Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 13, 2017

Lower semicontinuity of mass under C0 convergence and Huisken’s isoperimetric mass

Jeffrey L. Jauregui and Dan A. Lee

Abstract

Given a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed C0 Cheeger–Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. In other words, total mass is lower semicontinuous under such convergence. In order to prove this, we use Huisken’s isoperimetric mass concept, together with a modified weak mean curvature flow argument. We include a brief discussion of Huisken’s work before explaining our extension of that work. The results are all specific to three dimensions.

A Equivalence of definitions of isoperimetric mass

Recall Definition 11, in which miso(M,g) is defined. The following result is never used in the paper, but it is an interesting fact about isoperimetric mass.

Proposition 37.

Let (M,g) be a C0 asymptotically flat 3-manifold. We define an alternate version of isoperimetric mass as follows:

m~iso(M,g)=sup{Ωi}i=1(lim supimiso(Ωi,g)),

where the supremum is taken over all sequences {Ωi}i=1 of allowable regions for which |*Ωi| as i. Then

m~iso(M,g)=miso(M,g).

In other words, defining the isoperimetric mass using exhaustions is equivalent to using sequences whose perimeters become arbitrarily large.

Proof.

We only need to prove that m~iso(M,g)miso(M,g) since the other inequality is immediate. Let W be any allowable region that contains M. Let {Ωi}i=1 be a sequence of allowable regions such that |*Ωi| as i. Define Ωi:=WΩi, which are allowable regions. We will prove that

(A.1)lim supimiso(Ωi)lim supimiso(Ωi).

Note that |Ωi||Ωi| and |*Ωi||*Ωi|+|*W|. Using the fact that miso() is increasing with respect to volume and decreasing with respect to area, we can estimate:

miso(Ωi)=2|*Ωi|(|Ωi|-16π|*Ωi|3/2)
2|*Ωi|+|*W|(|Ωi|-16π(|*Ωi|+|*W|)3/2)
2|Ωi||*Ωi|+|*W|-13π(|*Ωi|+|*W|)1/2
2|Ωi||*Ωi|-O(|Ωi||*Ωi|-2)-13π|*Ωi|1/2-O(|*Ωi|-1/2),

where “big O” depends on W but not on i. Using the isoperimetric inequality to bound the volume |Ωi|, we now have

miso(Ωi)miso(Ωi)-O(|*Ωi|-1/2).

Inequality (A.1) now follows. From this inequality, we conclude that m~iso(M,g) can be computed using only sequences of regions that each contain W. The result now follows from a straightforward diagonalization argument, considering a sequence of sets W exhausting M.∎

Acknowledgements

The authors thank Gerhard Huisken and Felix Schulze for helpful discussions and the referee for thoughtful comments.

References

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Math. Monogr., Oxford University Press, New York 2000. Search in Google Scholar

[2] R. H. Bamler, A Ricci flow proof of a result by Gromov on lower boundsfor scalar curvature, Math. Res. Lett. 23 (2016), no. 2, 325–377. Search in Google Scholar

[3] R. Bartnik, Energy in general relativity, Tsing Hua lectures on geometry & analysis (Hsinchu 1990–1991), International Press, Cambridge (1997), 5–27. Search in Google Scholar

[4] R. Bartnik, Mass and 3-metrics of non-negative scalar curvature, Proceedings of the International Congress of Mathematicians. Vol. II (Beijing 2002), Higher Education Press, Beijing (2002), 231–240. Search in Google Scholar

[5] H. L. Bray, The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature, Ph.D. thesis, Stanford University, 1997. Search in Google Scholar

[6] H. L. Bray, Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Differential Geom. 59 (2001), no. 2, 177–267. Search in Google Scholar

[7] H. L. Bray and D. A. Lee, On the Riemannian Penrose inequality in dimensions less than eight, Duke Math. J. 148 (2009), no. 1, 81–106. Search in Google Scholar

[8] Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749–786. Search in Google Scholar

[9] O. Chodosh, M. Eichmair, Y. Shi and H. Yu, Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian 3-manifolds, preprint (2016), https://arxiv.org/abs/1606.04626. Search in Google Scholar

[10] T. H. Colding and W. P. Minicozzi II, Differentiability of the arrival time, Comm. Pure Appl. Math. 69 (2016), no. 12, 2349–2363. Search in Google Scholar

[11] C. B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Éc. Norm. Supér. (4) 13 (1980), no. 4, 419–435. Search in Google Scholar

[12] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681. Search in Google Scholar

[13] X.-Q. Fan, Y. Shi and L.-F. Tam, Large-sphere and small-sphere limits of the Brown–York mass, Comm. Anal. Geom. 17 (2009), no. 1, 37–72. Search in Google Scholar

[14] M. Gromov, Dirac and Plateau billiards in domains with corners, Cent. Eur. J. Math. 12 (2014), no. 8, 1109–1156. Search in Google Scholar

[15] G. Huisken, An isoperimetric concept for mass and quasilocal mass, Report 2/2006 – Mathematical aspects of general relativity, Oberwolfach Rep. 3, European Mathematical Society, Zürich (2006), 87–88. Search in Google Scholar

[16] G. Huisken, Private communication, (2015). Search in Google Scholar

[17] G. Huisken, An isoperimetric concept for the mass in general relativity, preprint (2009), https://video.ias.edu/node/234. Search in Google Scholar

[18] G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353–437. Search in Google Scholar

[19] T. Ilmanen, Generalized flow of sets by mean curvature on a manifold, Indiana Univ. Math. J. 41 (1992), no. 3, 671–705. Search in Google Scholar

[20] T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (1994), Paper No. 520. Search in Google Scholar

[21] J. L. Jauregui, On the lower semicontinuity of the ADM mass, Comm. Anal. Geom., to appear. Search in Google Scholar

[22] J. Metzger and F. Schulze, No mass drop for mean curvature flow of mean convex hypersurfaces, Duke Math. J. 142 (2008), no. 2, 283–312. Search in Google Scholar

[23] R. Schoen and S.-T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), 45–76. Search in Google Scholar

[24] F. Schulze, Nonlinear evolution by mean curvature and isoperimetric inequalities, J. Differential Geom. 79 (2008), no. 2, 197–241. Search in Google Scholar

[25] B. White, The size of the singular set in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 13 (2000), no. 3, 665–695. Search in Google Scholar

[26] E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), 381–402. Search in Google Scholar

Received: 2016-08-29
Revised: 2017-01-17
Published Online: 2017-05-13
Published in Print: 2019-11-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston