Rigidity results for initial data sets satisfying the dominant energy condition
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,
Abstract
Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes, we use approximations by manifolds with smooth boundary.
1 Introduction
An initial data set is a triplet
We say that
Let 𝑀 be a spacelike hypersurface in the Minkowski spacetime
The study of initial data sets under the dominant energy condition can be traced back to Schoen and Yau’s work [29], in which they proved the spacetime positive energy theorem using minimal surface techniques in dimension 3. Eichmair, Huang, Lee, and Schoen [13] later extended this work by proving a more general spacetime positive mass theorem up to dimension 7. Huang and Lee [21] subsequently proved the rigidity statement in this theorem. In a different approach, Witten [32] used spinors to prove the spacetime positive mass theorem, a method that generalizes to all higher dimensions for spin manifolds. Parker and Taubes [27] gave a mathematically rigorous formulation of Witten’s argument. Chruściel and Maerten [10] later provided a rigorous proof of the desired rigidity statement in all dimensions for spin manifolds, building on Beig and Chruściel’s proof [6] in dimension 3. Lee’s book [22] offers a comprehensive exposition on the positive mass theorem.
Instead of noncompact asymptotically flat manifolds, we consider compact manifolds with boundary. The condition of asymptotic flatness is replaced by suitable boundary conditions. In the interior, we still demand the dominant energy condition. First rigidity results have been obtained in this context by Eichmair, Galloway, and Mendes in [12] and by Glöckle in [16].
We consider the following class of manifolds with boundary, specializing Example 1.1.
Let
at each point on
for all tangent vectors
The case
The case
We denote by 𝑇 the future-oriented normal vector field to 𝑀 in
With this understood,
for all tangent vectors
for all tangent vectors
Our first result identifies this class of examples as an extremal case of initial data sets.
Let
The map
We do not assume that Σ is connected.
If Σ is not connected, the degree of
The quantity
There is also a variant of Theorem I for
The conclusion is now that there is a spacelike curve
For
Let
The map
The proof of Theorem I uses a boundary value problem for the Dirac operator.
Boundary value problems for the Dirac operator have been extensively studied thanks to the work of Hörmander [20], and the first named author and Ballmann [2, 3].
In the second part of this paper, we consider initial data sets of the form
We will employ the strategy developed by the second named author in [7], adapting it to initial data sets.
We write the given polytope in the form
If
Assume that
The Matching Angle Hypothesis is satisfied.
2 A boundary value problem for the Dirac operator in odd dimensions
Throughout this section, we assume that 𝑀 is a compact connected spin manifold of dimension
Let
Let
We define a complex vector bundle ℰ over 𝑀 by
The fiber of
Thus a section 𝑠 of ℰ can be considered as a field of homomorphisms.
Given an element
We denote by
where
Using the identity
For each
Let 𝒟 denote the Dirac operator on 𝑀, acting on sections of ℰ.
This Dirac operator is formally selfadjoint.
By [2, Corollary 7.23], the conditions
We denote by
By the trace theorem, the restriction map extends uniquely to a bounded linear map
We decompose the bundle
where
As usual, we denote by 𝜈 the outward unit normal along Σ.
Let 𝐻 denote the unnormalized mean curvature.
The sign convention is such that the boundary of a Euclidean ball has positive mean curvature.
We define an operator
where
Let
where the twist bundle
Along Σ, we have
where
Proof
Recall that
Along Σ, we have
Proof
This follows from Lemma 2.2. ∎
The following index computation generalizes the results in [7] and plays a key role in our arguments.
Suppose that 𝑀 has odd dimension
is a Fredholm operator.
Its Fredholm index equals the degree of
Proof
Note that 𝒟 and 𝒜 are formally self-adjoint, and 𝒜 is an adapted boundary operator for 𝒟.
It follows from Lemma 2.3 that 𝒜 is an odd operator in the sense that it interchanges the bundles
is Fredholm and
It remains to compute the index of the operator
where, for each point
where
Note that, even though
Recall that 𝒜 maps sections of
to sections of
where
are Dirac-type operators. Thus
We may compute
and, similarly,
Therefore,
Putting these facts together, the assertion follows. ∎
3 Spin geometry for initial data sets
Let 𝑀 be a compact spin manifold of dimension
and
where 𝑅 is the scalar curvature of 𝑔. In the first step, we use 𝑞 to define modified connections on 𝒮 and on ℰ.
