgements
- 2 Multiplicity one constructions with Dirichlet data
- 3 From phase transitions to Jacobi-Toda systems
- 4 Stable phase transitions ()
- 5 Phase transitions with bounded Morse index ()
- 6 Phase transitions with multiplicity one
- 7 Geometric applications
- A Mean curvature of normal graphs
Minimal surfaces—critical points of the area functional with respect to local deformations—are fundamental objects in Riemannian geometry due to their intrinsic interest and richness, as well as deep and surprising applications to the study of other geometric problems. Because many manifolds do not contain any area-minimizing hypersurfaces, one is quickly led to the study of surfaces that are only critical points of the area functional. Such surfaces are naturally constructed by min-max (i.e., mountain-pass) type methods. To this end, Almgren and Pitts [Pit81] have developed a far-reaching theory of existence and regularity (cf. [SS81]) of min-max (unstable) minimal hypersurfaces. In particular, their work implies that any closed Riemannian manifold contains at least one minimal hypersurface (in sufficiently high dimensions, may have a thin singular set). This result motivates a well-known question of Yau: “do all -manifolds contain infinitely many immersed minimal surfaces?” [Yau82].
Recently, there have been several amazing applications of Almgren–Pitts theory to geometric problems, including the proof of the Willmore conjecture by Marques–Neves [MN14] and the resolution of Yau’s conjecture for generic metrics by Irie–Marques–Neves [IMN17]. In spite of this, certain basic questions concerning the Almgren–Pitts construction remain unresolved: including whether or not the limiting minimal surfaces can arise with multiplicity (for a generic metric) as well as whether or not one-sided minimal surfaces can arise as limits of an “oriented” min-max sequence (see, however, [KMN16, MN16a]).
and its singular limit as . There is a well known expectation that, in limit, solutions to (1.1) produce minimal surfaces whose regularity reflects the solutions’ variational properties. In particular:
Under weaker assumptions on the sequence of solutions, one obtains different results. In general, solutions to (1.1) on a Riemannian manifold have a naturally associated -varifold obtained by “smearing out” their level sets of , weighted by the gradient,
Here, is a constant that is canonically associated with (see Section 1.3). A deep result of Hutchinson–Tonegawa [HT00, Theorem 1] ensures that limits to a varifold with a.e. integer density as . If, in addition, one assumes that the solutions are stable, Tonegawa–Wickramasekera [TW12] have shown that the limiting varifold is stable and satisfies the conditions of Wickramasekera’s deep regularity theory [Wic14]; thus the limiting varifold is a smooth stable minimal hypersurface (outside of a codimension singular set). In two dimensions, this was shown by Tonegawa [Ton05].
Guaraco’s approach has certain advantages when compared with Almgren–Pitts theory:
A key difficulty in the work of Almgren–Pitts is a lack of a Palais–Smale condition, which is usually fundamental in mountain pass constructions. On the other hand, the Allen–Cahn equation does satisfy the usual Palais–Smale condition for each (see [Gua18, Proposition 4.4]), so this aspect of the theory is much simpler.
In Almgren–Pitts theory, there is no “canonical” approximation of the limiting min-max surface by nearby elements of a sweepout. On the other hand, Allen–Cahn provides a canonical approximation built out of the function (which satisfies a PDE). It is thus natural to suspect that this might be useful when studying the geometric properties of the limiting surface.
For example, Hiesmayr [Hie17] and Gaspar [Gas17] have shown that index upper bounds for Allen–Cahn solutions directly pass to the limiting surface (we note that the Almgren–Pitts version of this result has been proven by Marques–Neves [MN16a]). Moreover, the second-named author has recently shown [Man17] that -parameter Allen–Cahn min-max on a surface produces a smooth immersed curve with at most one point of self-intersection; in general, Almgren–Pitts on a surface will only produce a geodesic net (cf. [Aie16]).
Our main contributions in this work are as follows:
We show (see Theorem 1.3 below) that the individual level sets of stable solutions to the Allen–Cahn equation on a -manifold with energy bounds satisfy a priori curvature estimates (similar to stable minimal surfaces). Using this, we are can avoid the regularity theory of Wickramasekera and Tonegawa–Wickramasekera entirely, making the whole theory considerably more self-contained.
