STABILITY OF THE POSITIVE MASS AND TORUS RIGIDTY THEOREMS UNDER INTEGRAL CURV ATURE BOUNDS BRIAN ALLEN EDWARD BRYDEN AND DEMETRE KAZARAS

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STABILITY OF THE POSITIVE MASS AND TORUS RIGIDTY THEOREMS UNDER

INTEGRAL CURVATURE BOUNDS

BRIAN ALLEN, EDWARD BRYDEN, AND DEMETRE KAZARAS

ABSTRACT. Work of D. Stern [Ste20] and Bray-Kazaras-Khuri-Stern [BKKS22] provide differential-

geometric identities which relate the scalar curvature of Riemannian 3-manifolds to global invariants in

terms of harmonic functions. These quantitative formulas are useful for stability results [Ste20, KKL21]

and show promise for more applications of this type. In this paper, we analyze harmonic maps to ﬂat

model spaces in order to address conjectures concerning the geometric stability of the positive mass

theorem and the Geroch conjecture. By imposing integral Ricci curvature and isoperimetric bounds, we

leverage the previously mentioned formulas to establish strong control on these harmonic maps. When

the mass of an asymptotically ﬂat manifold is sufﬁciently small or when a Riemannian torus has almost

non-negative scalar curvature, we upgrade the maps to diffeomorphisms and give quantitative H¨

older

closeness to the model spaces.

1. INTRODUCTION

The geometric control which follows from integral bounds on Ricci curvature is well understood.

This theory has been developed by Anderson [And90], Anderson-Cheeger [AC91], Colding [Col96,

Col97], Cheeger-Colding [CC97], Dai-Wang-Zhang [DWZ18], Petersen [Pet97a, Pet06], Petersen-

Sprouse [PS98], Petersen-Wei [PW01a, PW97a, PW01b], Gao [Gao90], and Yang [Yan92c, Yan92a,

Yan92b]. Readers unfamiliar with this convergence theory are encouraged to consult Petersen’s fan-

tastic survey article [Pet97b]. Speaking generally, methods in this area are based upon harmonic

coordinate systems, which directly relate the metric to its Ricci curvature by an elliptic PDE. Through

analysis of this equation, one is able to show that the class of manifolds with integral bounds on Ricci

(or Riemann) curvature and injectivity radius bounds (or volume growth of balls) enjoys a system of

harmonic coordinate charts in which the metric is controlled in Sobolev and H¨

older spaces. How-

ever, the arguments which obtain this control often proceed by contradiction, and the resulting metric

control is generally not explicitly quantitative in terms of given geometric bounds.

In this paper we take a new approach to the harmonic coordinate method described above. On

asymptotically ﬂat 3-manifolds and certain compact 3-manifolds, we study three distinct harmonic

functions which encode global geometric and topological information, but generally do not deﬁne

Date: October 11, 2022.

arXiv:2210.04340v1 [math.DG] 9 Oct 2022

2 BRIAN ALLEN, EDWARD BRYDEN, AND DEMETRE KAZARAS

a global system of coordinates. These harmonic functions are related to the scalar curvature of 3-

tori by Stern [Ste20] and the mass of asymptotically ﬂat 3-manifolds by Bray-Kazaras-Khuri-Stern

[BKKS22]. By using these breakthrough formulas in tandem with the geometric controls following

from integral Ricci curvature and isoperimetric bounds, we strongly control the regularity and behavior

of the harmonic functions. Next, when the appropriate quantities are chosen to be small enough,

we are able to show that these harmonic functions produce a global coordinate system. This global

coordinate system is the source of the topological stability as well as quantitative H¨

older stability for

both the positive mass theorem and scalar torus rigidity theorem. For the positive mass theorem, this

stability is explicitly computable in terms of integral Riemann curvature and isoperimetric bounds. In

the case of the scalar torus rigidity theorem, the stability remains implicitly quantitative due to the use

of a contradiction argument in the spirit of Petersen.

The positive mass theorem states that a complete, asymptotically ﬂat manifold with non-negative

scalar curvature has positive ADM mass. This was ﬁrst proved by Schoen-Yau [SY79b] using mini-

mal surfaces and later by Witten [Wit81] using spinors and the Lichnerowicz formula. The geometric

stability of the positive mass theorem was ﬁrst stated by Huisken-Illmanen [HI01] in terms of Gromov-

Hausdorff convergence and more recently by Lee-Sormani [LS14] in terms of Sormani-Wenger intrin-

sic ﬂat convergence. Recently, Bray-Kazaras-Khuri-Stern [BKKS22] gave a new proof of the positive

mass theorem in dimension 3with a formula which relates the mass and scalar curvature to harmonic

functions deﬁned on the asymptotically ﬂat manifold. This formula was then used by Kazaras-Khuri-

