# The Positive Mass Theorem with Arbitrary Ends

Martin Lesourd, Ryan Unger, and Shing-Tung Yau  Black Hole Initiative, Harvard
University, Cambridge, MA 02138<EMAIL_ADDRESS>Department of
Mathematics, Princeton University, Princeton, NJ 08544
<EMAIL_ADDRESS>Department of Mathematics, Harvard University,
Cambridge, MA 02138<EMAIL_ADDRESS>
###### Abstract.

We prove a Riemannian positive mass theorem for manifolds with a single
asymptotically flat end, but otherwise arbitrary other ends, which can be
incomplete and contain negative scalar curvature. The incompleteness and
negativity is compensated for by large positive scalar curvature on an
annulus, in a quantitative fashion. In the complete noncompact case with
nonnegative scalar curvature, we have no extra assumption and hence prove a
long-standing conjecture of Schoen and Yau.

###### Contents

  1. 1 Introduction
  2. 2 The Dirichlet problem for $\mu$-bubbles
  3. 3 Construction of the weight
  4. 4 Proof of Theorem 1.6 with an extra assumption
    1. 4.1 Construction of $\Sigma_{\infty}$
    2. 4.2 Asymptotics of $\Sigma_{\infty}$
    3. 4.3 Strong stability of $\Sigma_{\infty}$
    4. 4.4 Finishing the argument when $R_{g}>0$ far out
  5. 5 Proofs of the main theorems
  6. A Curvature estimate in homogeneously regular manifolds

##  1\. Introduction

The Riemannian positive mass theorem is a landmark result in geometric
analysis and the study of scalar curvature.

######  Theorem 1.1 (Schoen–Yau [31]).

Let $(M^{3},g)$ be a complete asymptotically Schwarzschild manifold with
finitely many ends which are all asymptotically Schwarzschild. Then the mass
of each end is nonnegative.

The positive mass theorem also involves a rigidity statement when the mass is
zero, but we will not concern ourselves with rigidity questions in this paper.
This theorem was generalized to the (now commonplace) general asymptotically
flat setting in [32] via the _density theorem_ , to dimensions $\leq 7$ by
Schoen in [28] by studying the _strong stability inequality_ , and finally to
all dimensions in [37] using $k$-slicings and the convenient compactification
technique of Lohkamp [21].
This theorem was furthermore generalized to initial data sets in [33] and to
dimensions $\leq 7$ by Eichmair–Huang–Lee–Schoen [9]. There are numerous other
contributions to this field, particularly the case of spin manifolds, which
was handled by Witten [42]. We note that an alternative approach to the higher
dimensional case has been proposed by Lohkamp [22, 23].

In this paper, we prove a generalization of this theorem to the setting of
manifolds with a distinguished asymptotically flat end and otherwise arbitrary
and possibly incomplete other ends. For precision, see Definition 1.9.

######  Theorem 1.2.

Let $(M^{n},g)$, $3\leq n\leq 7$, be a complete non-compact manifold with
nonnegative scalar curvature and suppose that $(M^{n},g)$ has at least one
asymptotically Schwarzschild end. Then the mass of the distinguished end is
nonnegative.

This theorem was long conjectured by Schoen and the third author [35, 36, 20],
where it was shown to imply [35] the following _Liouville theorem_.

######  Theorem 1.3.

Let $(M^{n},g)$, $n\geq 3$, be a complete, locally conformally flat manifold
with nonnegative scalar curvature. If $\Phi:M^{n}\to S^{n}$ a conformal map,
then $\Phi$ is injective and $\partial\Phi(M)$ has zero Newtonian capacity.

Recently, the present authors [20] carried out Lohkamp’s idea in this setting,
making use of good estimates on the Green’s function to simplify the
asymptotics, to show that Theorem 1.3 actually follows from the impossibility
of $R>0$ on $T^{n}\mathbin{\\#}X^{n}$ for arbitrary $X^{n}$. The present paper
gives a new proof completely in the spirit of what is outlined in [35].

######  Remark 1.4.

Manifolds with complicated ends satisfying the hypotheses of Theorem 1.2 can
be constructed using connected sums [16].

The difficulty in proving Theorem 1.2 is that some of the standard approaches
to Theorem 1.1 do not seem to apply. The usual procedure of using density
theorems to obtain harmonic asymptotics is hampered by the lack of control of
the PDEs on the arbitrary non-compact ends. That being said, it might be
possible to produce harmonic asymptotics on the asymptotically flat end via a
similar argument to the density theorem. That this can be done in a complete
way is not at all clear and it seems like the geometry on the arbitrary ends
can be altered significantly in this process. So this approach might work for
Theorem 1.2, but it is doubtful that it can yield our quantitative Theorem
1.6.

The reduction of Theorem 1.2 to a geometric statement about $R>0$ via a
Lokhamp-type compactification seems to be unavailable for the same reason.
This is somewhat tantalizing because the analog of the geometric statement
that proves Theorem 1.1 is known: $T^{n}\\#X^{n}$ admits no complete metric
with $R>0$ when $X^{n}$ is arbitrary by [4]; cf. [20] for a similar result
that applies to a topologically wider class of spaces under an additional
geometric assumption for $4\leq n\leq 7$. Finally, spinor based proofs also
fall short because they involve controlling boundary integrals in the
arbitrary ends, about which nothing is assumed.

In view of these obstacles, the only viable approach seems to be the original
proof [31, 28], which constructs a limiting hypersurface from solutions to the
Plateau problem in the asymptotically flat end on which the mass is assumed
negative. This method cannot be naively implemented however, because nothing
seems to stop the solutions from venturing out into the arbitrary ends, which
would ruin the asymptotic flatness and completeness of the limiting
hypersurface and conformal metric that is crucial for the rest of the
argument.

######  Remark 1.5.

When $n=3$, Richard Schoen has pointed out to us that Theorem 1.2 follows from
essentially the original argument in [31] as follows: The complete minimal
surface constructed must be conformally equivalent to $\mathbb{C}$ by [11, 34]
and the fact that the scalar curvature can be taken strictly positive on the
asymptotically Schwarzschild end (see Section 5 below). It follows that since
it is asymptotically planar, it can only enter the core a bounded amount. At
this point, the rest of the argument goes through unchanged. This does not
give our stronger Theorem 1.6 below, however.

To overcome the issues in dimensions $4\leq n\leq 7$, we minimize a version of
Gromov’s $\mu$-bubble functional [13, 14], which we construct to prevent
solutions from exploring the other ends. The $\mu$-bubble functional, which
has recently found great success in noncompact scalar curvature problems [4,
15, 27, 44, 45], is defined on Caccioppoli sets $\Omega$ by

(1.1)
$\mathcal{A}(\Omega)=\mathcal{H}^{n-1}(\partial^{*}\Omega)-\int_{M}(\chi_{\Omega}-\chi_{\Omega_{0}})h\,d\mathcal{H}^{n},$

where $h$ is a function tending to $\pm\infty$ at finite distance from the
region of interest and $\Omega_{0}$ is a reference set. This allows for a
trivial construction of barrier surfaces and stable $\mu$-bubbles (which have
prescribed mean curvature $h$) will be Yamabe positive as long as the
following condition is satisfied:

(1.2) $R_{g}+h^{2}-2|\nabla_{g}h|>0.$

So to use $\mu$-bubbles effectively, one has to construct $h$ which blows up
in the right way for the geometry to work out and yet still satisfy the
inequality (1.2). In our case, the function $h$ is identically zero in the
asymptotic region.

Based on this approach, we can actually prove something stronger and more
surprising than Theorem 1.2.

######  Theorem 1.6.

Let $(M^{n},g)$, $3\leq n\leq 7$, be an asymptotically Schwarzschild manifold,
not assumed to be complete or have nonnegative scalar curvature everywhere.
Let $U_{1}$ and $U_{2}$ be neighborhoods of infinity with
$\overline{U_{2}}\subset U_{1}$ and $D>0$. If furthermore

  1. (1)

$g$ has no points of incompleteness in the $D$-neighborhood of $U_{1}$,

  2. (2)

$R_{g}\geq 0$ in the $D$-neighborhood of $U_{1}$, and

  3. (3)

the scalar curvature satisfies the largeness assumption

(1.3)
$R_{g}>\frac{32}{D}\left(\frac{8}{D}+\frac{1}{\mathrm{dist}_{g}(U_{2},\partial
U_{1})}\right)$

on $\overline{U_{1}}\setminus U_{2}$,

then $m\geq 0$.

Figure 1. Theorem 1.6.

This theorem is proved in Section 4 under the additional assumption that
$R_{g}>0$ sufficiently far out on the end. This assumption is removed in
Section 5 and Theorem 1.2 is then obtained as a corollary.

######  Remark 1.7.

(1) This theorem allows for incompleteness and negative scalar curvature as
long as it is far enough away from the region in which the largeness condition
(1.3) holds.

(2) The fact that scalar curvature is only assumed to be nonnegative away from
the incompleteness is particularly surprising in view of the aforementioned
approaches to Theorem 1.1, which require nonnegative scalar curvature
everywhere.

(3) The condition (1.3) is only needed to enforce the condition (1.2) in our
construction.

(4) The prototypical $U_{i}$ is the region in $\mathcal{E}$ with $|x|>r_{i}$
and $r_{1}<r_{2}$. In this case, the largeness assumption (1.3) can be
replaced by a lower bound involving $D$, $r_{2}-r_{1}$, and the Lipschitz
constant of $|x|$ with respect to $g$, which in turn can be estimated using

(1.4) $|\nabla_{g}|x||^{2}\leq 1+\sum_{ij}|g^{ij}-\delta^{ij}|.$

See Remark 3.2.

(5) Theorem 1.6 is interesting to consider in light of the negative mass
Schwarzschild solution, which is incomplete and scalar flat and hence falls
short of Theorem 1.6 because of condition (3). But our theorem shows that any
perturbation which increases scalar curvature (and preserves the
asymptotically Schwarzschild nature) cannot “push” the incompleteness away, so
that condition (1) is not satisfied.

Definitions and conventions. We assume all manifolds are smooth and oriented.
When discussing minimal hypersurfaces, the dimensional restriction $n\leq 7$
coming from the regularity theory will be implicit.

We use the notation $A\lesssim B$ to denote $A\leq CB$, where the constant $C$
does not depend on $A$ or $B$.

######  Definition 1.8.

Let $(X,d)$ be a metric space and $(\overline{X},d)$ be its completion. For
example, $\overline{X}$ can be constructed by taking appropriate equivalence
classes of Cauchy sequences. A point in $\overline{X}\setminus X$ is called a
called a _point of incompleteness_ for $X$. A set $S\subset X$ is said to be
_complete_ if its closure in $X$ remains closed under the inclusion
$X\to\overline{X}$.

######  Definition 1.9.

Let $M^{n}$ be a noncompact manifold with distinguished end $\mathcal{E}$. We
say that $(M,g)$ possesses a _structure of infinity_ along $\mathcal{E}$ if
$\mathcal{E}$ possesses no points of incompleteness and there exists a
diffeomorphism

$\Phi:\mathcal{E}\to\mathbb{R}^{n}\setminus\text{ball}$

and the coordinate norm $|x|=((x^{1})^{2}+\cdots+(x^{n})^{2})^{1/2}$ diverges
as we go out along the end. The set $\mathring{M}=M\setminus\mathcal{E}$ is
called the _core_. Note that in our definition, the core is not assumed to be
compact.

We say that $(M^{n},g,\mathcal{E})$ is _asymptotically Schwarzschild_ of
_mass_ $m\in\mathbb{R}$ if in the coordinates $x^{i}$,

$g_{ij}-\left(1+\frac{m}{2|x|^{n-2}}\right)^{\frac{4}{n-2}}\delta_{ij}\in
C^{2,\alpha}_{n-1}(\mathcal{E})$

for some $\alpha\in(0,1)$. For the spaces $C^{k,\alpha}_{\tau}$, see
Definition 4.4.

We say that $(M^{n},g,\mathcal{E})$ is _asymptotically flat_ of _order_ $\tau$
if

$g_{ij}-\delta_{ij}\in W^{2,p}_{\tau}(\mathcal{E})$

for some $p>n$ and $R_{g}\in L^{1}(\mathcal{E})$. For the spaces
$W^{k,p}_{\tau}$, see Definition 4.20. We will always assume
$\tau>\frac{n-2}{2}$. In this case, the _ADM mass_ is defined by

(1.5)
$m=\lim_{\rho\to\infty}\frac{1}{2(n-1)|S^{n-1}|}\int_{|x|=\rho}(\partial_{i}g_{ij}-\partial_{j}g_{ii})\frac{x^{j}}{|x|}\,d\mathcal{H}^{n-1}.$

Acknowledgments. M.L. thanks the Gordon and Betty Moore and the John Templeton
foundations for supporting the research carried out at Harvard’s Black Hole
Initiative. R.U. thanks Prof. T. Bourni, Prof. G. De Philippis, and V. Giri
for discussions regarding Lemma 4.8 and Appendix A and Prof. R. Schoen for
Remark 1.5. S.-T.Y. acknowledges the support of NSF Grant DMS-1607871.

##  2\. The Dirichlet problem for $\mu$-bubbles

We first prove a general existence theorem for $\mu$-bubbles but in the
setting of the Dirichlet problem. A similar situation was already considered
by Yau in [43].

We always assume the dimension of the ambient manifold lies between $2$ and
$7$ so that we may apply the regularity theory for almost-minimizing
hypersurfaces [7, 24].

######  Theorem 2.1.

Let $(M^{n},g)$ be a compact orientable Riemannian manifold with boundary
$\partial M=\partial_{0}M\amalg\partial_{-}M$, where $\partial_{0}M$ and
$\partial_{-}M$ are closed hypersurfaces. Let
$\Gamma^{n-2}\subset\partial_{0}M$ be a smooth hypersurface which bounds a
compact set $P^{n}\subset\partial_{0}M$. Let $h$ be a real-valued smooth
function on $M^{\circ}$, extending smoothly to $\partial_{0}M$, such that
$h<H_{\partial_{0}M}$ along $\partial_{0}M$ and $h\to-\infty$ along
$\partial_{-}M$. Then there exists a hypersurface $\Sigma^{n-1}\subset M$ with
the following properties:

  1. (1)

$\partial\Sigma=\Gamma,$

  2. (2)

The interior of $\Sigma$ is smooth and contained in the interior of $M$, and

  3. (3)

The mean curvature of $\Sigma$ is equal to $h$ with respect to the outward
normal in Lemma 2.2 below.

We view $P$ as a top-dimensional chain in $\partial_{0}M$ and as a codimension
1 chain in $M$. Let $\Sigma^{n}\subset M$ be a smooth hypersurface with
oriented boundary equal to $\Gamma$.

