Lower semicontinuity of ADM mass under intrinsic flat convergence

Abstract

A natural question in mathematical general relativity is how the ADM mass behaves as a functional on the space of asymptotically flat 3-manifolds of nonnegative scalar curvature. In previous results, lower semicontinuity has been established by the first-named author for pointed $C^2$ convergence, and more generally by both authors for pointed $C^0$ convergence (all in the Cheeger-Gromov sense). In this paper, we show this behavior persists for the much weaker notion of pointed Sormani–Wenger intrinsic flat (${\mathcal {F}}$) volume convergence, under natural hypotheses. We consider smooth manifolds converging to asymptotically flat local integral current spaces (a new definition), using Huisken’s isoperimetric mass as a replacement for the ADM mass. Along the way we prove results of independent interest about convergence of subregions of ${\mathcal {F}}$-converging sequences of integral current spaces.

Appendix: perimeter and boundary mass

Here we recall the definition of the perimeter of a set in a Riemannian manifold, including the case in which the metric is only $C^0$. We also prove Lemma 23, giving the equality of perimeter and boundary mass.

We first recall some basic facts regarding the variation of a function. These concepts are typically stated in the setting of Euclidean space (see [2] for instance), but generally have analogs to smooth Riemannian manifolds (see [24] for instance, which we follow below).

Let (M, g) be a smooth Riemannian manifold (possibly with boundary) of dimension m, and let $f \in L^1(M,\mu _g)$ be a Borel function (where $\mu _g$ is the Riemannian volume measure induced by g). The variation of f is the quantity

$$\begin{aligned} |Df|_g(M) = \sup _{\phi } \left\{ \int \limits _M f{{\,\mathrm{div}\,}}_g\! \phi \; d\mu _g \;\Big |\; \phi \in \Gamma ^1_c(TM), |\phi |_g \le 1 \right\} \in [0,\infty ], \end{aligned}$$

(64)

where $\Gamma ^1_c(TM)$ denotes the space of $C^1$ vector fields with compact support on M. For example, if f happens to be $C^1$, then $|Df|_g(M) = \int _M |\nabla f|_g d\mu _g$. We say f has bounded variation (with respect to g) if $|Df|_g(M)$ is finite. In this case, there exists a finite Radon measure on M, denoted $|Df|_g$, and a $|Df|_g$-measurable vector field $\sigma _f$ on M, with $|\sigma _f|_g = 1$ a.e. (with respect to $|Df|_g$), so that

$$\begin{aligned} \int \limits _M f{{\,\mathrm{div}\,}}_g\! \phi \; d\mu _g = -\int \limits _M g(\sigma _f, \phi ) d|Df|_g \end{aligned}$$

(65)

for all $\phi \in \Gamma ^1_c(TM)$. Formula (65) can be viewed as defining the distributional gradient of a function f of bounded variation. Note that for any open set $U\subseteq M$,

$$\begin{aligned} |Df|_g(U) = \sup _{\phi } \left\{ \int \limits _U f{{\,\mathrm{div}\,}}_g\! \phi \; d\mu _g \;\Big |\; \phi \in \Gamma ^1_c(TU), |\phi |_g \le 1 \right\} , \end{aligned}$$

(66)

consistent with the notation in (64).

If f has bounded variation, it admits smooth approximations in the following sense (see [24, Proposition 1.4]; cf. [2, Theorem 3.9] in the Euclidean case): there exists a sequence $f_i$ of smooth functions on M of compact support, converging to f in $L^1(M,\mu _g)$, such that

$$\begin{aligned} |Df_i|_g(M) =\int \limits _M |\nabla f_i|_g d\mu _g\rightarrow |Df|_g(M) \end{aligned}$$

(67)

as $i \rightarrow \infty$.

We are interested in the following special case: let $E \subseteq M$ be a Borel set of finite $\mu _g$-measure, i.e. $\chi _E \in L^1(M,\mu _g)$. We say E has finite perimeter in M with respect to g if $\chi _E$ has bounded variation with respect to g. The perimeter of E is then defined to be $|D\chi _E|_g(M)$, which we will also denote in this appendix by $P_g(E)$.

