Minimal hypersurfaces and bordism of positive scalar curvature metrics

Abstract

Let (Y, g) be a compact Riemannian manifold of positive scalar curvature (psc). It is well-known, due to Schoen–Yau, that any closed stable minimal hypersurface of Y also admits a psc-metric. We establish an analogous result for stable minimal hypersurfaces with free boundary. Furthermore, we combine this result with tools from geometric measure theory and conformal geometry to study psc-bordism. For instance, assume $(Y_0,g_0)$ and $(Y_1,g_1)$ are closed psc-manifolds equipped with stable minimal hypersurfaces $X_0 \subset Y_0$ and $X_1\subset Y_1$. Under natural topological conditions, we show that a psc-bordism $(Z,{\bar{g}}) : (Y_0,g_0)\rightsquigarrow (Y_1,g_1)$ gives rise to a psc-bordism between $X_0$ and $X_1$ equipped with the psc-metrics given by the Schoen–Yau construction.

Appendix A

The main goal of this Appendix is to provide technical details we used in the main body of the paper. In Sect. A.1, we recall relevant facts on the minimal graph equation and provide the Schauder estimates we use in the proof of Main Lemma. Section A.2 is dedicated to Theorem 4. Here we recall necessary results on currents and state well-known facts on their compactness and regularity, adapted to our setting. Section A.3 describes a simple doubling method which is a convenient technical tool in the remaining sections. In Sect. A.4, we justify Step 2 from the proof of Claim 3. In Sect. A.5, we discuss regularity issues in dimension 8 and prove Theorem 5 for $n=7$.

1.1 The minimal graph equation

This section is concerned with local properties of hypersurfaces in Riemannian manifolds. Throughout this section we will consider the unit ball in Euclidian space $B=B_1(0)\subset \mathbb {R}^{n+1}$ equipped with a Riemannian metric g and a hypersurface $\Sigma ^n\subset B$. The balls of radius $s>0$ centered at $x\in \Sigma$ induced by g and $g|_\Sigma$ are denoted by $B^g_s(x)\subset B$ and $B^\Sigma _s(x)\subset \Sigma$, respectively. Assume there is a point $x_0\in \Sigma \cap B_{1/4}(0)$.

The following straight-forward Riemannian version of [6, Lemma 2.4] allows us to consider $\Sigma$ locally as a graph over $T_{x_0}\Sigma$.

Lemma 6

There is a constant $\mu _0>0$ so that if g satisfies

$$\begin{aligned} \begin{array}{c} {\displaystyle \sup _{x\in B}}\left| g_{ij}(x)-\delta _{ij}\right| \le \mu _0,\quad {\displaystyle \sup _{x\in B}}\left| \frac{\partial g_{ij}}{\partial x^k}(x)\right| \le \mu _0 \end{array} \end{aligned}$$

(A.1)

for $1\le i,j,k\le n+1$ in standard Euclidian coordinates, then the following holds:

If $s>0$ satisfies

$$\begin{aligned} \begin{array}{c} \mathrm {dist}^\Sigma (x_0,\partial \Sigma )\ge 3s,\quad \sup _{\Sigma }|A_g|^2\le \frac{1}{20 s^2}, \end{array} \end{aligned}$$

then there is an open subset $U\subset T_{x_0}\Sigma \subset \mathbb {R}^{n+1}$, a unit vector $\eta$ normal to $T_{x_0}\Sigma$, and a function $u:U\rightarrow \mathbb {R}$ such that

• (1) $\mathrm {graph}(u,\eta )=B_{2s}^\Sigma (x_0)$;

• (2) $|\nabla u|\le 1$ and $|\nabla \nabla u|\le \frac{1}{s\sqrt{2}}$ hold pointwise.

Moreover, the connected component of $B_s^{g}(x_0)\cap \Sigma$ containing $x_0$ lies in $B_{2s}^\Sigma (x_0)$.