######  Lemma 2.2.

With the notation as above, if $\Sigma$ is homologous to $P$, then $\Sigma\cup
P=\partial U$, where $U\subset M$ is a unique open set.

We denote this assignment by $\Sigma\mapsto U_{\Sigma}$.

###### Proof.

The chain $A=\llbracket\Sigma\rrbracket-\llbracket P\rrbracket$ is a cycle in
$M$. Using the Mayer–Vietoris sequence, it is easy to see that
$H_{0}(M\setminus A)$ has rank at most 2, and we want to rule out the case
when it has rank 1. In this case it would be possible to find a curve
intersecting $A$ exactly once. (Take a short segment orthogonal to $\Sigma$
and since $M\setminus A$ is connected, the endpoints can be joined up without
further intersections with $A$.) But $A$ is homologous to zero and
intersection number is a homological invariant. ∎

###### Proof of Theorem 2.1.

Let $\mathcal{C}$ be the class of smooth hypersurfaces considered in Lemma 2.2
and define

(2.1)
$\mathcal{F}(\Sigma)=\mathcal{H}^{n-1}(\Sigma)-\int_{U_{\Sigma}}h\,d\mathcal{H}^{n}.$

Let $\\{\Sigma_{i}\\}\subset\mathcal{C}$ be a sequence of hypersurfaces such
that

$\lim_{i}\mathcal{F}(\Sigma_{i})=\inf_{\mathcal{C}}\mathcal{F}.$

We show using a replacement argument that we can assume $\Sigma_{i}$ avoids a
collar neighborhood of $\partial_{-}M$. We can foliate a collar of
$\partial_{-}M$ by hypersurfaces
$S^{\tau}_{-}=\partial_{\mathrm{top}}\Omega^{\tau}_{-}$ whose normal $\eta$
(pointing into the bulk, out of $\Omega^{\tau}_{-}$) satisfies
$h<-\operatorname{div}\eta$ for $0<\tau\leq\tau_{0}$. If $\Sigma_{i}$ enters
this region, we perform a replacement and define
$\Sigma^{\prime}_{i}=\partial(U_{\Sigma}\setminus\Omega_{-}^{\tau})$. Then as
in Chodosh–Li [4], $\mathcal{F}(\Sigma^{\prime}_{i})<\mathcal{F}(\Sigma_{i})$.
So we may assume that $\Sigma_{i}$ avoids a collar of $\partial_{-}M$ and
hence we may assume $|h|\lesssim 1$.

The condition $h<H_{\partial_{0}M}$ allows for a similar replacement/barrier
argument near $\partial_{0}M$. See Yau [43].

Since $M$ is compact and $h$ is bounded in the region of interest,
$\mathcal{F}(\Sigma_{i})\gtrsim-1$. We obviously have
$\mathbf{M}(\Sigma_{i})+\mathbf{M}(P)\lesssim 1$ by previous observations.
Also, $\mathbf{M}(U_{\Sigma_{i}})\lesssim 1.$ So by the BV compactness
theorem, there exists a subsequence (still denoted by $i$) of the
$U_{\Sigma_{i}}$’s converging in the sense of Caccioppoli sets. The limiting
set will have smooth boundary away from the corner along $P$ and will have
mean curvature $h$. It also satisfies a stability inequality as shown below. ∎

######  Proposition 2.3.

If $h=0$ near $\partial_{0}M$, then the following second variation formula
holds for the functional $\mathcal{F}$ defined in (2.1). Let
$F:M\times(-\varepsilon,\varepsilon)\to M$ be a one-parameter family of
diffeomorphisms with variation vector field $X=\partial_{t}F|_{t=0}$ on
$\Sigma$ and acceleration $Z=\nabla_{\partial_{t}}\partial_{t}F|_{t=0}$ on
$\Sigma$. Then

(2.2)
$\delta^{2}_{X,Z}\mathcal{F}(\Sigma)=\int_{\Sigma}a_{X,Z}\,d\mathcal{H}^{n-1}+\int_{\Sigma}d_{X,Z}\,d\mathcal{H}^{n-1},$

where

(2.3) $\displaystyle a_{X,Z}$
$\displaystyle=\sum_{i=1}^{n-1}\left[|(\nabla_{e_{i}}X)^{\perp}|^{2}-\mathrm{Rm}(X,e_{i},X,e_{i})\right]-\sum_{i,j=1}^{n-1}\langle
e_{i},\nabla_{e_{j}}X\rangle\langle\nabla_{e_{i}}X,e_{j}\rangle$
$\displaystyle\quad+\operatorname{div}_{\Sigma}Z+(\operatorname{div}_{\Sigma}X)^{2},\quad\text{and}$
(2.4) $\displaystyle d_{X,Z}$ $\displaystyle=-h^{2}-X(h)\langle
X,\nu\rangle-\langle Z,\nu\rangle h-\langle X,\delta_{X}\nu\rangle h,$

where $\\{e_{i}\\}_{i=1}^{n-1}$ is an orthonormal frame on $T\Sigma$. The
first term satisfies the following integration by parts formula: If
$D\subset\Sigma$ is a subdomain and $X=\varphi\nu+\hat{X}$ and
$Z=\zeta\nu+\hat{Z}$, then

(2.5)
$\int_{D}a_{X,Z}\,d\mathcal{H}^{n-1}=\int_{D}b_{X,Z}\,d\mathcal{H}^{n-1}+\int_{\partial
D}c_{X,Z}\,d\mathcal{H}^{n-2},$

where

(2.6) $\displaystyle b_{X,Z}$
$\displaystyle=|\nabla_{\Sigma}\varphi|^{2}-(\mathrm{Ric}(\nu,\nu)+|A|^{2}-h^{2})\varphi^{2}+h(\zeta-2\nabla_{\hat{X}}\varphi+A(\hat{X},\hat{X})),\quad\text{and}$
(2.7) $\displaystyle c_{X,Z}$
$\displaystyle=\langle(\operatorname{div}_{\Sigma}\hat{X})\hat{X}-2\varphi
S(\hat{X})-\nabla_{\hat{X}}\hat{X}+\hat{Z},\eta\rangle,$

where $S$ is the shape operator and $\eta$ is the conormal in $\Sigma$.

###### Proof.

It is well known [39, 19] that the second variation of the area functional
alone is given by

$\delta_{X,Z}^{2}\mathcal{H}^{n-1}(\Sigma)=\int_{\Sigma}a_{X,Z}\,d\mathcal{H}^{n-1}.$

The first variation of the bulk term is given by [24]

$\delta_{X}\int_{U_{\Sigma}}h\,d\mathcal{H}^{n}=\int_{\Sigma}h\langle
X,\nu\rangle\,d\mathcal{H}^{n-1}.$

This shows that $\Sigma$ has mean curvature $h$. Varying this gives

$\int_{\Sigma}X(h)\langle X,\nu\rangle+h\langle Z,\nu\rangle+h\langle
X,\delta_{X}\nu\rangle-h\langle
X,\nu\rangle\langle\mathbf{H},X\rangle\,d\mathcal{H}^{n-1}.$

Since $h=0$ near $\partial\Sigma$, there is no boundary term involving $h$.
The integration by parts formula is also standard; a careful proof can be
found in [19]. ∎

We also recall

######  Lemma 2.4.

At points where $\hat{X}=0$, $\delta_{X}\nu=-\nabla_{\Sigma}\varphi$.

###### Proof.

Let $\delta_{X}\nu=\nabla_{\partial_{t}}\nu|_{t=0}$ be the variation. Using
$\langle\nu,\nu\rangle=1$ along the flow, it is easy to see that
$\langle\delta_{X}\nu,\nu\rangle=0$. A basis to the tangent space of
$\Sigma_{t}$ is given by $\partial_{x^{i}}F,$ where $x^{i}$ is a coordinate
system on $\Sigma$. We then compute, using the equality of mixed second
partials,

$\displaystyle\langle\partial_{x^{i}}F,\nabla_{\partial_{t}}\nu\rangle|_{t=0}$
$\displaystyle=-\langle\nabla_{\partial_{t}}\partial_{x^{i}}F,\nu\rangle|_{t=0}$
$\displaystyle=-\langle\nabla_{\partial_{x^{i}}}\partial_{t}F,\nu\rangle|_{t=0}$
$\displaystyle=-\langle\nabla_{i}X,\nu\rangle$
$\displaystyle=-\nabla_{i}\varphi.\qed$

For us, it is important that merely $\delta_{X}\nu\perp\Sigma$.

##  3\. Construction of the weight

In this section, we construct a general class of $\mu$-bubble weights in any
manifold admitting a structure of infinity. Let $U_{1}$ and $U_{2}$ be
neighborhoods of the distinguished infinity with compact closures in the
natural one-point compactification of $\mathcal{E}$ and such that
$\overline{U_{2}}\subset U_{1}$. We set $\rho=\mathrm{dist}(U_{2},\partial
U_{1})$, which is finite.

A model situation for this is the following: Let
$\mathcal{S}_{r}\subset\mathcal{E}$ be the coordinate sphere $|x|=r$. Consider
two radii $r_{1}<r_{2}$ and let $U_{i}$ be the set
$\\{x\in\mathcal{E}:|x|>r_{i}\\}$. Then $U_{1}\setminus\overline{U_{2}}$ is
the annulus $\\{r_{1}<|x|<r_{2}\\}$. In this case, $\rho$ is not necessarily
$r_{2}-r_{1}$, but see Remark 3.2.

We now state the main result of this section.

######  Proposition 3.1.

Let $(M^{n},g,\mathcal{E})$ possess a structure of infinity, let $U_{i}$ be as
above, and let $D>0$. For any $\varepsilon>0$ there exists a function $h\in
C^{0}(M,\overline{\mathbb{R}})$, smooth where it is finite, with the following
properties:

  1. (1)

$h=0$ in $U_{2}$,

  2. (2)

$h=-\infty$ outside the metric $D$-neighborhood of $U_{1}$, and

  3. (3)

$h$ satisfies the estimates

$h^{2}-2|\nabla_{g}h|\geq 0\quad\text{in $U_{2}\cup(M\setminus U_{1})$},$

and

$h^{2}-2|\nabla_{g}h|\geq-\sigma(\rho,D)+\eta(\varepsilon)\quad\text{in
$U_{1}\setminus U_{2}$},$

where

$\sigma(\rho,D)=\frac{32}{D}\left(\frac{8}{D}+\frac{1}{\rho}\right)$

In this context, we use the notation $\eta(\varepsilon)$ to simply mean a
quantity which tends to $0$ as $\varepsilon\to 0$.

######  Remark 3.2.

In an asymptotically flat manifold, the metric distance $\rho$ can be
estimated in terms of the coordinate distance $|x|$ using the pointwise
closeness of $g_{ij}$ to $\delta_{ij}$. It is easy to see then that as
$r_{i}\to\infty$, $\rho\to r_{2}-r_{1}$. Alternatively, we can define the
cutoff function below in terms of $|x|$, and then $\sigma$ will have to
involve an estimate for $|\nabla_{g}|x||$.

To prove Proposition 3.1 first recall the following elementary approximation
result.

######  Lemma 3.3.

Let $(M,g)$ be a compact Riemannian manifold and $f:M\to\mathbb{R}$ a
Lipschitz continuous function. For any $\varepsilon>0$ there exists a smooth
function $\overline{f}$ on $M$ such that
$\|f-\overline{f}\|_{W^{1,\infty}(M)}<\varepsilon.$ If $f$ is smooth on some
closed set $C$, we can further demand that $f=\overline{f}$ on $C$.

######  Corollary 3.4.

Let $U_{1}$ and $U_{2}$ be open sets in a Riemannian manifold $(M,g)$ with
compact closure and $\overline{U_{2}}\subset U_{1}$. Then there exists a
smooth function $\vartheta$, taking values in $[0,1]$, such that $\vartheta=0$
on $U_{2}$, $\vartheta=1$ on $U_{1}^{c}$, and

$|\nabla_{g}\vartheta|\leq\frac{2}{\mathrm{dist}_{g}(U_{2},\partial U_{1}).}$

###### Proof.

Apply the previous lemma to the distance function to $\partial U_{1}$ and
rescale. ∎

###### Proof of Proposition 3.1.

Our construction is motivated by consideration of solutions to the ODE
$2h^{\prime}=h^{2}$. Define

$\rho_{0}=\begin{cases}0&\text{in $U_{1}$}\\\ \mathrm{dist}_{g}(x,\partial
U_{1})&\text{in $M\setminus U_{1}$}\end{cases}.$

Let $\rho_{1}$ be a smoothing of this function with Lipschitz constant $2$. By
Lemma 3.3, we may assume $\rho_{1}$ to vanish in $U_{1}\setminus
N_{\varepsilon}(\partial U_{1})$, where $N_{\varepsilon}(\cdot)$ denotes the
metric $\varepsilon$-neighborhood. On $N_{\varepsilon}(\partial U_{1}),$
$\rho_{1}=\eta(\varepsilon).$ We also assume
$\sup_{M}|\rho_{0}-\rho_{1}|<\frac{1}{2}$.

Now define

$h_{1}=\frac{4}{\rho_{1}-\frac{D}{2}}.$

Note first that $h_{1}=-8/D$ in $U_{1}\setminus N_{\varepsilon}(\partial
U_{1})$ and $|h_{1}+8/D|=o(\varepsilon)$ in $N_{\varepsilon}(\partial U_{1})$.
Secondly, note that $h_{1}=-\infty$ in the region of distance $D$ away from
$U_{1}$ in $M\setminus U_{1}$. Thirdly, we compute

$|\nabla
h_{1}|=\left|\frac{-4}{(\rho_{1}-\frac{D}{2})^{2}}\nabla\rho_{1}\right|\leq\frac{8}{(\rho_{1}-\frac{D}{2})^{2}}=\frac{h^{2}_{1}}{2}.$

Next, we define $h=\vartheta h_{1}$, where $\vartheta$ is as in Corollary 3.4
and compute (in the region where $\vartheta$ is variable, i.e. $U_{1}\setminus
U_{2}$)

$\displaystyle h^{2}-2|\nabla h|$
$\displaystyle=\vartheta^{2}h^{2}_{1}-2|\vartheta\nabla
h_{1}+h_{1}\nabla\vartheta|$ $\displaystyle\geq-2\vartheta|\nabla
h_{1}|-2|h_{1}||\nabla\vartheta|$
$\displaystyle\geq-4\left(\frac{8}{D}+\eta(\varepsilon)\right)^{2}-2\left(\frac{8}{D}+\eta(\varepsilon)\right)\frac{2}{\rho}$
$\displaystyle=-\sigma(\rho,D)+\eta(\varepsilon).\qed$

######  Remark 3.5.

By making a small perturbation of $h$, leaving the necessary inequalities
satisfied, we can ensure $\partial\\{|h|<\infty\\}$ is a compact smooth
hypersurface without boundary. We make this assumption.