From the above approximation result, it can be shown that E can be approximated in volume and perimeter by smooth sets (cf. [2, Theorem 3.42]):

Lemma 37

Suppose (M, g) is a smooth, complete Riemannian manifold (possibly with boundary). Given a set $E \subseteq M$ of finite perimeter, there exists a sequence $E_i$ of open sets with smooth boundary in M, such that $\mu _g(E_i \triangle E) \rightarrow 0$ and $P_g(E_i) \rightarrow P_g(E)$ as $i \rightarrow \infty$. If E is precompact, the $E_i$ may be chosen to be precompact.

If the Riemannian metric g is only $C^0$, however, the above discussion no longer holds, because the divergence in (64) need not be well-defined. To work around this, we first show how the data $\sigma _f$ and $|Df|_g$ in (65) are related with respect to different smooth metrics g on M.

Lemma 38

Let f be a Borel function on a smooth manifold M (possibly with boundary), and let $g_1$ and $g_2$ be smooth Riemannian metrics on M. Then:

• (a) If f has compact support, then f has bounded variation with respect to $g_1$ if and only if f has bounded variation with respect to $g_2$.

• (b) If f has bounded variation with respect to both $g_1$ and $g_2$, then $|Df|_{g_1}$ and $|Df|_{g_2}$ are mutually absolutely continuous as Borel measures, and in this case,

• (c) the 1-forms $g_i(\sigma _f^i, \cdot )$ (for $i=1,2$) defined in (65) with respect to $g_1$ and $g_2$ are pointwise multiples of each other in $T^*M$ almost everywhere. (By (b), “almost-everywhere” can be taken with respect to $|Df|_{g_1}$ or $|Df|_{g_2}$.)

Proof

Let $W>0$ be the smooth function on M defined by

$$\begin{aligned} d\mu _{g_2} = W d\mu _{g_1}. \end{aligned}$$

From the characterization of divergence as the Lie derivative of the volume form, we have

$$\begin{aligned} {{\,\mathrm{div}\,}}_{g_2} (\phi ) = W^{-1} {{\,\mathrm{div}\,}}_{g_1}(W\phi ) \end{aligned}$$

(68)

for any $C^1$ vector field $\phi$ on M. Statement (a) follows from this and the definition of variation, using the fact that $g_1$ and $g_2$ have relative $C^1$ bounds on any compact set.

Now, assume that f has bounded variation with respect to $g_1$ and $g_2$. For $i=1,2$, let

$$\begin{aligned} \alpha _i(\cdot ) = g_i(\sigma ^i_f,\cdot ), \end{aligned}$$

which are $|Df|_{g_i}$-measurable 1-forms on M, of unit length with respect to $g_i$, $|Df|_{g_i}$-almost everywhere. Using (65) and (68), we have

$$\begin{aligned} -\int \limits _M \alpha _2(\phi ) d|Df|_{g_2}&= \int \limits _M f {{\,\mathrm{div}\,}}_{g_2}\!(\phi ) d\mu _{g_2} \nonumber \\&= \int \limits _M f {{\,\mathrm{div}\,}}_{g_1}\!(W\phi ) d\mu _{g_1} \end{aligned}$$

(69)

$$\begin{aligned}&=-\int \limits _M \alpha _1(\phi ) Wd|Df|_{g_1} \end{aligned}$$

(70)

for any $C^1$ vector field $\phi$ on M of compact support. We now prove (b) directly; clearly we need only show one direction. Suppose $A \subset M$ is a Borel set with $|Df|_{g_1}(A)=0$. Suppose first that A is compact. Since $|Df|_{g_1}$ is a Radon measure and is hence outer-regular, given any $\epsilon >0$, there exists a precompact open set $U_\epsilon \subset M$ containing A such that $|Df|_{g_1}(U_\epsilon ) < \epsilon .$ Let $C>0$ be a constant chosen so that $W|\cdot |_{g_1} \le C |\cdot |_{g_2}$ on tangent vectors based in $\overline{U_\epsilon }$. Then using (66) and (69), there exists a $C^1$ vector field $\phi$ supported in $U_\epsilon$ such that $|\phi |_{g_2}\le 1$ and

$$\begin{aligned} |Df|_{g_2}(U_\epsilon )&< \epsilon + \int \limits _M f{{\,\mathrm{div}\,}}_{g_2}\! \phi \; d\mu _{g_2}\\&= \epsilon + C\int \limits _M f{{\,\mathrm{div}\,}}_{g_1}\! \left( \frac{W\phi }{C}\right) \; d\mu _{g_1}\\&\le \epsilon + C |Df|_{g_1}(U_\epsilon )\\&\le \epsilon (1+C). \end{aligned}$$

Since $\epsilon$ was arbitrary and C can be chosen independently of $\epsilon$ as $\epsilon \rightarrow 0$, this shows $|Df|_{g_2}(A)=0$. If A is not compact, this argument together with a simple covering argument suffices to show $|Df|_{g_2}(A)=0$. This completes the proof of (b).

