Mass from an Extrinsic Point of View
We express the -th Gauss-Bonnet-Chern mass of an immersed submanifold of Euclidean space as a linear combination of two terms: the total (2)-th mean curvature and the integral, over the entire manifold, of the inner product between the (2 + 1)-th mean curvature vector and the position vector of the...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
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Інститут математики НАН України
2025
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| Цитувати: | Mass from an Extrinsic Point of View. Alexandre de Sousa and Frederico Girão. SIGMA 21 (2025), 018, 11 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860272610117419008 |
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| author | de Sousa, Alexandre Girão, Frederico |
| author_facet | de Sousa, Alexandre Girão, Frederico |
| citation_txt | Mass from an Extrinsic Point of View. Alexandre de Sousa and Frederico Girão. SIGMA 21 (2025), 018, 11 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We express the -th Gauss-Bonnet-Chern mass of an immersed submanifold of Euclidean space as a linear combination of two terms: the total (2)-th mean curvature and the integral, over the entire manifold, of the inner product between the (2 + 1)-th mean curvature vector and the position vector of the immersion. As a consequence, we obtain, for each , a geometric inequality that holds whenever the positive mass theorem (for the -th Gauss-Bonnet-Chern mass) holds.
|
| first_indexed | 2026-03-21T11:56:48Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 018, 11 pages
Mass from an Extrinsic Point of View
Alexandre DE SOUSA a and Frederico GIRÃO b
a) Escola de Ensino Fundamental e Médio Santa Luzia, Fortaleza, 60110-300, Brazil
E-mail: alexandre.ads@pm.me
b) Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, 60455-760, Brazil
E-mail: fred@mat.ufc.br
Received October 22, 2024, in final form March 03, 2025; Published online March 18, 2025
https://doi.org/10.3842/SIGMA.2025.018
Abstract. We express the q-th Gauss–Bonnet–Chern mass of an immersed submanifold of
Euclidean space as a linear combination of two terms: the total (2q)-th mean curvature and
the integral, over the entire manifold, of the inner product between the (2q + 1)-th mean
curvature vector and the position vector of the immersion. As a consequence, we obtain, for
each q, a geometric inequality that holds whenever the positive mass theorem (for the q-th
Gauss–Bonnet–Chern mass) holds.
Key words: Gauss–Bonnet–Chern mass; asymptotically Euclidean submanifolds; positive
mass theorem; Hsiung–Minkowski identities
2020 Mathematics Subject Classification: 83C99; 53C40; 51M16
1 Introduction
In this article, we explore the concept of mass from an extrinsic point of view. Our approach is
based on the introduction of a class of immersions in Euclidean space whose members we call
asymptotically Euclidean immersions (Definition 3.5) and on an integral identity (Theorem 3.6)
that can be seen as a version of the classical Hsiung-Minkowski formulas [16, 19] for a class of
non-compact immersed submanifolds. Through this identity, we deduce a geometric inequality
(Corollary 3.7) that must be satisfied whenever the positive mass conjecture for the GBC mass
(and ADM mass, in particular) is valid.
We also explore the necessary conditions for vector fields to generate the asymptotic charges
of interest (Proposition 2.7) and present two conjectures. In one of them, we conjecture that
asymptotically Euclidean spaces admit an asymptotically Euclidean isometric immersion (Con-
jecture 3.8); in the other, we conjecture that the aforementioned geometric inequality must hold
whenever an asymptotically Euclidean immersion satisfies a natural hypothesis (Conjecture 3.9).
If both conjectures are valid, the positive mass conjecture for the GBC mass (and ADM mass,
in particular) can be directly deduced from our results.
2 Mass of asymptotically Euclidean spaces
In this section, we establish the background needed for understanding our method. As far as we
are aware, the results presented in Proposition 2.7 and Corollary 2.8 are novel in the literature.