We see that if $R_{g}>0$ everywhere and $R_{g}>\sigma(\rho,D)$ on
$\overline{U_{1}}\setminus U_{2}$, then

(3.1) $R_{g}+h^{2}-2|\nabla_{g}h|>0$

for $\varepsilon$ sufficiently small, which we call the _(strict) $\mu$-bubble
condition_ for $h$. If merely

(3.2) $R_{g}+h^{2}-2|\nabla_{g}h|\geq 0,$

then $h$ satisfies the _weak $\mu$-bubble condition_.

##  4\. Proof of Theorem 1.6 with an extra assumption

###  4.1. Construction of $\Sigma_{\infty}$

We single out a coordinate $x^{n}$ on $\mathcal{E}$, split the coordinates as
$x=(x^{\prime},x^{n})$, and define

$\displaystyle Z_{\sigma}$ $\displaystyle=\\{x:|x^{\prime}|\leq\sigma\\}$
$\displaystyle C_{\sigma}$
$\displaystyle=\partial(Z_{\sigma}\cap\\{x:|x^{n}|\leq a_{0}\\})$
$\displaystyle\Gamma_{\sigma,a}$
$\displaystyle=\\{x:|x^{\prime}|=\sigma,x^{n}=a\\},$

where $a_{0}>0$ is a constant that will be chosen momentarily. Clearly
$Z_{\sigma}$ is mean convex for $\sigma$ sufficiently large and it is well
known that the coordinate “hyperplanes” $\\{x^{n}=a_{0}\\}$ are strictly mean
convex for sufficiently large $a_{0}$ under the assumption $m<0$. We prove
this in Lemma 4.14 below. We choose $a_{0}$ so that this is true. It follows
that $C_{\sigma}$ is a mean convex hypersurface (with corners that can be
smoothed out in a mean convex fashion).

Furthermore, we choose $P_{\sigma,a}$ to be the bottom “hemisphere” of
$C_{\sigma}\setminus\Gamma_{\sigma,a}$. Let $h$ be the function constructed in
Proposition 3.1. By construction, $h|_{C_{\sigma}}=0$. Let $\Sigma_{\sigma,a}$
be the hypersurface given by Theorem 2.1 with these choices. Let
$\mathcal{F}_{\sigma}$ be the functional defined by (2.1).

######  Lemma 4.1.

Fix $\sigma$. Then $a\mapsto\mathcal{F}_{\sigma}(\Sigma_{\sigma,a})$ is
Lipschitz continuous for $|a|\leq a_{0}$.

###### Proof.

Let $a_{1}<a_{2}$ and denote $\Sigma_{i}=\Sigma_{\sigma,a_{i}}$. Define a
competitor surface by

$\tilde{\Sigma}=\Sigma_{2}\cup\bigcup_{s=a_{1}}^{a_{2}}\Gamma_{\sigma,s}.$

(To be rigorous, one should push this in slightly so it lies in the interior,
and this only adds an $\varepsilon$ to the following inequalities that can be
sent to zero in the end, so we will ignore it.) Since $\Sigma_{1}$ is
minimizing and $\partial\tilde{\Sigma}=\Gamma_{\sigma,a_{1}}$,

$\mathcal{F}_{\sigma}(\Sigma_{1})\leq\mathcal{F}_{\sigma}(\tilde{\Sigma})=\mathcal{F}_{\sigma}(\Sigma_{2})+\mathcal{H}^{n-1}\left(\bigcup_{h=a_{1}}^{a_{2}}\Gamma_{\sigma,h}\right).$

From asymptotic flatness,

$\mathcal{H}^{n-1}\left(\bigcup_{s=a_{1}}^{a_{2}}\Gamma_{\sigma,s}\right)\lesssim|a_{2}-a_{1}|.$

Interchanging $a_{1}$ and $a_{2}$ and running the same argument gives
altogether

$|\mathcal{F}_{\sigma}(\Sigma_{1})-\mathcal{F}_{\sigma}(\Sigma_{2})|\lesssim|a_{2}-a_{1}|.\qed$

Figure 2.

######  Lemma 4.2.

$F(\sigma)=\min\\{F_{\sigma}(\Sigma_{\sigma,a}):|a|\leq a_{0}\\}$ is achieved
by an $a\neq\pm a_{0}$.

###### Proof.

Any surface with boundary $\Gamma_{\sigma,\pm a_{0}}$ can be pushed up or down
slightly to decrease its area, see the original proof of Schoen [28]. Since
our functionals $\mathcal{F}_{\sigma}$ are just the area functional in a
neighborhood of $C_{\sigma}$, this argument carries through to decrease
$\mathcal{F}_{\sigma}$. ∎

If $a(\sigma)$ is the minimizing $a$ produced by the previous lemma, we set
$\Sigma_{\sigma}=\Sigma_{\sigma,a(\sigma)}.$ We now study the limit
$\sigma\to\infty$.

It is now convenient to introduce some more notation. Let $H=\\{x^{n}=0\\}$ be
a hyperplane in $\mathbb{R}^{n}$ and $p:\mathbb{R}^{n}\to H$ the orthogonal
projection. For the rest of this section we delete from $M$ everything behind
one of the surfaces $S_{-}^{\tau}$ in the proof of Theorem 2.1. The point is
that each $\Sigma_{\sigma,a}$ we consider lies in this region, which is
compact if we one-point compactify $\mathcal{E}$. Let $R_{0}$ be a radius so
that $h=0$ for any $x\in\mathcal{E}$ with $|x|\geq R_{0}$. We define
$\mathcal{E}^{\prime}=\mathcal{E}\cap\\{|x^{\prime}|\geq R_{0}\\}$, so
$\mathcal{E}^{\prime}$ is the end minus a cylindrical region containing the
set where $h$ is variable. We can clearly define $p:\mathcal{E}^{\prime}\to
H^{\prime}:=H\setminus B_{R_{0}}(0)$, where we now view $H^{\prime}$ as a
submanifold of $M$.

######  Lemma 4.3.

The orthogonal projection $p:\Sigma\cap\\{R_{0}<|x^{\prime}|<\sigma\\}\to
H^{\prime}\cap\\{R_{0}<|x^{\prime}|<\sigma\\}$ is surjective.

###### Proof.

Delete from $\Sigma_{\sigma}$ everything outside of $\mathcal{E}^{\prime}$;
denote the resulting current by $\Sigma^{\prime}$. Note that
$\partial\Sigma^{\prime}=\Gamma_{\sigma,a(\sigma)}$ as currents in
$\mathcal{E}^{\prime}$. Let $T$ be the multiplicity 1 current in
$\mathcal{E}^{\prime}$ whose underlying set is given by
$\\{(x^{\prime},x^{n}):|x^{\prime}|\geq\sigma,x^{n}=a(\sigma)\\}.$ Then
$\partial(T+\Sigma^{\prime})=0$ and

$\partial p_{\\#}(T+\Sigma^{\prime})=p_{\\#}\partial(T+\Sigma^{\prime})=0.$

By the constancy theorem [39, Theorem 26.27], there exists $c\in\mathbb{R}$
such that

$p_{\\#}(T+\Sigma^{\prime})=c\llbracket H^{\prime}\rrbracket.$

Looking at radii larger than $\sigma$ implies $c=1$. Now it is easy to see
that if $\Sigma^{\prime}$ cannot miss some fiber $p^{-1}(x^{\prime})$ with
$|x^{\prime}|<\sigma$. ∎

###  4.2. Asymptotics of $\Sigma_{\infty}$

We work with the following functional spaces.

######  Definition 4.4 (Weighted Hölder spaces).

Let $\mathcal{E}$ be a Euclidean end of a manifold $(M^{n},g)$ admitting a
structure of infinity as per Definition 1.9. For $k\in\mathbb{N}_{0}$,
$\alpha\in(0,1)$, and $\tau\in\mathbb{R}$, we define the _weighted Hölder
space_ $C^{k,\alpha}_{\tau}(\mathcal{E})$ as the set of all functions $u\in
C^{k,\alpha}_{\mathrm{loc}}$ for which the norm

$\|u\|_{C^{k,\alpha}_{\tau}}=\sum_{i=0}^{k}\sup_{x\in\mathcal{E}}||x|^{i+\tau}\partial^{i}u|+\sup_{x\in\mathcal{E}}|x|^{k+\alpha+\tau}[\partial^{k}u]_{C^{\alpha}(B_{1}(x))}$

is finite. Note that with our notation, $\tau>0$ corresponds to _decay_ ,
which is opposite from [19]. Under pointwise multiplication,
$C^{2,\alpha}_{\mu}\cdot C^{2,\alpha}_{\nu}\subset C^{2,\alpha}_{\mu+\nu}$,
continuously. We will frequently use the fact that $|x|^{-\tau}\in
C^{k,\alpha}_{\tau}$ for any $k$ and $\alpha$. By definition,
$\partial(C^{k,\alpha}_{\tau})\subset C^{k-1,\alpha}_{\tau+1}$ continuously.

Finally, we use the notation $A*B$ to denote contractions and
$\mathbb{R}$-linear combinations of quantities. The decay rates can be easily
read off from this notation.

Now we study the convergence of the $\Sigma_{\sigma}$’s and the asymptotic
behavior of the limiting hypersurface.

######  Theorem 4.5.

There exists a sequence $\sigma_{j}\to\infty$ such that $\Sigma_{\sigma_{j}}$
converges in $C^{\infty}_{\mathrm{loc}}$ to a complete, smooth hypersurface
$\Sigma_{\infty}$ with mean curvature $h$. Furthermore, there exist constants
$a_{\infty},c_{\infty}$ such that for any $\varepsilon\in(0,1)$,
$\Sigma_{\infty}\cap\mathcal{E}^{\prime}$ can be written as a Euclidean graph
$x^{n}=u(x^{\prime})$ for some smooth function $u$ satisfying

$\displaystyle u(x^{\prime})$ $\displaystyle-a_{\infty}\in
C^{3,\alpha}_{\varepsilon}(H^{\prime})\quad\text{$n=3$},$ $\displaystyle
u(x^{\prime})$
$\displaystyle-\left(a_{\infty}+\frac{c_{\infty}}{|x^{\prime}|^{n-3}}\right)\in
C^{3,\alpha}_{n-3+\varepsilon}(H^{\prime})\quad\text{$n\geq 4$},$

where now $\alpha$ is the minimum of the $\alpha$ in Definition 1.9 and
$\frac{1}{n}$.

The proof occupies the rest of this section. The locally smooth convergence of
the hypersurfaces $\Sigma_{\sigma_{j}}$ for some sequence
$\sigma_{j}\to\infty$ follows from standard theory [7, 24, 39] as soon as we
note that a local mass bound, follows from the fact that it minimizes a
prescribed mean curvature-type functional in a compact subset of the core and
the area functional in the asymptotic region. Strictly speaking, we must also
show that the mass is locally bounded below so that $\Sigma_{\infty}\neq 0$.
This follows straightforwardly from Lemma 4.3.

The following is the closest we can get to harmonic asymptotics without a
density theorem. It says that we still have conformal flatness modulo strictly
lower order terms, which is crucial for our analysis.

######  Proposition 4.6.

Any asymptotically Schwarzschild metric $g$ can be written in the form

$g=\psi^{\frac{4}{n-2}}\overline{g},$

where

$\psi(x)=1+\frac{m}{2|x|^{n-2}}$

is the conformal factor of the Schwarzschild metric and

$\overline{g}_{ij}-\delta_{ij}\in C^{2,\alpha}_{n-1}.$

With respect to these metrics, we have the following decay rates for
components of various quantities in the asymptotic coordinate system:

| $g$ | $\overline{g}$
---|---|---
$\Delta\psi$ | $C^{1,\alpha}_{2n-2}$ | $C^{1,\alpha}_{2n-1}$
$\Gamma$ | $C^{1,\alpha}_{n-1}$ | $C^{1,\alpha}_{n}$
$\mathrm{Rm}$ | $C^{0,\alpha}_{n}$ | $C^{0,\alpha}_{n+1}$
$\mathrm{Rc}$ | $C^{0,\alpha}_{n}$ | $C^{0,\alpha}_{n+1}$
$R$ | $C^{0,\alpha}_{n+1}$ | $C^{0,\alpha}_{n+1}$

######  Remark 4.7.

$R_{g}\in C^{0,\alpha}_{n+1}$ implies $R_{g}\in L^{1}(\mathcal{E}).$ This
improved decay is not true for general asymptotically flat metrics of order
$n-2$.

###### Proof.

By assumption, $g_{ij}-\psi^{\frac{4}{n-2}}\delta_{ij}\in C^{2,\alpha}_{n-1},$
so

$\overline{g}_{ij}-\delta_{ij}=\psi^{-\frac{4}{n-2}}g_{ij}-\delta_{ij}\in\psi^{-\frac{4}{n-2}}\cdot
C^{2,\alpha}_{n-1}\subset C^{2,\alpha}_{n-1}.$

To compute $\Delta\psi$, we compute $\Delta|x|^{-(n-2)}$ explicitly to leading
order. The sign of this leading term will be particularly useful in Section 5.
Using elementary matrix expansions, we have

$\displaystyle g_{ij}$
$\displaystyle-\delta_{ij}-\frac{2m}{n-2}|x|^{-(n-2)}\delta_{ij}\in
C^{2,\alpha}_{n-1},$ $\displaystyle g^{ij}$
$\displaystyle-\delta^{ij}+\frac{2m}{n-2}|x|^{-(n-2)}\delta^{ij}\in
C^{2,\alpha}_{n-1},$ $\displaystyle\det({g_{ij}})$
$\displaystyle-1-\frac{2nm}{n-2}|x|^{-(n-2)}\in C^{2,\alpha}_{n-1},$
$\displaystyle\sqrt{\det(g_{ij})}$
$\displaystyle-1-\frac{nm}{n-2}|x|^{-(n-2)}\in C^{2,\alpha}_{n-1}.$

Multiplying the second and fourth lines gives

$\sqrt{\det(g_{ij})}g^{ij}-\delta_{ij}-m|x|^{-(n-2)}\delta_{ij}\in
C^{2,\alpha}_{n-1}.$

By the useful formula $\partial_{j}|x|^{-p}=-p|x|^{-(p+2)}x^{j}$, we compute

(4.1) $\displaystyle\Delta_{g}|x|^{-(n-2)}$ $\displaystyle=\frac{1}{\sqrt{\det
g}}\sum_{ij}\partial_{i}\left(\sqrt{\det
g}g^{ij}\partial_{j}|x|^{-(n-2)}\right)$
$\displaystyle=\frac{-(n-2)}{\sqrt{\det
g}}\sum_{ij}\partial_{i}\left(|x|^{-n}x^{i}+m|x|^{-(2n-2)}x^{i}+C^{2,\alpha}_{2n-2}\right)$
$\displaystyle=m(n-2)^{2}|x|^{-(2n-2)}+C^{1,\alpha}_{2n-1}.$

The same calculation for $\overline{g}$ gives
$\Delta_{\overline{g}}|x|^{-(n-2)}\in C^{1,\alpha}_{2n-1}$.

