Generalized Bernstein Theorem for Stable Minimal Plateau Surfaces Gaoming Wang

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Generalized Bernstein Theorem for Stable

Minimal Plateau Surfaces

Gaoming Wang

Abstract

In this paper, we consider a Generalized Bernstein Theorem for a

type of generalized minimal surfaces, namely minimal Plateau surfaces.

We show that if an orientable minimal Plateau surface is stable and has

quadratic area growth in R3, then it is ﬂat.

1 Introduction

When we study minimal surfaces, we usually focus on smooth minimal

surfaces. But in general, "non-smooth minimal surfaces" can be found

in some natural phenomena. Indeed, two types of singular points are

observed in soap ﬁlms and they are documented in Plateau’s work [Pla73].

Hence, a natural problem raised by Bernstein and Maggi [BM21] is,

Problem 1.1. To what extent may the classical theory of minimal sur-

faces be generalized to "non-smooth minimal surfaces"?

In [BM21], they deﬁned those "non-smooth minimal surfaces" as min-

imal Plateau surfaces.

To better illustrate our theorem here, let us deﬁne the minimal Plateau

surfaces ﬁrst. We deﬁne three cones in R3by

P:= {p= (x1, x2, x3)∈R3:x3= 0},

Y:= {p= (rcos θ, r sin θ, x3)∈R3:θ= 0,2π

3or 4π

3, r ≥0},

T:= {p=api+bpj:a, b ≥0,1≤i < j ≤4},

where piare deﬁned as

p1= (1,1,1), p2= (−1,−1,1), p3= (−1,1,−1), p4= (1,−1,−1).

For the following deﬁnition, we ﬁx an open set U⊂R3,α∈(0,1), and

a relatively closed subset Σ⊂U, we use reg(Σ) to denote the set of all

p∈Σsuch that Br(p)∩Σis a C1,α surface for r > 0small enough and

Br(p)⊂U.

Deﬁnition 1.2. ([BM21]) We say a relatively closed subset Σ⊂Uis a

Plateau surface if the following two conditions hold,

•For any p∈Σ, there is an r > 0and a C1,α diﬀeomorphism φ:

Br(p)→R3such that φ(Σ ∩Br(p)) = C∩Br(p)and Dφp∈O(3)

for some C=P, Y or T.

•Each connected component of reg(Σ) has constant mean curvature.

In addition, if each connected component of reg(Σ) has zero mean curva-

ture, then we say Σis a minimal Plateau surface in U.

arXiv:2210.11500v1 [math.DG] 20 Oct 2022

We note the tangent cone TpΣ := limρ→0+Σ−p

ρfor any p∈Σcan only

be isometric to P, Y or Tif Σis a minimal Plateau surface. In particular,

we use the following notations to denote Y-type and T-type singular sets.

ΣY:= {p∈Σ : TpΣis isometric to Y},

ΣT:= {p∈Σ : TpΣis isometric to T},

For a minimal Plateau surface Σin U, if ΣT=∅and ΣY6=∅, we

call it a minimal triple junction surface. (It was named as Y-surface in

[BM21].) Indeed, the author has studied the properties of minimal triple

junction surfaces in his thesis [Wan22b] and in [Wan22a] as a special case

of multiple junction surfaces.

Now we can state our main theorem.

Theorem 1.3. Suppose Σis a minimal Plateau surface in R3. We assume

Σis orientable, complete, stable and has at most quadratic area growth.

Then reg(Σ) is ﬂat.

Here, the related concepts are deﬁned in Section 2. As a corollary, we

can give a more precise description of minimal Plateau surfaces satisfying

Theorem 1.3.

Corollary 1.4. If Σis a minimal Plateau surface satisfying the conditions

in Theorem 1.3, then each component of reg(Σ) is an open subset of some

plane, ΣYis a union of disjoint line segments, rays, and straight lines.

In particular, if ΣT=∅, then we can write Σ = N×Rafter some

right motions in R3where Nis an embedded stationary network in R2.

If ΣT6=∅, there are only two possibilities. The ﬁrst one is ΣTcontains

only one point and Σis isometric to T. The second one is ΣTcontains

two points and we can glue two T-type sets to get Σas shown in Figure 1.

