NNSC-Cobordism of Bartnik Data in High Dimensions
In this short note, we formulate three problems relating to nonnegative scalar curvature (NNSC) fill-ins. Loosely speaking, the first two problems focus on: When are (n−1)-dimensional Bartnik data (Σⁿ⁻¹ᵢ, γᵢ, Hᵢ), i=1,2, NNSC-cobordant? If (𝕊ⁿ⁻¹, γₛₜd, 0) is positive scalar curvature (PSC) cobordant...
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| description | In this short note, we formulate three problems relating to nonnegative scalar curvature (NNSC) fill-ins. Loosely speaking, the first two problems focus on: When are (n−1)-dimensional Bartnik data (Σⁿ⁻¹ᵢ, γᵢ, Hᵢ), i=1,2, NNSC-cobordant? If (𝕊ⁿ⁻¹, γₛₜd, 0) is positive scalar curvature (PSC) cobordant to (Σⁿ⁻¹₁,γ₁, H₁), where (𝕊ⁿ⁻¹, γₛₜd) denotes the standard round unit sphere, then (Σⁿ⁻¹₁,γ₁, H₁) admits an NNSC fill-in. Just as Gromov's conjecture is connected with the positive mass theorem, our problems are connected with the Penrose inequality, at least in the case of n=3. Our third problem is on Λ(Σⁿ⁻¹, γ) defined below.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 030, 5 pages
NNSC-Cobordism of Bartnik Data
in High Dimensions
Xue HU † and Yuguang SHI ‡
† Department of Mathematics, College of Information Science and Technology,
Jinan University, Guangzhou, 510632, P.R. China
E-mail: thuxue@jnu.edu.cn
‡ Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences,
Peking University, Beijing, 100871, P.R. China
E-mail: ygshi@math.pku.edu.cn
Received January 22, 2020, in final form April 13, 2020; Published online April 20, 2020
https://doi.org/10.3842/SIGMA.2020.030
Abstract. In this short note, we formulate three problems relating to nonnegative scalar
curvature (NNSC) fill-ins. Loosely speaking, the first two problems focus on: When are
(n − 1)-dimensional Bartnik data
(
Σn−1
i , γi, Hi
)
, i = 1, 2, NNSC-cobordant? (i.e., there is
an n-dimensional compact Riemannian manifold
(
Ωn, g
)
with scalar curvature R(g) ≥ 0 and
the boundary ∂Ω = Σ1 ∪ Σ2 such that γi is the metric on Σn−1
i induced by g, and Hi is
the mean curvature of Σi in
(
Ωn, g
)
). If
(
Sn−1, γstd, 0
)
is positive scalar curvature (PSC)
cobordant to
(
Σn−1
1 , γ1, H1
)
, where
(
Sn−1, γstd
)
denotes the standard round unit sphere
then
(
Σn−1
1 , γ1, H1
)
admits an NNSC fill-in. Just as Gromov’s conjecture is connected with
positive mass theorem, our problems are connected with Penrose inequality, at least in the
case of n = 3. Our third problem is on Λ
(
Σn−1, γ
)
defined below.
Key words: scalar curvature; NNSC-cobordism; quasi-local mass; fill-ins
2020 Mathematics Subject Classification: 53C20; 83C99
Dedicate this paper to Professor Misha Gromov
on the occasion of his 75th birthday.
Bartnik data
(
Σn−1, γ,H
)
consists of an (n − 1)-dimensional orientable Riemannian mani-
fold
(
Σn−1, γ
)
and a smooth function H defined on Σn−1 which serves as the mean curvature
of Σn−1. One basic problem in Riemannian geometry is to study: under what conditions is it
that γ is induced by a Riemannian metric g with nonnegative scalar curvature, for example,
defined on Ωn, and H is the mean curvature of Σ in
(
Ωn, g
)
with respect to the outward unit
normal vector? Indeed, this problem was proposed by M. Gromov recently (see [8, Problem A]
and [9, Sections 3.3 and 3.6]).