The decay rates of the Christoffel symbols can be computed from the schematic
equation $\Gamma=g^{-1}*\partial g$ and for the Riemann and Ricci curvatures,
from $\mathrm{Rm}\sim\mathrm{Rc}=\partial\Gamma+\Gamma*\Gamma$. This also
yields a decay rate $R_{g}\in C^{0,\alpha}_{n}$ and $R_{\overline{g}}\in
C^{0,\alpha}_{n+1}$. We now improve the decay rate for $R_{g}$ using the
conformal structure. Using the standard formula for scalar curvature under a
conformal change (where $L$ is the conformal Laplacian),

$\displaystyle R_{g}$
$\displaystyle=\psi^{-\frac{n+2}{n-2}}L_{\overline{g}}\psi$
$\displaystyle=C^{2,\alpha}_{0}\cdot\left(-4\frac{n-1}{n-2}\Delta_{\overline{g}}\psi+R_{\overline{g}}\psi\right)$
$\displaystyle=C^{2,\alpha}_{0}\cdot(C^{1,\alpha}_{2n-1}+C^{0,\alpha}_{n+1}\cdot
C^{0,\alpha}_{0})\in C_{n+1}^{0,\alpha}.\qed$

We have an initial curvature estimate for $\Sigma_{\infty}$. This will
ultimately be improved in Lemma 4.19 below.

######  Lemma 4.8.

The asymptotically minimal hypersurface $\Sigma_{\infty}$ enjoys the curvature
decay

$|A|=O(|x^{\prime}|^{-1})$

with respect to the metric $g$.

###### Proof.

After rescaling, such a curvature estimate follows from work of Schoen–Simon,
see either [29] or [30]. To apply [29], we note that the injectivity radius is
controlled from below by [3] and the closeness of the metric to flat. In
general, it is not possible to obtain anything better than $C^{0,\alpha}$
control on the metric components in normal coordinates given a $C^{2,\alpha}$
bound [6]. However, we can introduce $C^{2,\alpha}$ harmonic coordinates about
every point of interest and these will satisfy the assumptions (1.2)-(1.6) in
[29].

We give another proof of this result in Appendix A via a “universal” curvature
estimate for area minimizing hypersurfaces in manifolds with bounded sectional
curvature. ∎

######  Lemma 4.9 (Comparison lemma [10, Lemma 2.4]).

In an asymptotically Schwarzschild manifold, there exists a constant $C=C(g)$
with the following property. For any hypersurface $\Sigma$, the second
fundamental forms relative to $g$ and $\delta$ satisfy

$|A_{g}-A_{\delta}|_{g}\leq C|x|^{-(n-1)}(|x||A_{g}|_{g}+1).$

For the mean curvature, we have the comparison

$|H_{g}-H_{\delta}|\leq C|x|^{-(n-1)}(|x||A_{g}|_{g}+1).$

For the proof, we require a simple lemma comparing the norms defined by the
metrics $g$ and $\delta$.

######  Lemma 4.10.

In an asymptotically Schwarzschild manifold, the norms defined by $g$ and
$\delta$ satisfy

$||X|_{g}-|X|_{\delta}|\lesssim|x|^{-(n-2)}|X|_{\delta}\lesssim|x|^{-(n-2)}|X|_{g},$

where $X$ is any tangent vector.

###### Proof.

From the decay of $g_{\ij}-\delta_{ij}$,

$\left|\frac{|X|^{2}_{g}}{|X|^{2}_{\delta}}-1\right|\lesssim|x|^{-(n-2)},$

Factoring the left-hand side and dividing through gives

$\left|\frac{|X|_{g}}{|X|_{\delta}}-1\right|\lesssim|x|^{-(n-2)}\left(\frac{|X|_{g}}{|X|_{\delta}}+1\right)^{-1},$

Clearly

$\frac{|X|_{g}}{|X|_{\delta}}\approx 1,$

so that

$\left|\frac{|X|_{g}}{|X|_{\delta}}-1\right|\lesssim|x|^{-(n-2)},$

which implies the claim. ∎

To prove Lemma 4.9, we use the formulas for a hypersurface $\Sigma$ defined as
a level set $\\{\zeta=0\\}$ for some smooth function $\zeta$ satisfying
$d\zeta\neq 0$. It is standard [26] that the normal with respect to $g$ is
given by

(4.2) $\nu_{g}=\frac{\nabla_{g}\zeta}{|d\zeta|_{g}},$

the second fundamental form is given by

(4.3) $A_{g}(X,Y)=\frac{\nabla^{2}_{g}\zeta(X,Y)}{|d\zeta|_{g}},\quad X,Y\in
T\Sigma,$

and the mean curvature is given by

(4.4) $H_{g}=\Pi^{ij}_{g}\frac{(\nabla^{2}_{g}\zeta)_{ij}}{|d\zeta|_{g}},$

where

(4.5) $\Pi_{g}=g^{-1}-\nu_{g}\otimes\nu_{g}$

is the projection matrix $T^{*}M\to T\Sigma$. We emphasize the dependence on
the metric in these formulas because we deal with three different metrics in
this section.

###### Proof of Lemma 4.9.

Consider $\Sigma$ as a level set $\\{\zeta=0\\}$. For $X$ and $Y$ tangent to
to $\Sigma$, we have

$(A_{g}-A_{\delta})(X,Y)=\left(\frac{1}{|d\zeta|_{g}}-\frac{1}{|d\zeta|_{\delta}}\right)\partial_{ij}\zeta
X^{i}Y^{j}-\Gamma^{k}_{ij}\frac{\partial_{k}\zeta}{|d\zeta|_{g}}X^{i}Y^{j}.$

Using Lemma 4.10,

$\left|\frac{1}{|d\zeta|_{g}}-\frac{1}{|d\zeta|_{\delta}}\right|=\frac{||d\zeta|_{\delta}-|d\zeta|_{g}|}{|d\zeta|_{g}|d\zeta|_{\delta}}\lesssim\frac{|x|^{-(n-2)}}{|d\zeta|_{g}}.$

Now we estimate

$\displaystyle|(A_{g}-A_{\delta})(X,Y)|$ $\displaystyle\leq
C|x|^{-(n-2)}\left|\frac{\partial_{ij}\zeta
X^{i}Y^{j}}{|d\zeta|_{g}}\right|+C|\Gamma||X|_{g}||Y|_{g}$
$\displaystyle\lesssim|x|^{-(n-2)}\left|A_{g}(X,Y)+\Gamma^{k}_{ij}\frac{\partial_{k}\zeta}{|d\zeta|_{g}}X^{i}Y^{j}\right|+|x|^{-(n-1)}|X|_{g}|Y|_{g}$
$\displaystyle\lesssim|x|^{-(n-2)}|A_{g}(X,Y)|+(|x|^{-(n-1)}+|x|^{-n(n-1)})|X|_{g}|Y|_{g}$
$\displaystyle\lesssim|x|^{-(n-1)}(|x||A_{g}(X,Y)|+|X|_{g}|Y|_{g}),$

which completes the proof. ∎

######  Corollary 4.11.

The mean curvature of $\Sigma_{\infty}$ is $O(|x^{\prime}|^{-(n-1)})$ relative
to the Euclidean metric.

###### Proof.

$H_{g}=0$ for $\Sigma_{\infty}$ on $\mathcal{E}$ and $|x||A_{g}|_{g}$ is
bounded. ∎

This decay on $H_{\delta}$ enables the use of Allard’s regularity theorem.

######  Proposition 4.12.

Let $\Sigma_{\infty}$ be the asymptotically minimal hypersurface considered in
Proposition 4.5. Choosing $R_{0}$ perhaps larger, we can write
$\Sigma_{\infty}$ as the graph of a function $u\in
C^{1,\gamma}_{0}(H^{\prime})$, where $\gamma=\frac{1}{n}$.

###### Proof.

We apply Allard’s regularity theorem in the form [39, Theorem 23.1] with $p=n$
and $T=\\{x^{n}=0\\}\cong\mathbb{R}^{n-1}$. The $L^{n}$ norm of the mean
curvature is estimated by

$\left(\int_{B_{\rho}(x)\cap\Sigma_{\infty}}|H_{\delta}|^{n}\,d\mathcal{H}^{n-1}\right)^{\frac{1}{n}}\rho^{\frac{1}{n}}\lesssim\rho^{-(n-2)}\leq\varepsilon_{n},$

where $\varepsilon_{n}$ is the constant appearing in Allard’s theorem. Here we
use $|H_{\delta}|\lesssim|x^{\prime}|^{-(n-1)}$, which is crucial. The excess
is estimated using [39, Lemma 22.2] and the height bound, which implies
$|x|\lesssim|x^{\prime}|$ on $\Sigma_{\infty}$. For details, see [18, 4.2.2].
∎

Now the strategy is to write $H_{g}=0$ as a prescribed curvature equation in
Euclidean space and to bootstrap this decay rate. We define
$\zeta(x^{\prime},x^{n})=u(x^{\prime})-x^{n}$, so that
$\Sigma_{\infty}=\\{\zeta=0\\}$. It is convenient to use Greek indices running
from $1$ to $n$ and Latin from $1$ to $n-1$. We see immediately that
$\partial_{\mu}\zeta=(\partial_{i}u,-1)$ and $\partial_{\mu\nu}\zeta$ is only
nonzero when $\mu\nu=ij$, in which case it equals $\partial_{ij}u$. Note that
with this definition, the normal coming from (4.2) points down, but this is of
no consequence since $H_{g}=0$ is independent of this choice.

######  Proposition 4.13.

The function $u$ satisfies

(4.6)
$\Pi_{\delta}^{ij}\partial_{ij}u=B^{ij}\partial_{ij}u+\Pi_{\overline{g}}*\Gamma_{\overline{g}}*\partial\zeta-2\frac{n-1}{n-2}\overline{g}^{\mu\nu}\partial_{\mu}\zeta\partial_{\nu}\log\psi,$

where, schematically,

$B=p+\left(\frac{1}{|d\zeta|^{2}_{\overline{g}}}-\frac{1}{|d\zeta|_{\delta}^{2}}\right)\partial
u*\partial
u+\frac{p*\partial\zeta*\partial\zeta}{|d\zeta|_{\overline{g}}^{2}}+\frac{p*p*\partial\zeta*\partial\zeta}{|d\zeta|^{2}_{\overline{g}}},$

and $p^{\mu\nu}=\overline{g}^{\mu\nu}-\delta^{\mu\nu}$.

###### Proof.

Using the conformal transformation properties of mean curvature [19, page 33],
we get

$H_{\overline{g}}=-2\frac{n-1}{n-2}\,\overline{\nu}(\log\psi),$

since $H_{g}=0.$ Using equations (4.2) and (4.4), we get

$\Pi_{\overline{g}}^{\mu\nu}(\partial_{\mu\nu}\zeta-\Gamma_{\overline{g}\,\mu\nu}^{\kappa}\partial_{\kappa}\zeta)=-2\frac{n-1}{n-2}\operatorname{grad}_{\overline{g}}\zeta(\log\psi)$

To obtain our conclusion, we simply have to compute
$\Pi_{\overline{g}}^{\mu\nu}\partial_{\mu\nu}\zeta-\Pi^{\mu\nu}_{\delta}\partial_{\mu\nu}\zeta$:

$\displaystyle\Pi^{ij}_{\overline{g}}\partial_{ij}u-\Pi_{\delta}^{ij}\partial_{ij}u$
$\displaystyle=\left(\delta^{ij}+p^{ij}-\frac{(\delta^{i\alpha}+p^{i\alpha})(\delta^{j\beta}+p^{j\beta})\partial_{\alpha}\zeta\partial_{\beta}\zeta}{|d\zeta|_{\overline{g}}^{2}}\right)\partial_{ij}u$
$\displaystyle\quad-\left(\delta^{ij}-\frac{\delta^{i\alpha}\delta^{j\beta}\partial_{\alpha}\zeta\partial_{\beta}\zeta}{|d\zeta|_{\delta}^{2}}\right)\partial_{ij}u$
$\displaystyle=p^{ij}\partial_{ij}u-\left(\frac{1}{|d\zeta|^{2}_{\overline{g}}}-\frac{1}{|d\zeta|_{\delta}^{2}}\right)\delta^{i\alpha}\delta^{j\beta}\partial_{\alpha}\zeta\partial_{\beta}\zeta\partial_{ij}u+\frac{p*\partial\zeta*\partial\zeta}{|d\zeta|_{\overline{g}}^{2}}*\partial^{2}u$
$\displaystyle\quad+\frac{p*p*\partial\zeta*\partial\zeta}{|d\zeta|^{2}_{\overline{g}}}*\partial^{2}u$
$\displaystyle=\left(p+\left(\frac{1}{|d\zeta|^{2}_{\overline{g}}}-\frac{1}{|d\zeta|_{\delta}^{2}}\right)\partial
u*\partial
u+\frac{p*\partial\zeta*\partial\zeta}{|d\zeta|_{\overline{g}}^{2}}+\frac{p*p*\partial\zeta*\partial\zeta}{|d\zeta|^{2}_{\overline{g}}}\right)*\partial^{2}u.$

∎

Applying these computational tricks to the function $\zeta(x)=x^{n}$ gives

######  Lemma 4.14.

In an asymptotically Schwarzschild manifold with negative mass, there exists
an $a_{0}>0$ such that for any $a\geq a_{0}$, the “hyperplane” $\\{x^{n}=a\\}$
is strictly mean convex with respect to the upward normal and $\\{x^{n}=-a\\}$
is strictly mean convex with respect to the downward normal.

###### Proof.

Using the above equations, and computing relative to the upward normal, the
mean curvature of a constant $x^{n}$ hyperplane is given by

$H_{g}=-(n-1)m\frac{x^{n}}{|x|^{n}}+O(|x|^{-n}).\qed$

We now redefine $\alpha$ to be the minimum of the original $\alpha$ and
$\frac{1}{n}$. So by Proposition 4.12, $u\in C^{1,\alpha}_{0}$. The following
is very useful (it will be used implicitly in almost every single line later)
and is easy to prove.

