• Corpus ID: 237259928

Dihedral rigidity for cubic initial data sets

@inproceedings{Tsang2021DihedralRF,
  title={Dihedral rigidity for cubic initial data sets},
  author={Tin-Yau Tsang},
  year={2021}
}
In this paper we pose and prove a spacetime version of Gromov’s dihedral rigidity theorem ([17],[24],[25]) for cubes when the dimension is 3 by studying the level sets of spacetime harmonic functions ([40],[7],[21]), extending the work of [9]. As a corollary, we also obtain an alternative proof of dihedral rigidity for prisms in hyperbolic space ([26]). We then discuss the relation between polyhedra and the spacetime positive mass theorem. This generalises the work of [32] and [25]. Finally, we… 
View PDF on arXiv
2 Citations
Background Citations
1
Methods Citations
1

2 Citations

On a spacetime positive mass theorem with corners

In this paper we consider the positive mass theorem for general initial data sets satisfying the dominant energy condition which are singular across a piecewise smooth surface. We find jump…

Rigid comparison geometry for Riemannian bands and open incomplete manifolds

. Comparison theorems are foundational to our understanding of the geometric features implied by various curvature constraints. This paper considers manifolds with a positive lower bound on either…

References

SHOWING 1-10 OF 46 REFERENCES

A polyhedron comparison theorem for 3-manifolds with positive scalar curvature

  • Chao Li
  • Mathematics
    Inventiones mathematicae
  • 2019
The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering…

On a spacetime positive mass theorem with corners

In this paper we consider the positive mass theorem for general initial data sets satisfying the dominant energy condition which are singular across a piecewise smooth surface. We find jump…

Spacetime Harmonic Functions and Applications to Mass

In the pioneering work of Stern [73], level sets of harmonic functions have been shown to be an effective tool in the study of scalar curvature in dimension 3. Generalizations of this idea, utilizing…

Interpreting Mass via Riemannian Polyhedra

We give an account of some recent development that connects the concept of mass in general relativity to the geometry of large Riemannian polyhedra, in the setting of both asymptotically flat and…

Dirac and Plateau billiards in domains with corners

Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these,…

Measuring Mass via Coordinate Cubes

  • P. Miao
  • Mathematics
    Communications in Mathematical Physics
  • 2020
Inspired by a formula of Stern that relates scalar curvature to harmonic functions, we evaluate the mass of an asymptotically flat 3-manifold along faces and edges of a large coordinate cube. In…

Hyperbolic Mass Via Horospheres

. We derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete…

The spacetime positive mass theorem in dimensions less than eight

We prove the spacetime positive mass theorem in dimensions less than eight. This theorem states that for any asymptotically flat initial data set satisfying the dominant energy condition, the ADM…

Spacetime Positive Mass Theorems for Initial Data Sets with Non-Compact Boundary

In this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under…

Schauder estimates by scaling

There are a number of methods which may be used to derive Schauder estimates for elliptic problems—see e.g. [Gia93], [Tru86] for some recent methods, and [GT83], [ADN59], [ADN64], [Mor66] for…