Now we will give a useful expression for the mean curvature of a graph. Let $U\subset \mathbb {R}^n$ be an open set with standard coordinates $x'=(x^1,\ldots ,x^n)$ and let g be a Riemannian metric on $U\times \mathbb {R}\subset \mathbb {R}^{n+1}$. For a function $u:U\rightarrow \mathbb {R}$, consider its graph

$$\begin{aligned} \{(x',u(x'))\in \mathbb {R}^{n+1}:x'\in U\}. \end{aligned}$$

For $i=1,\ldots ,n$, we have the tangential vector fields $E_i=\frac{\partial }{\partial x^i}+\frac{\partial u}{\partial x^i}\frac{\partial }{\partial x^{n+1}}$ and the upward-pointing unit vector field $\nu$ normal to the graph of u. Writing $h_{ij}=g(E_i,E_j)$ for the restriction metric, the mean curvature of the graph can be written

$$\begin{aligned} \begin{array}{lcl} H_g&{}= &{} h^{ij}g(\nu ,\nabla _{E_i}E_j)\\ \\ &{}= &{} \left( g^{ij}-\frac{\nabla ^i u\nabla ^j u}{1+|\nabla u|^2}\right) \Big [\frac{\partial ^2u}{\partial x_i\partial x_j}+\Gamma ^{n+1}_{ij}+\frac{\partial u}{\partial x_i}\Gamma ^{n+1}_{n+1\; j}+\frac{\partial u}{\partial x_j}\Gamma ^{n+1}_{n+1\; i}+\frac{\partial u}{\partial x_i} \frac{\partial u}{\partial x_j}\Gamma ^{n+1}_{n+1\; n+1} \\ \\ &{} &{} -\frac{\partial u}{\partial x_r}\left( \Gamma ^r_{ij}+\frac{\partial u}{\partial x_i}\Gamma ^r_{n+1\; j}+\frac{\partial u}{\partial x_j}\Gamma ^r_{i\; n+1}+\frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j}\Gamma ^r_{n+1\; n+1}\right) \Big ], \end{array} \end{aligned}$$

(A.2)

see [6, Section 7.1] for a detailed exposition in the 3-dimensional case.

Next, we will state a general version of the Schauder estimates for elliptic operators on Euclidian space. It is applied to the geometric setting in Sect. 3.

Theorem 7

[16, Corollary 6.3] Let $U\subset \mathbb {R}^n$ be an open set and let $\alpha \in (0,1)$. Suppose $u\in C^{2,\alpha }(U)$ satisfies a uniformly elliptic equation

$$\begin{aligned} Lu=a^{ij}(x)u_{ij}+b^i(x)u_i+c(x)u=0 \end{aligned}$$

with $a^{ij},b^i,c\in C^\alpha (U)$ and ellipticity constant $\lambda >0$. If $U'\subset \subset U$ with $\mathrm {dist}^{U}(U',\partial U)= d$, then there is a constant $C>0$, depending on $d,\lambda ,||a^{ij}||_{C^\alpha (U)}, ||b^i||_{C^\alpha (U)},||c||_{C^\alpha (U)},n$, and $\alpha$, such that

$$\begin{aligned} ||u||_{C^{2,\alpha }(U')}\le C||u||_{C^0(U)}. \end{aligned}$$

(A.3)

Corollary 3

Suppose the unit ball $B=B_1(0)\subset \mathbb {R}^{n+1}$ is equipped with a Riemannian metric g satisfying

$$\begin{aligned} \sup _{x\in B}\left| g_{ij}(x)-\delta _{ij}\right| \le \mu _0,\quad \sup _{x\in B}\left| \frac{\partial g_{ij}}{\partial x^k}(x)\right| \le \mu _0 \end{aligned}$$

in Euclidian coordinates for all $1\le i,j,k\le n+1$ where $\mu _0$ is the constant from Lemma 6. Let $C>0$ be given and set $r=\min (\frac{1}{8},\frac{1}{\sqrt{80C}})$. Assume that $\Sigma \subset B$ is a properly embedded minimal hypersurface with respect to g such that $\sup _B|A^g|^2\le C$ and there is a point $x_0\in B_{r}(0)\cap \Sigma$. Then there is a smooth function $u:U\rightarrow \mathbb {R}$ on $U\subset T_{x_0}\Sigma$ and a unit normal vector $\eta$ to $T_{x_0}\Sigma$ such that