We define a connection
on 𝒮.
We next define a homomorphism field
for every section 𝑠 of ℰ and every tangent vector 𝑋.
With this understood, we define a connection
Note that the connection
In order to derive a modified Weitzenböck formula involving
Let
Proof
Let
Therefore,
Consequently,
This proves a.
To show b, we extend the orthonormal basis to a neighborhood of 𝑝 such that
The well-known Weitzenböck formula gives
Proposition 3.3 (Modified Weitzenböck formula)
We have the following operator identity:
Proof
We compute, using Lemma 3.2,
Furthermore,
These two equations together with the standard Weitzenböck formula yield
For all
Proof
Note that
along Σ. Using this identity and integration by parts, we find
Similarly,
Substituting these equations into the modified Weitzenböck formula concludes the proof. ∎
We have
at each point on Σ.
Here,
Proof
Let us fix a point
Suppose that
Proof
We first observe that
at each point on Σ. Using Lemma 2.2, we obtain
at each point on Σ. Lemma 3.5 implies
at each point on Σ.
Moreover, for each
Putting
We conclude
Using (3.3) and (3.4) together with Corollary 3.4, the assertion follows. ∎
Suppose that 𝑀 has odd dimension
Moreover,
Proof
By Proposition 2.4, the operator 𝒟 subject to the boundary conditions
From this, the assertion follows easily. ∎
In the next step, we establish an analogue of Proposition 3.7 in even dimensions.
In the remainder of this section, we will assume that
We can find a smooth map
The map 𝜑 has degree 1.
For each
Proof
Let us fix a point
It is easy to see that 𝜑 induces a smooth map from
Since
If
In the following, we fix a smooth map
The trace norm of
Proof
By Lemma 3.8, we have
Let us denote the singular values of
We consider the bundles 𝒮 and ℰ over 𝑀 defined above.
Let
Suppose 𝑀 has even dimension
Moreover, we can find an element
Proof
Since the map
Moreover,
It follows from Lemma 3.9 that
Moreover, since
Suppose that 𝑀 has even dimension
Moreover, we can find an element
Proof
Consider an arbitrary sequence
Moreover, for each 𝑙, we can find an element
where 𝐶 is a constant that does not depend on 𝑙. Using a standard interpolation inequality, we can bound
where 𝐶 is a constant that does not depend on 𝑙. Putting these facts together, we obtain
where 𝐶 is a constant that does not depend on 𝑙. Using Young’s inequality, we conclude that
where 𝐶 is a constant that does not depend on 𝑙.
Since
After passing to a subsequence if necessary, we may assume that
Finally, on the boundary Σ, we have
4 Proof of Theorem I
Throughout this section, we consider an initial data set
We denote the second fundamental form of Σ in 𝑀 (viewed as a (1,1)-tensor) by ℎ.
For each
We begin by proving several preliminary statements.
The following statements hold.
There exists a
At each point
We have
We prove Proposition 4.1 first in odd and then in even dimensions.
4.1 Proof of Proposition 4.1 in odd dimensions
Suppose that
At each point
for all
Proof
Since 𝑠 is
for every vector field 𝑋 on 𝑀.
Using the boundary condition
on Σ. In the next step, we fix a tangential vector field 𝑋 along Σ and differentiate (4.2) along 𝑋. Using (4.1), we find for the left-hand side
and for the right-hand side
Thus (4.2) implies
To justify the last equality, we decompose
Let
Proof
This follows directly from the fact that
We define a linear subspace
The subspace
Proof
Let us fix a point
at the point 𝑝.
Since
at the point 𝑝.
Using Lemma 4.3, we conclude that
We have
Proof
It follows from Lemma 4.4 that 𝐿 is invariant under
The spinor representation of the Clifford algebra
Let 𝑝 be a point on the boundary Σ.
Let
Proof
Applying Lemma 4.2 with
for
In the next step, we take the absolute value on both sides. Using the triangle inequality, we obtain
On the other hand, our assumption implies that
for
for
is self-adjoint, while
In summary, we proved all the assertions in Proposition 4.1 in odd dimensions.
4.2 Proof of Proposition 4.1 in even dimensions
We now turn to the even-dimensional case of Proposition 4.1.