Figure 1: A ﬂat Σwith ΣT={p1, p2}

Theorem 1.3 can be viewed as an extension of our previous work

[Wan22a] in two aspects.

•We allow T-type singularities.

•We can remove the restriction on Γappearing in [Wan22a, Theorem

1]. See Appendix A for details.

Before the formal deﬁnition of Plateau surfaces in [BM21], Plateau pro-

posed well-known laws (known as Plateau’s laws) to describe the structure

of soap ﬁlms. The deﬁnition of Plateau surfaces is just the mathematical

way to describe those soap bubbles and soap ﬁlms obeying Plateau’s laws.

In 1976, Almgren [Alm76] started studying closed sets which minimize

Hausdorﬀ measures with respect to local Lipschitz deformations, proving

that they are smooth minimal surfaces out of a closed set of null area. In

the same year the work of Taylor [Tay76] appeared with the sharp regular-

ity in R3, and the singular set ΣYand ΣTappeared. Her result justiﬁed

Plateau’s laws mathematically. Using Taylor’s result, Choe [Cho89] could

obtain the regularity of the fundamental domains with the least boundary

area. In particular, singularities of Y-type and T-type appeared there.

Besides, Y-type and T-type singularities are also quite natural in the

sense of stationary varifolds. If we know a stationary integral 2-varifold V

is suﬃcent close to Yin a open ball B1(0) ⊂R3and we have an area bound

like kVk(B1)<2and a density bound Θ(kVk,0) ≥3

2, then by Simon’s

celebrated paper [Sim93], we know sptkVkis indeed a C1,α perturbation

of Yin a smaller ball B1

2(0). This result has been extended to the case

of polyhedral cones by Colombo, Edelen, and Spolaor [CES22]. Roughly

speaking, we can get if a stationary integral 2-varifold Vis suﬃcient close

to Tin B1(0) and has a suitable area and density bounds, then we can

conclude sptkVkis a C1,α perturbation of Tin B1

2(0).

Inspired by those minimizing properties and stationary properties for

minimal Plateau surfaces, we expect there are any other results which can

be extended to minimal Plateau surfaces to give some positive answers to

Problem 1.1. Of course, Bernstein and Maggi [BM21] showed they can

get the rigidity of the Y-shaped catenoid.

On the other hand, there are indeed some other properties held for

minimal triple junction surfaces compared with the usual minimal sur-

faces. For example, Mese and Yamada [MY06] showed they could solve

the singular version of the Plateau problem using mapping methods in

some cases.

These results motivate us to seek whether stable minimal Plateau sur-

faces have similar results like curvature estimate and generalized Bernstein

theorem for stable minimal surfaces. In my PhD thesis [Wan22b], we stud-

ied the triple junction surfaces intrinsically and showed we can solve some

particular type of elliptic partial diﬀerential equations on it. After that,

we ﬁnd several concepts like the Morse index are also well-deﬁned on triple

junction surfaces.

For usual stable minimal hypersurfaces, we know that the generalized

Bernstein Theorem is equivalent to some curvature estimates via a stan-

dard blow-up argument. But for stable minimal Plateau surfaces, things

get complicated. This is because even if we assume the second fundamen-

tal form of each surface is uniformly bounded, we cannot make sure any

sequence of minimal Plateau surfaces has a convergence subsequence. For

example, if we only consider the triple junction surfaces shown in Figure

2, we ﬁnd some of Σimay become degenerate after taking limits.

ΣiΣi, degenerate

Figure 2: Σiis degenerate after taking limit

Therefore, we cannot directly get a pointwise curvature estimate for

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GeneralizedBernsteinTheoremforStableMinimalPlateauSurfacesGaomingWangAbstractInthispaper,weconsideraGeneralizedBernsteinTheoremforatypeofgeneralizedminimalsurfaces,namelyminimalPlateausurfaces.WeshowthatifanorientableminimalPlateausurfaceisstableandhasquadraticareagrowthinR3,thenitisat.1Introductio...

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Generalized Bernstein Theorem for Stable Minimal Plateau Surfaces Gaoming Wang

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