Lee [KKL21] to show Gromov-Hausdorff convergence for a sequence of asymptotically ﬂat manifolds

with pointwise lower bounds on Ricci curvature whose mass is tending to zero. In this paper we also

use this formula to obtain quantitative C0,γ-stability, for some γ∈(0,1), under integral Ricci cur-

vature bounds and isoperimetric constants for metric balls. It is interesting to compare our methods

with those of Finster-Bray [BF02], Finster-Kath [FK02], and Finster [Fin09], which are based on a

spinorial mass formula due to Witten [Wit81]. In their work, an isoperimetric condition is used to

obtain a relation between the mass and the W1,2- and L∞-norms of the Riemann curvature. Stability

of the positive mass theorem in the asymptotically ﬂat setting has also been studied by Allen [All17],

Bryden [Bry20], Bryden-Khuri-Sormani [BKS21], Huang-Lee-Sormani [HLS17], Huang-Lee-Perales

[HLP22], Lee-Sormani [LS14], Sormani-Stavrov [SS19], and the references therein.

The main tool we use to study the geometric stability of the positive mass theorem is the follow-

ing formula relating the ADM mass to scalar curvature through an asymptotically linear harmonic

function.

STABILITY OF THE POSITIVE MASS AND TORUS RIGIDTY THEOREMS UNDER INTEGRAL CURVATURE BOUNDS 3

Theorem 1.1 (Theorem 1.2 of [BKKS22]).Let (M, g)be an orientable complete asymptotically ﬂat

manifold with no spherical classes in H2(M;Z), and let ube an asymptotically linear harmonic

function (see Section 4). Then, the ADM mass satisﬁes

(1) m(g)≥1

16πˆM

|∇2u|2

|∇u|+Rg|∇u|dVg,

where Rgdenotes the scalar curvature of (M, g).

The ﬁrst main result leverages controlled asymptotic falloff with good control on the local analysis

of Sobolev functions in order to show that ADM mass strongly controls geometry. In order to effec-

tively utilize the mass formula (1), we will need Sobolev and Poincar´

e inequalities on our manifold.

Crucially, these inequalities are known to be equivalent to isoperimetric constants (See Li [Li12]). Let

us now recall the isoperimetric constant central to our result.

Deﬁnition 1.2 (Deﬁnition 9.2 in [Li12]).Let us deﬁne INα(Mn, g)for 1≤α≤n

n−1as follows:

(2) INα(Mn, g) = inf (Areag(S)

min{Volg(Ω1),Volg(Ω2)}1

:M= Ω1∪S∪Ω2, ∂Ω1=S=∂Ω2).

INα(Mn, g)is called the Neumann α−isoperimetric constant of M.

With the above in hand, we can make precise what it means for a class of asymptotically ﬂat

Riemannian manifolds to have controlled asymptotic falloff and good control on the local analysis of

Sobolev functions, by deﬁning the following class of asymptotically ﬂat Riemannian manifolds.

Deﬁnition 1.3. Fix b, ¯m, Λ, κ > 0,τ > 1

2,p > 1,α∈[1,3

2]. An oriented connected complete asymp-

totically ﬂat 3-dimensional Riemannian manifold (M, g)is said to be homologically simple, (b, τ, ¯m)

asymptotically ﬂat, (Λ, α)Neumann-isoperimetrically bounded, and κcurvature-Lpbounded if

(1) there are no spherical classes in H2(M;Z)and ∂M =∅,

(2) there exists a coordinate chart Φ : M\Ω→R3\B1(0) such that on R3\B1(0) we have

|D(k)(gij −δij)|(x)≤b|x|−τ−k,

where k= 0,1,and 2,

(3) Rg≥0,and mADM(g)≤¯m,

(4) for all metric balls, Br(x)⊂M, we have

INα(Br(x)) ≥Λ,

(5) kRcgkLp≤κ.

4 BRIAN ALLEN, EDWARD BRYDEN, AND DEMETRE KAZARAS

FIGURE 1. The above short and stout bump is Neumann Λ-isoperimetrically bounded

if the ratio of its radius (in green) to its height (in blue) is bounded above and below.

By taking the radius and the height to be arbitrarily small, one obtains Neumann

Λ-isoperimetrically bounded spaces with arbitrarily small injectivity radius.

We will denote the family of such 3-dimensional asymptotically ﬂat Riemannian manifolds by M(b, τ, ¯m, Λ, α, κ, p).

In most of what follows, we will impose α=3

2and p= 3. In this case, we adopt the shorthand

M(b, τ, ¯m, Λ, κ) = Mb, τ, ¯m, Λ,3

2, κ, 3.

Remark 1.4. Ricci curvature has a unique character in dimension 3. In particular, all components

of a 3-manifold’s Riemann tensor can be algebraically expressed in terms of its Ricci curvature. This

means that condition (5) in Deﬁnition 1.3 is actually equivalent to a bound on kRmgk.

The ﬁrst main result of the paper gives quantitative stability of the positive mass theorem.