Suppose that
for all
At each point
for all
Proof
Analogous to the proof of Lemma 4.2. ∎
We have
Proof
Let us fix a point
In the next step, we take the absolute value on both sides. This gives
Since
Putting these facts together, the assertion follows. ∎
The map
Proof
Suppose that the assertion is false.
By Lemma 3.8, the map
Let
Proof
Analogous to the proof of Lemma 4.3. ∎
As above, we define a linear subspace
The subspace
Proof
Analogous to the proof of Lemma 4.4. ∎
We have
Proof
Since the map
Let 𝑝 be a point on the boundary Σ.
Let
Proof
The proof is analogous to the proof of Lemma 4.6. ∎
In summary, we proved all the assertions in Proposition 4.1 in even dimensions.
4.3 Existence of a local isometric immersion into
R
n
,
1
In this subsection, we construct a local isometric immersion of
We define a vector bundle 𝒯 over 𝑀 by
for
for every vector field 𝑋 and every section
The connection
Proof
Let 𝑋 be a vector field on 𝑀, and let
This gives
This completes the proof of Lemma 4.15. ∎
Suppose that
Proof
Let 𝑋 be a vector field on 𝑀. By assumption,
This implies
and
This completes the proof of Lemma 4.16. ∎
Suppose that 𝑝 is an arbitrary point in 𝑀.
Consider the set of all pairs
Proof
Suppose that the assertion is false.
Then there exists a non-zero pair
for all
The bundle 𝒯 admits a trivialization by
Proof
According to Proposition 4.1, there exists a
Let
where
for
for all sections
for all
for all
Since
It follows from (4.6) and (4.7) that the unit normal vector to the immersion 𝐹 in
If 𝑀 is simply connected, the isometric immersion
4.4 Proof of Theorem I in the simply connected case
In this subsection, we prove Theorem I in the special case that 𝑀 is simply connected.
In this case, there exists an isometric immersion 𝐹 of 𝑀 into the Minkowski spacetime
We define a vector bundle 𝒲 over 𝑀 by
where 𝑋 denotes a vector field on 𝑀, and
For each point
where 𝑋 denotes a vector field on 𝑀 and
The map
for all
Proof
We compute
for all
for all
Suppose that 𝑎 is a constant vector in
for all
Proof
Let 𝑋 be a section of
Using the identity
This implies
In the next step, we add (4.8) and (4.9).
Using the identity
This completes the proof of Lemma 4.22. ∎
For each point
Note that
By Proposition 4.1, there exists a
Replacing 𝑞 by
In the following, we fix sections 𝑠 and 𝑡 of
We denote by 𝒴 the set of all pairs
at each point on 𝑀.
Let
Clearly, 𝒴 is a linear subspace of
Suppose that
Proof
Let us fix an arbitrary point
for all
Suppose that
Proof
Fix an arbitrary point
for all
This gives
Let
Proof
Let us fix an arbitrary point
at the point 𝑝.
For abbreviation, let
at the point 𝑝.
Note that
at each point in 𝑀.
Since
We have
The projection
Proof
We consider the set
It follows from Lemma 4.27 that
We have
Proof
The first statement follows immediately from Proposition 4.28.
The second statement follows from the fact that 𝒴 has dimension 𝑛 and the projection
There exists a basis
Proof
It follows from Corollary 4.29 that the subspace
Case 1. The subspace
We claim that
Case 2. The subspace
This contradicts the fact that
By scaling, we may assume that
For each point
hence
For each point
The map 𝐵 can be interpreted as the unit normal to Σ within a hyperplane orthogonal to
In view of Lemma 4.26, we can find a linear transformation
Using Lemma 4.27, it follows that the map
Suppose that 𝑝 is a point on the boundary Σ with
Proof
Let
for each
for
Let 𝜋 denote the projection from
The composition of the isometric immersion
with the projection
Proof
We define a Riemannian metric
for
This completes the proof of Theorem I in the special case when 𝑀 is simply connected.
4.5 Proof of Theorem I in the general case
In this final subsection, we treat the case when 𝑀 is not simply connected.