More fundamentally, our curvature estimates (and sharp sheet separation estimates, which we will discuss below) allow us to study geometric properties of the limiting minimal surface using the “canonical” PDE approximations that exist prior to taking the limit. In particular, we will prove the multiplicity one conjecture of Marques–Neves [MN16a] in the Allen–Cahn setting (see Theorem 1.7 below) for min-max sequences on -manifolds. In fact, we prove an strengthened version of the conjecture which rules out (generically) stable components and one-sided surfaces.
As an application of our multiplicity one results we are able to give a new proof of Yau’s conjecture on infinitely many minimal surfaces in a -manifold, when the metric is bumpy (see Corollary 1.9 below). This has been recently proven using Almgren–Pitts theory by Irie–Marques–Neves [IMN17], for a slightly different class of metrics; their proof works in for and proves, in addition, that the minimal surfaces are dense. Our proof establishes several new geometric properties of the surfaces; in particular, we show that they are two-sided and that their area and Morse index behaves as one would expect, based on the theory of -widths [Gro03, Gut09, MN17, GG16].
We wish to emphasize two things:
Our results work at the level of sequences of critical points of the Allen–Cahn energy functional with uniform energy and Morse index bounds. At no point do we use any min-max characterization of the limiting surface; min-max is merely used as a tool to construct nontrivial sequences of critical points with energy and index bounds.
Our results highlight the philosophy that the solutions to Allen–Cahn provide a “canonical” approximation of the min-max surface.
Here, we recall certain useful notation that we will use below. In all that follows, is a smooth Riemannian manifold.
A function is a double-well potential if:
is non-negative and vanishes precisely at ;
satisfies , for , and ;
The standard double-well potential is , in which case the Allen–Cahn equation becomes .
The Allen–Cahn equation, (1.1), is the Euler–Lagrange equation for the energy functional
Depending on what we wish to emphasize, we will go back and forth between saying that a function is a solution of (1.1) on (or in a domain ) or a critical point of (resp. of ). The second variation of is easily computed (for ) to be
We are thus led to the notion of stability and Morse index (with respect to Dirichlet eigenvalues).
For a complete Riemannian manifold and open, we say that a critical point of is stable on if for all . More generally, we say has Morse index , denoted , if
where the maximum is taken over all subspaces . Sometimes we will write to emphasize the underlying set. Note that if and only if is stable on .
When is a solution of (1.1) and , we will write:
for the unit normal of the level set of through ;
for the second fundamental form of the level set of through ;
for the “Allen–Cahn” or “enhanced” second fundamental form of the level set:
One may check that
where represents the gradient in the directions orthogonal to ; in other words, strictly dominates the second fundamental form of the level sets.
Finally, we will often use Fermi coordinates centered on a hypersurface. To avoid confusion about which hypersurface the coordinates are associated to, we will define a function
where will denote a distinguished normal vector to . In this paper, is generally taken to be the upward pointing unit normal. Note that the pullback of the metric along has the form , which is the setting that most of our analysis will take place below.
1.2. Main results
Curvature estimates for stable solutions of (1.1) on -manifolds
We start this section by discussing the concept of stability applied to minimal surfaces, since that guides some aspects of our work in the Allen–Cahn setting.
We recall that a two-sided minimal surface with normal vector is said to be stable if it satisfies
for . Here, we briefly recall the well-known curvature estimates of Schoen [Sch83] for stable minimal surfaces. If is a complete, two-sided stable minimal surface, then the second fundamental form of , , satisfies
Observe that (1.4) readily implies a stable Bernstein theorem: “a complete two-sided stable minimal surfaces in without boundary must be a flat plane.” On the other hand, the stable Bernstein theorem (proven in [FCS80, dCP79, Pog81]) implies (1.4) by a well known blow-up argument: if (1.4) failed for a sequence of stable minimal surfaces , then by choosing a point of (nearly) maximal curvature and rescaling appropriately (cf. [Whi16]), we can produce a sequence of minimal surfaces in manifolds that are converging on compact sets to with the flat metric, and so that , uniformly bounded on compact sets, and . The second fundamental form bounds yield local bounds for the surfaces , which may then be upgraded to bounds for all . Thus, passing to a subsequence, the surfaces converge smoothly to a complete stable minimal surface without boundary in . Because the convergence occurs in , the we see that , so is non-flat. This contradicts the stable Bernstein theorem.
