Applications to Positive Scalar Curvature and General Relativity | Gaoming Wang

The conformal Laplacian

Mon, 29 Jun 2026 00:00:00 +0000

Let \((X^n,g)\) be a closed Riemannian manifold, \(n\geq3\). We use the conformal Laplacian

\[ L_g:=-c_n\Delta^g+R_g, \qquad c_n:=\frac{4(n-1)}{n-2}. \]

Here \(R_g\) is the scalar curvature of \(g\). With our sign convention,

\[ \int_X uL_gu\,d\mu_g = \int_X c_n|\nabla u|^2+R_gu^2\,d\mu_g. \]

\[ \widehat g=u^{\frac{4}{n-2}}g, \qquad u\gt{}0, \]

then this is the same as writing \(\widehat g=e^{2f}g\) with

\[ e^{2f}=u^{\frac{4}{n-2}}, \qquad f=\frac{2}{n-2}\log u. \]

For a general conformal change \(\widehat g=e^{2f}g\), the scalar curvature formula is

\begin{equation} \tag{6.1.1} \label{eq:psc-scalar-exp-transform} R_{\widehat g} = e^{-2f}\bigl( R_g-2(n-1)\Delta^gf-(n-1)(n-2)|\nabla f|^2 \bigr). \end{equation}

We now plug in \(f=\frac{2}{n-2}\log u\). First,

\[ \nabla f=\frac{2}{n-2}\frac{\nabla u}{u}, \qquad |\nabla f|^2 = \frac{4}{(n-2)^2}\frac{|\nabla u|^2}{u^2}, \]

and

\[ \Delta^g f = \frac{2}{n-2}\Delta^g\log u = \frac{2}{n-2} \left( \frac{\Delta^gu}{u} - \frac{|\nabla u|^2}{u^2} \right). \]

Therefore

\[ \begin{aligned} R_{\widehat g} &= u^{-\frac{4}{n-2}} \biggl[ R_g - \frac{4(n-1)}{n-2} \left( \frac{\Delta^gu}{u} - \frac{|\nabla u|^2}{u^2} \right) - \frac{4(n-1)}{n-2} \frac{|\nabla u|^2}{u^2} \biggr] \\ &= u^{-\frac{4}{n-2}} \left( R_g-\frac{4(n-1)}{n-2}\frac{\Delta^gu}{u} \right). \end{aligned} \]

The two \(|\nabla u|^2/u^2\) terms cancel exactly: the first comes from \(-2(n-1)\Delta^g f\), and the second comes from \(-(n-1)(n-2)|\nabla f|^2\). Since \(c_n=\frac{4(n-1)}{n-2}\), we get

\begin{equation} \tag{6.1.2} \label{eq:psc-conformal-laplacian-transform} R_{\widehat g} = u^{-\frac{n+2}{n-2}} \bigl(-c_n\Delta^gu+R_gu\bigr) = u^{-\frac{n+2}{n-2}}L_gu . \end{equation}

Lemma 6.1.1

If the first eigenvalue of \(L_g\) is positive, then \(X\) admits a metric of positive scalar curvature in the conformal class of \(g\).

Proof. Let \(u\gt{}0\) be the first eigenfunction:

\[ L_gu=\lambda_1u, \qquad \lambda_1\gt{}0. \]

For \(\widehat g=u^{4/(n-2)}g\), formula (6.1.2) gives

\[ R_{\widehat g} = \lambda_1u^{-\frac{4}{n-2}}\gt{}0. \]

◻

Lemma 6.1.2

Let \(X^n\) be closed, \(n\geq3\). If \(R_g\geq0\) and \(R_g\gt{}0\) somewhere, then \(X\) admits a metric of positive scalar curvature.

Proof. We show that \(\lambda_1(L_g)\gt{}0\). For every nonzero \(\phi\),

\[ \int_X \phi L_g\phi\,d\mu_g = \int_X c_n|\nabla\phi|^2+R_g\phi^2\,d\mu_g\geq0. \]

Thus \(\lambda_1\geq0\). If \(\lambda_1=0\), let \(u\gt{}0\) be a first eigenfunction. Then

\[ 0=\int_X c_n|\nabla u|^2+R_gu^2\,d\mu_g . \]

Both terms are nonnegative, so \(\nabla u\equiv0\) and \(R_gu^2\equiv0\). Since \(u\gt{}0\), this forces \(R_g\equiv0\), contradicting the assumption that \(R_g\gt{}0\) somewhere. Hence \(\lambda_1\gt{}0\), and Lemma 6.1.1 applies. ◻

Conformal descent on a stable minimal hypersurface

Mon, 29 Jun 2026 00:00:00 +0000

Let \((N^{n+1},g_N)\) be a closed Riemannian manifold and let \(\Sigma^n\subset N\) be a closed, two-sided, stable minimal hypersurface. Denote the induced metric on \(\Sigma\) by \(g\). The stability inequality is

\begin{equation} \tag{6.2.1} \label{eq:psc-stability-hypersurface} \int_\Sigma |\nabla\phi|^2\,d\mu_g \geq \int_\Sigma \bigl(|A|^2+\mathrm{Ric}_N(\nu,\nu)\bigr)\phi^2\,d\mu_g, \qquad \phi\in C^\infty(\Sigma). \end{equation}

The Gauss equation, using \(H=0\), gives

\begin{equation} \tag{6.2.2} \label{eq:psc-gauss-scalar} R_N = R_\Sigma+2\mathrm{Ric}_N(\nu,\nu)+|A|^2. \end{equation}

Combining (6.2.1) and (6.2.2),

\begin{equation} \tag{6.2.3} \label{eq:psc-sy-rearrangement} \int_\Sigma 2|\nabla\phi|^2+R_\Sigma\phi^2\,d\mu_g \geq \int_\Sigma (R_N+|A|^2)\phi^2\,d\mu_g . \end{equation}

This is the basic Schoen–Yau rearrangement.

Proposition 6.2.1

Assume \(R_N\gt{}0\) and \(n\geq3\). Then every closed, two-sided, stable minimal hypersurface \(\Sigma^n\subset N^{n+1}\) admits a metric of positive scalar curvature.

Proof. Since \(\Sigma\) is compact and \(R_N\gt{}0\) on \(N\), there is a number \(\kappa\gt{}0\) such that \(R_N\geq\kappa\) on \(\Sigma\). From (6.2.3),

\[ \int_\Sigma 2|\nabla\phi|^2+R_\Sigma\phi^2\,d\mu_g \geq \kappa\int_\Sigma \phi^2\,d\mu_g . \]

Because \(c_n=4(n-1)/(n-2)\geq2\), we also have

\[ \int_\Sigma c_n|\nabla\phi|^2+R_\Sigma\phi^2\,d\mu_g \geq \kappa\int_\Sigma \phi^2\,d\mu_g . \]

Thus the first eigenvalue of the conformal Laplacian \(L_g\) is positive. The claim follows from Lemma 6.1.1. ◻

When \(n=2\), the same rearrangement gives useful topological information directly. Taking \(\phi\equiv1\) in (6.2.3) gives

\[ \int_\Sigma R_\Sigma\,d\mu_g = 2\int_\Sigma K_\Sigma\,d\mu_g \gt{}0. \]

Hence \(\chi(\Sigma)\gt{}0\) by Gauss–Bonnet. In particular a closed orientable stable minimal surface in a three-manifold with \(R_N\gt{}0\) is a union of two-spheres.

Geroch’s Conjecture by Dimension Reduction

Mon, 29 Jun 2026 00:00:00 +0000

The following theorem is the form of the Geroch conjecture used in the positive mass theorem. The dimension restriction comes only from the regularity theory for area-minimizing hypersurfaces: an area-minimizing hypersurface in an \(n\)-manifold is smooth when \(n\leq7\).

Theorem 6.3.1

Let \(3\leq n\leq7\). Let \(X^n\) be a closed oriented smooth manifold. If there is a continuous map

\[ F:X\to \mathbb{T}^n \]

of nonzero degree, then \(X\) admits no metric of positive scalar curvature.

Proof. We argue by induction on \(n\). The base \(n=2\) is Gauss–Bonnet: if a closed oriented surface maps to \(\mathbb{T}^2\) with nonzero degree, then it has genus at least one, so it cannot carry a metric with positive Gaussian curvature.