In this case, the isometric immersion
for every tangent vector
As above, we define a Riemannian metric
for
5 Approximating a compact convex polytope by smooth domains
Throughout this section, we fix an integer
For each
Let 𝑔 be a Riemannian metric which is defined on an open set containing Ω, and let 𝑞 be a symmetric
For each
If 𝜆 is sufficiently large, then
In the remainder of this section,
We define a map
The map
Proposition 5.2 (cf. [7, Proposition 3.9])
Let
We define a function
Suppose that the Matching Angle Hypothesis is satisfied. Then
Proof
It follows from [7, Proposition 3.10] that
In the remaining part of this section, we will estimate the
Consider three pairwise distinct elements
For each
for all pairwise distinct elements
Proof
We adapt the proof of [7, Lemma 3.13]. Let us consider an arbitrary point
By definition of
we obtain
at the point 𝑥.
Moreover,
at the point 𝑥. Putting these facts together, we obtain
at the point 𝑥.
Since
By assumption,
at the point 𝑦. Consequently,
at the point 𝑥. Putting these facts together, we conclude that
for each point
On the other hand, the intersection
This completes the proof of Lemma 5.5. ∎
Assume that the Matching Angle Hypothesis holds.
Let us fix an exponent
for all pairwise distinct elements
Proof
The corresponding estimate for
Let us fix an exponent
for all pairwise distinct elements
Proof
The corresponding estimate for
For each
Proof
Combining Lemma 5.5, Lemma 5.6, and Lemma 5.7, we conclude that
for all pairwise distinct elements
For each
Proof
Let us consider an arbitrary sequence
This implies
On the other hand, it follows from Proposition 5.8 that
Putting these facts together, the assertion follows. ∎
6 Proof of Theorem II
Throughout this section, we assume that Ω, 𝑔, 𝑞 satisfy the assumptions in Theorem II.
In particular, we assume that Ω satisfies the Matching Angle Hypothesis.
We consider the spinor bundle 𝒮 and the twisted spinor bundle
Let 𝑈 denote a Euclidean ball such that the closure of 𝑈 is contained in the interior of Ω.
Consider a sequence
Suppose that
for every nonnegative smooth function 𝐹 on
Proof
Note that the hypersurface
There exists a uniform constant 𝐶 (independent of 𝑙) such that
for every smooth function 𝐹 on
Proof
Note that the hypersurface
We have
for every smooth function 𝐹 on
For each section
where 𝐶 is a constant that does not depend on 𝑙.
Proof
By the triangle inequality, we can bound
We now apply Proposition 6.1 to the function
Putting these facts together, the assertion follows. ∎
For each section
where 𝐶 is a constant that does not depend on 𝑙.
Proof
We apply Proposition 6.2 to the function
Hence the assertion follows from Corollary 6.4. ∎
For each section
Proof
We apply Proposition 6.3 to the function
Hence the assertion follows from Corollary 6.4. ∎
The following statements hold.
There exists a
Each of the boundary faces is totally geodesic with respect to 𝑔.
Along each of the boundary faces, we have
Having established Proposition 6.7, the arguments in Section 4.3 imply that
In the remainder of this section, we will give the proof of Proposition 6.7. We first treat the odd-dimensional case, and then the even-dimensional case.
6.1 Proof of Proposition 6.7 in odd dimensions
Suppose that
Moreover,
Putting these facts together, we conclude that
if 𝑙 is sufficiently large. Using this estimate together with Corollary 6.4, we obtain
if 𝑙 is sufficiently large.
Since
By scaling, we may arrange that
and
Passing to a subsequence if necessary, the sequence
We have
Proof
Using Corollary 6.5, we obtain
Since
The assertion now follows from (6.1) together with the fact that
We have
Here,
Proof
Recall that
For each
Let
Proof
The proof is analogous to the proof of Lemma 4.3. ∎
As above, we define
The subspace
Proof
The proof is analogous to the proof of Lemma 4.4. ∎
We have
Let
Proof
Recall that
on the boundary face
and let 𝑋 be a tangent vector to that boundary face. Arguing as in the proof of Lemma 4.2, we obtain
Since
Note that
From this, the assertion follows. ∎
6.2 Proof of Proposition 6.7 in even dimensions
Suppose that
Moreover, for each 𝑙, we can find an element
Putting these facts together, we conclude that
if 𝑙 is sufficiently large. Using this estimate together with Corollary 6.4, we obtain
if 𝑙 is sufficiently large.
Since
By scaling, we may arrange that
and
We have
Proof
The proof is analogous to the proof of Proposition 6.8. ∎
We have
Proof
Recall that
For each
on
on
Suppose that
Proof
By assumption, we can find a point
at the point 𝑝. This implies
at the point 𝑝. On the other hand, the Matching Angle Hypothesis implies that
at the point 𝑝.