On the other hand, when n = 3, for each Bartnik data
(
Σ2, γ,H
)
may be associated with
certain quasi-local masses, for instance, when the Gaussian curvature K of γ is positive, (S2, γ)
can be isometrically embedded into R3 with mean curvature H0 (with respect to the outward unit
normal vector of the embedded image in R3), with this embedding we may define Brown–York
mass for
(
S2, γ,H
)
[4, 5] as
mBY
(
S2; γ,H
)
=
1
8π
ˆ
S2
(H0 −H) dσγ .
This paper is a contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov
on his 75th Birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Gromov.html
mailto:thuxue@jnu.edu.cn
mailto:ygshi@math.pku.edu.cn
https://doi.org/10.3842/SIGMA.2020.030
https://www.emis.de/journals/SIGMA/Gromov.html
2 X. Hu and Y. Shi
If
(
S2, γ,H
)
admits an NNSC fill-in and H > 0, it was shown that mBY
(
S2; γ,H
)
≥ 0 [22].
There are several pieces of interesting work on NNSC fill-ins relating to positivity of Brown–
York mass (for instance see [13, 14]). Obviously, positivity of Brown–York mass is one necessary
condition for the existence of such a fill-in, but it is far from sufficient. It was shown that
for Bartnik data
(
S2, γ,H
)
with positive Gaussian curvature and H > 0, let H0 be the mean
curvature of isometric embedding of
(
S2, γ
)
in R3, if mBY
(
S2; γ,H
)
= 0 and H 6= H0 then there
is a constant ε depending only on
(
S2, γ,H
)
such that for any H̃ > H − ε,
(
S2, γ, H̃
)
admits no
NNSC fill-ins [14, Theorem 3].
If K > −κ2 where κ is a constant, then
(
S2, γ
)
can be isometrically embedded into the hy-
perbolic space with constant sectional curvature −κ2, and we can make use of such embedding
to define a generalized Brown–York mass, moreover if H > 0 we were able to prove its posi-
tivity [25]. Clearly, this positivity of generalized Brown–York mass is also a kind of necessary
condition for the Bartnik data with K > −κ2 and H > 0 to admit NNSC fill-ins.
For Bartnik data
(
Σ2, γ,H
)
, we can define its Hawking mass as following:
mH(Σ, γ,H) =
√
Area(Σ)
16π
(
1− 1
16π
ˆ
Σ
H2 dσγ
)
.
It should be interesting to explore similar relation between Hawking mass or other quasi-local
masses of the Bartnik data with its NNSC fill-ins. Unfortunately, it is not easy to obtain a lower
bound of the Hawking mass which depends only on
(
Σ2, γ
)
.
In the investigation of above Gromov’s NNSC fill-in problem, we often need to deal with
NNSC-cobordisms of Bartnik data which may have its own interests. More specifically, given
Bartnik data
(
Σn−1
i , γi, Hi
)
, i = 1, 2, we say
(
Σn−1
1 , γ1, H1
)
is NNSC-cobordant to
(
Σn−1
2 , γ2, H2
)
if there is an orientable n-dimensional manifold
(
Ωn, g
)
with ∂Ωn = Σn−1
1 ∪ Σn−1
2 , R(g) ≥ 0,
γi = g|Σi, i = 1, 2, H1 is the mean curvature of Σn−1
1 in
(
Ωn, g
)
with respect to inward unit
normal vector, and H2 is the mean curvature of Σn−1
2 in
(
Ωn, g
)
with respect to outward unit
normal vector. Our first problem is:
Problem 1. Given Bartnik data
(
Σn−1
i , γi, Hi
)
, i = 1, 2, when are they NNSC-cobordant?