As such, before discussing curvature estimates for stable solution to Allen–Cahn, we must discuss the stable Bernstein theorem for complete solutions on . In general, it is not known if there are stable solutions to Allen–Cahn on with non-flat level sets. However, under the additional assumption of quadratic energy growth, i.e.,
then it follows from the work of Ambrosio–Cabre [AC00] (see also [FMV13]) that has flat level sets. We note that the corresponding stable Bernstein theorem on is known to hold without any energy growth assumption; see the works of Ghoussoub–Gui [GG98] and Ambrosio–Cabre [AC00].
As such, one may expect that the blow-up argument described above may be used to prove curvature estimates. However, there is a fundamental difficulty present in the Allen–Cahn setting: if are stable solutions of Allen–Cahn
on , then if their curvature (we will make this precise below) is diverging, then if we rescale by a factor in a blow-up argument this changes to . If converges to a non-zero constant, then standard elliptic regularity implies the rescaled functions limit smoothly to an entire stable solution of Allen–Cahn on . The smooth convergence guarantees that this solution will have non-flat level sets. If the original functions had uniformly bounded energy, we can show that the limit has quadratic area growth, which contradicts the aforementioned Bernstein theorem. However, if still converges to zero, we must argue differently. In this case, we have a sequence of solutions to Allen–Cahn whose level sets are uniformly bounded in a -sense. This can be used to show that the level sets converge to a plane (possibly with multiplicity) in the -sense. If the level sets behaved precisely like minimal surfaces, we could upgrade this -convergence using elliptic regularity, to conclude that the limit was not flat. However, in this situation, the level sets themselves do not satisfy a good PDE, so this becomes a significant obstacle.
Recently, a fundamental step in understanding this issue has been undertaken by Wang–Wei [WW17]. They have developed a technique for gaining geometric control of solutions to Allen–Cahn whose level sets are converging with Lipschitz bounds. Using this (and the -dimensional stable Bernstein theorem) they have proven curvature estimates for individual level sets of stable solutions on two-dimensional surfaces. Moreover, they have shown that if one cannot upgrade bounds to convergence, then by appropriately rescaling the height functions of the nodal sets, one obtains a nontrivial solution to the a system of PDE’s known as the Toda system (see [WW17, Remark 14.1]). Finally, their proof of curvature estimates in -dimensions points to the crucial observation that it is necessary to use stability to upgrade the regularity of the convergence of the level sets.
This brings us to our first main result here, which is an extension of the Wang–Wei curvature estimates to dimensions.
Our -dimensional curvature estimates can be roughly stated as follows (see Theorem 4.5 for a slightly more refined statement and the proof)
For a complete Riemannian metric on and a stable solution to (1.1) with , the enhanced second fundamental form of satisfies
as long as is sufficiently small.
We emphasize that Wang–Wei’s -dimensional estimates [WW17, Theorem 3.7] do not require the energy bound (see also [Man17, Theorem 4.13] for the Riemannian modifications of this result). Note that we cannot expect to prove estimates with a constant that tends to as (which was the case in [WW17]) since—unlike geodesics—minimal surfaces do not necessarily have vanishing second fundamental form.
We note that due to our curvature estimates, it is not hard to see that stable (and more generally, uniformly bounded index) solutions to the Allen–Cahn equation (with uniformly bounded energy) in a -manifold limit to a surface that has vanishing (weak) mean curvature. Standard arguments thus show that the surface is smooth. Thus, our estimates show that it is possible to completely avoid the regularity results of Wickramasekera and Wickramasekera–Tonegawa [Wic14, TW12] in the setting of Allen–Cahn min-max on a -manifold (cf. [Gua18]).
We briefly remark on the possibility of extending curvature estimates to higher dimensions:
For , the Allen–Cahn stable Bernstein result is not known (even with an energy growth condition).
Even if the stable Bernstein theorem were to be established in dimensions , we note that our proof currently uses the dimension restriction in one other place: we use a logarithmic cutoff function in the proof of our sheet separation estimates (Propositions 4.2 and 4.3).
On the other hand, we remark that the curvature estimate for minimizing solutions, can be proven using to the “multiplicity one” nature of minimizers [HT00, Theorem 2], together with [WW17, Section 15] (or Remark 3.6).
