Sobolev stability of the Positive Mass Theorem and Riemannian Penrose Inequality using inverse mean curvature flow
Tóm tắt
We study the Sobolev stability of the Positive Mass Theorem and the Riemannian Penrose Inequality in the case where a region of a sequence of manifolds
$$M^3_i$$
can be foliated by a smooth solution of inverse mean curvature flow (IMCF) which is uniformly controlled for time
$$t \in [0,T]$$
. In particular, we consider a sequence of regions of manifolds
$$U_T^i\subset M_i^3$$
, foliated by a IMCF,
$$\Sigma _t$$
, such that if
$$\partial U_T^i = \Sigma _0^i \cup \Sigma _T^i$$
and
$$m_H(\Sigma _T^i) \rightarrow 0$$
then
$$U_T^i$$
converges in
$$W^{1,2}$$
to a flat annulus or in the hyperbolic setting it converges to a annulus portion of hyperbolic space. If instead
$$m_H(\Sigma _T^i)-m_H(\Sigma _0^i) \rightarrow 0$$
and
$$m_H(\Sigma _T^i) \rightarrow m >0$$
then we show that
$$U_T^i$$
converges in
$$W^{1,2}$$
to a topological annulus portion of the Schwarzschild metric or in the Hyperbolic case to a topological annulus portion of the Anti-de Sitter Schwarzschild metric.
Tài liệu tham khảo
Abbott, L.F., Deser, S.: Stability of gravity with a cosmological constant. Nucl. Phys. B 195(1), 76–96 (1982)
Allen, B.: Inverse mean curvature flow and the stability of the PMT and RPI under \(L^2\) convergence. Ann. Henri Poincaré 19(4), 1283–1306 (2018)
Allen, B.: Stability of the PMT and RPI for asymptotically hyperbolic manifolds foliated by IMCF. J. Math. Phys. 59, 082501 (2018)
Allen, B.: Long Time Existence of Inverse Mean Curvature Flow in Metrics Conformal to Warped Product Manifolds. arXiv:1708.02535 [math.DG] (2017)
Allen, B.: Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem. arXiv:1807.08822 [math.DG] (2018)
Allen, B., Sormani, C.: Contrasting Various Notions of Convergence in Geometric Analysis. arXiv:1803.06582 [math.MG] (2018)
Andersson, L., Cai, M., Galloway, G.J.: Rigidity and positivity of mass for asymptotically hyperbolic manifolds. Ann. Henri Poincaré 9(1), 1–33 (2008)
Ashtekar, A., Megnon, A.: Asymptotically anti-de Sitter space–times. Class. Quantum Gravity 1(4), L39–L44 (1984)
Bray, H.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differ. Geom. 59(2), 177–267 (2001)
Bray, H., Finster, F.: Curvature estimates and the positive mass theorem. Commun. Anal. Geom. 2, 291–306 (2002)
Brendle, S., Hung, P.-K., Wang, M.-T.: A Minkowski inequality for hypersurfaces in the anti-de Sitter–Schwarzschild manifold. Commun. Pure Appl. Math. 69(1), 124–144 (2016)
Bryden, E.: Stability of the Positive Mass Theorem for Axisymmetric Manifolds. arXiv:1806.02447 [math.DG] (2018)
Cabrera Pacheco, A.J.: On the Stability of the Positive Mass Theorem for Asymptotically Hyperbolic Graphs. arXiv:1803.01899 [math.DG] (2018)
Chruściel, P., Herzlich, M.: The mass of asymptotically hyperbolic Riemannian manifolds. Pac. J. Math. 2, 393–443 (2003)
Corvino, J.: A note on asymptotically flat metrics on \(\mathbb{R}^{3}\) which are scalar-flat and admit minimal spheres. Proc. Am. Math. Soc. 12, 3669–3678 (2005). (electronic)
Dahl, M., Gicquaud, R., Sakovich, A.: Penrose type inequalities for asymptotically hyperbolic graphs. Ann. Henri Poincaré 14(5), 1135–1168 (2013)
Dahl, M., Gicquaud, R., Sakovich, A.: Asymptotically hyperbolic manifolds with small mass. Commun. Math. Phys. 325, 757–801 (2014)
de Lima, L.L., Girão, F.: An Alexandrov–Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality. Ann. Henri Poincaré 17(4), 979–1002 (2016)
Ding, Q.: The inverse mean curvature flow in rotationally symmetric spaces. Chin. Ann. Math. Ser. B 32(1), 27–44 (2011)
Finster, F.: A level set analysis of the Witten spinor with applications to curvature estimates. Math. Res. Lett. 1, 41–55 (2009)
Finster, F., Kath, I.: Curvature estimates in asymptotically flat manifolds of positive scalar curvature. Commun. Anal. Geom. 5, 1017–1031 (2002)
Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32, 299–314 (1990)
Gerhardt, C.: Inverse curvature flows in hyperbolic space. J. Differ. Geom. 89, 487–527 (2011)
Geroch, R.: Energy extraction. Ann. N. Y. Acad. Sci. 224, 108–117 (1973)
Gibbons, G.W., Hawking, S.W., Horowitz, G.T., Perry, M.J.: Positive mass theorems for black holes. Commun. Math. Phys. 88(3), 295–308 (1983)
Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)
Huang, L.-H., Lee, D., Sormani, C.: Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space. J. fur die Riene und Ang. Math. (Crelle’s Journal) 727, 1–299 (2015)
Hung, P.-K., Wang, M.-T.: Inverse mean curvature flows in the hyperbolic 3-space revisited. Calc. Var. PDE 54, 119–126 (2015)
Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59, 353–437 (2001)
Lee, D.: On the near-equality case of the positive mass theorem. Duke Math. J. 1, 63–80 (2009)
Lee, D., Sormani, C.: Near-equality in the Penrose inequality for rotationally symmetric Riemannian manifolds. Ann. Henri Poinc. 13, 1537–1556 (2012)
Lee, D., Sormani, C.: Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds. J. fur die Riene und Ang. Math. (Crelle’s Journal) 686, 187–220 (2014)
LeFloch, P., Mardare, C.: Definition and stability of Lorentzian manifolds with distributional curvature. Portugaliae Mathematica 64, 535–574 (2007)
LeFloch, P., Sormani, C.: The nonlinear stability of spaces with low regularity. J. Funct. Anal. 268(7), 2005–2065 (2015)
Neves, A.: Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds. J. Differ. Geom. 84, 191–229 (2010)
Petersen, P.: Convergence theorems in riemannian geometry. In: Grove, K., Petersen, P. (eds.) Comparison Geometry, vol. 30, pp. 167–202. MSRI Publications, Cambridge University Press, Cambridge (1997)
Petersen, P., Wei, G.: Relative volume comparison with integral curvature bounds. Geom. Funct. Anal. 7, 1031–1045 (1997)
Sakovich, A., Sormani, C.: Almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds with spherical symmetry. Gen. Relativ. Gravit. 49, 125 (2017)
Scheuer, J.: The inverse mean curvature flow in warped cylinders of non-positive radial curvature. Adv. Math. 306, 1130–1163 (2017)
Scheuer, J.: Inverse curvature flows in Riemannian warped products. J. Funct. Anal. 276(4), 1097–1144 (2019)
Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–46 (1979)
Urbas, J.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. A. 205, 355–372 (1990)
Wang, X.: The mass of asymptotically hyperbolic manifolds. J. Differ. Geom. 2, 273–299 (2001)
Zhou, H.: Inverse mean curvature flows in warped product manifolds. J. Geom. Anal. 28(2), 1749–1772 (2018)