######  Lemma 4.15 (_Hilfsaussage_ [18, page 82]).

Assume $u\in C^{1,\alpha}_{0}(H^{\prime})$. Given a function $f\in
C^{2}_{\tau}(\mathcal{E})$, its restriction to the graph of $u$,
$\tilde{f}(x^{\prime})=f(x^{\prime},u(x^{\prime}))$, lies in
$C^{1,\alpha}_{\tau}(H^{\prime})$. The same holds for $f\in
C^{1,\alpha}_{\tau}(\mathcal{E})$.

######  Lemma 4.16.

The assumption $u\in C^{1,\alpha}_{0}$ implies the following decay rates:

  1. (1)

$\partial\zeta\in C^{0,\alpha}_{0}$

  2. (2)

$\frac{1}{|d\zeta|^{2}_{\overline{g}}}\in C^{0,\alpha}_{0}$

  3. (3)

$\frac{1}{|d\zeta|^{2}_{\overline{g}}}-\frac{1}{|d\zeta|_{\delta}^{2}}\in
C^{0,\alpha}_{n-1}$

  4. (4)

$\delta^{ij}-\Pi_{\delta}^{ij}\in C^{0,\alpha}_{2}$

  5. (5)

$\Pi_{\overline{g}}\in C^{0,\alpha}_{0}$

  6. (6)

$\Gamma_{\overline{g}}(x^{\prime},u(x^{\prime}))\in C^{1,\alpha}_{n}$

###### Proof.

(1) follows from Proposition 4.12. (2) follows (1) and the decay of the
metric. (3) follows from the simple calculation

$\frac{1}{|d\zeta|_{\overline{g}}^{2}}-\frac{1}{|d\zeta|^{2}_{\delta}}=\frac{|d\zeta|_{\delta}^{2}-|d\zeta|^{2}_{\overline{g}}}{|d\zeta|_{\delta}^{2}|d\zeta|^{2}_{\overline{g}}}=-\frac{p(d\zeta,d\zeta)}{|d\zeta|_{\delta}^{2}|d\zeta|^{2}_{\overline{g}}}\in
C^{0,\alpha}_{n-1}.$

(4) follows from the definition,

$\delta^{ij}-\Pi_{\delta}^{ij}=\frac{\partial_{i}u\partial_{j}u}{1+|\partial
u|^{2}},$

and $\partial u\in C^{0,\alpha}_{1}$. Points (5) and (6) follow using these
same ideas and the _Hilfsaussage_. ∎

To obtain the desired asymptotic expansion for $u$, we use the following
theorem of Meyers (for a proof, see [19, Appendix A]).

######  Theorem 4.17 (Meyers [25]).

Let $(M^{n-1},g)$ be an asymptotically flat manifold. Suppose $Lu=f$, where
$u$ is a bounded smooth function and
$Lu=a^{ij}\partial^{2}_{ij}u+b^{i}\partial_{i}u+cu$ is a linear elliptic
operator satisfying the standard conditions $a^{ij}-\delta^{ij}\in
C^{k,\alpha}_{\varepsilon}$, $b^{i}\in C^{k,\alpha}_{1+\varepsilon}$, $c\in
C^{k,\alpha}_{n-1+\varepsilon}$, and $f\in C^{k,\alpha}_{n-1+\varepsilon}$ for
some $\varepsilon\in(0,1)$. Then there exist constants $h_{\infty},c_{\infty}$
such that

$\displaystyle u(x)$ $\displaystyle-a_{\infty}\in
C^{k+2,\alpha}_{\varepsilon}\quad\text{$n=3$},$ $\displaystyle u(x)$
$\displaystyle-\left(a_{\infty}+\frac{c_{\infty}}{|x|^{n-3}}\right)\in
C^{k+2,\alpha}_{n-3+\varepsilon}\quad\text{$n\geq 4$}.$

We emphasize that the theorem is stated here for $(n-1)$-dimensional
manifolds. We can finally prove the main result of this section.

###### Proof of Proposition 4.5.

We rewrite (4.6) in the form

(4.7)
$(\Pi_{\delta}^{ij}-B^{ij})\partial_{ij}u+(\Pi_{\overline{g}}*\Gamma_{\overline{g}}+\partial\log\psi)*\partial
u=\Pi_{\overline{g}}*\Gamma_{\overline{g}}+\overline{g}*\partial_{n}\log\psi,$

neglecting numerical constants multiplying the various terms. Comparing with
Meyers’ theorem, the only thing that needs to be checked is that
$\partial_{n}\log\psi\in C^{1,\alpha}_{n-1+\varepsilon}$. Naively, we only get
a decay rate of $n-1$, but the $x^{n}$ derivative enjoys extra decay when
restricted to the graph:

$\partial_{n}\psi=c\frac{u}{|x|^{n}}\in C^{1,\alpha}_{n}.$

Applying Myers’ theorem with $k=0$ gives a constant $a_{\infty}$ so that
$v:=u-a_{\infty}\in C^{2,\alpha}_{n-3}$. To improve the decay further we wish
to use the theorem with $k=1$, but this requires improving the decay of the
coefficients first. It is easy to see that the (newfound) decay of
$\partial^{2}u$ now gives $C^{1,\alpha}$ decay for every quantity in Lemma
4.16 which involves $u$. So we may apply Meyers’ theorem with $k=1$ and
conclude our result. ∎

######  Corollary 4.18.

When $n\geq 4$, the induced metric $\gamma$ on $\Sigma_{\infty}$ is
asymptotically flat with zero mass. More precisely, the projection
$\Sigma_{\infty}\to H^{\prime}$ induces coordinates $y^{i}$ on
$\Sigma_{\infty}$ so that

$\gamma_{ij}=\delta_{ij}+C^{2,\alpha}_{n-2}(H^{\prime}).$

###### Proof.

We use the standard graphical coordinates $y^{i}$ associated to
$x^{\prime}\mapsto(x^{\prime},u(x^{\prime}))$. The coordinate vector fields
are given by

$\frac{\partial}{\partial y^{i}}=\frac{\partial}{\partial
x^{i}}+\frac{\partial u}{\partial x^{i}}\frac{\partial}{\partial x^{n}},$

so we compute

$\displaystyle\gamma(\partial_{y^{i}},\partial_{y^{j}})$
$\displaystyle=g(\partial_{y^{i}},\partial_{y^{j}})$
$\displaystyle=\psi^{\frac{4}{n-2}}(x^{\prime},u(x^{\prime}))\left(\delta_{ij}+\partial_{i}u\partial_{j}u+(\overline{g}-\delta)(\partial_{y^{i}},\partial_{y^{j}})\right)$
$\displaystyle=(1+C^{2,\alpha}_{n-2})^{\frac{4}{n-2}}\left(\delta_{ij}+C^{2,\alpha}_{n-2}\cdot
C^{2,\alpha}_{n-2}+C^{2,\alpha}_{n-1}\right)$
$\displaystyle=\delta_{ij}+C^{2,\alpha}_{n-2}.$

Now the ADM mass integrand 1.5 associated to $\gamma$ decays like
$O(|y|^{-(n-1)})$, whereas the coordinate spheres have area growing like
$O(|y|^{n-2})$ (recall $\dim\Sigma_{\infty}=n-1$), so the mass is zero. ∎

We remark for later that Theorem 4.5 gives improved decay of the second
fundamental form.

######  Lemma 4.19.

The asymptotically minimal hypersurface $\Sigma_{\infty}$ enjoys the curvature
decay

$|A|=O(|x^{\prime}|^{-(n-1)})$

with respect to the metric $g$.

###### Proof.

By Lemma 4.9 and Lemma 4.8, it suffices to show that
$|A_{\delta}|=O(|x^{\prime}|^{-(n-1)})$. Since $\partial u$ is bounded,
$|A_{\delta}|\approx|\partial^{2}u|=O(|x^{\prime}|^{-(n-1)})$. ∎

###  4.3. Strong stability of $\Sigma_{\infty}$

By construction, the $\mu$-bubbles $\Sigma_{\sigma,a}$ satisfy the stability
inequality with respect to variations which vanish on the boundary. But as a
result of our height picking, Lemma 4.2, we gain an additional _vertical
stability_ property. This leads to an improved stability inequality on
$\Sigma_{\infty}$ which holds not only for compactly supported functions (and
limits thereof), but functions of the form $\alpha+\varphi$, where $\varphi$
can be approximated by compactly supported functions and $\alpha$ is a
possibly nonzero constant. This is called the _strong stability inequality_
[28] and to state it precisely we need one more definition.

######  Definition 4.20 (Weighted Sobolev Spaces).

Let $\mathcal{E}$ be a Euclidean end of a manifold $(M^{n},g)$ admitting a
structure of infinity per definition 1.9. For $p\geq 1$ an
$\tau\in\mathbb{R}$, we define the _weighted Lebesgue space_ $L^{p}_{\tau}$ as
the set of all functions $u\in L^{p}_{\mathrm{loc}}$ for whic the norm

$\|u\|_{L^{p}_{\tau}}=\left(\int_{M}|u|^{p}r^{\tau p-n}\,dx\right)^{1/p}$

is finite. For $k\in\mathbb{N}$, we define the _weighted Sobolev space_
$W^{k,p}_{\tau}$ as the set of all functions $u\in W^{k,p}_{\mathrm{loc}}$ for
which the norm

$\|u\|_{W^{k,p}_{\tau}}=\sum_{i=0}^{k}\|\nabla^{i}u\|_{L^{p}_{\tau+i}}$

is finite. The space of compactly supported smooth functions,
$C^{\infty}_{c}$, is dense in $W^{k,p}_{\tau}$.

######  Theorem 4.21 (Strong stability).

Let $\varphi$ be a Lipschitz function on $\Sigma_{\infty}$ and suppose there
exists a constant $\alpha\in\mathbb{R}$ such that

(4.8)
$\varphi-\alpha\in\begin{cases}W^{1,2}_{\frac{n-3}{2}}(\Sigma_{\infty})&\text{if
$n\geq 4$}\\\ C^{1}_{c}(\Sigma_{\infty})&\text{if $n=3$}\end{cases}.$

Then

(4.9)
$\int_{\Sigma_{\infty}}|\nabla\varphi|^{2}+\frac{1}{2}R_{\Sigma_{\infty}}\varphi^{2}-\frac{1}{2}(R_{M}+h^{2}+2\nu(h))\varphi^{2}\,d\mathcal{H}^{n-1}\geq
0.$

######  Corollary 4.22.

Let $L_{\gamma}$ be the conformal Laplacian of the induced metric $\gamma$ on
$\Sigma_{\infty}$. Then

$\int_{\Sigma_{\infty}}\frac{1}{2}(R_{M}+h^{2}-2|\nabla
h|)\varphi^{2}\,d\mathcal{H}^{n-1}\leq\int_{\Sigma_{\infty}}\varphi\,L_{\gamma}\varphi\,d\mathcal{H}^{n-1}$

for any $\varphi$ satisfying (4.8).

First, we record the stability inequality satisfied by the approximating
hypersurfaces $\Sigma_{j}=\Sigma_{\sigma_{j}}.$

######  Proposition 4.23 (Stability of the approximators).

Let $\alpha\in\mathbb{R}$, $R\geq R_{0}$, and $\varphi$ be a function on
$\Sigma_{\infty}$ that equals $\langle\nu_{\infty},\alpha\partial_{n}\rangle$
for $|x|\geq R$. Then there exists a vector field $X$ on $M$ with the
following properties:

  1. (1)

$X=\alpha\partial_{n}$ for $|x|\geq 10R$,

  2. (2)

$\varphi=\langle X,\nu_{\infty}\rangle$ on $\Sigma_{\infty}$,

  3. (3)

$\delta^{2}_{X}\mathcal{F}_{\sigma_{j}}(\Sigma_{j})\geq 0$ for all $j$
sufficiently large under the flow of $X$; see Proposition 2.3,

  4. (4)

If $b_{\infty}$ and $d_{\infty}$ denote the integrands (2.6) and (2.4)
associated to $\Sigma_{\infty}$ and the flow of $X$, then

(4.10) $\displaystyle b_{\infty}$
$\displaystyle=|\nabla\varphi|^{2}-(\mathrm{Ric}(\nu_{\infty},\nu_{\infty})+|A|^{2})\varphi^{2}$
(4.11) $\displaystyle d_{\infty}$
$\displaystyle=-(h^{2}+\nu_{\infty}(h))\varphi^{2}$

everywhere on $\Sigma_{\infty}$.

###### Proof.

We seek a vector field $X$ on $M$ with the following properties:

  1. (i)

$X=\alpha\partial_{n}$ for $|x|\geq 10R$,

  2. (ii)

$\varphi=\langle X,\nu_{\infty}\rangle$ everywhere on $\Sigma_{\infty}$,

  3. (iii)

$\hat{X}=0$ on $\Sigma_{\infty}$ for $|x|\leq R$, and

  4. (iv)

$Z=0$ on $\Sigma_{\infty}$ for $|x|\leq R$.

Let $\theta_{1}$ be a smooth cutoff function $\equiv 1$ for $|x|\leq R$ and
$\equiv 0$ for $|x|\geq 10R$ with the property that there exists another
smooth cutoff $\theta_{2}$ satisfying $\theta_{1}^{2}+\theta_{2}^{2}=1$
everywhere. Define a vector field $X_{1}=\theta_{1}\varphi\nu_{\infty}$ on
$\Sigma_{\infty}$. Extend it by normal geodesics to a neighborhood of
$\Sigma_{\infty}\cap\\{|x|\leq 10R\\}$ and arbitrarily beyond it. Then set
$\tilde{X}_{1}=\theta_{1}X_{1}$, globally. In the asymptotic region, we simply
set $\tilde{X}_{2}=\alpha\theta_{2}^{2}\partial_{n}$, which is a globally
defined vector field. It is easy to see that this vector field has the desired
properties (i)-(iv).

Now we explain how these imply (1)-(4). Clearly (1) follows from (i) and (2)
from (ii). (3) follows from the height picking procedure and our choice of
$a(\sigma_{j})$, Lemma 4.2. Finally, (4) follows from plugging in (iii) and
(iv) into equations (2.6) and (2.4) and using the fact that $h=0$ for $|x|\geq
R\geq R_{0}$. ∎

Taking $\alpha=0$ in Proposition 4.23 gives Theorem 4.21 for compactly
supported $\varphi$, which is all we require when $n=3$. The next step in the
proof of Theorem 4.21 is a careful analysis of the limit $j\to\infty$ in
$\delta^{2}_{X}\mathcal{F}_{\sigma_{j}}(\Sigma_{j})\geq 0.$

######  Lemma 4.24.

For a fixed choice of $X$, the second variation integrands satisfy

$a_{j}=O(|x^{\prime}|^{-(n-1)})$

uniformly in $j$ (also allowing $j=\infty$).