Strong sheet separation estimates for stable solutions
A key ingredient in the proof of our curvature estimates is showing that distinct sheets of the nodal set of a stable solution to the Allen–Cahn equation remain sufficiently far apart. This aspect was already present in the work of Wang–Wei. For our applications to the case of uniformly bounded Morse index (and thus min-max theory), we must go beyond the sheet separation estimates proven in [WW17]. We prove in Proposition 4.3 that distinct sheets of nodal sets of a stable solution to the Allen–Cahn equation must be separated by a sufficiently large distance so that the location of the nodal sets becomes “mean curvature dominated.”
In particular, as a consequence of these estimates, we show in Theorem 5.1 that if a sequence of stable solutions to the Allen–Cahn equation converge with multiplicity to a closed two-sided minimal surface , then there is a positive Jacobi field along (which implies that is stable). It is interesting to compare this to the examples constructed by del Pino–Kowalczyk–Wei–Yang of minimal surfaces in -manifolds with positive Ricci curvature that are the limit with multiplicity of solutions to the Allen–Cahn equation [dPKWY10]. Note that such a minimal surface cannot admit a positive Jacobi field, so the point here is that Allen–Cahn solutions are not stable. (In fact, our Theorem 5.1 implies that they have diverging Morse index.) Note that the separation between the sheets of the examples constructed in [dPKWY10] satisfy
while we prove in Proposition 4.3 that stability implies that the separation satisfies
We emphasize that the improved separation estimates here are not contained in the work of Wang–Wei [WW17] and are fundamental for the subsequent applications of our results.
The multiplicity one-conjecture for limits of the Allen–Cahn equation in -manifolds
In their recent work [MN16a], Marques–Neves make the following conjecture:
Conjecture 1.5 (Multiplicity one conjecture).
For generic metrics on , , two-sided unstable components of closed minimal hypersurfaces obtained by min-max methods must have multiplicity one.
In [MN16a], Marques–Neves confirm this in the case of a one parameter Almgren–Pitts sweepout. The one parameter case had been previously considered for metrics of positive Ricci curvature by Marques–Neves [MN12] and subsequently by Zhou [Zho15]. See also [Gua18, Corollary E] and [GG16, Theorem 1] for results comparing the Allen–Cahn setting to Almgren–Pitts setting which establish multiplicity one for hypersurfaces obtained by a one parameter Allen–Cahn min-max method in certain settings. We also note that Ketover–Liokumovich–Song [Son17, KL17, KLS] have proven multiplicity (and index) estimates for one parameter families in the Simon–Smith [Smi82] variant of Almgren–Pitts in -manifolds.
We recall the following standard definition:
We say that a metric on a Riemannian manifold is bumpy if there is no immersed closed minimal hypersurface with a non-trivial Jacobi field.
We are able to prove a strong version of the multiplicity one conjecture (when ) for minimal surfaces obtained by Allen–Cahn min-max methods with an arbitrary number of parameters. Such a method was set up by Gaspar–Guaraco [GG16].
Indeed, we prove that for any metric on a closed -manifold, the unstable components of such a surface are multiplicity one. Moreover, for a generic metric, we show that each component of the surface occurs with multiplicity one (not just the unstable components). Finally, we are able to show for generic metrics on a -manifold, , the minimal surfaces constructed by Allen–Cahn min-max methods are two-sided. For a one-parameter Almgren–Pitts sweepoints in a -manifold with positive Ricci curvature, this was proven by Ketover–Marques–Neves [KMN16]. More precisely, our main results here are as follows (see Theorem 5.1 and Corollary 7.1 for the full statements).
We emphasize once again that our theorem applies generally to sequences of Allen–Cahn solutions with uniformly bounded energy and Morse index. Thus, unlike the proofs in the Almgren–Pitts setting, we do not need to make use of any min-max characterization of the limiting surface to rule out multiplicity.
Theorem 1.7 (Multiplicity and two-sidedness of minimal surfaces constructed via Allen–Cahn min-max).
Let denote a smooth embedded minimal surface constructed as the limit of solutions to the Allen–Cahn equation on a -manifold with uniformly bounded index and energy. If occurs with multiplicity or is one-sided, then it carries a positive Jacobi field (on its two-sided double cover, in the second case).
Note that positive Jacobi fields do not occur when is bumpy or when has positive Ricci curvature. Thus, in either of these cases, each component of is two-sided and occurs with multiplicity one.
