RIGIDITY OF MASS-PRESERVING 1-LIPSCHITZ MAPS FROM INTEGRAL CURRENT SPACES INTO Rn GIACOMO DEL NIN AND RAQUEL PERALES

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RIGIDITY OF MASS-PRESERVING 1-LIPSCHITZ MAPS FROM

INTEGRAL CURRENT SPACES INTO Rn

GIACOMO DEL NIN AND RAQUEL PERALES

Abstract. We prove that given an n-dimensional integral current space and

a 1-Lipschitz map, from this space onto the n-dimensional Euclidean ball, that

preserves the mass of the current and is injective on the boundary, then the map

has to be an isometry. We deduce as a consequence a stability result with respect

to the intrinsic ﬂat distance, which implies the stability of the positive mass

theorem for graphical manifolds as originally formulated by Huang–Lee–Sormani.

1. Introduction

Integral currents and ﬂat distance are classical notions from Geometric Measure

Theory, being employed, for instance, by Federer–Fleming to solve the Plateau prob-

lem [15]. Ambrosio–Kirchheim [3] deﬁned integral currents in metric measure spaces

and applying this work Sormani–Wenger [28] deﬁned integral current spaces and the

intrinsic ﬂat distance. The intrinsic ﬂat distance between two compact oriented Rie-

mannian manifolds of the same dimension Mi, endowed with their canonical currents

[[Mi]], is deﬁned as the inﬁmum of the ﬂat distances between the push-forwards of

both currents,

inf ndF(ϕ1][[M1]], ϕ2][[M2]])o,

where the inﬁmum runs over all complete metric spaces Zand distance preserving

embeddings ϕi:Mi→Z.

Gromov suggested using intrinsic ﬂat convergence to study manifolds of non-

negative scalar curvature [16]. In addition, Sormani presented open problems and

examples, concerning these type of manifolds, emphasizing the suitability of in-

trinsic ﬂat convergence [26]. The intrinsic ﬂat distance has shown to be an ade-

quate notion to study several stability problems, such as the stability of the positive

mass theorem and the stability of tori with almost non-negative scalar curvature.

Sakovich–Sormani [24] obtained an intrinsic ﬂat stability result for the positive mass

theorem for complete rotationally symmetric asymptotically hyperbolic manifolds.

Huang–Lee [17] showed stability of the positive mass theorem with respect to the

Federer–Fleming ﬂat distance for a class of asymptotically ﬂat graphical manifolds.

Motivated by this work, Huang–Lee–Sormani [19] studied the stability of the pos-

itive mass theorem with respect to the intrinsic ﬂat distance. The proof of their

ﬁrst result, [19, Theorem 1.3], consisted in reducing it to the following rigidity of

mass-preserving 1-Lipschitz maps into Euclidean space.

Problem. Let R > 0 and (ΩR, dR, TR) be an n-dimensional integral current space,

and let ψ: (ΩR, dR)→(Rn+1, dEuc) be a 1-Lipschitz function with the following

properties:

(1) ψ]TR= [[BR×{0}]], where BRdenotes a ball of radius Rin the n-dimensional

Euclidean space

Date: April 5, 2023.

arXiv:2210.06406v3 [math.DG] 4 Apr 2023

2 GIACOMO DEL NIN AND RAQUEL PERALES

(2) M(TR) = M(ψ]TR) = ωnRn

(3) ψ|set(∂TR): set(∂TR)→Rn+1 is bi-Lipschitz onto its image, ∂BR× {0}.

Then ψshould be an isometry between (ΩR, dR) and (BR, dEuc). Hence, as integral

current spaces, (ΩR, dR, TR) = (BR, dEuc,[[BR]]).

When dealing with Euclidean n-currents in Rnassociated with connected open

sets, and if we further assume that ψis C1, a positive answer to the Problem is

a consequence of the area formula and classical rigidity for maps with gradient in

SO(n). For completeness, and since the metric case uses similar ideas, we report

the proof at the beginning of Section 3.

A positive answer to the Problem was assumed by Huang–Lee-Sormani [19, Proof

of Theorem 1.3] without providing a proof or reference, see the corrigendum written

by them [20], and so the main contribution of this manuscript is to provide a proof

of this Problem.

Theorem 1.1 (Rigidity).Let (X, d, T )be an n-dimensional integral current space,

and let ψ:X→Rnbe a 1-Lipschitz function with the following properties:

(1) ψ]T= [[B1]];

(2) M(T) = M(ψ]T);

(3) ψis injective on set(∂T ), and ψ(set(∂T )) ⊆∂B1.

Then ψis an isometry between (X, d)and (B1, dEuc). Hence, as integral current

spaces, (X, d, T )=(B1, dEuc,[[B1]]).