They are fundamental ingredients for the employment of our line of action, since they establish
sufficient conditions for a vector field to generate the asymptotic charges we are interest in.
We begin by remembering the concept of an asymptotically Euclidean end.
Definition 2.1 (asymptotically Euclidean end). An asymptotically Euclidean end of order τ ,
with τ > 0, is a Riemannian manifold (En, g), n ≥ 3, for which there exists a diffeomorphism
mailto:alexandre.ads@pm.me
mailto:fred@mat.ufc.br
https://doi.org/10.3842/SIGMA.2025.018
2 A. de Sousa and F. Girão
Ψ: E → Rn \ B1(0), introducing coordinates in E, say Ψ(x) =
(
x1, x2, . . . , xn
)
, such that, in
these coordinates, the following asymptotic condition holds:
|gij − δij |+ ρ|gij,k|+ ρ2|gij,kl| = O(ρ−τ ), as ρ→ ∞, (2.1)
for all i, j, k, l ∈ {1, 2, . . . , n}, where the gij ’s are the coefficients of g with respect to the coordi-
nates Ψ(x), gij,k = ∂gij/∂x
k, gij,kl = ∂gij/∂x
k∂xl, and ρ = |Ψ(x)| denotes the distance function
to the origin with respect to the Euclidean metric induced on the end.
Our model mass concept is the ADM mass of an end (E, g), introduced by Arnowitt, Deser
and Misner in [2].
Definition 2.2 (ADM mass). The ADM mass of an asymptotically Euclidean end (En, g) is
defined by
mADM(E, g) =
1
2(n− 1)ωn−1
lim
ρ→∞
∫
Sρ
(gij,i − gii,j)ν
j dSρ, (2.2)
where ωn−1 is the volume of the unit sphere of dimension n− 1, Sρ is the Euclidean coordinate
sphere of radius ρ, dSρ is the volume form induced on Sρ by the Euclidean metric, and ν is the
outward pointing unit normal to Sρ (with respect to the Euclidean metric).
It is known that if τ > (n − 2)/2 and the scalar curvature of (E, g) is integrable, then
the limit (2.2) exists, is finite, and is a geometric invariant, that is, two coordinate systems
satisfying (2.1) yield the same value for it [4, 5] (see also [30]).
A complete Riemannian manifold (Mn, g), n ≥ 3, is said to be asymptotically Euclidean
of order τ if there exists a compact subset K of M such that M \ K has finitely many con-
nected components and, for any connected component E of M \K, it occurs that (E, g) is an
asymptotically Euclidean end of order τ .
If (Mn, g) is an asymptotically Euclidean Riemannian manifold of order τ > (n− 2)/2 whose
scalar curvature is integrable, then its ADM mass, denoted by mADM(M, g), is defined as the
sum of the ADM masses of its ends.
One of the most important results in mathematical general relativity is the positive mass
theorem (PMT):
Theorem 2.3. If (Mn, g) is an asymptotically flat Riemannian manifold of order τ > (n−2)/2
whose scalar curvature is nonnegative and integrable, then each of its ends has nonnegative ADM
mass. Moreover, if the ADM mass of at least one of its ends is zero, then (M, g) is isometric
to the Euclidean space (Rn, δ).
The PMT was settled by Schoen and Yau when n ≤ 7 [34, 35, 36] and when (M, g) is con-
formally Euclidean [37], and by Witten when M is spin [40] (see also [33]). The general cases
of the PMT were treated by Lohkamp [23, 24, 25, 26] and by Schoen and Yau [38]. Proofs
for the case when (M, g) is an Euclidean graph (without the rigidity statement) were given by
Lam [20, 21] for graphs of codimension one (see also [8]) and by Mirandola and Vitório [31] for
graphs of arbitrary codimension. The case of Euclidean hypersurfaces (not necessarily graphs),
including the rigidity statement, was treated in [18]. Note that, since Euclidean graphs (of arbi-
trary codimension) and Euclidean hypersurfaces are spin, these cases also follow from Witten’s
proof.