Our proof here is modeled on the study of bounded index minimal hypersurfaces in a Riemannian manifold. Indeed, Sharp has shown that minimal hypersurfaces in for with uniformly bounded area and index are smoothly compact away from finitely many points where the index can concentrate [Sha17] (see also White’s proof [Whi87] of the Choi–Schoen compactness theorem [CS85]). A crucial point there is to prove that higher multiplicity of the limiting surface produces a positive Jacobi field (even across the points of index concentration (where the convergence of the hypersurfaces need not occur smoothly). This can be handled via an elegant argument of White, based on the construction of a local foliation by minimal surfaces to use as a barrier for the limiting surfaces (cf. [Whi15]).
In the minimal surface setting, the existence of the foliation is a simple consequence of the implicit function theorem. However, in the Allen–Cahn setting, the singular limit limit complicates this argument. Instead, we construct barriers by a more involved fixed point method in Theorem 2.2. Our arguments here are modeled on the work of Pacard [Pac12] (with appropriate extension to the case of Dirichlet boundary conditions), but there is a significant technical obstruction here: we do not know that the level sets of the solution Allen–Cahn converge smoothly, but only in . To apply the fixed point argument, we need some control on higher derivatives. By an observation of Wang–Wei [WW17, Lemma 8.1], we control one higher derivative of the level sets, but only by a constant that is (see (2.4)). This complicates the proof of Theorem 2.2. We show how the barriers can be used to bound the Jacobi fields along the points of index concentration in the process of the proof of Theorem 5.1 by carrying out a new sliding plane type argument for the Allen–Cahn equation on Riemannian manifolds.
Index lower bounds
Lower semicontinuity of the Morse index along the singular limit of a sequence of solutions to the Allen–Cahn equation is proven by Hiesmayr [Hie17] (for two-sided surfaces) and Gaspar [Gas17]. On the other hand, upper semicontinuity of the index does not hold in general (cf. Example 6.2). Here, we establish upper semicontinuity of the index, in all dimensions, under the a priori assumption that the limiting surface is multiplicity one.
Theorem 1.8 (Upper semi-continuity of the index in the multiplicity one case).
Suppose that a smooth embedded minimal hypersurface is the multiplicity one limit as of a sequence of solutions to the Allen–Cahn equation. Then for sufficiently small,
Applications related to Yau’s conjecture on infinitely many minimal surfaces
Finally, we mention an immediate application of our multiplicity estimates. A well known conjecture of Yau posits that any closed -manifold admits infinitely many immersed minimal surfaces. By considering the -widths introduced by Gromov [Gro03] (see also [Gut09]), Marques–Neves proved [MN17] that a closed Riemannian manifold (for ) with positive Ricci curvature admits infinitely many minimal surfaces. Moreover, by an ingenious application of the Weyl law for the -widths proven by Liokumovich–Marques–Neves [LMN16], Irie–Marques–Neves [IMN17] have recently shown that the set of metrics on a closed Riemannian manifold (with ) with the property that the set of minimal surfaces is dense in the manifold is generic (see also [MNS17]). Thus, the preceding works have proven Yau’s conjecture (in dimensions ) for generic metrics and metrics with positive Ricci curvature.
We note that both arguments [MN17, IMN17] to prove the existence of infinitely many minimal surfaces are indirect, and do not rule out the -widths being achieved with higher multiplicity. Here, due to Corollary 7.1, we may give a “direct” proof (for ) of Yau’s conjecture for bumpy metrics
Corollary 1.9 (Yau’s conjecture for bumpy metrics and geometric properties of the minimal surfaces).
Let denote a closed -manifold with a bumpy metric. Then, there is and a smooth embedded minimal surfaces for each positive integer so that
each component of is two-sided,
the area of satisfies ,
the index of is satisfies , and
the genus of satisfies .
In particular, thanks to the index estimate, all of the are geometrically distinct.
We emphasize that each of the bullet points in the preceding corollary do not follow from the work of Irie–Marques–Neves [IMN17]. Some of these properties were conjectured by Marques and Neves in [Mar14, p. 24], [Nev14, p. 17], [MN16b, Conjecture 6.2]. In particular, they conjectured that a generic Riemannian manifold contains an embedded two-sided minimal surface of each positive Morse index.
Remark 1.10 (Yau’s conjecture for -manifolds with positive Ricci curvature).