• (1) $\mathrm {graph}(u,\eta )=B^\Sigma _{2r}(x_0)$;

• (2) $|\nabla u|\le 1$ and $|\nabla \nabla u|\le \frac{1}{s\sqrt{2}}$ hold pointwise;

• (3) for each $k\ge 1$ and $\alpha \in (0,1)$ there is a constant $C^\prime >0$, depending only on $n,k,\alpha ,$ and $||g||_{C^{k,\alpha }(B)}$, such that

  $$\begin{aligned} ||u||_{C^{k,\alpha }(U)}\le C^\prime . \end{aligned}$$

  (A.4)

Moreover, the connected component of $B_r(x_0)\cap \Sigma$ containing $x_0$ is contained in $B^\Sigma _{2r}(x_0)$.

Proof

The choice of radius r allows us to apply Lemma 6 to obtain an open subset $U\subset T_{x_0}\Sigma \subset \mathbb {R}^{n+1}$, a unit vector $\eta$ normal to $T_{x_0}\Sigma$, and a smooth function $u:U\rightarrow \mathbb {R}$ such that $\mathrm{graph}(u,\eta ) =B_{2s}^\Sigma (x_0)$, $|\nabla u|\le 1$, and $|\nabla \nabla u|\le \frac{1}{s\sqrt{2}}$ on U. Since $\Sigma$ is minimal, u solves the equation $H=0$. Now since $||u||_{C^{1,\alpha }(U)}$ is bounded for any fixed $\alpha \in (0,1)$, one can inspect the expression A.2 to see that u solves a linear elliptic equation with coefficients bounded in $C^\alpha$ in terms of $\mu _0$ and r. This allows us to apply Theorem 7 to obtain the estimate $||u||_{C^{2,\alpha }(U')}\le C||u||_{C^0(U)}$ for some $C>0$ depending only on $\mu _0$ and r. Standard elliptic estimates [16, Section 6] give a similar estimate in the $C^{k,\alpha }$-norm for any k. $\square$

1.2 Details on Theorem 4

Let us recall some basic notions from theory of integer multiplicity currents. The main reference for this material is [14, Chapter 4].

For an open subset $U\subset \mathbb {R}^{n+k}$, let $\Omega ^n(U)$ denote the space of all n-forms on $\mathbb {R}^{n+k}$ with compact support in U. An n -current on U is a continuous linear functional $T:\Omega ^n(U)\rightarrow \mathbb {R}$ and collection of such T for a vector space $\mathcal {D}_{n}(U)$. The boundary of an n-current T is the $(n-1)$-current $\partial T$ defined by

$$\begin{aligned} (\partial T)(\omega )=T(d\omega ),\quad \quad \omega \in \Omega ^{n-1}(U). \end{aligned}$$

The mass of $T\in \mathcal {D}_{n}(U)$ is given by $\mathbf {M}(T)=\sup \{T(\omega ):\omega \in \Omega ^{n}(U),|\omega |\le 1\}$. For example, if T is given by integration along a smooth oriented submanifold M, then $\mathbf {M}(T)=\mathrm {Vol}(M)$.

Let $\mathcal {H}^n$ denote the n-dimensional Hausdorff measure on $\mathbb {R}^{n+k}$. A current $T\in \mathcal {D}_n(U)$ is called integer multiplicity rectifiable if it takes the form

$$\begin{aligned} T(\omega )=\int _M\omega (\xi (x))\theta (x) d\mathcal {H}^n(x),\quad \quad \omega \in \Omega ^n(U), \ \ \ \hbox {where} \end{aligned}$$

(A.5)

• (1) $M\subset U$ is $\mathcal {H}^n$-measurable and countably n-rectifiable, see [14, Section 3.2.14];

• (2) $\theta :M\rightarrow \mathbf {Z}$ is locally $\mathcal {H}^n$-integrable;

• (3) for $\mathcal {H}^n$-almost every $x\in M$, $\xi :M\rightarrow \Lambda ^nT\mathbb {R}^{n+k}$ takes the form $\xi (x)=e_1\wedge \cdots \wedge e_n$ where $\{e_i\}_{i=1}^n$ form an orthonormal basis for the approximate tangent space $T_xM$, see [14, Section 3.2.16].