From (b), by the Radon–Nikodym theorem, $d|Df|_{g_1} = h d|Df|_{g_2}$ as measures, for a positive Borel function h on M. Combining this with (70), we have

$$\begin{aligned} \int \limits _M\left( \alpha _2(\phi )-\alpha _1(\phi ) Wh\right) d|Df|_{g_2} =0 \end{aligned}$$

(71)

for any $C^1$ vector field $\phi$ of compact support. This implies that $\alpha _1$ and $\alpha _2$ are pointwise multiples of each other a.e. (with respect to $|Df|_{g_1}$ or $|Df|_{g_2}$). $\square$

The previous lemma allows us to compare the measures $|Df|_g$ with respect to different smooth metrics g that are related by a $C^0$ bound:

Lemma 39

Suppose $g_1$ and $g_2$ are smooth Riemannian metrics on M of dimension m, satisfying

$$\begin{aligned} \Lambda ^{-1} |\cdot |_{g_1} \le |\cdot |_{g_2} \le \Lambda |\cdot |_{g_1} \end{aligned}$$

(72)

on tangent vectors, for some constant $\Lambda \ge 1$. Then

$$\begin{aligned} \Lambda ^{-m-1} |Df|_{g_1} \le |Df|_{g_2} \le \Lambda ^{m+1}|Df|_{g_1} \end{aligned}$$

as Borel measures for any function f on M of bounded variation with respect to both $g_1$ and $g_2$.

Proof

Continuing with the notation in the proof of the previous lemma, consider the 1-forms $\alpha _1$, $\alpha _2$ that are multiples of each other pointwise a.e. and have unit length with respect to $g_1$ and $g_2$, respectively. From (72), this implies

$$\begin{aligned} \Lambda ^{-1} \alpha _1(\cdot ) \le \alpha _2(\cdot ) \le \Lambda \alpha _1(\cdot ) \end{aligned}$$

as 1-forms a.e. From (72) and the definition of W, we have

$$\begin{aligned} \Lambda ^{-m} \le W \le \Lambda ^m. \end{aligned}$$

From these bounds and (71), it follows that

$$\begin{aligned} \Lambda ^{-m-1} \le h \le \Lambda ^{m+1} \end{aligned}$$

a.e., and from this, the claim follows. $\square$

Corollary 40

Suppose $g_1$ and $g_2$ are smooth Riemannian metrics on M, satisfying (72). Let $E \subset M$ be a precompact Borel set. Then

$$\begin{aligned} \Lambda ^{-m-1} P_{g_1}(E) \le P_{g_2}(E) \le \Lambda ^{m+1} P_{g_1}(E). \end{aligned}$$

Proof

Since E is precompact, $|D\chi _E|_{g_1}(M)$ and $|D\chi _E|_{g_1}(M)$ are either both finite or both infinite, by Lemma 38(a). In the former case, the result follows from the previous Lemma with $f=\chi _E$, and in the latter it is trivial. $\square$

At last we can define perimeter with respect to a $C^0$ Riemannian metric g on M. Suppose $E \subset M$ is a precompact Borel set. We say E has finite perimeter with respect to g if E has finite perimeter with respect to any smooth Riemannian metric on M (and hence all such metrics, by Lemma 38(a)). In this case, define

$$\begin{aligned} P_g(E) = \lim _{i \rightarrow \infty } P_{g_i}(E), \end{aligned}$$

for any sequence of smooth Riemannian metrics $\{g_i\}$ on M, such that $g_i \rightarrow g$ in $C^0$. Corollary 40 implies that $P_g(E)$ is well-defined, i.e., is independent of the sequence.

Corollary 41

The smoothing result in Lemma 37 holds if g is merely $C^0$, provided the set E is precompact.

This follows from Corollary 40 as well.