Assume now \(3\leq n\leq7\), and suppose the theorem is known in dimension \(n-1\). Let \(g\) be a positive-scalar-curvature metric on \(X\). Write

\[ \theta_1,\ldots,\theta_n\in H^1(\mathbb{T}^n;\mathbb{Z}) \]

for the standard degree-one cohomology classes, and set

\[ \omega_i:=F^*\theta_i\in H^1(X;\mathbb{Z}). \]

Since \(\deg F\neq0\),

\[ \omega_1\smile\cdots\smile\omega_n\neq0 \qquad\text{in }H^n(X;\mathbb{Z}). \]

Let \(\alpha=\omega_n\). Its Poincare dual is a nonzero integral homology class in \(H_{n-1}(X;\mathbb{Z})\). Choose an area-minimizing integral current \(\Sigma\) in this class. Since \(n\leq7\), regularity theory implies that \(\Sigma\) is a smooth embedded closed hypersurface, possibly with several components and integer multiplicities. It is oriented, two-sided, minimal, and stable.

By Proposition 6.2.1, every component of \(\Sigma\) of dimension at least three admits a positive-scalar-curvature metric; in the surface case, Remark 6.2.2 says that every component is a two-sphere.

Now compute the cup product on \(\Sigma\). Since \([\Sigma]\) is Poincare dual to \(\omega_n\),

\[ \int_\Sigma \omega_1\smile\cdots\smile\omega_{n-1} = \int_X \omega_1\smile\cdots\smile\omega_n \neq0. \]

Therefore at least one connected component \(\Sigma_0\) has

\[ \int_{\Sigma_0} \omega_1\smile\cdots\smile\omega_{n-1}\neq0. \]

Equivalently, the map

\[ F_0:=(F_1,\ldots,F_{n-1})|_{\Sigma_0}: \Sigma_0\to\mathbb{T}^{n-1} \]

has nonzero degree.

If \(n-1=2\), this contradicts the fact that \(\Sigma_0\) is a two-sphere. If \(n-1\geq3\), then \(\Sigma_0\) carries a positive-scalar-curvature metric by conformal descent, contradicting the induction hypothesis in dimension \(n-1\). This proves the theorem. ◻

Corollary 6.3.2

For \(3\leq n\leq7\) and for every closed oriented \(n\)-manifold \(Y\), the connected sum

\[ \mathbb{T}^n\#Y \]

admits no metric of positive scalar curvature.

Proof. There is a degree-one map

\[ \mathbb{T}^n\#Y\to\mathbb{T}^n \]

obtained by collapsing \(Y\) minus a ball to the connected-sum point and using the identity map on the torus side. Theorem 6.3.1 applies. ◻

The Positive Mass Theorem and the Reduction to Geroch’s Conjecture

Mon, 29 Jun 2026 00:00:00 +0000

We now explain how the Riemannian positive mass theorem is reduced to Corollary 6.3.2. We state the time-symmetric version, which is the one governed by scalar curvature.

The time-symmetric, or Riemannian, positive mass theorem was first proved by Schoen–Yau by the minimal-hypersurface method [SY79b, SY81]. In the smooth dimension-reduction form of their argument, the regularity theory for area-minimizing hypersurfaces gives the theorem for \(3\leq n\leq7\). Witten then gave a spinorial proof for spin asymptotically flat manifolds, without this regularity dimension restriction [Wit81].

The later history is largely about removing the non-spin dimension restriction. Schoen–Yau proposed an all-dimensional singular minimal-slicing approach [SY19], while Lohkamp’s cut-off/compactification observation relates the Riemannian PMT to Geroch torus rigidity; this is the reduction used below. The generic-regularity work culminating in Chodosh–Mantoulidis–Schulze–Wang pushes the minimal-hypersurface/Lohkamp route to dimension \(11\) [CMSW25]. Bi–Hao–He–Shi–Zhu then proved the Riemannian PMT up to dimension \(19\) [BHH+26], and Brendle–Wang subsequently gave a dimension descent scheme which closes the Riemannian theorem in arbitrary dimension [BW26a]. They also derived the spacetime positive energy theorem in arbitrary dimension from the Riemannian theorem and Jang-type arguments [BW26b].

There are also alternative proofs and stability directions which are useful to keep mentally separate from the reduction below. In dimension three, Bray–Kazaras–Khuri–Stern gave a harmonic-function proof of the Riemannian PMT [BKKS22]. Stability versions ask whether small ADM mass forces the geometry to be close to Euclidean space. Lee–Sormani proved pointed intrinsic-flat stability for rotationally symmetric asymptotically flat manifolds [LS14]. Huang–Lee–Sormani proved pointed intrinsic-flat stability for graphical hypersurfaces in Euclidean space under natural technical hypotheses [HLS17]. Dong–Song proved a three-dimensional stability theorem in the general asymptotically flat setting: if the mass of a chosen end tends to zero, then after removing subsets whose boundary areas tend to zero, the remaining spaces converge to Euclidean 3-space in the pointed measured Gromov–Hausdorff topology [DS25]. The proof below records the classical smooth Schoen–Yau/Lohkamp/Geroch mechanism, because that is the version whose topology is most visible from stable minimal hypersurfaces.

Theorem 6.4.1

Let \(3\leq n\leq7\). Let \((M^n,g)\) be a complete one-ended asymptotically flat Riemannian manifold with nonnegative scalar curvature. Then the ADM mass is nonnegative. If the mass is zero and the usual rigidity hypotheses hold, then \((M,g)\) is isometric to Euclidean space.

For a manifold with several asymptotically flat ends, the same conclusion is applied to one end at a time after the standard reduction which compactifies or fills the other ends without changing the sign of the chosen mass. We focus on the one-ended case because it contains the topological argument.

Recall the ADM mass of an end with asymptotically flat coordinates \(x\):

\[ m_{\operatorname{ADM}}(g) = \frac{1}{2(n-1)\omega_{n-1}} \lim_{r\to\infty} \int_{S_r} (\partial_jg_{ij}-\partial_ig_{jj})\nu^i\,d\sigma . \]

The nonnegativity part is the one whose topology is most transparent.

Definition 6.4.2

Fix an asymptotically flat end and write its coordinate chart as

\[ x=(x^1,\ldots,x^n):M\setminus K\to \mathbb{R}^n\setminus B_R, \qquad r=|x|. \]

The symbol \(\delta\) denotes the Euclidean metric in these coordinates. For a tensor \(T\) on the end and a number \(\tau\gt{}0\), we use

\[ \|T\|_{L^p_{-\tau}}^p := \int_{M\setminus K} \bigl|(1+r)^\tau T\bigr|^p(1+r)^{-n}\,dx, \]

and

\[ \|T\|_{W^{k,p}_{-\tau}} := \sum_{j=0}^k \|\partial^jT\|_{L^p_{-\tau-j}}. \]

Thus \(T\in W^{k,p}_{-\tau}\) means that \(T\) decays like \(r^{-\tau}\) and its \(j\)-th coordinate derivatives decay like \(r^{-\tau-j}\) in weighted \(L^p\) sense.

In particular, an AF metric of Sobolev type \((p,q)\) means

\[ g_{ij}-\delta_{ij}\in W^{2,p}_{-q}, \qquad p\gt{}n, \qquad q\gt{}\frac{n-2}{2}, \]

together with the integrability condition \(R_g\in L^1\) when the ADM mass is used. The condition \(p\gt{}n\) gives enough Sobolev embedding to read the metric and first derivatives pointwise, while \(q\gt{}\frac{n-2}{2}\) is the decay threshold which makes the ADM mass stable under the density deformation.

If one starts instead from the classical pointwise AF assumption of order \(\tau\),

\[ \partial^\alpha(g_{ij}-\delta_{ij}) = O(r^{-\tau-|\alpha|}), \qquad |\alpha|\leq2, \]

then the weighted Sobolev hypothesis follows with every smaller decay rate \(q\lt{}\tau\). Indeed, for \(|\alpha|\leq2\),

\[ \begin{aligned} \|\partial^\alpha(g-\delta)\|_{L^p_{-q-|\alpha|}}^p &\leq C\int_R^\infty r^{p(q+|\alpha|)} r^{-p(\tau+|\alpha|)} \frac{dr}{r} \\ &= C\int_R^\infty r^{-p(\tau-q)-1}\,dr \lt{}\infty . \end{aligned} \]

Thus a \(C^2\) AF end of order \(\tau\gt{}\frac{n-2}{2}\) gives the Sobolev assumption needed below after choosing \(q\) with \(\frac{n-2}{2}\lt{}q\lt{}\tau\). Many analytic statements of the density theorem take this Sobolev condition as the definition of asymptotic flatness.