It then follows from basic measure theory that $a_{\infty}\in
L^{1}(\Sigma_{\infty})$ and that

(4.12)
$\int_{\Sigma_{\infty}}a_{\infty}\,d\mathcal{H}^{n-1}+\int_{\Sigma_{\infty}}d_{\infty}\,d\mathcal{H}^{n-1}\geq
0.$

For details in taking this limit, see [18, 4.4.2 _Stabilität von $\Sigma$,
Teil 1_].

###### Proof of Lemma 4.24.

The proofs in Section 4.2 in fact show that
$\Sigma_{\sigma}\cap\mathcal{E}^{\prime}$ can be represented as a graph
$u_{\sigma}$ as in Proposition 4.12, with a uniform $C^{1,\alpha}_{0}$ bound.

Since $a_{j}$ is 2-homogeneous in $X$ ($Z$ scales quadratically), we may
assume $\alpha=1$ for simplicity. In the asymptotic region, the flow of $X$ is
simply vertical translation: $F(x,t)=x+te_{n}$ for $|x|\geq 10R$. Therefore,
the acceleration vector field (with respect to $g$) is
$\nabla_{\partial_{n}}^{g}\partial_{n}$. Using the transformation formula for
the connection under a conformal change [2, Theorem 1.159],

(4.13) $\displaystyle Z$
$\displaystyle=\Gamma^{\mu}_{\overline{g}\,nn}\partial_{\mu}+\frac{2}{n-2}\left(2\partial_{n}\log\psi\partial_{n}-\overline{g}_{nn}\overline{g}^{\mu\nu}\partial_{\mu}\log\psi\partial_{\nu}\right)$
$\displaystyle=C^{1,\alpha}_{n}\cdot\partial_{\mu}+C^{1,\alpha}_{n-1}\cdot\partial_{\mu}$

From this it is apparent that
$\operatorname{div}_{\Sigma}^{g}Z=O(|x^{\prime}|^{-n})$. (In fact, it appears
to decay faster but this is all we shall require.)

Next, we analyze
$\operatorname{div}_{\Sigma_{j}}^{g}\partial_{n}=\operatorname{div}^{g}\partial_{n}-\langle\nabla_{\nu_{j}}\partial_{n},\nu_{j}\rangle$.
Indeed,

$\operatorname{div}_{g}\partial_{n}=\frac{1}{\sqrt{\det
g}}\partial_{n}\sqrt{\det g}=\Gamma^{\mu}_{g,n\mu}=O(|x^{\prime}|^{-(n-1)}).$

The term $\langle\nabla_{\nu_{j}}\partial_{n},\nu_{j}\rangle$ is also on the
level of a Christoffel symbol of $g$, hence
$(\operatorname{div}_{\Sigma_{j}}^{g}\partial_{n})^{2}$ decays rapidly enough
for our purposes.

The terms involving $\langle
e_{i},\nabla_{e_{j}}X\rangle\langle\nabla_{e_{i}}X,e_{j}\rangle$ are on the
the level of $\Gamma_{g}^{2}$, hence also decay rapidly enough. This same
argument works for
$|(\nabla_{e_{i}}X)^{\perp}|^{2}=\langle\nabla_{e_{i}}X,\nu_{j}\rangle^{2}$.

Finally, the Riemann tensor term is $O(|x^{\prime}|^{-n})$ by Proposition 4.6.
∎

######  Lemma 4.25.

Along the domains $D_{\sigma}=\Sigma_{\infty}\cap Z_{\sigma}$, we have the
limit

$\lim_{\sigma\to\infty}\int_{\partial
D_{\sigma}}c_{\infty}\,d\mathcal{H}^{n-2}=0.$

From now on we may drop the $\infty$ subscript when there is no risk for
confusion since we are not working with the $\Sigma_{j}$’s any more.

###### Proof.

Since $\mathcal{H}^{n-2}(\partial D_{\sigma})=O(\sigma^{n-2})$, we merely
require that $c=O(|x^{\prime}|^{-(n-1)})$. For reference, the integrand is
given by

$c=\langle
Z,\eta\rangle+(\operatorname{div}_{\Sigma}^{g}\partial_{n})\langle\partial_{n},\eta\rangle-\langle\nabla^{g}_{\partial_{n}}\partial_{n},\eta\rangle-2\langle\partial_{n},\nu\rangle
A(\hat{\partial_{n}},\eta),$

where $\hat{\partial_{n}}$ is the tangential component and $\eta$ is the
conormal to $\partial D_{\sigma}$ in $\Sigma$. The first term has sufficient
decay per (4.13). The second and third terms also decay to order $n-1$, as
shown in the previous lemma. The last term has the desired decay rate by Lemma
4.19. ∎

We can now prove the strong stability inequality.

###### Proof of Theorem 4.21.

From the previous work, (4.9) holds for any $\varphi$ which equals
$\langle\nu,\alpha\partial_{n}\rangle$ outside a compact set. Indeed, we
already remarked that in this case, (4.12) holds. By measure theory,

$\displaystyle\int_{\Sigma_{\infty}}a_{\infty}\,d\mathcal{H}^{n-1}$
$\displaystyle=\lim_{\sigma\to\infty}\int_{D_{\sigma}}a_{\infty}\,d\mathcal{H}^{n-1}$
$\displaystyle=\lim_{\sigma\to\infty}\int_{D_{\sigma}}b_{\infty}\,d\mathcal{H}^{n-2}+\lim_{\sigma\to\infty}\int_{\partial
D_{\sigma}}c_{\infty}\,d\mathcal{H}^{n-2}$
$\displaystyle=\int_{\Sigma_{\infty}}b_{\infty}\,d\mathcal{H}^{n-2}$

by the integration by parts formula (2.5) and Lemma 4.25, as long as
$b_{\infty}\in L^{1}(\Sigma_{\infty})$. We now justify this. By Proposition
4.6 and Lemma 4.19,

$(\mathrm{Ric}(\nu,\nu)+|A|^{2})\varphi^{2}=O(|x^{\prime}|^{-n}),$

hence is summable. For the upward pointing normal, we have

$\langle\nu,\alpha\partial_{n}\rangle=\alpha\frac{\psi^{\frac{4}{n-2}}}{|d\zeta|_{\overline{g}}}.$

It is easy to see that the derivatives of this function are
$O(|x^{\prime}|^{-(n-1)})$, so $|\nabla\varphi|^{2}\in L^{1}(\Sigma_{\infty})$
as well.

On $\Sigma_{\infty}$, the traced Gauss equation reads [19, Corollary 2.7]

$\mathrm{Ric}(\nu,\nu)+|A|^{2}=\frac{1}{2}(R_{M}-R_{\Sigma_{\infty}}-|A|^{2}+h^{2}).$

The scalar curvature of $M$ is integrable over $\Sigma_{\infty}$ by
Proposition 4.6 and the same calculation using Corollary 4.18 shows that
$R_{\Sigma_{\infty}}$ is integrable as well. Putting this into the stability
inequality shows

(4.14)
$\int_{\Sigma_{\infty}}|\nabla\varphi|^{2}+Q\varphi^{2}\,d\mathcal{H}^{n-1}\geq
0,$

where

$-2Q=R_{M}-R_{\Sigma_{\infty}}+|A|^{2}+h^{2}+2\nu(h),$

for any $\varphi$ which equals $\langle\nu,\alpha\partial_{n}\rangle$ outside
of a compact set.

Passing from such “compactly supported”
$\varphi-\langle\nu,\alpha\partial_{n}\rangle$ to the general case is now as
in [9, page 25], by density. ∎

###  4.4. Finishing the argument when $R_{g}>0$ far out

We are finally ready to prove Theorem 1.6, with a modified assumption (2) that
will be taken care of in the next section.

######  Proposition 4.26.

Let $(M^{n},g)$ be an asymptotically Schwarzschild manifold. Let $U_{1}$ and
$U_{2}$ be neighborhoods of infinity with $\overline{U_{2}}\subset U_{1}$ and
$D>0$. If furthermore

  1. (1)

$g$ has no points of incompleteness in the $D$-neighborhood of $U_{1}$,

  2. (2)

$R_{g}\geq 0$ in the $D$-neighborhood of $U_{1}$ and $R_{g}>0$ for $|x|$
sufficiently large, and

  3. (3)

the scalar curvature satisfies the largeness assumption

$R_{g}>\sigma(\rho,D)$

on $\overline{U_{1}}\setminus U_{2}$,

then $m\geq 0$.

###### Proof of Proposition 4.26 when $n=3$.

We utilize the logarithmic cutoff trick (this step requires only quadratic
area growth), i.e. plug

$\varphi_{\sigma}(x^{\prime})=\begin{cases}1&\text{if
$|x^{\prime}|\leq\sigma$}\\\ 2-\frac{\log|x^{\prime}|}{\log\sigma}&\text{if
$\sigma<x^{\prime}<\sigma^{2}$}\\\ 0&\text{if
$\sigma^{2}<|x^{\prime}|$}\end{cases}$

into the stability inequality (4.9). The kinetic term is then

$\int_{\Sigma_{\infty}}|\nabla\varphi_{\sigma}|^{2}\,d\mathcal{H}^{2}\approx\int_{\sigma}^{\sigma^{2}}\frac{1}{r^{2}}\,r\,dr=\frac{2}{\log\sigma}\to
0,$

so we have

$\int_{\Sigma_{\infty}}K\,d\mathcal{H}^{2}>0,$

where $K$ is the Gauss curvature of $\Sigma_{\infty}$ and we get strict
positivity because the strict $\mu$-bubble condition holds sufficiently far
out by assumption (2). It is standard that the geodesic curvature of $\partial
D_{\sigma}$ is $\sigma^{-1}+O(\sigma^{-2})$ [5, page 255] and hence

$\int_{D_{\sigma}}k_{g}\,d\mathcal{H}^{1}=2\pi+O(\sigma^{-1}).$

By the Gauss–Bonnet theorem

$\int_{D_{\sigma}}K\,d\mathcal{H}^{2}=2\pi(\chi(D_{\sigma})-1)+O(\sigma^{-1}).$

Since we always have $\chi(D_{\sigma})\leq 1$ for a connected surface with
boundary, this gives a contradiction. ∎

###### Proof of Proposition 4.26 when $4\leq n\leq 7$.

Here we follow [9]. We seek a conformal factor $w>0$, $w\to 1$ at $\infty$, so
that $w^{\frac{4}{n-3}}\gamma$ is scalar flat. This is equivalent to solving
$L_{\gamma}v=-R_{\Sigma_{\infty}}$ where $v=w-1$, with $v\to 0$ at $\infty$.
The scalar curvature of $\Sigma_{\infty}$ is $O(|x^{\prime}|^{-n})$ (which is
integrable!) and lies in $L^{p}_{n-1-\varepsilon}$ for any
$\varepsilon\in(0,1)$ and $p\geq 1$, which we can fix to be $>n$. We now claim
that

$L_{\gamma}:W^{2,p}_{n-3-\varepsilon}\to L^{p}_{n-1-\varepsilon}$

is an isomorphism.

Suppose $\varphi\in W^{2,p}_{n-2-\varepsilon}$ solves $L_{\gamma}w=0$. Then by
elliptic regularity and Sobolev embedding, $\varphi\in
C^{1}_{n-3-\varepsilon}\cap C^{\infty}_{\mathrm{loc}}\subset
W^{1,2}_{\frac{n-3}{2}}.$ Hence according to Lemma 4.22 and the weak
$\mu$-bubble condition satisfied by $h$,

$\int_{\Sigma_{\infty}}\frac{1}{2}(R_{M}+h^{2}-2|\nabla h|)\varphi^{2}=0.$

Since $h$ satisfies the strict $\mu$-bubble condition for large
$|x^{\prime}|,$ $\varphi$ must vanish for large $|x^{\prime}|$. Since
$\varphi$ solves a homogeneous elliptic equation, strong unique continuatation
[5, Remark 6.2] implies it vanishes identically. Now by standard Fredholm-type
theorems in asymptotic analysis [19, Theorems A.40, A.42], $L_{\gamma}$ is an
isomorphism.

Let $v$ be the solution so obtained and set $w=v+1$. Using the fact that the
conformal Laplacian has a positive principal eigenfunction with positive
eigenvalue on each $D_{\sigma}$ for $\sigma$ sufficiently large, it is an easy
matter to verify that $w>0$ via the maximum principle. By standard decay
theorems (such as Theorem 4.17), $w^{\frac{4}{n-3}}\gamma$ is asymptotically
flat and has mass

$-\frac{2}{(n-3)|S^{n-2}|}\int_{\Sigma_{\infty}}w\,L_{\gamma}w\,\mathcal{H}^{n-1}.$

By Lemma 4.22, this is strictly negative since $w\to 1$ at $\infty$. We have
thus obtained a contradiction the positive mass theorem in dimension $n-1$. ∎

##  5\. Proofs of the main theorems

We formalize the observation made in Step 1 of [31].

######  Lemma 5.1.

Let $(M^{n},g)$ be a asymptotically Schwarzschild manifold with negative mass
and nonnegative scalar curvature on $\mathcal{E}$. For $r_{0}>0$ sufficiently
large (depending on $g$), there exists a conformal factor $\varphi$ with the
following properties:

  1. (1)

The conformal metric $\tilde{g}=\varphi^{\frac{4}{n-2}}g$ is asymptotically
Schwarzschild,

  2. (2)

$R_{\tilde{g}}>0$ for $|x|\geq 2r_{0}$, and

  3. (3)

$\varphi$ is constant equal to $1+O(r_{0}^{-(n-2)})$ for $|x|\leq r_{0}$ and
on $M\setminus\mathcal{E}$.

###### Proof.

By (4.1),

$\Delta_{g}|x|^{-(n-2)}=m(n-2)^{2}|x|^{-(2n-2)}+O(|x|^{-(2n-1)}),$

which shows the existence of a $\sigma_{0}>0$ such that
$-\Delta_{g}|x|^{-(n-2)}>0$ for any $|x|\geq\sigma_{0}$. Let
$t_{0}=-m/8r_{0}^{n-2}>0$ for $r_{0}\geq\sigma_{0}$.

