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

Jeffrey L. Jauregui; Dan A. Lee

doi:10.1515/crelle-2017-0007

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

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2017-0007

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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 supi→∞⁡miso⁢(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′:=W∪Ωi, which are allowable regions. We will prove that

(A.1)lim supi→∞⁡miso⁢(Ωi′)≥lim supi→∞⁡miso⁢(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′|-16⁢π⁢|∂*⁡Ωi′|3/2)

≥2|∂*⁡Ωi|+|∂*⁡W|⁢(|Ωi|-16⁢π⁢(|∂*⁡Ωi|+|∂*⁡W|)3/2)

≥2⁢|Ωi||∂*⁡Ωi|+|∂*⁡W|-13⁢π⁢(|∂*⁡Ωi|+|∂*⁡W|)1/2

≥2⁢|Ωi||∂*⁡Ωi|-O⁢(|Ωi|⋅|∂*⁡Ωi|-2)-13⁢π⁢|∂*⁡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.

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Received: 2016-08-29

Revised: 2017-01-17

Published Online: 2017-05-13

Published in Print: 2019-11-01

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

Abstract

A Equivalence of definitions of isoperimetric mass

Proposition 37.

Proof.

Acknowledgements

References

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