There are two different ideas which should not be conflated. The original Schoen–Yau proof of the positive mass theorem uses a noncompact minimal hypersurface produced by a limiting Plateau argument. The more topological shortcut described below—cutting off a negative-mass end, compactifying it to a torus, and then contradicting torus rigidity—is usually attributed to Lohkamp’s compactification observation; see [Loh99]. Modern accounts phrase this as follows: using Lohkamp’s idea, one can reduce the Riemannian positive mass theorem to the impossibility of \(R\gt{}0\) on \(\mathbb{T}^n\#X^n\); see the discussion in [LUY24].

Theorem 6.4.3

Let \((M^n,g)\) be complete, one-ended, and asymptotically flat, with \(R_g\geq0\) and negative ADM mass \(m\lt{}0\). Assume the usual decay for which the ADM mass and the weighted elliptic theory are valid, including \(R_g\in L^1\); for instance one may work with the Sobolev AF condition in Definition 6.4.2. Then, for every sufficiently small \(\varepsilon\gt{}0\), there is a complete asymptotically flat metric \(g_\varepsilon\) with the following properties:

\[ R_{g_\varepsilon}\geq0,\qquad |m_{\operatorname{ADM}}(g_\varepsilon)-m|\lt{}\varepsilon, \]

and on some exterior coordinate region \(\{r\geq R_\varepsilon\}\),

\[ g_\varepsilon = u_\varepsilon^{\frac{4}{n-2}}\delta, \qquad \Delta^\delta u_\varepsilon=0, \qquad u_\varepsilon = 1+a_\varepsilon r^{2-n}+O_\infty(r^{1-n}). \]

Moreover

\[ m_{\operatorname{ADM}}(g_\varepsilon)=2a_\varepsilon. \]

Thus, if \(\varepsilon\lt{}-m/2\), then \(a_\varepsilon\lt{}0\). In the positive-mass contradiction argument, we may therefore replace the original metric by one which is conformally flat and scalar-flat near infinity, with negative harmonic mass coefficient.

Here the notation in the conclusion is as follows. The function \(u_\varepsilon\) is a positive harmonic function on the exterior Euclidean coordinate region, and \(a_\varepsilon\) is the coefficient of its leading \(r^{2-n}\) term. The notation \(O_\infty(r^\beta)\) is a shorthand for “big-\(O\) with all derivatives”. More precisely, if \(E=O_\infty(r^\beta)\), then for every multi-index \(\alpha\) there is a constant \(C_\alpha\) such that, in the chosen exterior coordinates,

\[ |\partial^\alpha E| \leq C_\alpha r^{\beta-|\alpha|} \qquad\text{for }r\text{ large}. \]

Thus \(O_\infty(r^{1-n})\) means not only that the error itself is \(O(r^{1-n})\), but also that every coordinate derivative has the corresponding differentiated decay; for example

\[ \partial^\alpha O_\infty(r^{1-n}) = O(r^{1-n-|\alpha|}). \]

This is stronger than ordinary \(O(r^{1-n})\) notation and is the form needed when differentiating the expansion in the ADM mass or scalar-curvature computations. The equality \(m_{\operatorname{ADM}}(g_\varepsilon)=2a_\varepsilon\) is the standard ADM normalization for a conformally flat end \(u_\varepsilon^{4/(n-2)}\delta\).

Proof. This is the standard density step in the positive mass theorem; see [Bar86, LLU23]. We spell out the mechanism because it is the analytic part which precedes Lohkamp’s compactification.

Choose a large radial cut-off \(\chi_\lambda\) on the asymptotically flat end, with \(\chi_\lambda=1\) for \(r\leq\lambda\) and \(\chi_\lambda=0\) for \(r\geq2\lambda\). Let \(g_{\mathrm{Euc}}=\delta\) be the Euclidean metric in the AF coordinates and set

\[ g_\lambda = \chi_\lambda g+(1-\chi_\lambda)g_{\mathrm{Euc}} \]

on the end, extending it by \(g\) on the compact part. Thus \(g_\lambda=g\) on a large compact set and \(g_\lambda=\delta\) for \(r\geq2\lambda\). The only scalar curvature error is in the annulus \(\{\lambda\leq r\leq2\lambda\}\). Define

\[ V_\lambda:=R_{g_\lambda}-\chi_\lambda R_g . \]

The basic estimate is

\begin{equation} \tag{6.4.1} \label{eq:density-error-small} \|V_\lambda\|_{L^p_{-q'-2}} + \|V_\lambda\|_{L^{n/2}} + \|V_\lambda\|_{L^{2n/(n+2)}} \to0 \end{equation}

for every \(q'\lt{}q\) with \(\frac{n-2}{2}\lt{}q'\lt{}n-2\). The point is that the linear part of the scalar curvature at the Euclidean metric is

\[ DR_\delta(k)=\partial_i\partial_jk_{ij}-\partial_j\partial_jk_{ii}. \]

When \(k_\lambda=\chi_\lambda(g-\delta)\), the terms with two derivatives on \(g-\delta\) are exactly \(\chi_\lambda DR_\delta(g-\delta)\); the remaining terms contain at least one derivative of \(\chi_\lambda\) and are supported in the annulus. Since \(|\nabla^k\chi_\lambda|\leq C\lambda^{-k}\) there, and \(g-\delta\in W^{2,p}_{-q}\), these commutator terms go to zero in the weighted norms above. The nonlinear terms in \(R_{g_\lambda}\) contain either \((g_\lambda-\delta)\partial^2g_\lambda\) or \((\partial g_\lambda)^2\), and have the same decay. This proves (6.4.1).

For \(\lambda\) large, solve the conformal correction equation

\begin{equation} \tag{6.4.2} \label{eq:density-conformal-correction} -c_n\Delta^{g_\lambda} w_\lambda+V_\lambda w_\lambda=0, \qquad w_\lambda\to1 \quad\text{on the chosen end}, \end{equation}

where \(c_n=\frac{4(n-1)}{n-2}\). The weighted Fredholm estimate for the Laplacian on an asymptotically flat end, together with the smallness of \(V_\lambda\) in (6.4.1), gives a unique solution with

\[ w_\lambda-1\in W^{2,p}_{-q'},\qquad \|w_\lambda-1\|_{W^{2,p}_{-q'}}\to0 . \]

Equivalently, writing \(w_\lambda=1+\eta_\lambda\), the equation is

\[ \bigl(-c_n\Delta^{g_\lambda}+V_\lambda\bigr)\eta_\lambda = -V_\lambda. \]

The operator on the left is a small perturbation of the AF Laplacian \(-c_n\Delta^{g_\lambda}:W^{2,p}_{-q'}\to L^p_{-q'-2}\), which is an isomorphism for \(0\lt{}q'\lt{}n-2\). Thus a Neumann-series/Fredholm argument solves for \(\eta_\lambda\) and gives the norm estimate above. This is the only analytic input in the deformation step. Since \(p\gt{}n\), weighted Sobolev embedding gives \(\|w_\lambda-1\|_{C^0}\to0\); after increasing \(\lambda\) we therefore have \(\frac12\leq w_\lambda\leq2\). Define

\[ \tilde g_\lambda:=w_\lambda^{\frac{4}{n-2}}g_\lambda . \]

The conformal scalar curvature formula gives

\[ \begin{aligned} R_{\tilde g_\lambda} &= w_\lambda^{-\frac{n+2}{n-2}} \bigl(-c_n\Delta^{g_\lambda}w_\lambda+R_{g_\lambda}w_\lambda\bigr)\\ &= w_\lambda^{-\frac{n+2}{n-2}} (R_{g_\lambda}-V_\lambda)w_\lambda = \chi_\lambda R_g\,w_\lambda^{-\frac{4}{n-2}} \geq0. \end{aligned} \]

Thus the scalar curvature sign is preserved. Since \(w_\lambda\) stays uniformly bounded above and below and \(g_\lambda\) is uniformly equivalent to \(g\), the new metric is complete.