In [12], a new mass (actually, a family of masses) for asymptotically Euclidean manifolds,
called the Gauss–Bonnet–Chern mass, was introduced. For a positive integer q < n/2, consider
the q-th Gauss–Bonnet curvature, denoted L(q), and defined by
L(q) =
1
2q
δ
a1a2...a2q−1a2q
b1b2...b2q−1b2q
q∏
s=1
R b2s−1b2s
a2s−1a2s = P ijkl(q) Rijkl,
Mass from an Extrinsic Point of View 3
where R is the Riemann curvature tensor of (M, g) and P(q), which has the same symmetries
of R (see [12, Section 3]), is given by
P ijkl(q) =
1
2q
δ
a1a2···a2q−3a2q−2ij
b1b2···b2q−3b2q−2b2q−1b2q
(
q−1∏
s=1
R b2s−1b2s
a2s−1a2s
)
gb2q−1kgb2ql.
Definition 2.4 (GBC mass). The q-th GBC mass of an asymptotically Euclidean end (En, g)
is defined by
mq(E, g) = c(n, q) lim
ρ→∞
∫
Sρ
P ijkl(q) gjk,lνi dSρ, (2.3)
where
c(n, q) =
(n− 2q)!
2q−1(n− 1)!ωn−1
and Sρ, dSρ, ν and ωn−1 are as in the definition of the ADM mass.
Note that L(1) is just the scalar curvature and, as observed in [12], m1 coincides with the
ADM mass.
In the same article, it is shown that if τ > τq and L(q) is integrable, then the limit (2.3)
exists, is finite, and is a geometric invariant, where here and throughout the text,
τq =
(n− 2q)
(q + 1)
.
As in the q = 1 case, if (Mn, g) is an asymptotically Euclidean manifold of order τ > τq whose
q-th Gauss–Bonnet curvature is integrable, then its q-th GBC mass, denoted by mq(M, g), is
defined as the sum of the q-th GBC masses of its ends.
The following is a version of the PMT for the GBC mass.
Conjecture 2.5. Let n and q be integers such that n ≥ 3 and 0 < q < n/2. If (Mn, g) is
an asymptotically Euclidean Riemannian manifold of order τ > τq whose q-th Gauss–Bonnet
curvature L(q) is nonnegative and integrable, then the q-th GBC mass of each of its ends is
nonnegative. Moreover, if the GBC mass of at least one of its ends is zero, then (M, g) is
isometric to the Euclidean space (Rn, δ).
Conjecture 2.5, without the rigidity statement, was proved for graphs of codimension one
in [12]; the case of graphs of arbitrary codimension and flat normal bundle was done by Li, Wei
and Xiong when q = 2 [22] and by the authors when 0 < q < n/2 [10, 11]. This conjecture,
including the rigidity statement, is known to be true for conformally flat manifolds [13].
2.1 Mass in terms of the Lovelock tensor
Let (En, g), n ≥ 3, be an asymptotically Euclidean end of order τ > τ1, and let G be its Einstein
tensor, that is,
G = Ric− 1
2
(Sc)g,
where Ric and Sc denote, respectively, the Ricci tensor and the scalar curvature of (E, g).
Throughout the text, we denote by X the vector field given by
X = xi
∂
∂xi
, (2.4)
where Ψ(x) =
(
x1, x2, . . . , xn
)
is a coordinate system satisfying (2.1).
4 A. de Sousa and F. Girão
Is is known that the ADM mass of (E, g) can be computed as follows (see [3, 6, 7]):
mADM(E, g) = − 1
(n− 1)(n− 2)ωn−1
lim
ρ→∞
∫
Sρ
G(X, νg) dS
g
ρ , (2.5)
where ωn−1 and Sρ are as in (2.2), νg is the outward unit normal vector to Sρ with respect
to the metric g, and dSgρ is the volume form induced on Sρ by g. The equivalence between
formulas (2.2) and (2.5) can be shown by reducing the general case to the case of harmonic
asymptotics via a density theorem (see, for example, [17]). Proofs without the use of a density
theorem were given in [15, 29].