Theorem 1.5. Let b, ¯m, Λ, κ > 0, τ > 1

2, and γ∈(0,1

2)be given parameters. For any ε > 0there

exists δ=δ(b, τ, ¯m, Λ, κ, ε, γ)>0such that the following holds: if (M, g)lies in M(b, τ, ¯m, Λ, κ)

and mADM(g)≤δ, then Mis diffeomorphic to R3and

kg−gEkC0,γ (R3)< ε(3)

where gEdenotes the ﬂat metric on R3.

Remark 1.6. The H¨

older norm in (3) can be computed using distances measured with respect to either

gor gE– since these metrics are C0-close, this choice is immaterial. Also, an interesting consequence

of Theorem 1.5 is the following topological stability: if a manifold (M, g)∈ M(b, τ, ¯m, Λ, κ)has

sufﬁciently small mass, then Mis diffeomorphic to R3.

Next, we turn to the second main application of our methods. The Geroch conjecture asserts that

a Riemannian 3-torus with non-negative scalar curvature must be ﬂat. This rigidity theorem was ﬁrst

proven by Schoen-Yau [SY79a] and soon after in higher dimensions by Gromov-Lawson [GL80] using

different methods. The stability problem associated to the Geroch conjecture was ﬁrst considered by

Gromov [Gro14] and made more precise by Sormani [Sor17], who observed the necessity of an extra

geometric constraint known as a minA condition. A minA condition is a lower bound on the area

STABILITY OF THE POSITIVE MASS AND TORUS RIGIDTY THEOREMS UNDER INTEGRAL CURVATURE BOUNDS 5

of all closed minimal surfaces within the manifold, and is used to eliminate pathological bubbling

examples ﬁrst observed by Basilio-Sormani [BS19]. See the survey paper by Sormani [Sor22] for a

recent discussion of this conjecture as well as many other conjectures involving scalar curvature.

Various special cases of these conjectures have been studied by Allen [All21], Allen-Bryden [AB21],

Allen-Hernandez-Vazquez-Parise-Payne-Wang [AHP+18], Cabrera Pacheco-Ketterer-Perales [PKP20],

Chu-Lee [CL22], and Lee-Naber-Neumeyer [LNN21]. In this paper, we establish quantitative stabil-

ity and H¨

older convergence in the context of integral bounds on the Ricci curvature. We are able to

avoid bubbling by requiring a uniform lower bound on the Neumann isoperimetric constant of metric

balls. The present work suggests that a lower bound on the Neumann isoperimetric constant is an

analytically effective – yet still geometric – replacement to a minA condition when considering the

stability of 3-dimensional rigidity theorems related to scalar curvature.

The main ingredient we use to establish quantitative stability of the Geroch conjecture is a formula

relating the scalar curvature of a closed 3-manifold to S1-valued harmonic maps developed by D.

Stern.

Theorem 1.7 (Theorem 1.1 of [Ste20]).Let (M, g)be a closed oriented Riemannian 3-manifold, and

suppose u:M→ S1is a non-constant harmonic map. Then

4πˆS1

χ(Σθ)dθ ≥ˆM

|∇du|2

|du|+Rg|du|dVg.

(4)

where χ(Σθ)is the Euler characteristic of a regular level set Σθ=u−1(θ)and du denotes the 1-form

u∗dθ.

Remark 1.8. For closed manifolds M, recall that there is a bijection between H1(M;Z)and homo-

topy classes of maps from Mto a circle. For each such homotopy class of maps, there is an harmonic

representative, unique up to rotations of the circle. Consequently, the ﬁrst integral cohomology group

of a manifold provides the harmonic maps appearing in Theorem 1.7.

We can now deﬁne the class of manifolds which we will consider for our second main result.

Deﬁnition 1.9. Given parameters Λ, V, κ > 0, we denote by N(Λ, V, κ)the collection of closed

connected oriented Riemannian 3-manifolds (M, g)which satisfy the following:

(1) (M, g)is (Λ,3

2)-Neumann-isoperimetrically bounded,

(2) 1

V≤Volg(M)≤V,

(3) kRcgkL3≤κ,

(4) the second integral homology H2(M;Z)contains no spherical classes,

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STABILITYOFTHEPOSITIVEMASSANDTORUSRIGIDTYTHEOREMSUNDERINTEGRALCURVATUREBOUNDSBRIANALLEN,EDWARDBRYDEN,ANDDEMETREKAZARASABSTRACT.WorkofD.Stern[Ste20]andBray-Kazaras-Khuri-Stern[BKKS22]providedifferential-geometricidentitieswhichrelatethescalarcurvatureofRiemannian3-manifoldstoglobalinvariantsintermsof...

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STABILITY OF THE POSITIVE MASS AND TORUS RIGIDTY THEOREMS UNDER INTEGRAL CURV ATURE BOUNDS BRIAN ALLEN EDWARD BRYDEN AND DEMETRE KAZARAS

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