Let $\zeta$ be a smooth function on $(0,\infty)$ with the following
properties:

$\zeta(t)=\begin{cases}t&\text{for $t<t_{0}$}\\\ \tfrac{3}{2}t_{0}&\text{for
$t>2t_{0}$}\end{cases},\quad\zeta^{\prime}\geq
0,\quad\text{and}\quad\zeta^{\prime\prime}\leq 0.$

On $\mathcal{E}$, we define $\varphi$ by

$\varphi(x)=1+\zeta\left(-\frac{m}{4|x|^{n-2}}\right).$

If $|x|\leq r_{0}$, then $\varphi(x)=1+\frac{3}{2}t_{0}=1+O(r_{0}^{-(n-2)})$
and we extend it by this constant to all of $M$. If $|x|\geq 2r_{0}$, then

$\varphi(x)=1-\frac{m}{4|x|^{n-2}}.$

From this is is apparent that $\tilde{g}$ is asymptotically Schwarzschild with
mass $\frac{m}{2}<0$. Finally, the scalar curvature is given by the usual
formula

$R_{\tilde{g}}=\varphi^{-\frac{n+2}{n-2}}\left(-4\frac{n-1}{n-2}\Delta_{g}\varphi+R_{g}\varphi\right).$

The Laplacian term has the correct sign, as is easily checked:

$-\Delta_{g}\varphi(x)=\frac{m}{4}\zeta^{\prime\prime}|\nabla|x|^{-(n-2)}|^{2}+\frac{m}{4}\zeta^{\prime}\Delta_{g}|x|^{-(n-2)}\geq
0$

everwhere, and $>0$ for $|x|\geq 2t_{0}$. ∎

This allows us to remove the strict positivity assumption in Proposition 4.26.

###### Proof of Theorem 1.6.

We apply the conformal transformation of Lemma 4.1 to $(M,g)$ to bring it into
the form of Proposition 4.26. Indeed, since $\overline{U_{1}}\setminus U_{2}$
is compact and $R_{g}$ is continuous, $R_{g}-\sigma(\rho,D)$ has a positive
minimum there. It follows that if we choose $r_{0}$ sufficiently large in
Lemma 4.1, then $R_{\tilde{g}}>\sigma(\rho,D)$ on $\overline{U_{1}}\setminus
U_{2}$ as well. Furthermore, distances will be distorted an arbitrarily small
amount. So if the original mass is negative, we have a counterexample to
Proposition 4.26. ∎

###### Proof of Theorem 1.2.

If $(M,g)$ is complete and has nonnegative scalar curvature everywhere, apply
Lemma 4.1 with some $r_{0}$ to make the scalar curvature strictly positive on
the end. We may then take $U_{i}$ as in Remark 1.7 (4) sufficiently far out
that the scalar curvature is strictly positive on the compact set
$\overline{U_{1}}\setminus U_{2}$. Now $\rho$ is fixed and since for fixed
$\rho$,

$\lim_{D\to\infty}\sigma(\rho,D)=0,$

we can find $D$ sufficiently large that $R_{\tilde{g}}>\sigma(D,\rho)$ on
$\overline{U_{1}}\setminus U_{2}$. But $(M,g)$ is complete with nonnegative
scalar curvature globally, so we satisfy all the hypotheses of Theorem 1.6
with this choice of $\rho$ and $D$. It follows that the original mass is
nonnegative. ∎

##  Appendix A Curvature estimate in homogeneously regular manifolds

In this appendix we give an independent proof Lemma 4.8.

######  Definition A.1.

Let $K\geq 0$ and $i_{0}>0$ be real numbers. We say that a complete Riemannian
manifold $(M^{n},g)$ is _$(K,i_{0})$ -homogeneously regular_ if its sectional
curvatures are bounded in absolute value by $K$ and injectivity radius bounded
below by $i_{0}$. In the case of a noncompact manifold, we use the same
definition, with the understanding that we work at a distance at least
$2i_{0}$ away from any incomplete points; see Definition 1.8.

In the rest of this section, $K$ and $i_{0}$ are fixed and all manifolds are
assumed to be $(K,i_{0})$-homogeneously regular. We work with geodesic balls
$B_{\rho}(x_{0})^{g}$ and use the phrase “$\Sigma$ is a minimal/minimizing
submanifold of $B_{\rho}^{g}(x_{0})$” to mean that $\Sigma\subset
B_{\rho}^{g}(x_{0})$ and that $\partial\Sigma\subset\partial
B_{\rho}^{g}(x_{0})$.

######  Theorem A.2.

Let $2\leq n\leq 7$, $K\geq 0$, and $i_{0}>0$. There exist universal constants
$\rho_{0}\in(0,i_{0})$ and $C$ depending only on $n,K$, and $i_{0}$ with the
following property. Let $(M^{n},g)$ be a $(K,i_{0})$-homogeneously regular
Riemannian manifold, $x_{0}\in M$, and $\rho\in(0,\rho_{0})$. If $\Sigma$ is
an area minimizing hypersurface in $B_{\rho}^{g}(x_{0})$ with
$x_{0}\in\Sigma$, then

$|A_{\Sigma}(x_{0})|^{2}\leq\frac{C}{\rho^{2}}.$

In our definition of homogeneously regular, we can only obtain coordinates in
which the metric is $C^{1,\alpha}$-controlled, see Lemma A.5 below. Therefore,
it might seem like this is a regularity theorem for $C^{1,\alpha}$ parametric
elliptic functionals, but G. De Philippis has pointed out that this is still
essentially perturbative about the flat space regularity theory (due to the
freedom of choosing $\rho_{0}$ very small). This can be formalized using the
work of Tamanini [40]. In this appendix, we give an alternative proof via a
contradiction argument, in the spirit of White [41] (for an Allard-type
result) and Simon [38] (for a universal density bound).

######  Theorem A.3 (Allard-type theorem).

Let $n\in\mathbb{N}$, $K\geq 0$, and $i_{0}>0$. There exist universal
constants $\rho_{0}\in(0,i_{0})$, $\varepsilon_{0}>0$, $C$, and $\tau>0$
depending only on $n,K$, and $i_{0}$ with the following property. Let
$(M^{n},g)$ be a $(K,i_{0})$-homogeneously regular Riemannian manifold,
$x_{0}\in M$, $\rho\in(0,\rho_{0})$, and $\Sigma$ a minimal hypersurface in
$B_{\rho}^{g}(x_{0})$ with $x_{0}\in\Sigma$. If

$\Theta(x_{0},\rho)\leq 1+\varepsilon_{0},$

then

$\sup_{\Sigma\cap B_{\tau\rho}^{g}(x_{0})}|A|^{2}\leq\frac{C}{\rho^{2}}.$

Note that this does not follow directly from Allard’s theorem [1] because it
is not possible to bound the mean curvature of $\Sigma$ with respect to the
underlying Euclidean structure.

######  Proposition A.4 (Universal density bound).

Let $2\leq n\leq 7$, $K\geq 0$, and $i_{0}>0$. There exists a universal
constant $\delta_{0}\in(0,1)$ depending only on $n,K$, and $i_{0}$ with the
following property. Let $(M^{n},g)$ be a $(K,i_{0})$-homogeneously regular
Riemannian manifold, $x_{0}\in M$, and $\rho\in(0,\rho_{0})$. If $\Sigma$ is
an area minimizing hypersurface in $B_{\rho}^{g}(x_{0})$ with
$x_{0}\in\Sigma$, then

$\Theta(x_{0},\delta_{0}\rho)\leq 1+\varepsilon_{0},$

where $\varepsilon_{0}$ is as in Theorem A.3.

Combining Theorems A.3 and A.4 immediately gives Theorem A.2. To prove these
theorems, we recall a basic result about existence of harmonic coordinate
charts.

######  Lemma A.5 (Hebey–Herzlich [17]).

Let $(M^{n},g)$ be $(K,i_{0})$-homogeneously regular and $\alpha\in(0,1)$.
There exists a constant $Q_{0}$ and a number $r_{0}$ (depending only on
$K,i_{0}$, $n$, and $\alpha$) such that for any $x_{0}\in M$, there exists a
harmonic coordinate chart $x^{i}$ on $B_{r_{0}}^{g}(x_{0})$ sending $x_{0}$ to
the origin and satisfying

(A.1) $Q_{0}^{-1}\delta_{ij}\leq g_{ij}\leq Q_{0}g_{ij}\quad\text{as quadratic
forms},$

and

(A.2) $\|\partial_{k}g_{ij}\|_{C^{0,\alpha}(U)}\leq Q_{0},$

where $U\subset\mathbb{R}^{n}$ is the domain of definition of the harmonic
coordinates.

For a metric $g$, we define the area ratio at a point $x_{0}\in M$ and scale
$r$ according to

$\Theta_{\Sigma}^{g}(x_{0},r)=\frac{\mathcal{H}^{n-1}_{g}(\Sigma\cap
B_{r}^{g}(x_{0}))}{\omega_{n}r^{n}}.$

The following is easily proved using the Hessian comparison theorem [5, page
234].

######  Lemma A.6 (Monotonicity formula).

On any minimal hypersurface in a $(K,i_{0})$-homogeneously regular manifold,

(A.3) $\frac{d}{dr}\left(e^{(n-1)\sqrt{K}r}\Theta(x_{0},r)\right)\geq 0$

for $r<i_{0}$.

Using this, we can bound the area ratio at nearby points; compare [8, Lemma
D.7].

######  Lemma A.7.

Assume the hypotheses of Theorem A.3 and also let $\varepsilon>0$. Then there
exist constants $\beta,\eta>0$ and $\rho_{0}\in(0,\min\\{r_{0},i_{0}\\})$
(depending only on $n,K,i_{0}$, and $\varepsilon$ and where $r_{0}$ is the
harmonic radius from Lemma A.5 with an otherwise arbitrary choice of $\alpha$)
with the property that if

$\Theta(x_{0},\rho)\leq 1+\varepsilon$

for some $\rho\in(0,\rho_{0})$, then

$1-\eta\leq\Theta(y,\sigma)\leq 1+2\varepsilon$

for every $y\in B^{g}_{\beta\rho}(x_{0})\cap\Sigma$ and
$\sigma<(1-\beta)\rho$. With all other parameters fixed, $\lim_{\varepsilon\to
0}\beta=\lim_{\varepsilon\to 0}\eta=\lim_{\varepsilon\to 0}\rho_{0}=0$.

###### Proof.

Using the monotonicity at $y\in B^{g}_{\beta}(x_{0})$ as well as the inclusion
$B_{(1-\beta)\rho}(y)\subset B_{\rho}(x_{0})$ we estimate

$\displaystyle\Theta(y,\sigma)$ $\displaystyle\leq
e^{(n-1)\sqrt{K}((1-\beta)\rho-\sigma)}\Theta(y,(1-\beta)\rho)$
$\displaystyle\leq\frac{e^{(n-1)\sqrt{K}((1-\beta)\rho-\sigma)}}{(1-\beta)^{n}}(1+\varepsilon)$
$\displaystyle\leq 1+2\varepsilon$

for $\beta$ and $\rho_{0}$ well chosen.

For the lower bound, we note that monotonicity implies

$\Theta(y,\sigma)\geq e^{-(n-1)\sqrt{K}\sigma}.$

If $\sigma$ is sufficiently small, then we can write this as

$\Theta(y,r)\geq 1-\eta(\varepsilon).\qed$

###### Proof of Theorem A.3.

By Lemma A.7 it suffices to show that there exist constants $\varepsilon_{0}$
and $C$ such that

$\sup_{y\in
B_{\beta\rho}^{g}(x_{0})\cap\Sigma,\sigma<(1-\beta)\rho}\Theta_{\Sigma}^{g}(y,\sigma)\leq
1+\varepsilon_{0}$

implies

$\sup_{\sigma<\beta\rho}\left(\sigma^{2}\sup_{B_{\beta\rho-\sigma}(x_{0})}|A_{\Sigma}|^{2}\right)\leq
C,$

where $\rho<\rho_{0}(\varepsilon_{0})$. Indeed, if this is true we may take
$\tau=\frac{\beta}{2}$ and $\sigma=\tau\rho$.

Arguing by contradiction, we infer the existence of $(K,i_{0})$-homogeneously
regular manifolds $(M^{n}_{j},g_{j},x_{j})$, numbers $\varepsilon_{j}\to 0$,
minimal hypersurfaces $\Sigma_{j}\subset B^{g_{j}}_{\rho_{j}}(x_{j})$,
$\rho_{j}\in(0,\rho_{0}(\varepsilon_{j}))$, such that

(A.4) $\sup_{y\in
B^{g_{i}}_{\beta_{j}\rho_{j}}(x_{j})\cap\Sigma_{j},\sigma<(1-\beta_{j})\rho_{j}}\Theta_{j}(y,\sigma)\leq
1+\varepsilon_{j},$

but

$\gamma_{j}^{2}:=\sup_{\sigma<\beta_{j}\rho_{j}}\left(\sigma^{2}\sup_{B_{\beta_{j}\rho_{j}-\sigma}(x_{j})\cap\Sigma_{j}}\right)\to\infty.$

We find $\sigma_{j}<\beta_{j}\rho_{j}$ and
$y_{j}\in\overline{B}_{\beta_{j}\rho_{j}-\sigma_{j}}(x_{j})$ such that

$\gamma_{j}^{2}=\sigma_{j}^{2}\sup_{B_{\beta_{j}\rho_{j}-\sigma_{j}}(x_{j})\cap\Sigma_{j}}|A_{j}|^{2}=\sigma_{j}^{2}|A_{j}(y_{j})|^{2}.$

From the definitions, we see

$\sup_{B_{\beta_{j}\rho_{j}-\frac{\sigma_{j}}{2}}(x_{j})\cap\Sigma_{j}}|A_{j}|^{2}\leq
4|A_{j}(y_{j})|^{2}$

and by the inclusion $B_{\frac{\sigma_{j}}{2}}(y_{j})\subset
B_{\beta_{j}\rho_{j}-\frac{\sigma_{j}}{2}}(x_{j})$,

$\sup_{B_{\frac{\sigma_{j}}{2}}(y_{j})}|A_{j}|^{2}\leq 4|A_{j}(y_{j})|^{2}.$

By Lemma A.5 and our choice of $\rho_{0}$, we can introduce harmonic
coordinates $y_{j}^{i}$ on $B_{\frac{\sigma_{j}}{2}}^{g_{j}}(y_{j})$ (which we
can now view as a subset of $\mathbb{R}^{n}$) sending $y_{j}$ to $0$ and
satisfying (A.1) and (A.2). Let $\lambda_{j}=|A_{j}(y_{j})|^{-1}=o(j)$ and
rescale $(B^{g_{j}}_{\frac{\sigma_{j}}{2}}(y_{j}),g_{j})$ by $\lambda_{j}$.
Since $\frac{\sigma_{j}}{2}\lambda_{j}^{-1}\to\infty,$
$\bigcup_{j}B^{g_{j}}_{\frac{\sigma_{j}}{2}}(y_{j})=\mathbb{R}^{n}$.
Furthermore, the components of the rescaled metrics are given by
$(g_{j})_{kl}(\frac{y}{\lambda_{j}})$, so by (A.1) and (A.2) (sub-)converge in
$C^{1,\alpha}_{\mathrm{loc}}(\mathbb{R}^{n})$ to the Euclidean metric. By
essentially the same calculation leading to Lemma 4.9, Proposition 4.13, and
Schauder theory [12, Chapter 6], the rescaled hypersurfaces
$\lambda_{j}^{-1}\Sigma_{j}$ (sub-)converge in $C^{2,\alpha}_{\mathrm{loc}}$
to a complete, smooth minimal hypersurface in $\mathbb{R}^{n}$. (The
coefficients of the minimal graph-type equations satisfied by the
$\lambda_{j}^{-1}\Sigma_{j}$’s converge in $C^{0,\alpha}_{\mathrm{loc}}$ to
their Euclidean values, which suffices for this application of Schauder
theory.) The limiting hypersurface $\Sigma$ has $|A(0)|=1$.