By using surgery arguments (see [10, 21]), it is not difficult to show that if Bartnik data(
Σn−1
i , γi, Hi
)
, i = 1, 2 can be filled in with positive scalar curvature metrics, then
(
Σn−1
1 , γ1,−H1
)
is NNSC-cobordant to
(
Σn−1
2 , γ2, H2
)
. Another possible relevant notion to this is so called “PSC-
concordant”. Namely, two PSC-metrics γ0 and γ1 on Σn−1 are said to be PSC-concordant if there
is a PSC-metric g on the cylinder Σ×I which are the product γ0 +dt2 near Σ×{0} and γ1 +dt2
near Σ×{1} (see [28]), in that case,
(
Σn−1, γ0, 0
)
is NNSC-cobordant to
(
Σn−1, γ1, 0
)
. By index
theory, it is known that there are countable infinity distinct PSC-concordant classes for S4k−1,
for any positive integer k ≥ 2. When two PSC-metrics γ0 and γ1 are isotopic, i.e., they can be
connected by a continuous path γt, t ∈ [0, 1], and for each t ∈ [0, 1], γt is a PSC-metric. Then
we may use quasi-spherical metric to show that if H1 is not too large then
(
S2, γ0, H0
)
is NNSC-
cobordant to
(
S2, γ1, H1
)
, here H0 can be any given smooth positive function (see [1, 22, 23]).
On the other hand, when H1 is large enough we are able to show
(
S2, γi, Hi
)
, i = 0, 1, cannot
be NNSC-cobordant [2].
Let γ0 be a Riemannian metric on S2 with its first eigenvalue λ1(−∆0 + K) > 0, here ∆0
is the Laplacian operator of γ0, then it was shown in [18] that
(
S2, γ0, 0
)
is NNSC-cobordant
to
(
S2, γrou, H
)
provided mH
(
S2, γrou, H
)
>
√
Area(S2,γ0)
16π , here γrou denotes the round metric
on S2. For a generalization to the case of Bartnik data with constant mean curvature surfaces
see [6, Theorem 1.1], and higher-dimensional analogues see [7, Theorems 1.1 and 1.2], and [19,
Proposition 2.1]. An NNSC fill-in by a conformal blow-down argument which may have deep
NNSC-Cobordism of Bartnik Data in High Dimensions 3
relation to Problem 1 please see the proof of Theorem 1.2 in [11]. For deep discussion on PSC-
concordant relation for two PSC-metrics on a manifold from topological point of view, please
see [29, 30] and references therein.
As we mentioned above, one obstruction of the above NNSC fill-in problem is from pos-
itivity of certain quasi-local mass (for instance, Brown–York mass, see [22, 26]). It may be
reasonable to think that there may be a potential obstruction of NNSC-cobordism problem
which is from Penrose-type inequality (for Penrose inequality, see [3, 12], for local Penrose
inequality, see [15, 20, 24, 27]). For instance, we observed that if
(
S2, γ2, H2
)
is with posi-
tive Gaussian curvature and H2 > 0, and
(
Σ2
1, γ1, H1
)
is NNSC-cobordant to
(
S2, γ2, H2
)
, then
mBY
(
S2; γ2, H2
)
≥ mH
(
Σ2
1, γ1, H1
)
provided mH
(
Σ2
1, γ1, H1
)
≤ 0 [2].
To our knowledge, even the following simple case is still unknown:
Problem 2. Given Bartnik data
(
Sn−1, g1, H
)
and
(
Sn−1, g0, 0
)
, both are with positive scalar
curvature, what is the largest inf
Sn−1
H so that
(
Sn−1, g0, 0
)
is NNSC-cobordant to
(
Sn−1, g1, H
)
?
Remark 1.
• By the arguments of [26, Theorem 1.4] and some gluing technique, we are able to show
that for any PSC-metric g1 on Sn−1, no matter whether g1 is PSC-concordant to g0 or
not, there is a constant H so that
(
Sn−1, g0, 0
)
is NNSC-cobordant to
(
Sn−1, g1, H
)
and
the ambient manifold bounded by these Bartnik data is diffeomorphic to Sn−1 × [0, 1]
provided g0 is the standard round metric on Sn−1 [2].
• If g0 is the standard round metric on Sn−1, then by gluing arguments, the largest inf
Sn−1
H
in Problem 2 is the corresponding number for
(
Sn−1, g1, H
)
to admit NNSC fill-ins.1
• As we know,
´
Σ2 H dµ1 and
´
Σ2 H
2 dµ1 are closely related to Brown–York mass and Haw-
king mass respectively, they are also involved in classical Minkowski’s inequality for a con-
vex surface and Willmore functional for a surface in R3, so, it may also be interesting to
ask what the possible largest values of
´
Sn−1 H dµ1 and
´
Sn−1 H
2 dµ1 are, especially for
n = 3.