On the region \(r\geq2\lambda\), we have \(g_\lambda=\delta\) and \(V_\lambda=0\), so (6.4.2) becomes

\[ \Delta^\delta w_\lambda=0 . \]

Therefore \(\tilde g_\lambda\) is harmonically flat there:

\[ \tilde g_\lambda=w_\lambda^{\frac{4}{n-2}}\delta, \qquad w_\lambda=1+A_\lambda r^{2-n}+O_\infty(r^{1-n}). \]

Here the “conformal mass formula” is the following elementary consequence of the ADM boundary integral. If an AF metric \(g_0\) has mass \(m_{\rm ADM}(g_0)\) and

\[ \hat g=u^{\frac{4}{n-2}}g_0, \qquad u=1+A r^{2-n}+O_\infty(r^{1-n}), \]

then

\begin{equation} \tag{6.4.3} \label{eq:conformal-mass-adds-two-A} m_{\operatorname{ADM}}(\hat g) = m_{\operatorname{ADM}}(g_0)+2A . \end{equation}

Indeed, in the flat coordinates of the end, \(u^{4/(n-2)}=1+\frac{4A}{n-2}r^{2-n}+O_\infty(r^{1-n})\), so the extra contribution to the ADM integrand is

\[ \partial_j\bigl((u^{4/(n-2)}-1)\delta_{ij}\bigr) - \partial_i\bigl((u^{4/(n-2)}-1)\delta_{jj}\bigr) = -(n-1)\partial_i(u^{4/(n-2)}), \]

and the boundary integral gives \(2A\).

Applying (6.4.3) with \(g_0=g_\lambda\) gives

\[ m_{\operatorname{ADM}}(\tilde g_\lambda) - m_{\operatorname{ADM}}(g_\lambda) = 2A_\lambda . \]

The equation for \(w_\lambda\) then identifies this coefficient. Integrating

\[ -c_n\Delta^{g_\lambda}w_\lambda+V_\lambda w_\lambda=0 \]

over a large coordinate ball and letting the radius tend to infinity, using that \(g_\lambda=\delta\) near infinity, gives

\[ 0 = -c_n\lim_{r\to\infty}\int_{S_r}\partial_\nu w_\lambda\,d\sigma + \int_M V_\lambda w_\lambda\,d\mu_{g_\lambda}. \]

Since \(\partial_\nu w_\lambda=-(n-2)A_\lambda r^{1-n}+O(r^{-n})\) and \(c_n=\frac{4(n-1)}{n-2}\), this becomes

\[ 2A_\lambda = -\frac{1}{2(n-1)\omega_{n-1}} \int_M V_\lambda w_\lambda\,d\mu_{g_\lambda}. \]

Hence

\[ m_{\operatorname{ADM}}(\tilde g_\lambda) - m_{\operatorname{ADM}}(g_\lambda) = -\frac{1}{2(n-1)\omega_{n-1}} \int_M V_\lambda w_\lambda\,d\mu_{g_\lambda}, \]

Here \(g_\lambda\) is exactly Euclidean at infinity, so \(m_{\operatorname{ADM}}(g_\lambda)=0\); the integral above is precisely what produces the new harmonic mass coefficient.

To see that this new mass is close to the original mass, one uses the standard mass-continuity lemma in the density theorem. The construction gives \(\tilde g_\lambda-g\to0\) in \(W^{2,p}_{-q'}\) for every \(\frac{n-2}{2}\lt{}q'\lt{}n-2\), and the scalar curvatures also converge in \(L^1\):

\[ R_{\tilde g_\lambda}-R_g = \chi_\lambda R_g \bigl(w_\lambda^{-\frac{4}{n-2}}-1\bigr) + (\chi_\lambda-1)R_g . \]

The first term goes to zero because \(w_\lambda\to1\) uniformly and \(R_g\) is integrable on the AF end; the second goes to zero because its support escapes to infinity. The ADM boundary integral is continuous under this \(W^{2,p}_{-q'}\) convergence together with \(L^1\) scalar-curvature convergence: this is the usual Bartnik density mass lemma, obtained by writing the mass integrand as the Euclidean linearization of scalar curvature plus quadratic terms, whose tails are integrable when \(q'\gt{}\frac{n-2}{2}\). Hence

\[ m_{\operatorname{ADM}}(\tilde g_\lambda)\to m_{\operatorname{ADM}}(g)=m. \]

Choosing \(\lambda\) large, the mass of \(\tilde g_\lambda\) therefore differs from \(m\) by less than \(\varepsilon\).

It remains only to identify the coefficient \(A_\lambda\) with the ADM mass. Set \(v=w_\lambda^{4/(n-2)}\), so \(\tilde g_{\lambda,ij}=v\delta_{ij}\) near infinity. Since

\[ v = 1+\frac{4A_\lambda}{n-2}r^{2-n}+O(r^{1-n}), \qquad \partial_\nu v=-4A_\lambda r^{1-n}+o(r^{1-n}), \]

we compute

\[ \partial_j\tilde g_{\lambda,ij} - \partial_i\tilde g_{\lambda,jj} = -(n-1)\partial_i v. \]

Substituting in the ADM formula yields

\[ m_{\operatorname{ADM}}(\tilde g_\lambda) = \frac{1}{2(n-1)\omega_{n-1}} \lim_{r\to\infty} \int_{S_r}4A_\lambda(n-1)r^{1-n}\,d\sigma = 2A_\lambda . \]

Taking \(g_\varepsilon=\tilde g_\lambda\) and \(a_\varepsilon=A_\lambda\) proves the theorem. ◻

Lemma 6.4.4

Assume that on an exterior coordinate region \(\{r\geq R_0\}\) the metric is

\[ \begin{gathered} g=u^{\frac{4}{n-2}}\delta,\qquad R_g\geq0,\qquad a\lt{}0,\\ u=1+a r^{2-n}+o(r^{2-n}),\qquad \nabla u=-(n-2)a r^{1-n}\partial_r+o(r^{1-n}). \end{gathered} \]

Then, after increasing \(R_0\) if necessary, one can replace \(u\) outside a large compact set by a smooth positive function \(\bar u\) such that

\[ \bar u=u \quad\text{near the compact core},\qquad \bar u\equiv c\gt{}0 \quad\text{near infinity},\qquad \Delta^\delta\bar u\leq0, \]

with strict inequality somewhere. Therefore

\[ \bar g=\bar u^{\frac{4}{n-2}}\delta \]

has nonnegative scalar curvature on the end, has positive scalar curvature somewhere in the transition region, and is exactly flat near infinity. Since \(\bar u=u\) on a full neighborhood of the inner matching sphere, this replacement glues smoothly to the original metric on the compact part.

Proof. The conformal scalar curvature formula on the flat background gives

\[ R_g = -c_nu^{-\frac{n+2}{n-2}}\Delta^\delta u, \qquad c_n=\frac{4(n-1)}{n-2}. \]

Since \(R_g\geq0\), the conformal factor is superharmonic:

\[ \Delta^\delta u\leq0. \]

The negative coefficient \(a\lt{}0\) says that \(u\) approaches \(1\) from below. More precisely, for \(r\) sufficiently large,

\[ \partial_r u=-(n-2)a r^{1-n}+o(r^{1-n})\gt{}0 . \]

After enlarging the compact core, choose \(\varepsilon\gt{}0\) and radii \(R_1\lt{}R_2\) so large that

\[ u\lt{}1-3\varepsilon \quad\text{on a collar of }S_{R_1}, \qquad u\gt{}1-\varepsilon \quad\text{for }r\geq R_2, \]

and such that \(\partial_r u\gt{}0\) throughout \(R_1\leq r\leq R_2\). Thus the transition set

\[ 1-3\varepsilon\lt{}u\lt{}1-\varepsilon \]

is contained in a compact annulus in the end, and \(|\nabla u|\neq0\) somewhere inside this transition set.