Remark

The above definition of integer multiplicity rectifiable currents can also be extended to Riemannian manifolds (M, g) – one defines the mass of a current using the Hausdorff measure given by the metric g.

The regular set $\mathrm {reg}(T)$ of a rectifiable n-current T is given by the set of points $x\in \mathrm {spt}(T)$ for which there exists an oriented n-dimensional oriented $C^1$-submanifold $M\subset U$, $r>0$, and $m\in \mathbf {Z}$ satisfying

$$\begin{aligned} T|_{B_r(x)}(\omega )=m\cdot \int _{M\cap B_r(x)}\omega ,\quad \quad \forall \omega \in \Omega ^n(U). \end{aligned}$$

The singular set $\mathrm {sing}(T)$ is given by $\mathrm {spt}(T){\setminus }\mathrm {reg}(T)$. The abelian group of n-dimensional integral flat chains on U is given by

$$\begin{aligned} \mathcal {F}_n(U)=\{R+\partial S:R\in \mathcal {D}_n(U)\text { and } S\in \mathcal {D}_{n+1}(U)\text { are rectifiable}\}. \end{aligned}$$

Now we consider subsets $B\subset A\subset U$. We have the group of integral flat cycles

$$\begin{aligned} \mathcal {C}_n(A,B)=\{T\in \mathcal {F}_n(U):\mathrm {spt}(T)\subset A,\mathrm {spt}(\partial T)\subset B,\text { or }n=0\} \end{aligned}$$

and the subgroup of integral flat boundaries

$$\begin{aligned} \mathcal {B}_n(A,B)=\{T+\partial S:T\in \mathcal {F}_n(U),\mathrm {spt}(T)\subset B, S\in \mathcal {F}_{n+1}(U),\mathrm {spt}(S)\subset A\}. \end{aligned}$$

The quotient groups $\mathbf {H}_n(A,B)=\mathcal {C}_n(A,B)/\mathcal {B}_n(A,B)$ are the n-dimensional integral current homology groups.

There is a natural transformation between the integral singular homology functor and the integral current homology functor which induces an isomorphism $H_n(A,B;\mathbf {Z})\cong \mathbf {H}_n(A,B)$ in the category of local Lipschitz neighborhood retracts, see [14, Section 4.4.1]. This isomorphism can be combined with a basic compactness result for rectifiable currents to find volume minimizing representatives of homology classes.

Lemma 7

Let $(M,{\bar{g}})$ be a compact $(n+1)$-dimensional Riemannian manifold with boundary and consider an integral homology class $\alpha \in H_n(M,\partial M;\mathbf {Z})$. Let $\tilde{\alpha }\in \mathbf {H}_n(M,\partial M)$ be the image of $\alpha$ under the isomorphism $H_n(M,\partial M;\mathbf {Z})\rightarrow \mathbf {H}_n(M,\partial M)$. Then there exists a homologically volume minimizing integer multiplicity rectifiable current $T\in {\tilde{\alpha }}$.

Proof

By the Nash embedding theorem there is an isometric embedding $\iota :M\rightarrow \mathbb {R}^{n+k}$ for some sufficiently large k. Let ${\hat{M}}$ be the image of this embedding and set ${\hat{\alpha }}=\iota _*{\tilde{\alpha }}\in \mathbf {H}_n({\hat{M}},\partial {\hat{M}})$. Applying the compactness result in [14, Section 5.1.6], we obtain a homologically volume minimizing current ${\hat{T}}\in \mathcal {C}_n({\hat{M}},\partial {\hat{M}})$ representing ${\hat{\alpha }}$. Since $\iota$ is an isometry, $(\iota ^{-1})_*{\hat{T}}$ is the desired current. $\square$

Since Lemma 7 guarantees the existence of homologically volume minimizing representative for the homology class $\alpha$ from the hypothesis of Theorem 4, the final ingredient is regularity theory for volume minimizing rectifiable currents with free boundary. The following is a regularity theorem due to M. Grünter [17, Theorem 4.7] adapted to the context of an ambient Riemannian metric. See [15, 21, 28] for Riemannian adaptations of similar results.