For an orientable closed null-cobordant Riemannian manifold
(
Σn−1, γ
)
, define Λ
(
Σn−1, γ
)
by
Λ
(
Σn−1, γ
)
= sup
{ˆ
Σ
H dµγ
∣∣∣ (Σn−1, γ,H
)
admits an NNSC fill-in
}
.
In the case of n = 3 and H > 0, the above Λ was introduced in [16, 17], and also some
interesting properties were discussed therein. An open problem on an estimate of Λ
(
Σn−1, γ
)
was proposed in [9, p. 31], and a partial result in the case of H > 0 was obtained in [26,
Theorem 1.3].
Suppose
(
S2, γ
)
is a 2-dimensional surface with positive Gaussian curvature, then it can be
isometrically embedded into R3, let H0 be the mean curvature of the embedding image with
respect to the outward unit normal vector, then we have:
Problem 3. Is Λ
(
S2, γ
)
=
´
S2 H0 dµγ?
The affirmative answer implies the positivity of Brown–York mass without assumption of
positivity of the mean curvature.
1We are grateful to the referee for pointing this fact to us.
4 X. Hu and Y. Shi
Acknowledgements
The authors would like to thank Dr. Georg Frenck for his comments to clarify some notions in
this note, and are also deeply grateful to anonymous referees for invaluable suggestions on the
exposition. The research of the first and the second author was partially supported by NSFC
11701215, NSFC 11671015 and 11731001 respectively.
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References
|
| id | nasplib_isofts_kiev_ua-123456789-210580 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:03:33Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Hu, Xue Shi, Yuguang 2025-12-12T10:16:59Z 2020 NNSC-Cobordism of Bartnik Data in High Dimensions. Xue Hu and Yuguang Shi. SIGMA 16 (2020), 030, 5 pages 1815-0659 2020 Mathematics Subject Classification: 53C20; 83C99 arXiv:2001.05607 https://nasplib.isofts.kiev.ua/handle/123456789/210580 https://doi.org/10.3842/SIGMA.2020.030 In this short note, we formulate three problems relating to nonnegative scalar curvature (NNSC) fill-ins. Loosely speaking, the first two problems focus on: When are (n−1)-dimensional Bartnik data (Σⁿ⁻¹ᵢ, γᵢ, Hᵢ), i=1,2, NNSC-cobordant? If (𝕊ⁿ⁻¹, γₛₜd, 0) is positive scalar curvature (PSC) cobordant to (Σⁿ⁻¹₁,γ₁, H₁), where (𝕊ⁿ⁻¹, γₛₜd) denotes the standard round unit sphere, then (Σⁿ⁻¹₁,γ₁, H₁) admits an NNSC fill-in. Just as Gromov's conjecture is connected with the positive mass theorem, our problems are connected with the Penrose inequality, at least in the case of n=3. Our third problem is on Λ(Σⁿ⁻¹, γ) defined below. The authors would like to thank Dr. Georg Frenck for his comments to clarify some notions in this note, and are also deeply grateful to anonymous referees for invaluable suggestions on the exposition. The research of the first and the second author was partially supported by NSFC 11701215, NSFC 11671015, and 11731001, respectively. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications NNSC-Cobordism of Bartnik Data in High Dimensions Article published earlier |
| spellingShingle | NNSC-Cobordism of Bartnik Data in High Dimensions Hu, Xue Shi, Yuguang |
| title | NNSC-Cobordism of Bartnik Data in High Dimensions |
| title_full | NNSC-Cobordism of Bartnik Data in High Dimensions |
| title_fullStr | NNSC-Cobordism of Bartnik Data in High Dimensions |
| title_full_unstemmed | NNSC-Cobordism of Bartnik Data in High Dimensions |
| title_short | NNSC-Cobordism of Bartnik Data in High Dimensions |
| title_sort | nnsc-cobordism of bartnik data in high dimensions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210580 |
| work_keys_str_mv | AT huxue nnsccobordismofbartnikdatainhighdimensions AT shiyuguang nnsccobordismofbartnikdatainhighdimensions |