In the main body of the paper, we use the notation $|\partial ^*E|_g$ to denote $P_g(E)$, though we do not require the notion of the reduced boundary $\partial ^* E$ itself and so do not discuss it here.

We move on to the proof of Lemma 23, which will be deduced from the following lemma relating perimeter and boundary mass in $C^0$ Riemannian manifolds.

Lemma 42

Let (M, g) be a connected, oriented $C^0$ Riemannian manifold (without boundary) of dimension m, and let $E \subseteq M$ be a precompact Borel set. Let $T_E$ be the integer rectifiable m-current on $(M,d_g)$ given by integration over E. Then ${\mathbb {M}}(\partial T_E)$ is finite if and only if E has finite perimeter with respect to g, and in this case $\Vert \partial T_E\Vert = |D \chi _E|_g$, so in particular

$$\begin{aligned} P_g(E) = {\mathbb {M}}(\partial T_E). \end{aligned}$$

Proof

This proof uses [3, Theorem 3.7], which implies the analogous result on Euclidean space.

Let $\epsilon > 0$, and let $p \in M$. Using a g-orthogonal basis of $T_pM$ along with the continuity of g, we can find a coordinate system $(x^i)$ about p on a small precompact neighborhood $U \subset M$ of p such that on U:

$$\begin{aligned} (1+\epsilon )^{-2} \delta _{ij}&\le g_{ij} \le (1+\epsilon )^2 \delta _{ij} \end{aligned}$$

(73)

$$\begin{aligned} (1+\epsilon )^{-1} d_\circ (\cdot , \cdot )&\le d_g(\cdot , \cdot ) \le (1+\epsilon ) d_\circ (\cdot , \cdot ) \end{aligned}$$

(74)

on U. We may shrink U if necessary so that it is convex with respect to $\delta _{ij}$; then $d_\circ$, the metric on U induced by the Riemannian metric $\delta _{ij}$, can be regarded as the restriction of the Euclidean metric to U.

First, suppose ${\mathbb {M}}(\partial T_E) < \infty$ so that $\Vert \partial T_E\Vert \llcorner U$ is a finite Borel measure on U (with the mass measure taken as usual with respect to d). Since U is open this is the same measure as $\Vert \partial T_{E \cap U} \llcorner U\Vert$. Working on U, using the fact that the mass measure’s metric dependence comes solely from Lipschitz constants, (74) implies that $\Vert \partial T_{E \cap U} \llcorner U\Vert _\circ$ is a finite Borel measure on U, and moreover

$$\begin{aligned} \Vert \partial T_{E \cap U} \llcorner U\Vert _\circ \le (1+\epsilon )^{m-1}\Vert \partial T_{E \cap U} \llcorner U\Vert , \end{aligned}$$

where the $\circ$ subscript means taken with respect to the Euclidean metric on U.

Now we regard U as lying in ${\mathbb {R}}^m$. Using the same steps in the proof of [3, Theorem 3.7] (only restricting to the open set U instead of ${\mathbb {R}}^m$), we see $|D\chi _{E \cap U}|_{\circ }$ is a finite measure on U, and moreover that $|D\chi _{E \cap U}|_{\circ } \le \Vert \partial T_{E \cap U} \llcorner U\Vert _\circ$. Using (73) and Lemma 39, we have $|D\chi _{E \cap U}|_g \llcorner U \le (1+ \epsilon )^{m+1} |D\chi _{E \cap U}|_{\circ }$ as Borel measures on U. Since U is open, the former is the same as $|D\chi _E|_g \llcorner U$. Combining the inequalities, we arrive at

$$\begin{aligned} |D \chi _{E }|_g \llcorner U \le (1+ \epsilon )^{2m} \Vert \partial T_{E }\Vert \llcorner U. \end{aligned}$$

Since M may be covered by such open neighborhoods U, and $|D \chi _E|_g$ and $\Vert \partial T_E\Vert$ are Borel measures (with the latter finite), we find

$$\begin{aligned} |D \chi _{E}|_g \le (1+ \epsilon )^{2m} \Vert \partial T_{E }\Vert \end{aligned}$$

as Borel measures on M. Since $\epsilon$ was arbitrary, we have $|D \chi _{E}|_g \le \Vert \partial T_{E }\Vert$.