As we will see in a moment, a formula similar to (2.5) holds for the GBC mass. To state this
result, we need to recall the so-called Lovelock curvature tensors.
Let (Mn, g), n ≥ 3, be a Riemannian manifold and let q < n/2 be a positive integer. The
q-th Lovelock curvature tensor, denoted by G(q), is defined by
G(q)ij = − 1
2q+1
gikδ
kb1b2...b2q−1b2q
ja1a2...a2q−1a2q
q∏
s=1
R
a2s−1a2s
b2s−1b2s
. (2.6)
Note that G(1) is just the Einstein tensor.
Proposition 2.6. The Lovelock curvature tensor G(q) satisfies the following:
(i) It is symmetric, that is,
G(q)ij = G(q)ji. (2.7)
(ii) It is divergent-free, that is,
∇iG(q)ij = 0. (2.8)
(iii) Its trace satisfies the equation
trg G(q) = −n− 2q
2
L(q). (2.9)
Proof. Identities (2.7) and (2.8) follow from [28, Theorem 1] (see also [27]). Identity (2.9) is
a straightforward computation. ■
Let (En, g), n ≥ 3, be an asymptotically Euclidean end of order τ > τq, where q < n/2 is
a positive integer. It was shown in [39] that the q-th GBC mass of (E, g) can be computed as
follows:
mq(E, g) = −b(n, q) lim
ρ→∞
∫
Sρ
G(q)(X, νg) dS
g
ρ , (2.10)
where X, νg and dSg are as in (2.5) and
b(n, q) =
(n− 2q − 1)!
2q−1(n− 1)!ωn−1
.
Note that, when q = 1, formulas (2.10) and (2.5) coincide.
The following proposition establishes sufficient conditions for a vector field to generate the
GBC mass.
Mass from an Extrinsic Point of View 5
Proposition 2.7. Let (En, g), n ≥ 3, be an asymptotically Euclidean end of order τ > τq, where
q < n/2 is a positive integer. Let Ψ(x) = (x1, x2, . . . , xn) be coordinates in E satisfying (2.1)
and let X be the vector field given by (2.4). If Y is a vector field on E such that(
Y i −Xi
)
(x) = O
(
ρ−τ+1
)
, i = 1, 2, . . . , n,
as ρ→ ∞, then
mq(E, g) = −b(n, q) lim
ρ→∞
∫
Sρ
G(q)(Y, νg) dS
g
ρ .
Proof. Let Ψ(x) =
(
x1, x2, . . . , xn
)
be coordinates on E satisfying (2.1). We can use these
coordinates to compare νg, the unit normal to Sρ with respect to g, and dSgρ , the volume form
induced on Sρ by g, with their Euclidean counterparts νδ and dSδρ, respectively.
It holds
νig − νiδ = O(ρ−τ ), as ρ→ ∞,
and
dSgρ = (1 + w) dSδρ
for some function w : E → R satisfying
w = O(ρ−τ ), as ρ→ ∞.
Furthermore, the components of the Riemman curvature tensor satisfy
Rlijk = O
(
ρ−τ−2
)
, as ρ→ ∞.
Together with (2.6), this gives
G(q)ij = O
(
ρ−q(τ+2)
)
, as ρ→ ∞.