Putting these facts together, we conclude that
The map
Proof
Suppose that the assertion is false.
By Lemma 3.8, the map
for
Let
Proof
The proof is analogous to the proof of Lemma 4.3. ∎
We again define
The subspace
Proof
The proof is analogous to the proof of Lemma 4.4. ∎
We have
Proof
Note that
By Lemma 6.19, 𝐿 is invariant under
Let
Proof
The proof is analogous to the proof of Lemma 6.13. ∎
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: DFG-SPP 2026
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2403981
Funding statement: Christian Bär was supported by DFG-SPP 2026 “Geometry at Infinity” and by the University of Augsburg. Simon Brendle was supported by the National Science Foundation under grant DMS-2403981 and by the Simons Foundation. Tsz-Kiu Aaron Chow was supported by the Croucher Foundation. Bernhard Hanke was supported by DFG-SPP 2026 “Geometry at Infinity”, by Columbia University, by the NYU Courant Institute, by the Isaac Newton Institute and by the University of Potsdam.
A A metric characterization of the ball
Let
𝑔 is flat.
The second fundamental form of
There exists a point in
Proof
For
where
Let
We define
Case 1. Suppose that
In the next step, we fix a point
In the following, we identify
If 𝑀 is not simply connected, then there exists an isometry of
Case 2. Suppose that
B An auxiliary estimate
Let
for every nonnegative function
Proof
Suppose the assertion is false.
Then we can find a sequence of nonnegative functions
Using the triangle inequality, we obtain
After passing to a subsequence, the sequence
Let
for all
Let us fix an arbitrary point
containing 𝑝.
Let
at each point in
Finally, we send
References
[1] C. Bär, Metrics with harmonic spinors, Geom. Funct. Anal. 6 (1996), no. 6, 899–942. 10.1007/BF02246994Search in Google Scholar
[2] C. Bär and W. Ballmann, Boundary value problems for elliptic differential operators of first order, Surveys in differential geometry. Vol. XVII, International Press, Boston (2012), 1–78. 10.4310/SDG.2012.v17.n1.a1Search in Google Scholar
[3] C. Bär and W. Ballmann, Guide to elliptic boundary value problems for Dirac-type operators, Arbeitstagung Bonn 2013, Progr. Math. 319, Birkhäuser/Springer, Cham (2016), 43–80. 10.1007/978-3-319-43648-7_3Search in Google Scholar
[4] C. Bär, S. Brendle, B. Hanke and Y. Wang, Scalar curvature rigidity of warped product metrics, SIGMA Symmetry Integrability Geom. Methods Appl. 20 (2024), Paper No. 035. Search in Google Scholar
[5] C. Bär, P. Gauduchon and A. Moroianu, Generalized cylinders in semi-Riemannian and Spin geometry, Math. Z. 249 (2005), no. 3, 545–580. 10.1007/s00209-004-0718-0Search in Google Scholar
[6] R. Beig and P. T. Chruściel, Killing vectors in asymptotically flat space-times. I. Asymptotically translational Killing vectors and the rigid positive energy theorem, J. Math. Phys. 37 (1996), no. 4, 1939–1961. 10.1063/1.531497Search in Google Scholar
[7] S. Brendle, Scalar curvature rigidity of convex polytopes, Invent. Math. 235 (2024), no. 2, 669–708. 10.1007/s00222-023-01229-xSearch in Google Scholar
[8] S. Brendle and Y. Wang, On Gromov’s rigidity theorem for polytopes with acute angles, preprint (2023), https://arxiv.org/abs/2308.08000; to appear in J. reine angew. Math. Search in Google Scholar
[9] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971/72), 119–128. 10.4310/jdg/1214430220Search in Google Scholar
[10] P. T. Chruściel and D. Maerten, Killing vectors in asymptotically flat space-times. II. Asymptotically translational Killing vectors and the rigid positive energy theorem in higher dimensions, J. Math. Phys. 47 (2006), no. 2, Article ID 022502. 10.1063/1.2167809Search in Google Scholar
[11] M. do Carmo and E. Lima, Immersions of manifolds with non-negative sectional curvatures, Bol. Soc. Brasil. Mat. 2 (1971), no. 2, 9–22. 10.1007/BF02584681Search in Google Scholar