Theorem 8

Let $S\subset \mathbb {R}^{n+1}$ be an n-dimensional smooth submanifold, $U\subset \mathbb {R}^{n+1}$ an open set with $\partial S\cap U=\emptyset$, and g a Riemannian metric on U with bounded injectivity radius and sectional curvature. Suppose $T\in \mathcal {F}_n(U)$ with $\mathrm {spt}(\partial T)\subset S$ satisfies $\mathbf {M}_g(T)\le \mathbf {M}_g(T+R)$ for all open $W\subset \subset U$ and all $R\in \mathcal {F}_n(U)$ with $\mathrm {spt}(R)\subset W$ and $\mathrm {spt}(\partial R)\subset S$. Then we have

• $\mathrm {sing}(T)=\emptyset$ if $n\le 6;$

• $\mathrm {sing}(T)$ is discrete for $n=7;$

• $\mathrm {dim}_{\mathcal {H}}(\mathrm {sing}(T))\le n-7$ if $n>7,$

where $\mathrm {dim}_{\mathcal {H}}(A)$ denotes the Hausdorff dimension of a subset $A\subset U$.

We will briefly explain how Theorem 4 follows from Theorem 8. Let T be the volume minimizing representative of ${\bar{\alpha }}$ from Theorem 4. For a point $x\in \mathrm {spt}(T)$, set $\phi =\exp _x^{{\bar{g}}}$ and consider

$$\begin{aligned} U= & {} \phi ^{-1}(B^{{\bar{g}}}_{r'}(x))\subset T_x M,\quad S= \phi ^{-1}(\partial M\cap B^{{\bar{g}}}_{r}(x)),\\ T'= & {} (\phi ^{-1})_*T\in \mathcal {D}_n(U),\quad g=(\phi ^{-1})_*{\bar{g}}, \end{aligned}$$

where $0<r'<r\le \mathrm {inj}({\bar{g}})$. By Theorem 8, the singular set of $T'$ is empty and so there is a neighborhood V of $0\in U$ such that $T'|_V$ is given by an integer multiple of integration along a $C^1$-submanifold $M\subset V$. Locally, M can be written as the graph of a $C^1$-function which weakly solves the minimal surface equation. Standard elliptic PDE methods imply that M is smooth, see, for instance the proof of Lemma 8 below.

1.3 Doubling minimal hypersurfaces with free boundary

In this section we consider the reflection of a free boundary stable minimal hypersurface over its boundary. To fix the setting, let $(M,{\bar{g}})$ be an $(n+1)$-dimensional compact oriented Riemannian manifold with boundary $\partial M$ and restriction metric $g={\bar{g}}|_{\partial M}$. Assume that there is a neighborhood of the boundary on which ${\bar{g}}=g_{\partial M}+dt^2$. The double of $(M,{\bar{g}})$ is the smooth closed manifold $M_\mathcal {D}$ given by $M_\mathcal {D}=M\cup _{\partial M}(-M).$ Notice that the double $M_\mathcal {D}$ comes equipped with an involution $\iota :M_\mathcal {D}\rightarrow M_\mathcal {D}$ which interchanges the two copies of M and fixes the doubling locus $\partial M\subset M_\mathcal {D}$. Since ${\bar{g}}$ splits as a product near the boundary, one can also form the smooth doubling of ${\bar{g}}$, denoted by ${\bar{g}}_\mathcal {D}$, by setting ${\bar{g}}_\mathcal {D}={\bar{g}}$ on M and ${\bar{g}}_\mathcal {D}=\iota _*{\bar{g}}$ on $-M$.