Thus, we have∣∣∣∣∫
Sρ
G(q)(Y, νg) dS
g
ρ −
∫
Sρ
G(q)(X, νg) dS
g
ρ
∣∣∣∣
≤
∫
Sρ
∣∣G(q)ij
(
Y i −Xi
)∣∣(∣∣νig − νiδ
∣∣+ ∣∣νiδ∣∣)(1 + w) dSδρ
≤ C(n)ρ−[q(τ+2)+(τ−1)−(n−1)](1 + ρ−τ )2,
where C(n) is a constant that depends only on the dimension. Using that τ > τq, it follows that
lim
ρ→∞
∣∣∣∣∫
Sρ
G(q)(Y, νg) dS
g
ρ −
∫
Sρ
G(q)(X, νg) dS
g
ρ
∣∣∣∣ = 0. (2.11)
The proposition follows from (2.10) and (2.11). ■
The specialization of this proposition to the case of gradient fields is a key ingredient in the
development of our extrinsic approach.
6 A. de Sousa and F. Girão
Corollary 2.8. Under the same hypothesis as the ones in Proposition 2.7, if there exists
f ∈ C∞(E) such that
∂
∂xi
(
f − ρ2
2
)
= O
(
ρ−τ+1
)
, i = 1, 2, . . . , n, (2.12)
as ρ→ ∞, then
mq(E, g) = −b(n, q) lim
ρ→∞
∫
Sρ
G(q)
(
∇f, νg
)
dSgρ .
Proof. Write gij = δij + θij and take Y = ∇f . We have
Y i −Xi = gij
∂f
∂xj
− ∂
∂xi
(
ρ2
2
)
= δij
∂f
∂xj
+ θij
∂f
∂xj
− ∂
∂xi
(
ρ2
2
)
=
∂
∂xi
(
f − ρ2
2
)
+ θij
∂f
∂xj
. (2.13)
Note that (2.12) is equivalent to
∂f
∂xi
= xi +O
(
ρ−τ+1
)
, as ρ→ ∞. (2.14)
Thus, since
θij = O(ρ−τ ), as ρ→ ∞, (2.15)
from (2.14) and (2.15) we find
θij
∂f
∂xj
= O
(
ρ−τ+1
)
, as ρ→ ∞. (2.16)
The corollary follows from Proposition 2.7, (2.13) and (2.16). ■
3 Imersions in Euclidean spaces
Let ψ : Mn ↬ Rd, d > n, be a smooth immersion and let δ̄ = ⟨·, ·⟩ be the canonical Euclidean
metric on Rd. All quantities related to this ambient space will also be denoted with an overbar,
unless otherwise indicated.
Let B (which is normal-vector valued) be the second fundamental form of the immersion and
let p be an integer such that 0 < p ≤ n; if p is even, then the p-th mean curvature S(p) ∈ C∞(M)
is defined by
S(p) =
1
p!
δ
b1...bp
a1...ap
〈
Ba1
b1
, Ba2
b2
〉
· · ·
〈
B
ap−1
bp−1
, B
ap
bp
〉
,
and, if p is odd, then the p-th mean curvature S(p) ∈ Γ
(
TM⊥) is defined by
S(p) =
1
p!
δ
b1...bp
a1...ap
〈
Ba1
b1
, Ba2
b2
〉
· · ·
〈
B
ap−2
bp−2
, B
ap−1
bp−1
〉
B
ap
bp
,
where ⟨·, ·⟩ denotes the Euclidean metric on Rd. We also set S(0) = 1 and S(n+1) = 0.
Next, we turn to the so-called Newton transformations. The 0-th Newton transformation is
defined as
T(0) = g,
Mass from an Extrinsic Point of View 7
where g = ψ∗δ̄ denotes the induced metric. If p ≥ 2 is even, then the p-th Newton transforma-
tion T(p) is defined by
T(p)ij =
1
p!
gikδ
kb1b2...bp
ja1a2...ap
〈
Ba1
b1
, Ba2
b2
〉
· · ·
〈
B
ap−1
bp−1
, B
ap
bp
〉
, (3.1)
and, if p is odd, then the p-th Newton transformation T(p) is defined by
T(p)ij =
1
p!
gikδ
kb1b2...bp−1
ja1a2...ap−1
〈
Ba1
b1
, Ba2
b2
〉
· · ·
〈
B
ap−2
bp−2
, B
ap−1
bp−1
〉
B
ap
bp
.