We note that assumption (3), or a similar one, is really necessary to rule out

several counterexamples, see Example 4.1.

There are other rigidity results by Cecchini–Hanke–Schick [13], Besson–Courtois-

Gallot [6], Li [21], Li–Wang [22], Connell–Dai–N´u˜nez-Zimbr´on–Perales–Su´arez-Serrato–

Wei [14] of volume preserving 1-Lipschitz functions deﬁned on spaces with no bound-

ary that satisfy lower curvature bounds, such as scalar curvature bounds, sectional

curvature bounds, Alexandrov spaces, Ricci limits and RCD spaces, respectively.

Rigidity results for spaces with boundary have been obtained by Burago–Ivanov

[8,9] where boundary rigidity and minimal ﬁllings are studied. See also the ref-

erences within. Furthermore, a similar result to Theorem 1.1 has been recently

proven by Basso, Creutz and Soultanis [5], of which we became aware while in the

ﬁnal stages of the completion of this manuscript. There, condition (3) is replaced

by M(∂T ) = M(∂[[B1]]). They include interesting examples of the necessity of the

hypotheses. Furthermore, in their work B1can be taken to be any convex set in

Euclidean space. After this, we realized that Theorem 1.1 also holds for any convex

set in Euclidean space.

Theorem 1.1 directly implies a positive answer to the Problem above, and there-

fore the original proof of [19, Theorem 1.3] is valid. See Section 5for a sketch of

this proof. We note that there is a diﬀerent and rigorous proof of [19, Theorem 1.3]

provided by Huang, Lee and the second named author [18, Theorem 3.2] that does

not rely on the existence of such 1-Lipschitz function nor on solving the Problem

above. The proof uses an intrinsic ﬂat compactness result, [18, Theorem 3.4], which

is an easy corollary of [1, Theorem 4.2] of Allen and the second named author. The

latter extends the work of both of them and Sormani to manifolds with boundary.

One can also ﬁnd a proof of [19, Theorem 1.3] in [1, Section 7] that applies [1, The-

orem 4.2], under the added assumption that the manifolds are entire. To obtain a

full proof of [19, Theorem 1.3], in [18] the manifolds with non-empty inner bound-

ary are extended to manifolds homeomerphic to balls in Euclidean space and the

LIPSCHITZ-MASS RIGIDITY 3

homeomorphisms are carefully constructed to ensure they are C1, so that one can

apply [18, Theorem 3.4].

From Theorem 1.1 we derive a stability property for the Plateau problem.

Theorem 1.2 (Stability).Let (Xj, dj, Tj)be a sequence of n-dimensional integral

current spaces that converge in the intrinsic ﬂat sense to (X, d, T ). Assume that a

sequence of 1-Lipschitz maps ψj:Xj→Rn+1 is given, S∞

j=1 ψj(Xj)is contained in

a compact set, and let ψ: (X, d)→Rn+1 be an Arzel`a-Ascoli intrinsic ﬂat limit of

such ψ0

js. If

(1) ψ](∂T ) = [[∂BR× {0}]]

(2) lim infj→∞ M(Tj)≤M([[BR× {0}]])

(3) ψis injective in set(∂T )and ψ(∂T )⊂∂BR× {0}.

Then, (X, d, T )equals (BR, dEuc,[[BR]]).

We remark that under the conditions of Theorem 1.2 (X, d) could have a diﬀerent

topology than each (Xj, dj) (see Remark 4.2). Additionally, Theorem 1.2 is stronger

than [1, Theorem 4.2] when the limit space is expected to be (B1, dEuc,[[BR]]). On the

other hand [1, Theorem 4.2] allows limit spaces to be diﬀerent to (BR, dEuc,[[BR]]).

See Remark 4.3 for more details.

Since Theorem 1.1 also holds for any convex set in Euclidean space, Theorem

1.2 also holds for convex sets and this implies that the stability with respect to

the intrinsic ﬂat distance of fundamental domains of 3-dimensional tori with scalar

curvature converging to zero, as originally proven (but unpublished) by Cabrera

Pacheco, Ketterer and the second named author [10, Theorem 5.5], holds. This was

used to prove stability of 3-dimensional graphical tori in [10, Theorem 1.4]. We

remark that there is a rigorous and published version of the latter [11](c.f. [12,

Theorem 1.4]), with a proof in the same spirit as the proof of the stability of the

positive mass theorem for entire manifolds that appears in [1, Section 7].

We now give a brief outline of the paper. In Section 2we recall some preliminaries

on currents and integral current spaces. In Section 3we prove the rigidity result

of Theorem 1.1. In Section 4we provide an example showing that condition (3) in

Theorem 1.1 cannot be dropped and another one showing that in Theorem 1.2 the

topology of the sequence can be diﬀerent to the limit space. Finally, in Section 5we

prove Theorem 1.2 and discuss as an application the stability of the positive mass

theorem stated by Huang–Lee–Sormani.

