40 research outputs found
Large isoperimetric surfaces in initial data sets
We study the isoperimetric structure of asymptotically flat Riemannian
3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0.
Refining an argument due to H. Bray we obtain an effective volume comparison
theorem in Schwarzschild. We use it to show that isoperimetric regions exist in
(M, g) for all sufficiently large volumes, and that they are close to centered
coordinate spheres. This implies that the volume-preserving stable constant
mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye
as perturbations of large centered coordinate spheres minimize area among all
competing surfaces that enclose the same volume. This confirms a conjecture of
H. Bray. Our results are consistent with the uniqueness results for
volume-preserving stable constant mean curvature surfaces in initial data sets
obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian.
The additional hypotheses that the surfaces be spherical and far out in the
asymptotic region in their results are not necessary in our work.Comment: 29 pages. All comments welcome! This is the final version to appear
in J. Differential Geo
The Jang Equation Reduction of the Spacetime Positive Energy Theorem in Dimensions Less Than Eight
We extend the Jang equation proof of the positive energy theorem due to Schoen and Yau (Commun Math Phys 79(2):231-260, 1981) from dimension n = 3 to dimensions 3 ≤ n <8. This requires us to address several technical difficulties that are not present when n = 3. The regularity and decay assumptions for the initial data sets to which our argument applies are weaker than those in Schoen and Yau (Commun Math Phys 79(2):231-260, 1981
Topological censorship from the initial data point of view
We introduce a natural generalization of marginally outer trapped surfaces,
called immersed marginally outer trapped surfaces, and prove that three
dimensional asymptotically flat initial data sets either contain such surfaces
or are diffeomorphic to R^3. We establish a generalization of the Penrose
singularity theorem which shows that the presence of an immersed marginally
outer trapped surface generically implies the null geodesic incompleteness of
any spacetime that satisfies the null energy condition and which admits a
non-compact Cauchy surface. Taken together, these results can be viewed as an
initial data version of the Gannon-Lee singularity theorem. The first result is
a non-time-symmetric version of a theorem of Meeks-Simon-Yau which implies that
every asymptotically flat Riemannian 3-manifold that is not diffeomorphic to
R^3 contains an embedded stable minimal surface. We also obtain an initial data
version of the spacetime principle of topological censorship. Under physically
natural assumptions, a 3-dimensional asymptotically flat initial data set with
marginally outer trapped boundary and no immersed marginally outer trapped
surfaces in its interior is diffeomorphic to R^3 minus a finite number of open
balls. An extension to higher dimensions is also discussed.Comment: v2: Appendix added, Theorem 5.1 improved, other minor changes. To
appear in J. Diff. Geo
Effective versions of the positive mass theorem
The study of stable minimal surfaces in Riemannian -manifolds
with non-negative scalar curvature has a rich history. In this paper, we prove
rigidity of such surfaces when is asymptotically flat and has horizon
boundary. As a consequence, we obtain an effective version of the positive mass
theorem in terms of isoperimetric or, more generally, closed volume-preserving
stable CMC surfaces that is appealing from both a physical and a purely
geometric point of view. We also include a proof of the following conjecture of
R. Schoen: An asymptotically flat Riemannian -manifold with non-negative
scalar curvature that contains an unbounded area-minimizing surface is
isometric to flat .Comment: All comments welcome! The final version has appeared in Invent. Mat