Note that, when p is odd, the p-th Newton transformation is normal-vector valued. Furthermore,
by antisymmetry, it follows that T(p) ≡ 0 when p > n.
The relation between the trace of Newton transformations and the higher-order mean curva-
tures is a fact well-known in the literature (see [14, Lemma 2.2], for example).
Proposition 3.1. Let p be an integer such that 0 ≤ p ≤ n. The Newton transformation T(p)
satisfy
trgT(p) = (n− p)S(p). (3.2)
Moreover, if p is even, then
T(p)ijB
ij = (p+ 1)S(p+1).
By a direct application of the Gauss equation, we obtain that the q-th Lovelock tensor (2.6) of
the induced metric g = ψ∗δ̄ and the 2q-th Newton transformation (3.1) of an immersion contain
the same information.
Proposition 3.2. Let q be a positive integer such that q < n/2. It holds
Gq = −(2q)!
2
T(2q). (3.3)
The next lemma is an infinitesimal version of a Pohozaev–Schoen-type integral identity pre-
sented in [9, Propositon 3.2].
Lemma 3.3. If K and V are, respectively, a symmetric bilinear form and a vector field on M ,
then the following identity holds:
divg(ιVK) = ιV (divgK) +
1
2
g(K,LV g). (3.4)
Throughout the text, we denote by Z̄ the vector field on Rd given by
Z̄ = x̄α
∂
∂x̄α
, 1 ≤ α ≤ d,
where
(
x̄1, x̄2, . . . , x̄d
)
is the standard coordinate system on Rd. It is known that it is a conformal
Killing gradient field and that it satisfies the following identity:
Z̄ =
∇ρ̄2
2
,
where ρ̄ is the distance function to origin on Rd.
The following proposition can be interpreted as an infinitesimal version of the integral iden-
tity of Theorem 3.6. It is inspired by a infinitesimal version of the flux formula presented
in [1, equation (8.4)].
8 A. de Sousa and F. Girão
Proposition 3.4. Let Y = ψ∗Z̄⊤, where Z̄⊤ denotes the tangent part of the vector field Z̄.
It holds
divg(ιYG(q)) = −(2q)!
2
[
(n− 2q)S2q + (2q + 1)
〈
S2q+1, Z̄
〉]
. (3.5)
Proof. Since Y = ψ∗Z̄⊤, we have
ψ∗Y = Z̄⊤ = Z̄ − Z̄⊥
and
LY g = LY ψ∗δ̄ = ψ∗(Lψ∗Y δ̄
)
= ψ∗(L(Z̄−Z̄⊥)δ̄
)
= ψ∗(LZ̄ δ̄)− ψ∗(LZ̄⊥ δ̄
)
. (3.6)
Denote by BZ̄ the symmetric bilinear form on M defined by
BZ̄(V,W ) =
〈
B(V,W ), Z̄
〉
.
For any V,W ∈ Γ(TM),(
LZ̄⊥ δ̄
)
(V,W ) =
〈
DV Z̄
⊥,W
〉
+
〈
V,DW Z̄
⊥〉 = −2
〈
B(V,W ), Z̄
〉
= −2BZ̄(V,W ),
that is,
LZ̄⊥ δ̄ = −2BZ̄ . (3.7)
We also have
LZ̄ δ̄ = 2δ̄. (3.8)
Thus, (3.6)–(3.8) yield
g(G(q),LY g) = 2 trgG(q) + 2g(G(q), BZ̄). (3.9)
Identity (3.5) follows from (2.8), (3.2)–(3.4) and (3.9). ■
3.1 Asymptotically Euclidean imersions
Next, we describe our main object of study: a class of smooth immersions that place certain
smooth manifolds Mn into some Euclidean space Rd, where d > n, in a special way.