Finally, we need to show that $\Theta_{\Sigma}(0,R)=1$ for every $R$. This
follows from the upper bound (A.4) and the lower bound in Lemma A.7. Therefore
$\Sigma$ is a cone, but as it is smooth, it must be a hyperplane. This
contradicts $|A(0)|=1$. ∎

###### Proof of Proposition A.4.

We argue again by contradiction. There exist Riemannian manifolds
$(M^{n}_{j},g_{j},x_{j})$, numbers $\delta_{j}\to 0$, and area minimizing
hypersurfaces $\Sigma_{j}\subset B_{\rho_{j}}^{g_{j}}(x_{j})$ with
$x_{j}\in\Sigma_{j}$ such that

$\Theta_{j}(x_{j},\delta_{j}\rho_{j})\geq 1+\varepsilon_{0}.$

By the monotonicity formula,

$\Theta_{j}(x_{j},\rho)\geq
e^{-(n-1)\sqrt{K}(1-\delta_{j})\rho_{j}}(1+\varepsilon_{0})\geq
1+\frac{\varepsilon_{0}}{2}$

for any $\rho\in(\delta_{j}\rho_{j},\rho_{j})$, after possibly adjusting
$\rho_{0}(\varepsilon_{0})$. We now rescale with scale factor
$\lambda_{j}=(\delta_{j}\rho_{j})^{-1/2}=o(j)$. By the same argument as in the
proof of Theorem A.3, the rescaled metrics will (sub-)converge (in harmonic
coordinates) to the Euclidean metric. By a slight modification of the proof of
[39, Theorem 34.5] (using the fact that the area functionals with respect to
the rescaled metrics are converging to the flat one in the obvious way), one
can show that $\lambda_{j}^{-1}\Sigma_{j}$ converge to some (Euclidean) area
minimizing current $T$ in the weak topology and as varifolds. Furthermore, $T$
will be multiplicity one and is carried by a smooth hypersurface $\Sigma$
(here we use $n\leq 7$). By varifold convergence, we have

$\Theta_{\Sigma}(0,R)\geq 1+\frac{\varepsilon_{0}}{2}$

for every $R>0$. But $\Sigma$ is a smooth hypersurface, so its $(n-1)$-density
at the origin is 1. This is a contradiction. ∎

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Now, an EHPO procedure using random search with $KR$ trials will convince
$\mathcal{B}_{*}$ of $p$ with probability $Q^{R}$, since all $R$ independently
sampled groups of $K$ trials must “hit” and each hit happens with probability
$Q$. Thus, the expected time it will take the fastest strategy to convince us
of $p$ is $Q^{-R}\cdot KR$. Similarly, the fastest strategy to convince us of
$\lnot p$ takes expected time $Q_{\lnot}^{-R}\cdot KR$.

Suppose now, by way of contradiction, that the $t$-non-deceptiveness axiom is
violated, and there are strategies that can achieve both of these in time at
most $t$. That is,

$Q^{-R}\cdot KR\leq t\hskip 20.00003pt\text{and}\hskip
20.00003ptQ_{\lnot}^{-R}\cdot KR\leq t.$

From here, it’s fairly easy to conclude that

$Q+Q_{\lnot}\geq 2\left(\frac{KR}{t}\right)^{1/R}.$

Combining this with our conclusion above gives

$\left(\frac{KR}{t}\right)^{1/R}\leq\frac{1}{1+\exp(-\gamma K)}.$

It’s clear that we can cause this to be violated by setting $R$ to be large
enough. Observe that

$\frac{1}{1+\exp(-\gamma K)}\leq\exp(-\exp(-\gamma K)),$

so

$\left(\frac{KR}{t}\right)^{1/R}\leq\exp(-\exp(-\gamma K)).$

Taking the root of both sides gives

$\left(\frac{KR}{t}\right)^{\frac{1}{R\exp(-\gamma K)}}\leq\frac{1}{e}.$

To simplify this expression, let $\beta$ denote

$\beta=R\exp(-\gamma K).$

So that

$\left(\frac{\beta K}{t\exp(-\gamma K)}\right)^{1/\beta}\leq\frac{1}{e}.$

Finally, we set $R$ such that

$\beta=\sqrt{\frac{t\exp(-\gamma K)}{K}}.$

To give

$\left(\frac{1}{\beta}\right)^{1/\beta}\leq\frac{1}{e}.$

But this is impossible, as the minimum of $x^{x}$ occurs above $1/e$. This
setting of $R$ gives

$R=\beta\exp(\gamma K)=\sqrt{\frac{t\exp(\gamma K)}{K}}=O(\sqrt{t}).$

This shows that, for this task, if we run our constructed EHPO with
$R=O(\sqrt{t})$ assigned in this way, it will be guaranteed to be $t$-non-
deceptive.

###  E.3 Defense Experiments

In this section we provide more information about the implementation of a
random-search-based defense to hyperparameter deception in Wilson et al. [72],
which we discuss in Section 5.

####  E.3.1 Our Implemented Defense Algorithm

The defense we implement in our experiments is a bit different than what we
describe in our theoretical results in Section 5. In particular, in practice
it is easier to implement subsampling rather than resampling.

Protocol of Selecting Hyper-HPs. As partially illustrated in Figure 1 and
elaborated on in Appendix B, Wilson et al. [72]’s choice of hyper-HPs does not
capture the space where Adam effectively simulates Heavy Ball. In Wilson et
al. [72]i, Adam-specific HPs like numerical variable $\epsilon=10^{-8}$ [42]
are treated as constants, leading to a biased HP-search space.

In contrast, we select the hyper-HPs of $\epsilon$ following a dynamic
searching protocol:

Inspired by [11], we start from a wide range $\epsilon\in[10^{-12},10^{12}]$
as a wide search space. We iteratively select powers-of-10 grids that are
uniformly spaced in the logarithmic scale of the current range. For instance,
the selected grids for the prior range would be
$\\{10^{-12},10^{-11},\cdots,10^{11},10^{12}\\}$. We perform a single run on
each grid selected, and shrink the range towards grids where the best
performance are achieved. The shrinkage follows the policy of either $\times
10$ to the lower boundary or $\times 0.1$ to the upper boundary. For example,
for the prior range, we found the best performance is achieved on grid
$10^{11}$, so we multiply the lower boundary $10^{-12}$ with $10$ and shift
the range to $\epsilon\in[10^{-11},10^{12}]$. Our protocol terminates with
$\epsilon\in[10^{10},10^{12}]$ as the final hyper-HPs that we use for our
defended random search EHPO.

Scaling Learning Rate $\eta$. Note that directly applying the hyper-HP of
$\epsilon\in[10^{10},10^{12}]$ to Adam would lead to extremely slow
convergence, since essentially large $\epsilon$ indicates a small effective
learning rate $\eta$. Similar to Choi et al. [11], we explore a shifted hyper-
HPs for the $\eta$, scaled proportionally with $\epsilon$. Specifically, note
that a large $\epsilon$ would make the update of Adam approach the update rule
in the Heavy Ball method; for any randomly selected
$\epsilon\in[10^{10},10^{12}]$, we perform the random search of
$\eta/\epsilon$ instead of $\eta$ itself in the space of $[0.5,0.7]$, which is
the search space of HB’s learning rate shown in [72].

####  E.3.2 Experimental setup

We follow the setup from [72], where the details are specified in Section
B.2.1.

####  E.3.3 Code

The code for running these experiments can be found at
https://github.com/pasta41/deception.

####  E.3.4 Associated results and logs

In line with our notion of a log, we provide heatmaps of our logs in Figures
13, 14 and that correspond with our results in Section 5. We note that the
performance of Heavy Ball for some random seeds is very bad (e.g., $10\%$ test
accuracy). The performance varies widely – also nearing $92\%$ for different
random seeds. We affirm that this is the search space that yields the best
results for Heavy Ball ($~{}92\%$).

The results for Heavy Ball exhibit large variance. This illustrates a strength
of our defensed: it actually helps with robustness against potentially making
the wrong conclusion about Heavy Ball’s performance (more generally), due to
not making conclusions off of a single result (and perhaps using a random seed
for which performance is particularly bad). We make a different claim about
relative algorithm performance than Wilson et al. [72] about Heavy Ball (i.e.,
we do not claim that it is better than Adam); but we do not reach this
conclusion for the wrong reason (i.e., that we got one bad Heavy Ball result
for a particular random seed).

Figure 10: Heatmap logs of SGD defended random search. Redder rows indicate
higher test accuracy.

Figure 11: Heatmap logs of Heavy Ball (HB) defended random search. Redder rows
indicate higher test accuracy.

Figure 12: Heatmap logs of Adam defended random search. Redder rows indicate
higher test accuracy.

Figure 13: Heatmap logs of test accuracy of VGG-16 on CIFAR-10 for Adam vs.
SGD using our defended random search EHPO. Red indicates higher test accuracy
for the given random seed.

Figure 14: Heatmap logs of test accuracy of VGG-16 on CIFAR-10 for Adam vs.
Heavy Ball (HB) using our defended random search EHPO. Red indicates higher
test accuracy for the given random seed.

##  Appendix F Section 6 Appendix: Additional Notes on Conclusion

###  F.1 Additional Practical Takeaways

In our conclusion in Section 6, we note the following practical takeaways:

  * •

Researchers should have their own notion of skepticism, appropriate to their
specific task. There is no one-size-fits-all defense solution. Our results are
_broad insights_ about defended EHPO: A defended EHPO is _always possible_ ,
but finding an efficient one will depend on the task.

  * •

Researchers should make explicit how they choose hyper-HPs. What is reasonable
is ultimately a function of what the ML community accepts. Being explicit,
rather than eliding hyper-HP choices, is essential for helping decide what is
reasonable. As a heuristic, we recommend setting hyper-HPs such that they
include HPs for which the optimizers’ performance starts to degrade, as we do
above.

  * •

Avoiding hyperparameter deception is just as important as reproducibility. We
have shown that reproducibility [34, 57, 29, 7, 64] is only part of the story
for ensuring reliability. While necessary for guarding against brittle
findings, it is not sufficient. We can replicate results—even statistically
significant ones—that suggest conclusions that are altogether wrong.

We elaborate here that our defended random search EHPO indicates a particular
form of skepticism that may (or may not) be appropriate to different ML tasks.
That is, we suggest a defended EHPO, but do not claim that that EHPO is
optimal or suited for all tasks. Even though it may not be optimal, the
guarantees it affords would translate to other tasks (so long as the
assumption is maintained that there is an upper bound on how much the hyper-
HPs can control the HPs). So, while we do not necessarily encourage
practitioners to use our particular defended EHPO, we do not discourage it
either. The main take away is that practitioners should develop their own
notion of skepticism (appropriate to their particular task) and be explicit
about the assumptions they rely on when selecting hyper-HPs. The way one
chooses hyper-HPs should be defensible.

When in doubt, as a heuristic we recommend using a search space that includes
where an algorithm’s performance starts to degrade (to be assured that a
maximum, even if a local one, has been found). We refer to our dynamic two-
phase protocol (which we describe in detail in Appendix E.3.1) as an example
of how to do this. We first do a broad (but coarse) search. We used grid
search for that initial sweep. Random search may be a better choice for some
tasks. We were familiar with Wilson et al. [72] (and many have written about
it), and felt confident that grid search would capture the space well based on
the results that others have also reported on this task. We then used this
first sweep to determine which hyper-HPs we should use for our second, finer-
grained sweep. We apply our more expensive, defended EHPO for this second
sweep, using the hyper-HPs we selected from the first sweep. In other words,
we spent a bit of time/our compute budget justifying to ourselves that we were
picking reasonable hyperparameters – instead of just picking one grid or range
for random search to sample, and hoping that our results would be
representative of other search spaces.

###  F.2 Broader Impact

As we suggest in Section 5, our work can be considered as related to (but
orthogonal with) with prior studies on reproducibility as advocating for more
robust scientific practices in ML research. In particular, our work
complements prior empirical studies that shine a light on reliability issues
in ML—issues that relate particularly to traditionally underspecified choices
in hyperparameter optimization [11, 65, 48]. In contrast to this prior work,
which illustrates the issue with experiments, we provide a theoretical
contribution that enables ML practitioners and researchers to defend against
unreliable, inconsistent HPO. We provide a theoretically-backed mechanism to
promote and facilitate more trustworthy norms and practices in ML research.

More broadly, our work can be understood as a mechanism for dealing with
_measurement bias_ —the misalignment between what one intends to measure and
what they are actually measuring—for overall ML algorithm performance. While
alleviating measurement bias is by no means novel to more mature branches of
science [28], including other fields of computing [54], until recently it has
been under-explored in ML. Beginning in the last couple of years, measurement
bias is now coming under increased scrutiny with respect to the origins of
empirical gains in ML [53, 22]. In current work, it is often difficult to
disentangle whether the concluded measured performance gains are due to
properties of the training algorithm or to fortuitous HP selection. Our
formalization, rather than allowing HPO choices to potentially obscure
empirical results, provides confidence in the conclusions we can draw about
overall algorithm performance.

Our work also highlights how there is a human element, not a just statistical
one, to bias in ML pipelines: Practitioners make decisions about HPO that can
heavily influence performance (e.g., choice of hyper-hyperparameters). The
human element of biasing solution spaces has been discussed in sociotechnical
writing [23, 66, 14], in AI [52], in the context of “p-hacking” or “fishing
expeditions" for results that fit a desired pattern [25], and was also the
focus of Professor Isbell [40]’s NeurIPS 2020 keynote. Formalizing the process
for how to draw conclusions from HPO, as we do here, has the potential to
alleviate the effects of this type of human bias in ML systems.

Lastly, our insights concerning robustness also extend to growing areas in ML
that use learning to guide hyperparemeter selection, such as meta-learning and
neural architecture search [17, 75, 38, 8, 39]. While the assisting learning
agents in those methods guide choosing hyperparameters for the trained output
model, their own hyperparameters tend to be either manually tuned or chosen
with more traditional HPO strategies, like grid search [74]. In other words,
these processes can exhibit the bias problem discussed above and are therefore
potentially subject to hyperparameter deception, which can be mitigated by the
work we present here.