A similar argument, but under the hypothesis that E has finite perimeter with respect to g, together with the reverse inequality $\Vert \partial T_{E \cap U} \llcorner U\Vert _{\circ } \le |D \chi _{E \cap U}|_{\circ }$ coming from [3, Theorem 3.7], completes the proof. $\square$

Proof of Lemma 23

Assume ${\mathbb {M}}(\partial (T \llcorner E)) < \infty$. First, we claim that $\Vert \partial (T\llcorner E)\Vert$ is supported in $X\setminus K$. Since E contains K in its interior $\mathring{E}$, it suffices to show $\Vert \partial (T\llcorner E)\Vert (\mathring{E})=0$. Let f be a Lipschitz function on X that is supported in $\mathring{E}$, and let $\pi _1, \ldots , \pi _{m-1}$ be Lipschitz functions on X. Recalling the definition of the boundary and the locality of metric currents with locally finite mass in Definition 2.1 and Lemma 2.1 of [21], since $\partial T=0$, we have $T(\chi _E,f,\pi _1, \ldots , \pi _{m-1})=0$, i.e., $\partial (T \llcorner E)(f, \pi _1, \ldots , \pi _{m-1})=0$. Since $\partial (T \llcorner E)$ has finite mass and thus can be regarded as an Ambrosio–Kirchheim current, the continuity property of currents in [3, Theorem 3.5(ii)] implies $\partial (T \llcorner E)(\chi _B, \pi _1, \ldots , \pi _{m-1})=0$ for any Borel set $B \subseteq \mathring{E}$. Finally, by the characterization of mass in [3, eqn. (2.6)], it follows that $\Vert \partial (T\llcorner E)\Vert (\mathring{E})=0$.

Now, near each point of p in $X\setminus K$, thanks to local compatibility, we can run the exact same argument used in the proof of Lemma 42 to see that on $X \setminus K$, $\Vert \partial (T\llcorner E)\Vert$ is equal to the measure $|D\chi _{E\setminus K}|_g$ as defined in the $C^0$ Riemannian manifold $(X\setminus K, g)$. Hence we have ${\mathbb {M}}(\partial (T\llcorner E))$ is equal to the perimeter of $E \setminus K$ in the $C^0$ Riemannian manifold $(X \setminus K, g)$, which is how we defined $|\partial ^* E|_g$ in the discussion following Definition 14. $\square$

We close with a lemma dealing with the boundary mass of the intersection of two open sets.

Lemma 43

Let (M, g) be a connected, oriented $C^0$ Riemannian manifold (without boundary) of dimension m. Let $U, V \subseteq M$ be precompact open subsets of M, and let $T_U, T_V,$ and $T_{U \cap V}$ be the integer rectifiable m-currents on $(M,d_g)$ given by integration over U, V,  and $U \cap V$ respectively. Assume $T_U, T_V,$ and $T_{U \cap V}$ have finite boundary mass, i.e., they are integral currents. Then the following inequality holds:

$$\begin{aligned} {\mathbb {M}}((\partial T_U) \llcorner V) + {\mathbb {M}}((\partial T_V )\llcorner U) \le {\mathbb {M}}(\partial T_{U \cap V}). \end{aligned}$$

Proof

In the following, we use Lemma 42 and the fact that for an open set U,  the support of $|D\chi _U|$ is disjoint from U.

$$\begin{aligned} {\mathbb {M}}((\partial T_U)\llcorner V) + {\mathbb {M}}((\partial T_V)\llcorner U)&=\Vert (\partial T_U)\llcorner V\Vert (M) + \Vert (\partial T_V)\llcorner U\Vert (M) \\&= \Vert \partial T_U \Vert (V) + \Vert \partial T_V\Vert (U) \\&= |D\chi _U | (V) +|D\chi _V | (U) \\&= |D\chi _U | (V\setminus U) +|D\chi _V | (U\setminus V) \\&= |D\chi _{U\cap V} | (V\setminus U) +|D\chi _{U\cap V} | (U\setminus V) \\&\le |D\chi _{U\cap V} |(M) \\&= {\mathbb {M}}(\partial T_{U\cap V}). \end{aligned}$$

Note that the fifth equality holds because, for example, $\chi _U$ and $\chi _{U\cap V}$ are the same function on the open set V, and the total variation measure is defined locally. $\square$