Definition 3.5 (asymptotically Euclidean immersion). Let ψ : Mn ↬ Rd, d > n, be a smooth
immersion of a smooth manifold Mn, n ≥ 3. We say that ψ is an asymptotically Euclidean
immersion of order τ , for some τ > 0, if the following conditions hold:
(i) the Riemannian manifold
(
M,ψ∗δ̄
)
is complete and asymptotically Euclidean of order τ ;
(ii) if E ⊂ M is such that (E,ψ∗δ̄) is an asymptotically Euclidean end of order τ , then,
as ρ→ ∞,
∂
∂xi
(
ψ∗ρ̄2 − ρ2
)
(x) = O
(
ρ−τ+1
)
, i = 1, 2, . . . , n,
for any coordinate system Ψ(x) =
(
x1, x2, . . . , xn
)
on E satisfying (2.1).
The following theorem is the main result of this article. It can be seen as a version of the
Hsiung–Minkowski formulas [16, 19] to asymptotically Euclidean immersions.
Mass from an Extrinsic Point of View 9
Theorem 3.6. Let n and q be integers such that n ≥ 3 and 0 < q < n/2. If ψ : Mn ↪→
(
Rd, δ̄
)
,
d > n, is an asymptotically Euclidean immersion of order τ > τq such that L(q), the q-th Gauss-
Bonnet curvature of the induced metric ψ∗δ̄, is integrable, then the functions S2q and
〈
S2q+1, Z̄
〉
are integrable and the q-th Gauss–Bonnet–Chern mass of
(
M,ψ∗δ̄
)
satisfies the following integral
identity:
mq
(
M,ψ∗δ̄
)
= a(n, q)
[
(n− 2q)
∫
M
S2q dM + (2q + 1)
∫
M
⟨S2q+1, Z̄⟩dM
]
,
where dM is the Riemannian measure on
(
M,ψ∗δ̄
)
and
a(n, q) =
(2n)!(n− 2q − 1)!
2q(n− 1)!ωn−1
.
Proof. Let E be one of the ends of M . Consider the function f : M → R, with
f =
ψ∗ρ̄2
2
.
Since ψ : Mn ↬ Rd is an asymptotically Euclidean immersion of order τ > τq, the function f
satisfies (2.12), and hence, by Corollary 2.8,
mq
(
E,ψ∗δ̄
)
= −b(n, q) lim
ρ→∞
∫
Sρ
G(q)(∇f, ν) dSρ, (3.10)
where the geometric quantities in the integral are computed with respect the induced metric ψ∗δ̄.
Note that
∇f = ψ∗Z̄⊤.
Thus, applying equation (3.10) to all the ends of M together with the divergence theorem and
identity (3.5), we find
mq
(
M,ψ∗δ̄
)
= a(n, q)
∫
M
[
(n− 2q)S2q + (2q + 1)
〈
S2q+1, Z̄
〉]
dM. (3.11)
It remains to show that both S2q and
〈
S2q+1, Z̄
〉
are integrable, so the right hand side of (3.11)
can be broken in two. The integrability of S2q follows from (2.9), (3.2), (3.3) and the integrability
of L(q). Once we know that S(2q) is integrable, the integrability of
〈
S2q+1, Z̄
〉
follows from the
fact that the left hand side of (3.11) is finite. ■
An immediate consequence of this theorem is the following.
Corollary 3.7. Under the same hypothesis as the ones in Theorem 3.6, mq
(
M,ψ∗δ̄
)
≥ 0 if and
only if
(n− 2q)
∫
M
S2q dM + (2q + 1)
∫
M
〈
S2q+1, Z̄
〉
dM ≥ 0.
A famous theorem by Nash [32] states that any Riemannian manifold (M, g) can be isomet-
rically immersed in some Euclidean space
(
Rd, δ̄
)
. Encouraged by Nash’s theorem, we make the
following conjecture.