[12] M. Eichmair, G. J. Galloway and A. Mendes, Initial data rigidity results, Comm. Math. Phys. 386 (2021), no. 1, 253–268. 10.1007/s00220-021-04033-xSearch in Google Scholar PubMed PubMed Central
[13] M. Eichmair, L.-H. Huang, D. A. Lee and R. Schoen, The spacetime positive mass theorem in dimensions less than eight, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 1, 83–121. 10.4171/jems/584Search in Google Scholar
[14] C. Fefferman and D. H. Phong, Lower bounds for Schrödinger equations, Conference on partial differential equations (Saint Jean de Monts 1982), Société Mathématique de France, Paris (1982), Conf. No. 7. 10.5802/jedp.249Search in Google Scholar
[15] S. R. Ghorpade, A. R. Patil and H. K. Pillai, Decomposable subspaces, linear sections of Grassmann varieties, and higher weights of Grassmann codes, Finite Fields Appl. 15 (2009), no. 1, 54–68. 10.1016/j.ffa.2008.08.001Search in Google Scholar
[16] J. Glöckle, Initial data rigidity via Dirac–Witten operators, preprint (2023), https://arxiv.org/abs/2304.02331. Search in Google Scholar
[17] M. Gromov, Dirac and Plateau billiards in domains with corners, Cent. Eur. J. Math. 12 (2014), no. 8, 1109–1156. 10.2478/s11533-013-0399-1Search in Google Scholar
[18] M. Gromov, Four lectures on scalar curvature, Perspectives in scalar curvature. Vol. 1, World Scieintific, Hackensack (2023), 1–514. 10.1142/9789811273223_0001Search in Google Scholar
[19] M. Gromov, Convex polytopes, dihedral angles, mean curvature and scalar curvature, Discrete Comput. Geom. 72 (2024), no. 2, 849–875. 10.1007/s00454-024-00657-7Search in Google Scholar
[20] L. Hörmander, The analysis of linear partial differential operators. III, Grundlehren Math. Wiss. 274, Springer, Berlin 1994. Search in Google Scholar
[21] L.-H. Huang and D. A. Lee, Equality in the spacetime positive mass theorem, Comm. Math. Phys. 376 (2020), no. 3, 2379–2407. 10.1007/s00220-019-03619-wSearch in Google Scholar
[22] D. A. Lee, Geometric relativity, Grad. Stud. Math. 201, American Mathematical Society, Providence 2019. Search in Google Scholar
[23] C. Li, A polyhedron comparison theorem for 3-manifolds with positive scalar curvature, Invent. Math. 219 (2020), no. 1, 1–37. 10.1007/s00222-019-00895-0Search in Google Scholar
[24] C. Li, Dihedral rigidity of parabolic polyhedrons in hyperbolic spaces, SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No. 099. Search in Google Scholar
[25] C. Li, Correction to: A polyhedron comparison theorem for 3-manifolds with positive scalar curvature, Invent. Math. 228 (2022), no. 1, 535–538. 10.1007/s00222-022-01104-1Search in Google Scholar
[26] C. Li, The dihedral rigidity conjecture for 𝑛-prisms, J. Differential Geom. 126 (2024), no. 1, 329–361. 10.4310/jdg/1707767340Search in Google Scholar
[27] T. Parker and C. H. Taubes, On Witten’s proof of the positive energy theorem, Comm. Math. Phys. 84 (1982), no. 2, 223–238. 10.1007/BF01208569Search in Google Scholar
[28] R. Sacksteder, On hypersurfaces with no negative sectional curvatures, Amer. J. Math. 82 (1960), 609–630. 10.2307/2372973Search in Google Scholar
[29] R. Schoen and S. T. Yau, Proof of the positive mass theorem. II, Comm. Math. Phys. 79 (1981), no. 2, 231–260. 10.1007/BF01942062Search in Google Scholar
[30] T.-Y. Tsang, Dihedral rigidity for cubic initial data sets, preprint (2021), https://arxiv.org/abs/2108.08942. Search in Google Scholar
[31] J. Wang, Z. Xie and G. Yu, On Gromov’s dihedral extremality and rigidity conjectures, preprint (2021), https://arxiv.org/abs/2112.01510. Search in Google Scholar
[32] E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), no. 3, 381–402. 10.1007/BF01208277Search in Google Scholar
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