Choose a smooth function \(\Psi:\mathbb{R}\to\mathbb{R}\) with the following properties:

\[ \Psi(t)=t\quad\text{for }t\leq1-3\varepsilon,\qquad \Psi(t)=c\quad\text{for }t\geq1-\varepsilon, \]

where \(c\gt{}0\), and

\[ 0\leq\Psi'\leq1,\qquad \Psi''\leq0, \]

with \(\Psi''\lt{}0\) somewhere in \((1-3\varepsilon,1-\varepsilon)\). Such a function is obtained by choosing a smooth nonincreasing function \(\theta:[1-3\varepsilon,1-\varepsilon]\to[0,1]\) which equals \(1\) near the left endpoint and \(0\) near the right endpoint, and then setting

\[ \Psi(t) = 1-3\varepsilon+\int_{1-3\varepsilon}^t\theta(s)\,ds \]

on the transition interval, with the two constant/identity extensions above.

Now set

\[ \bar u:=\Psi(u) \]

on the end, and keep \(\bar u=u\) on the compact core. This is smooth across the inner matching region because \(\Psi(t)=t\) wherever \(u\leq1-3\varepsilon\). It is positive because \(u\gt{}0\) and \(c\gt{}0\), and it is constant equal to \(c\) near infinity because \(\Psi\) is constant for \(t\geq1-\varepsilon\).

By the chain rule,

\[ \Delta^\delta\bar u = \Psi''(u)|\nabla u|^2+\Psi'(u)\Delta^\delta u \leq0. \]

The inequality is strict somewhere in the transition annulus: there \(|\nabla u|\neq0\) at some point where \(\Psi''(u)\lt{}0\). This proves the desired superharmonic cut-off. Notice that no spherical symmetry is used here; the only inputs are conformal flatness, superharmonicity, and the negative mass asymptotic.

Finally use the conformal scalar curvature formula with flat background metric. For any positive function \(\phi\),

\[ R_{\phi^{4/(n-2)}\delta} = -c_n\phi^{-\frac{n+2}{n-2}}\Delta^\delta\phi, \qquad c_n=\frac{4(n-1)}{n-2}. \]

Applying this to \(\phi=\bar u\) gives

\[ R_{\bar g} = -c_n\bar u^{-\frac{n+2}{n-2}}\Delta^\delta\bar u \geq0, \]

and it is positive somewhere. Since \(\bar u\) is constant near infinity, \(\bar g\) is flat there. ◻

Proposition 6.4.5

If a one-ended asymptotically flat manifold \((M^n,g)\), \(3\leq n\leq7\), has nonnegative scalar curvature and negative ADM mass, then some closed manifold of the form

\[ \mathbb{T}^n\#Y \]

admits a positive-scalar-curvature metric.

Proof. First apply Theorem 6.4.3, choosing the deformation so that the mass remains negative. Then apply Lemma 6.4.4 to the harmonically flat end. We obtain a new complete metric, still denoted by \(g\), with the following properties: \(R_g\geq0\), \(R_g\gt{}0\) somewhere in a compact annulus in the chosen end, and on \(\{r\geq R_2\}\) the metric is

\[ g=c^{\frac{4}{n-2}}\delta \]

for a constant \(c\gt{}0\).

Choose a coordinate radius \(R\) with \(R_0\lt{}R\lt{}R_2\), and let \(K_R:=M\setminus\{r\gt{}R\}\) be the compact manifold obtained by cutting the chosen end at the coordinate sphere \(S_R=\partial B_R\). Then choose \(L\gt{}R_2\) and let

\[ Q_L=[-L,L]^n\subset\mathbb{R}^n. \]

Since every point of \(\partial Q_L\) has Euclidean radius at least \(L\gt{}R_2\), a whole collar of \(\partial Q_L\) lies in the exactly flat region. Remove from \(M\) the part of the chosen end outside \(Q_L\). Equivalently, the remaining compact manifold with corners is

\[ W=K_R\cup_{S_R}(Q_L\setminus B_R), \]

where \(Q_L\setminus B_R\) is read in the asymptotic coordinate chart. The outer boundary of this fundamental domain is the boundary of the cube \(Q_L\).

Now identify opposite faces of \(\partial Q_L\) by the translations

\[ (x^1,\ldots,x^i=L,\ldots,x^n) \sim (x^1,\ldots,x^i=-L,\ldots,x^n), \qquad i=1,\ldots,n. \]

These translations are isometries for the constant flat metric \(c^{4/(n-2)}\delta\). Hence the metric descends smoothly across the identified faces. There is no corner singularity: the cube is only a fundamental domain for the standard smooth quotient \(Q_L/\!\sim\,=\mathbb{T}^n\), and the metric is the translation-invariant flat metric in a neighborhood of the boundary faces.

Let \(X\) be the closed quotient. We next identify its topology. Form the closed manifold

\[ Y:=K_R\cup_{S_R}B_R. \]

On the other hand, after the opposite faces of \(Q_L\) are identified, \(Q_L\) becomes \(\mathbb{T}^n\), and the image of the interior ball \(B_R\) is an embedded ball in this torus. Therefore

\[ X = K_R\cup_{S_R}\bigl((Q_L\setminus B_R)/\!\sim\bigr) \simeq Y\#\mathbb{T}^n . \]

This is the promised torus connected sum.

The scalar curvature statement is local, so it survives the quotient. Thus the induced metric on \(X\) has \(R\geq0\) everywhere and \(R\gt{}0\) somewhere. By Lemma 6.1.2, the closed manifold \(X\) admits a metric with \(R\gt{}0\) everywhere. Since \(X\simeq\mathbb{T}^n\#Y\), this proves the proposition. ◻

Proof. Suppose, to the contrary, that the mass is negative. By Proposition 6.4.5, some \(\mathbb{T}^n\#Y\) admits a positive-scalar-curvature metric. This contradicts Corollary 6.3.2. Hence the ADM mass is nonnegative. ◻

The equality case is less topological but fits the same philosophy. If an asymptotically flat manifold with \(R_g\geq0\) has zero mass and is not Euclidean, one uses a conformal/deformation argument to produce a new asymptotically flat metric with \(R\geq0\) and strictly negative mass. This contradicts the nonnegativity just proved. In the original Schoen–Yau proof this is combined with the strong maximum principle and the regularity theory for the minimizing hypersurfaces.

The proof has three moving parts.

Stability plus the Gauss equation gives

\[ \int_\Sigma 2|\nabla\phi|^2+R_\Sigma\phi^2 \geq \int_\Sigma (R_N+|A|^2)\phi^2. \]

This is the whole reason positive scalar curvature descends to a stable minimal hypersurface.

A nonzero-degree map to a torus supplies cohomology classes whose cup product remains nonzero after passing to a Poincare-dual minimizing hypersurface.
A negative-mass end can be flattened to make a closed positive-scalar-curvature metric on \(\mathbb{T}^n\#Y\), contradicting Geroch.

The Horowitz–Myers Conjecture

Mon, 29 Jun 2026 00:00:00 +0000

The Horowitz–Myers conjecture [HM98] is a positive mass statement for spaces with negative cosmological constant. The point of this section is to explain the Riemannian version proved by Brendle–Hung [BH24], and then to outline how their systolic inequality proves it.

From AdS energy to a Riemannian inequality

Recall first the analogy with the positive mass theorem. For asymptotically flat initial data, Euclidean space is the reference geometry and the expected lower bound for the ADM mass is \(0\). With a negative cosmological constant, the natural reference geometry is anti-de Sitter space. In spacetime language the Einstein equation is

\[ \mathrm{Ric}_{\mathbf g}-\frac12 R_{\mathbf g}\mathbf g+\Lambda\mathbf g =8\pi T. \]

For AdS geometry one takes \(\Lambda\lt{}0\); in our normalization,

\[ \Lambda=-\frac{n(n-1)}2. \]

On a time-symmetric spacelike slice, the second fundamental form vanishes and the Hamiltonian constraint becomes a scalar-curvature condition. In vacuum it is

\[ R_g=-n(n-1), \]

and under the dominant energy condition it becomes

\[ R_g\geq -n(n-1). \]

For asymptotically hyperbolic data with spherical conformal infinity, the ground state is pure AdS, whose time-symmetric slice is hyperbolic space. The Horowitz–Myers phenomenon begins when the conformal infinity is toroidal. In that case the product hyperbolic end is not the expected lowest-energy geometry. One circle direction at infinity can fill in smoothly in the interior, producing the AdS soliton on \(\mathbb R^2\times T^{n-2}\). This metric has negative mass relative to the product hyperbolic end. The conjecture says that, among metrics with the same toroidal asymptotics and \(R_g\geq -n(n-1)\), the AdS soliton has the least possible mass.