Conjecture 3.8. If (Mn, g), n ≥ 3, is an asymptotically Euclidean manifold of order τ > 0,
then there exists an asymptotically Euclidean isometric immersion ψ : M → Rd of order τ (in
the sense of Definition 3.5).
10 A. de Sousa and F. Girão
Let n and q be integers such that n ≥ 3 and 0 < q < n/2. Suppose that Conjecture 3.8 is
true for every asymptotically Euclidean manifold (Mn, g) of order τ > τq (or at least for those
whose Gauss–Bonnet curvature L(q) is nonnegative and integrable). Then, by Theorem 3.6,
Conjecture 2.5 would be a direct consequence of the following conjecture.
Conjecture 3.9. Let n and q be integers such that n ≥ 3 and 0 < q < n/2. Let ψ : Mn → Rd,
d > n, be an asymptotically Euclidean immersion of order τ > τq for which S2q is integrable and
nonnegative. It holds
(n− 2q)
∫
M
S2q dM + (2q + 1)
∫
M
〈
S2q+1, Z̄
〉
dM ≥ 0,
with the equality holding if and only if
(
M,ψ∗δ̄
)
is isometric to Euclidean space.
Acknowledgements
This study was partially financed by the Coordenação de Aperfeiçoamento de Pessoal de Nı́vel
Superior - Brasil (CAPES) - Finance Code 001. This work was partially done while Alexandre
de Sousa was a CAPES Fellow at the Mathematics Institute of Federal University of Alagoas
(IM/UFAL), whose members he would like to thank for the hospitality. Frederico Girão was
partially supported by CNPq, grant number 307239/2020-9.
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1 Introduction
2 Mass of asymptotically Euclidean spaces
2.1 Mass in terms of the Lovelock tensor
3 Imersions in Euclidean spaces
3.1 Asymptotically Euclidean imersions
References
|
| id | nasplib_isofts_kiev_ua-123456789-212873 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:56:48Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | de Sousa, Alexandre Girão, Frederico 2026-02-13T13:49:03Z 2025 Mass from an Extrinsic Point of View. Alexandre de Sousa and Frederico Girão. SIGMA 21 (2025), 018, 11 pages 1815-0659 2020 Mathematics Subject Classification: 83C99; 53C40; 51M16 arXiv:2403.06782 https://nasplib.isofts.kiev.ua/handle/123456789/212873 https://doi.org/10.3842/SIGMA.2025.018 We express the -th Gauss-Bonnet-Chern mass of an immersed submanifold of Euclidean space as a linear combination of two terms: the total (2)-th mean curvature and the integral, over the entire manifold, of the inner product between the (2 + 1)-th mean curvature vector and the position vector of the immersion. As a consequence, we obtain, for each , a geometric inequality that holds whenever the positive mass theorem (for the -th Gauss-Bonnet-Chern mass) holds. This study was partially financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior- Brasil (CAPES) - Finance Code 001. This work was partially done while Alexandre de Sousa was a CAPES Fellow at the Mathematics Institute of Federal University of Alagoas (IM/UFAL), whose members he would like to thank for the hospitality. Frederico Girão was partially supported by CNPq, grant number 307239/2020-9. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Mass from an Extrinsic Point of View Article published earlier |
| spellingShingle | Mass from an Extrinsic Point of View de Sousa, Alexandre Girão, Frederico |
| title | Mass from an Extrinsic Point of View |
| title_full | Mass from an Extrinsic Point of View |
| title_fullStr | Mass from an Extrinsic Point of View |
| title_full_unstemmed | Mass from an Extrinsic Point of View |
| title_short | Mass from an Extrinsic Point of View |
| title_sort | mass from an extrinsic point of view |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212873 |
| work_keys_str_mv | AT desousaalexandre massfromanextrinsicpointofview AT giraofrederico massfromanextrinsicpointofview |