Acknowledgements. GDN received funding from the European Research Coun-

cil (ERC) under the European Union’s Horizon 2020 research and innovation pro-

gramme, grant agreement No 757254 (SINGULARITY). RP acknowledges support

from CONACyT Ciencia de Frontera 2019 CF217392 grant.

2. Metric currents and Integral current spaces

We recall that Ambrosio and Kirchheim developed the theory of currents in com-

plete metric spaces [3] and that Sormani–Wenger developed the notion of integral

current spaces and intrinsic ﬂat distance [28]. We assume the reader to be famil-

iar with these works. Nonetheless, we will brieﬂy deﬁne some concepts and results

needed in the proofs of our main theorems.

We denote by B=B(0,1) the open Euclidean unit ball in Rn, and by Bror B(r)

the open ball of radius r. Given a complete metric space X, we denote by In(X)

the space of all n-dimensional integral currents in X.

4 GIACOMO DEL NIN AND RAQUEL PERALES

2.1. Currents in Euclidean space. Given an Hk-rectiﬁable set E⊂Rn, a simple

unit k-vector ﬁeld τon E, and a multiplicity function θ:E→R, we denote by

[E, τ, θ] the Euclidean k-current given by

[E, τ, θ](ω) = ˆE

hω(x), τ(x)iθ(x)dHk(x),

for every ωsmooth, compactly supported k-form in Rn. We will denote for simplicity

[B, τstd,1] (where τstd =e1∧. . . ∧enis the standard orientation of Rn) by [[B]].

Given a Lipschitz map f:Rn→Rn, and an n-current T= [E, τstd, θ], we recall

that the pushforward of Tby f, denoted by f]T, equals [f(E), τstd,˜

θ], where

(2.1) ˜

θ(y) = X

z∈f−1(y)

sign(det(df|z))θ(z) for Hn-a.e. y.

It is a consequence of the area formula that the expression above is well-deﬁned,

since the cardinality of f−1(y) is ﬁnite for Hn-a.e. y, and fis diﬀerentiable at

Hn-a.e. point by Rademacher’s theorem.

2.2. Structure of rectiﬁable currents. Given two complete metric spaces X, Y ,

a Lipschitz function f:X→Y, and an n-current Tin X, the pushforward f]Tis a

well-deﬁned n-current in Y[3, Deﬁnition 2.4], and we have M(f]T)≤Lip(f)nM(T),

where M(T) denotes the mass of the current T.

We have the following structure theorem for rectiﬁable currents in metric spaces,

which we can also take as a deﬁnition.

Theorem 2.1 ([3, Theorem 4.5]).Every rectiﬁable n-current Tin a complete metric

space (X, d)can be represented as

(2.2) T=X

(fi)][[θi]] with M(T) = X

M((fi)][[θi]])

for a countable collection of bi-Lipischitz maps fi:Ki→X, with Kicompact in

Rn,fi(Ki)pairwise disjoint and θi∈L1(Ki;R\ {0}). If Tis integral, then θiare

integer-valued.

Given a Lipschitz curve γ: [0,1] →X, with (X, d) a complete metric space, we

denote by [[γ]] the associated integral 1-current with weight equal to 1. In the case

of integral 1-currents we can say more, as we have the following structure result,

proved by Bonicatto–Del Nin–Pasqualetto.

Theorem 2.2 (Decomposition of integral 1-currents [7, Theorem 5.3]).Every in-

tegral 1-current Tin a complete metric space (X, d)can be written as T=Pi[[γi]],

where γiare either injective Lipschitz curves or injective Lipschitz loops, and so that

M(T) = X

M([[γi]]),M(∂T ) = X

M(∂[[γi]]).

2.3. Area factor. For a rectiﬁable n-current [E, τ, θ] in a Euclidean space we have

the following formula for its total mass:

M(T) = ˆE

|θ|dHn.

However, for a rectiﬁable current in a metric space an extra factor appears, the

so-called area factor λ, which we will deﬁne now. We remark that the following

material is taken from Ambrosio–Kirchheim [3].

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RIGIDITYOFMASS-PRESERVING1-LIPSCHITZMAPSFROMINTEGRALCURRENTSPACESINTORnGIACOMODELNINANDRAQUELPERALESAbstract.Weprovethatgivenann-dimensionalintegralcurrentspaceanda1-Lipschitzmap,fromthisspaceontothen-dimensionalEuclideanball,thatpreservesthemassofthecurrentandisinjectiveontheboundary,thenthemaphast...

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