The Brendle–Hung mass inequality

Let \(\gamma\) be a flat metric on \(S^1\times T^{n-2}\). The model end is

\[ \bar g=r^{-2}dr^2+r^2\gamma \]

on \((r_0,\infty)\times S^1\times T^{n-2}\). An asymptotically Horowitz–Myers end has an expansion

\[ g=\bar g+r^{2-n}Q+o(r^{2-n}), \]

where \(Q\) is a symmetric \(2\)-tensor on \(S^1\times T^{n-2}\). The quantity

\[ \int_{S^1\times T^{n-2}} n\operatorname{tr}_\gamma Q\,dV_\gamma \]

is the mass term in the normalization used here.

There is also a systolic quantity built into the toroidal end. Let \(\xi:S^1\times T^{n-2}\to S^1\) be the circle projection and let \(\Xi\) be the pullback of the volume form on \(S^1\). Define

\[ \sigma = \inf\left\{ \operatorname{length}_\gamma(\alpha): \alpha\subset S^1\times T^{n-2}\text{ closed and } \int_\alpha \Xi\neq0 \right\}. \]

Thus \(\sigma\) is the length of the shortest loop which winds nontrivially in the distinguished \(S^1\)-direction.

Theorem 6.5.1

Let \(3\leq n\leq7\), and let \(\gamma\) be a flat metric on \(S^1\times T^{n-2}\). Let

\[ \bar{g}=r^{-2}dr^2+r^2\gamma \]

on \((r_0,\infty)\times S^1\times T^{n-2}\), and let \(Q\) be a symmetric \(2\)-tensor on \(S^1\times T^{n-2}\). Suppose \((N,g_N)\) is a smooth \(n\)-manifold such that:

\(N\setminus E\cong (r_0,\infty)\times S^1\times T^{n-2}\) for some compact set \(E\subset N\);
the \(T^{n-2}\)-projection on the end extends smoothly to a map \(N\to T^{n-2}\);
on the end,

\[ |g_N-\bar g-r^{2-n}Q|_{\bar g}=o(r^{-n}), \qquad |\bar D(g_N-\bar g-r^{2-n}Q)|_{\bar g}=o(r^{-n}); \]

\(R_{g_N}\geq -n(n-1)\).

Then

\[ \int_{S^1 \times T^{n-2}} n\,\operatorname{tr}_{\gamma}(Q)\,dV_\gamma \geq - \int_{S^1 \times T^{n-2}} \left( \frac{4\pi}{n\sigma} \right)^n dV_{\gamma}. \]

The left-hand side is the Horowitz–Myers mass term. The right-hand side is the mass of the corresponding AdS soliton. The rigidity statement, proved by Brendle–Hung in a subsequent work [BH25], says that equality forces the metric to be locally isometric to a Horowitz–Myers metric.

The systolic boundary inequality

The mass inequality is proved by cutting off the end and applying a sharp boundary inequality to the resulting compact manifold. We state that inequality in the compact form in which it is used.

Let \(M\) be a compact, connected, orientable \(n\)-manifold with nonempty boundary. Suppose

\[ \xi:\partial M\to S^1,\qquad \theta=(\theta_1,\ldots,\theta_{n-2}):\partial M\to T^{n-2} \]

are smooth maps such that \((\xi,\theta):\partial M\to S^1\times T^{n-2}\) has nonzero degree. Let \(\Xi=\xi^*(d\theta_{S^1})\), where \(d\theta_{S^1}\) is the volume form of the circle. Similarly, let \(\Theta_j=\theta_j^*(d\theta_j)\) denote the pulled-back volume forms from the circle factors of \(T^{n-2}\). Define \(\sigma\) to be the shortest length of a closed curve \(\alpha\subset\partial M\) with \(\int_\alpha\Xi\neq0\). Let \(H_{\partial M}\) be the mean curvature of \(\partial M\) with respect to the outward unit normal \(\eta\).

Theorem 6.5.2

Let \(3\leq n\leq7\), let \(\beta\gt{}n\), and let \(\varphi\in C^\infty(M)\). If

\[ -2 \Delta^M \varphi - \frac{\beta-n+1}{\beta-n} |\nabla^M \varphi|^2 + R_M + \beta(\beta-1) \geq 0, \]

then

\[ 2 \sigma^{\beta} \inf_{\partial M} \left( \langle \nabla^M \varphi, \eta \rangle + H_{\partial M} - (\beta-1) \right) \leq \left( \frac{4\pi}{\beta} \right)^\beta. \]

For the mass theorem, one only needs the scalar-curvature consequence obtained by taking \(\varphi\equiv0\) and letting \(\beta\to n\).

Corollary 6.5.3

With the same topological notation, if \(R_M\geq -n(n-1)\), then

\[ 2 \sigma^{n} \inf_{\partial M} \left( H_{\partial M} - (n-1) \right) \leq \left( \frac{4\pi}{n} \right)^n. \]

The AdS soliton and sharpness

The sharp example is the Horowitz–Myers, or AdS soliton, metric. Geometrically one should picture the \(S^1\)-factor at infinity as a polar-angle direction which collapses smoothly in the interior. Thus the underlying manifold is \(\mathbb R^2\times T^{n-2}\), while the conformal infinity is \(S^1\times T^{n-2}\).

For simplicity, assume that \((S^1,g_{S^1})\) has length \(4\pi/n\). On \((1,\infty)\times S^1\times T^{n-2}\), set

\[ g=\frac{1}{\rho^2(1-\rho^{-n})}d\rho^2 +\rho^2(1-\rho^{-n})g_{S^1}+\rho^2g_{T^{n-2}}. \]

With the substitution

\[ \rho=\left(\cosh \frac{n\tilde\rho}{2}\right)^{2/n}, \]

this becomes

\[ g=d\tilde\rho^2+ \left( \cosh \frac{n \tilde{\rho}}{2} \right)^{4/n} \left[ \left( \tanh \frac{n\tilde{\rho}}{2} \right)^2 g_{S^1} +g_{T^{n-2}} \right]. \]

Near \(\tilde\rho=0\), the first two directions look like

\[ d\tilde\rho^2+\left(\frac{n\tilde\rho}{2}\right)^2g_{S^1}. \]

The choice \(\operatorname{length}(S^1)=4\pi/n\) is exactly the no-cone-angle condition, so the metric extends smoothly across the collapsed circle. The resulting metric has scalar curvature

\[ R_g=-n(n-1). \]

To compare it with the asymptotic model, define \(r\) by

\[ \rho^{n/2}=r^{n/2}\left(1+\frac14 r^{-n}\right). \]

Then

\[ \begin{aligned} g={}&r^{-2}dr^2 +r^2\left(1+\frac14r^{-n}\right)^{4/n-2} \left(1-\frac14r^{-n}\right)^2g_{S^1} +r^2\left(1+\frac14r^{-n}\right)^{4/n}g_{T^{n-2}}\\ ={}&r^{-2}dr^2 +r^2\left(1-\frac{n-1}{n}r^{-n}\right)g_{S^1} +r^2\left(1+\frac1n r^{-n}\right)g_{T^{n-2}} +o(r^{-n}). \end{aligned} \]

Thus the tensor \(Q\) is

\[ Q=-\frac{n-1}{n}g_{S^1}+\frac{1}{n}g_{T^{n-2}}, \]

and

\[ \operatorname{tr}_\gamma Q=-\frac1n. \]

Since \(\sigma=4\pi/n\), the right-hand side in Theorem 6.5.1 is exactly the mass of this metric. Hence the inequality is sharp.

How the Brendle–Hung proof works

We keep the proof in three parts. The first part converts the boundary inequality into the mass inequality. The second part proves the boundary inequality by dimension reduction. The final part is the two-dimensional endpoint, where the sharp constant is computed.

Step 1: from the boundary inequality to the mass inequality.