We note that because the multiplicity-one property also holds even for non-bumpy metrics of positive Ricci curvature, we may also give a “direct” proof of Yau’s conjecture for a -manifold with positive Ricci curvature (this was proven by Marques–Neves [MN17] in dimensions using Almgren–Pitts theory). We obtain, exactly as in Corollary 7.2, the new conclusions that the surfaces are two-sided, have , and .
Moreover, approximating the metric by a sequence of bumpy metrics and passing to the limit (the limit occurs smoothly and with multiplicity one due to the positivity of the Ricci curvature, cf. [Sha17]), we find that there is a sequence (we do not know if this is the same sequence as ) with these properties and additionally satisfies the genus bound
It is interesting to observe that when is the round -sphere, combining our bound with work of Savo [Sav10] implies that
as long as is sufficiently large to guarantee that . Similar conclusions can be derived in certain other -manifolds embedded in Euclidean spaces by [ACS16].
There has been significant activity concerning the index of the minimal surfaces constructed in [MN17], but before the present work, all that was known was that: for a bumpy metric of positive Ricci curvature, there are closed embedded minimal surfaces of arbitrarily large Morse index [LZ16, CKM17, Car17], albeit without information on their area.
1.3. One-dimensional heteroclinic solution,
Recall that the one-dimensional Allen-Cahn equation with is:
for a function of one variable. It’s not hard to see that (1.5) admits a unique bounded solution with the properties
We call this the one-dimensional heteroclinic solution, and denote it as . It’s also standard to see that the heteroclinic solution satisfies:
as , for some fixed that depends on . Moreover,
where also depends on ; it is explicitly given by
Finally, we also define
which is clearly a solution of .
1.4. Organization of the paper
In Section 2 we construct solutions of (1.1) that resemble multiplicity-one heteroclinic solutions with prescribed Dirichlet data centered on nondegenerate minimal submanifolds-with-boundary , , with only relatively low regularity. (See (2.3)-(2.4) and the footnote.) To do so, we adopt an elegant argument of Pacard [Pac12] to account for Dirichlet boundary conditions and the low regularity assumption on .
In Section 3 we continue to work in all dimensions and make precise the dependence of the regularity of the nodal set of bounded energy and bounded curvature solutions of (1.1) on the distance between its different sheets. The dependence is essentially modeled by a Toda system. (See, e.g., (3.17) and Remark 3.6.) We start off following [WW17] but keep track of all errors in terms of sheet distances; this affords us the freedom to bootstrap the estimates later.
Restricting to dimensions, in Section 4 we use the stability of Allen–Cahn solutions to bootstrap the estimates until they become sharp. Specifically, we combine three things: (i) an estimate on the height function of (following an observation of Wang–Wei [WW17, (19.7)]), (ii) a subtle application of Moser’s Harnack inequality, and (iii) the nonexistence of nontrivial entire stable critical points of the Toda system on (cf. the stable Bernstein problem for minimal surfaces in ).
In Section 5 we study solutions of (1.1) with bounded energy and Morse index in dimensions. We use our strong sheet separation estimates from Section 4 to construct, in the presence of multiplicity, positive Jacobi fields on the limiting minimal surface away from finitely many points. Then, a “sliding plane” argument that relies on our constructions and estimates from Section 2 gives us precise enough control on the height of across these finitely many points, to allow us to extend the Jacobi field to the entire limiting surface.
O.C. was supported in part by the Oswald Veblen fund and NSF Grant no. 1638352. He would like to thank Simon Brendle and Michael Eichmair for their continued support and encouragement, as well as Costante Bellettini, Guido De Philippis, Daniel Ketover, and Neshan Wickramasekera for their interest and for enjoyable discussions. C.M. would like to thank Rick Schoen, Rafe Mazzeo, and Yevgeniy Liokumovich for helpful conversations on topics addressed by this paper. Both authors would like to thank Fernando Codá Marques and André Neves for their interest in this work. This work originated during the authors’ visit to the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) during the “Advances in General Relativity Workshop” during the summer of 2017, which they would like to acknowledge for its support.
2. Multiplicity one constructions with Dirichlet data
Notice that this solution is “centered” on the hyperplane. One may just as easily center it on any hyperplane in by a suitable translation and rotation.
The question of centering approximate heteroclinic solutions on arbitrary minimal has been well-studied in the compact setting; see, e.g., [PR03] for the boundary-less case and the geometrically natural case of Neumann conditions at the boundary when , , or see [Pac12] for a more general survey with a faster construction than the one in [PR03], albeit only presented in the boundary-less case.
In this section we establish a corresponding existence theorem similar in spirit to those in [PR03, Pac12], except we prescribe Dirichlet data. This will be a crucial ingredient in a “sliding” argument that we need to employ in Section 5.
The setup is as follows.
Define , , , to be the standard Hölder spaces after rescaling by , i.e., whose Banach norm is
Next, suppose that is an -dimensional manifold with nonempty boundary, over which we take a topological cylinder
whose coordinates we label . Let be a smooth metric on , given in coordinates (Fermi coordinates) by
We require that
whose second fundamental form is uniformly bounded :
We furthermore assume that there are -coordinate charts on so that the induced metric is -close and -close to the Euclidean metric in the sense that
where run through the coordinates on in the given coordinate chart.
for small . Here, , run through all Fermi coordinates .
Again, , , run through all Fermi coordinates.
We also require that carries no nontrivial Jacobi fields with Dirichlet boundary conditions in the following quantitative sense:
denotes the Jacobi operator on . (Note that our sign convention for the Jacobi operator differs from the one in [Pac12].)
Let’s also fix , and define cutoff functions , with , so that
as well as . We further require that the be even functions. We set .
For , set
where the corresponds to , , respectively, and is as in (1.9). This is a truncation of the one-dimensional heteroclinic solution, , which coincides with near and with away from .
For subsets , let’s define
to be given by
We note two things:
does not appear in the projection notation, but it will clear from the context or it will not be relevant.
Our normalization is such that .
We point out the following trivial lemma:
and both lift to linear maps
Viewed as linear maps over these Hölder spaces, we have
For , we define to be the map
Finally, we introduce the modified Hölder norm:
Recall that is as in (2.1).
The main result of this section is:
If , and we’re given boundary data
, , on ,
, , on ,
and a metric for which the setup above holds, there exist
, , ,
, , , ,
, , ,
so that satisfies
Moreover, the solution map
is Lipschitz continuous, with Lipschitz constant , as a map
This follows along the lines of [Pac12, Section 3], provided one makes the necessary modifications to account for (possibly nonzero, but small) Dirichlet data as well as the important fact that our Fermi coordinate regularity is constrained by the weaker assumptions (2.3)-(2.4). This lower regularity situation makes certain aspects of Theorem 2.2 delicate, so we describe the proof in detail below.
2.2. Linear scheme
In this section we generalize linear estimates found in [Pac12, Section 3] to allow Dirichlet boundary conditions, possibly with nonzero data. The operators we’ll study are:
Lemma 2.3 (cf. [Pac12, Lemma 3.7]).
Assume that satisfies and on . Then .
The result follows from [Pac12, Lemma 3.7] after an odd reflection of across . ∎
The next results that need to be adapted pertain to and functions satisfying
where is as in (2.13).
Lemma 2.4 (cf. [Pac12, Proposition 3.1]).
If , , and on , then
Here, , .
Lemma 2.5 (cf. [Pac12, Proposition 3.2]).
There exists depending on , , , such that for all , all with on , and all with on , there exists a unique function , also with on , such that
When this follows from the functional analytic argument already found in [Pac12, Proposition 3.2] applied, instead, to .
When , this follows by extending to , , and applying the previous existence result with zero boundary data to solve . ∎
Finally, [Pac12] deals with .
Lemma 2.6 (cf. [Pac12, Proposition 3.3]).
If , then
The interior estimate follows from interior Schauder theory, since is smooth. The boundary estimate on the regular portion of follows from boundary Schauder theory, because is at those points. Finally, the estimate at the corners of follows from the boundary theory as well. This is because we can carry out odd reflections across since the angles at the corners are all . ∎
We also derive an improved estimate for functions satisfying on a strip of height , and on its lateral boundary. Recall the definition of the norm in (2.16).
Lemma 2.7 (cf. [Pac12, (3.26)]).
If , , and
Here, , .
Considering Lemma 2.6, it suffices to check that
Since on , on , and , Schauder’s interior estimates estimates on and Schauder’s boundary estimates near imply:
We use the same barrier argument as in [Pac12, Remark 3.2], paying closer attention to the boundary and to the regularity. Define