The goal is to prove Theorem 6.5.1:

\[ \int_{S^1\times T^{n-2}} n\operatorname{tr}_{\gamma}Q\,dV_{\gamma} \geq -\int_{S^1\times T^{n-2}} \left(\frac{4\pi}{n\sigma}\right)^n dV_{\gamma}. \]

The compact input is Corollary 6.5.3. Thus we need to cut off the end of \(N\) in such a way that the boundary mean curvature sees the mass aspect.

Choose \(u\) and a constant \(\mu\) on \(S^1\times T^{n-2}\) by

\[ \Delta^\gamma u+\frac n2\operatorname{tr}_{\gamma}Q+\mu=0, \qquad \int_{S^1\times T^{n-2}}u\,dV_{\gamma}=0. \]

Equivalently,

\[ \int_{S^1\times T^{n-2}} \left(n\operatorname{tr}_{\gamma}Q+2\mu\right)dV_{\gamma}=0. \]

For large \(\widehat r\), cut off the end by the graph

\[ \widehat N = N\cap\{r\leq \widehat r+\widehat r^{3-n}\widehat u\}, \qquad \widehat u=u\circ\pi. \]

The reason for using this graph, rather than the coordinate torus \(\{r=\widehat r\}\), is that the graph makes the first nontrivial term in the mean curvature constant. The two estimates one needs are

\[ D^2\widehat u = D_{\gamma}^2u -r^{-1}(dr\otimes d\widehat u+d\widehat u\otimes dr) +O(r^{-n-1}) \]

and

\[ D^2r=rg-\frac n2 r^{3-n}Q+o(r^{1-n}). \]

Substituting these into the level-set formula

\[ H_{\partial\widehat N} = \frac{\operatorname{tr}_{\partial\widehat N}D^2 (r-\widehat r^{3-n}\widehat u)} {|D(r-\widehat r^{3-n}\widehat u)|} \]

gives

\[ H_{\partial\widehat N} =(n-1)+\widehat r^{-n}\mu+o(\widehat r^{-n}). \]

Therefore Corollary 6.5.3 applied to \(\widehat N\) gives

\[ 2\widehat\sigma^n\widehat r^{-n}\mu \leq \left(\frac{4\pi}{n}\right)^n+o(1), \]

where \(\widehat\sigma\) is the boundary systole on \(\partial\widehat N\). Since

\[ \widehat r^{-2}g_{\partial\widehat N}\to\gamma, \qquad \frac{\widehat\sigma}{\widehat r}\to\sigma, \]

we pass to the limit and recover the mass inequality. Equivalently, one may argue by contradiction: if the mass integral were too negative, then for some \(\varepsilon\gt{}0\),

\[ 2(1-\varepsilon)^{n+1}\sigma^n\mu \geq \left(\frac{4\pi}{n}\right)^n, \]

whereas the boundary inequality gives

\[ 2(1-\varepsilon)\widehat\sigma^n\widehat r^{-n}\mu \leq \left(\frac{4\pi}{n}\right)^n \]

for large \(\widehat r\). Hence \(\widehat\sigma/\widehat r\leq (1-\varepsilon)\sigma\), contradicting \(\widehat\sigma/\widehat r\to\sigma\).

Step 2: dimensional reduction for the systolic inequality.

The goal is now Theorem 6.5.2. The proof reduces the \(n\)-dimensional boundary inequality to a two-dimensional one by repeatedly taking free-boundary hypersurfaces which preserve the relevant cohomological information.

The inductive object is a pair \((\Sigma^k,\varphi_k)\), where \(\Sigma^k\) has boundary and carries the nontrivial topological information coming from \(\Xi,\Theta_1,\ldots,\Theta_{k-2}\). The pair is required to satisfy

\[ E_k(\varphi_k,g_k):= -2\Delta^{\Sigma^k}\varphi_k -\frac{\beta-k+1}{\beta-k} |\nabla^{\Sigma^k}\varphi_k|^2 +R_{\Sigma^k}+\beta(\beta-1)\geq0, \]

and the boundary term is

\[ B_k:=\partial_\eta\varphi_k+H_{\partial\Sigma^k}. \]

Starting with \(\Sigma^n=M\) and \(\varphi_n=\varphi\), the reduction step is:

Reduction proposition.

Assume \(E_k(\varphi_k,g_k)\geq0\) and \(B_k\geq0\). Then there is a compact free boundary hypersurface \(\Sigma^{k-1}\subset\Sigma^k\), stable for the weighted area functional

\[ \mathcal A_k(S)=\int_S e^{\varphi_k}\,dA_{g_k}, \]

and a function \(\varphi_{k-1}\) on \(\Sigma^{k-1}\), such that

\[ E_{k-1}(\varphi_{k-1},g_{k-1})\geq0 \]

and

\[ \partial_\eta\varphi_{k-1} +H_{\partial\Sigma^{k-1}} = \partial_\eta\varphi_k+H_{\partial\Sigma^k}. \]

The hypersurface is chosen in the homology class detected by the remaining forms, so the relevant systole cannot decrease in the direction needed for the final inequality.

Let us indicate why the differential inequality is preserved. Write \(\bar\Sigma=\Sigma^k\), \(\widetilde\Sigma=\Sigma^{k-1}\), and \(\bar\varphi=\varphi_k|_{\widetilde\Sigma}\). All geometric quantities below are computed in the original metric \(g_k\). If \(\widetilde\nu\) is the unit normal of \(\widetilde\Sigma\subset\bar\Sigma\), then the second variation of \(\mathcal A_k\) gives, for every smooth test function \(\zeta\),

\[ \begin{aligned} 0\leq{}& \int_{\widetilde\Sigma} e^{\bar\varphi} \left( |\widetilde\nabla\zeta|^2 -\bigl(\overline{\operatorname{Ric}}(\widetilde\nu,\widetilde\nu) +|\widetilde A|^2\bigr)\zeta^2 +\bar{\nabla}^2\bar\varphi(\widetilde\nu,\widetilde\nu)\zeta^2 \right)\\ &\quad -\int_{\partial\widetilde\Sigma}e^{\bar\varphi} A_{\partial\bar\Sigma}(\widetilde\nu,\widetilde\nu)\zeta^2 . \end{aligned} \]

Here \(\widetilde A\) is the second fundamental form of \(\widetilde\Sigma\subset\bar\Sigma\). The usual Gauss equation and the free-boundary relation give

\[ |\widetilde A|^2+ \overline{\operatorname{Ric}}(\widetilde\nu,\widetilde\nu) = \frac12\left(\bar R-\widetilde R +|\widetilde A|^2+\widetilde H^2\right) \geq \frac12\left(\bar R-\widetilde R+\widetilde H^2\right), \]

and

\[ A_{\partial\bar\Sigma}(\widetilde\nu,\widetilde\nu) = H_{\partial\bar\Sigma}-H_{\partial\widetilde\Sigma}. \]

Thus the stability inequality may be weakened to

\[ \begin{aligned} 0\leq{}& \int_{\widetilde\Sigma} e^{\bar\varphi} \left( |\widetilde\nabla\zeta|^2 -\frac12(\bar R-\widetilde R+\widetilde H^2)\zeta^2 +\bar{\nabla}^2\bar\varphi(\widetilde\nu,\widetilde\nu)\zeta^2 \right)\\ &\quad -\int_{\partial\widetilde\Sigma}e^{\bar\varphi} (H_{\partial\bar\Sigma}-H_{\partial\widetilde\Sigma})\zeta^2 . \end{aligned} \]

By the first-eigenfunction argument for this Robin problem, there is a positive function \(\omega\) on \(\widetilde\Sigma\) and a number \(\lambda\geq0\) such that

\[ \begin{aligned} &-\operatorname{div}^{\widetilde\Sigma} (e^{\bar\varphi}\widetilde\nabla\omega) -\frac12(\bar R-\widetilde R+\widetilde H^2) e^{\bar\varphi}\omega +e^{\bar\varphi} \bar{\nabla}^2\bar\varphi(\widetilde\nu,\widetilde\nu)\omega\\ &\qquad\qquad =\lambda e^{\bar\varphi}\omega\geq0, \end{aligned} \]

with boundary condition

\[ \partial_\eta\omega -(H_{\partial\bar\Sigma}-H_{\partial\widetilde\Sigma})\omega=0. \]

Define

\[ \widetilde\varphi=\varphi_{k-1}:=\bar\varphi+\log\omega. \]

The Schoen–Yau rearrangement of the stability inequality gives

\[ \begin{aligned} &-2\widetilde\Delta\widetilde\varphi+\widetilde R -|\widetilde\nabla\widetilde\varphi|^2 \\ &\qquad\geq -2\bar\Delta\bar\varphi+\bar R -|\bar\nabla\bar\varphi|^2 +|\widetilde\nabla\bar\varphi -\widetilde\nabla\widetilde\varphi|^2 . \end{aligned} \]

The only algebraic point is

\[ |a-b|^2+p|a|^2 \geq \frac{p}{1+p}|b|^2. \]

Taking \(p=1/(\beta-k)\) gives

\[ |\widetilde\nabla\bar\varphi -\widetilde\nabla\widetilde\varphi|^2 +\frac{1}{\beta-k} |\widetilde\nabla\bar\varphi|^2 \geq \frac{1}{\beta-k+1} |\widetilde\nabla\widetilde\varphi|^2 . \]

Therefore

\[ \begin{aligned} &-2\widetilde\Delta\widetilde\varphi+\widetilde R -\left(1+\frac{1}{\beta-k+1}\right) |\widetilde\nabla\widetilde\varphi|^2+\beta(\beta-1)\\ &\qquad\geq -2\bar\Delta\bar\varphi+\bar R -\left(1+\frac{1}{\beta-k}\right) |\bar\nabla\bar\varphi|^2+\beta(\beta-1)\geq0. \end{aligned} \]

This is exactly the inequality \(E_{k-1}\geq0\). The boundary equality follows directly from the Neumann condition for \(\omega\):

\[ \partial_\eta\widetilde\varphi+H_{\partial\Sigma^{k-1}} = \partial_\eta\bar\varphi+H_{\partial\Sigma^k}. \]

Iterating the reduction either stops early, in which case the boundary infimum is already nonpositive and the desired inequality is immediate, or reaches a surface \((\Sigma^2,\varphi_2)\). In the latter case the boundary terms are the same along the construction and, if \(\sigma_2\) is the corresponding systole on \(\partial\Sigma^2\), then \(\sigma\leq\sigma_2\). The two-dimensional estimate therefore gives

\[ \begin{aligned} &2\sigma^\beta \inf_{\partial M} \left(\partial_\eta\varphi+H_{\partial M}-(\beta-1)\right)\\ &\qquad\leq 2\sigma_2^\beta \inf_{\partial\Sigma^2} \left(\partial_\eta\varphi_2 +H_{\partial\Sigma^2}-(\beta-1)\right) \leq \left(\frac{4\pi}{\beta}\right)^\beta . \end{aligned} \]

Step 3: the two-dimensional endpoint and the monotonicity.

The dimension reduction leaves a sharp inequality on a surface. This is the only place where the numerical constant \((4\pi/\beta)^\beta\) is produced explicitly.

Theorem 6.5.4

Let \(\Sigma\) be a compact connected orientable surface with nonempty boundary. Let \(K\) be its Gaussian curvature, \(\kappa\) the geodesic curvature of \(\partial\Sigma\), and \(\eta\) the outward unit normal. Suppose

\[ -2\Delta\psi-\frac{\beta-1}{\beta-2}|\nabla\psi|^2 +2K+\beta(\beta-1)\geq0. \]

Then:

If \(\Sigma\) is diffeomorphic to a disk, then

\[ 2|\partial\Sigma|^\beta \inf_{\partial\Sigma} \left(\partial_\eta\psi+\kappa-(\beta-1)\right) \leq \left(\frac{4\pi}{\beta}\right)^\beta . \]

If \(\Sigma\) is not diffeomorphic to a disk, then

\[ \inf_{\partial\Sigma} \left(\partial_\eta\psi+\kappa-(\beta-1)\right)\leq0. \]

The second case is proved by reducing further to a free-boundary geodesic. The main new estimate is the disk case.

For the disk case define the parallel domains

\[ \Omega_s=\{x\in\Sigma:d(x,\partial\Sigma)\gt{}s\}, \qquad \gamma_s=\partial\Omega_s, \qquad L(s)=\mathcal H^1(\gamma_s), \]

and let \(l=\sup\{s:\Omega_s\neq\emptyset\}\). Put

\[ F(s)=\tanh\frac{\beta s}{2}, \qquad G(s)=\left(\cosh\frac{\beta s}{2}\right)^{\frac{2(\beta-1)}{\beta}}. \]

For almost every \(s\), the curve \(\gamma_s\) is piecewise smooth. If \(\Gamma(s)\) denotes its total curvature, then the comparison geometry gives

\[ L'(s)\leq -\Gamma(s). \]

Since the connected components of \(\Omega_s\) are disks, Gauss–Bonnet gives

\[ 2\pi\leq \Gamma(s)+\int_{\Omega_s}K. \]

Now define

\[ I(s)=2\pi-(\beta-1)F(l-s)L(s) +\int_{\Omega_s}(\Delta\psi-K). \]

The fundamental monotonicity statement is

\[ I'(s)-(\beta-1)F(l-s)I(s)\geq0 \qquad\text{for a.e. }s\in(0,l). \]

The computation is short enough to record the structure. Differentiate \(I\), use \(L'(s)\leq-\Gamma(s)\), use Gauss–Bonnet to replace \(\Gamma(s)\), and use the scalar inequality to control the integral over \(\gamma_s\). The last point is the elementary estimate

\[ \frac{\beta-1}{2(\beta-2)}|\nabla\psi|^2 +\frac{(\beta-1)(\beta-2)}{2}F^2(l-s) \geq (\beta-1)F(l-s)|\nabla\psi|. \]

These inequalities combine to give

\[ I'(s)\geq(\beta-1)F(l-s)I(s). \]

Since

\[ (\log G)'(s)=(\beta-1)F(s), \]

the equivalent formulation is

\[ J(s):=G(l-s)I(s) \qquad\Longrightarrow\qquad J'(s)\geq0. \]

This is the monotonicity one should remember.

Finally \(J(0)\leq J(l)\). Using Gauss–Bonnet on the original disk, this gives

\[ 2\pi \geq G(l)\left( \int_{\partial\Sigma}(\partial_\eta\psi+\kappa) -(\beta-1)F(l)|\partial\Sigma| \right). \]

Let \(\sigma=|\partial\Sigma|\) in the disk case. Write

\[ a:=\inf_{\partial\Sigma} \left(\partial_\eta\psi+\kappa-(\beta-1)\right). \]

The preceding inequality controls the boundary average, hence also the infimum:

\[ a \leq \frac{1}{\sigma}\int_{\partial\Sigma} (\partial_\eta\psi+\kappa-(\beta-1)) \leq \frac{2\pi}{\sigma G(l)}-(\beta-1)(1-F(l)). \]

Multiplying by \(2\sigma^\beta\) gives

\[ 2\sigma^\beta a \leq \frac{4\pi\sigma^{\beta-1}}{G(l)} -2(\beta-1)(1-F(l))\sigma^\beta . \]

Since \(0\leq F(l)\lt{}1\), we have

\[ 1-F(l)\geq \frac12(1-F^2(l)) =\frac12G(l)^{-\frac{\beta}{\beta-1}}. \]

Therefore

\[ 2\sigma^\beta a \leq \frac{4\pi\sigma^{\beta-1}}{G(l)} -(\beta-1)\frac{\sigma^\beta} {G(l)^{\frac{\beta}{\beta-1}}}. \]

Now set

\[ x=\frac{\sigma}{G(l)^{1/(\beta-1)}}. \]

The right-hand side becomes

\[ 4\pi x^{\beta-1}-(\beta-1)x^\beta . \]

This one-variable expression is maximized at \(x=4\pi/\beta\), and its maximum is

\[ \left(\frac{4\pi}{\beta}\right)^\beta . \]

This proves Theorem 6.5.4. Step 2 gives Theorem 6.5.2, Corollary 6.5.3 follows by taking \(\varphi\equiv0\) and \(\beta\to n\), and Step 1 proves the Horowitz–Myers mass inequality.