Lemma 8

Let $(M,{\bar{g}})$ be a compact oriented Riemannian manifold with boundary with ${\bar{g}}=g+dt^2$ near $\partial M$. If $\Sigma \subset M$ be a properly embedded minimal hypersurface with free boundary, then double of $\Sigma$, given by $\Sigma _{\mathcal {D}}= \Sigma \cup _{\partial \Sigma }\iota (\Sigma )$ is a smooth minimal hypersurface of $(M_\mathcal {D},{\bar{g}}_\mathcal {D})$. Moreover, if $\Sigma$ is stable, then so is $\Sigma _\mathcal {D}$.

Proof

First, we will show that $\Sigma _\mathcal {D}$ is a smooth hypersurface. Clearly, $\Sigma _\mathcal {D}$ is smooth away from the doubling locus $\partial \Sigma \subset M_\mathcal {D}$. Let $x_0\in \partial \Sigma$ and let $r>0$ be less than the injectivity radius of ${\bar{g}}_{\mathcal {D}}$. Set $\phi =\exp _{x_0}^{{\bar{g}}_\mathcal {D}}$ and consider

$$\begin{aligned} {\hat{\Sigma }}=\phi ^{-1} (\Sigma \cap B_{r}(x_0)),\quad {\hat{\Sigma }}_{\mathcal {D}}=\phi ^{-1} (\Sigma _{\mathcal {D}}\cap B_{r}(x_0)), \quad {\hat{g}}=\phi ^*{\bar{g}}_\mathcal {D} \end{aligned}$$

and $\nu$, the unit normal vector field to ${\hat{\Sigma }}$ with respect to ${\hat{g}}$. Evidently, ${\hat{\Sigma }}$ is a minimal hypersurface in $T_{x_0}M_\mathcal {D}$ with free boundary contained in $T_{x_0}\partial M\subset T_{x_0}M_\mathcal {D}$ with respect to ${\hat{g}}$. We choose an orthonormal basis for $T_{x_0}M_\mathcal {D}$ so that, writing $x\in T_{x_0}M$ as $(x^1,\ldots ,x^{n+1})$ in this basis,

• (1) $T_{x_0}\partial {\hat{\Sigma }}=\{(x^1,\ldots ,x^{n-1},0,0)\}$;

• (2) $T_{x_0}{\hat{\Sigma }}=\{(x^1,\ldots ,x^n,0)\}$;

• (3) $T_{x_0}\partial M=\{(x^1,\ldots ,x^{n-1},0,x^{n+1})\}$.

This can be accomplished since $\Sigma$ meets $\partial M$ orthogonally. In these coordinates, the involution $\iota$ now takes the form $(x^1,\ldots ,x^n,x^{n+1})\mapsto (x^1,\ldots ,-x^n,x^{n+1})$. Notice that, because the second fundamental form of $\partial M$ vanishes, $\phi ^{-1}(\partial M\cap B_{r}(x_0))$ is contained in the hyperplane $\{(x^1,\ldots ,x^{n+1}):x^n=0\}$.

For a radius $r^\prime <r$, we consider the n-dimensional ball

$$\begin{aligned} B_{r^\prime }^n(0)=\{x\in T_{x_0}M:x^{n+1}=0,||x||< r^\prime \}, \end{aligned}$$

the n-dimensional half-ball $B_{r^\prime ,+}^n(0)=\{x\in B_{r^\prime }^n(0):x^n\ge 0\},$ and the cylinder

$$\begin{aligned} C_{r^\prime }(0)=\{x\in T_{x_0}M:(x^1,\ldots ,x^n,0)\in B_{r^\prime }^n(0)\}. \end{aligned}$$

For small enough $r^\prime$, we may write ${\hat{\Sigma }}\cap C_{r^\prime }(0)$ as the graph of a function

$$\begin{aligned} u:B_{r^\prime ,+}^n(0)\rightarrow \mathbb {R},\quad \mathrm {graph}(u,\partial _{x^{n+1}})={\hat{\Sigma }}\cap C_{r^\prime }(0). \end{aligned}$$

Now we may form the doubling of u to a function $u_\mathcal {D}:B_{r^\prime }^n(0)\rightarrow \mathbb {R}$, setting

$$\begin{aligned} u_\mathcal {D}(x^1,\ldots ,x^n)={\left\{ \begin{array}{ll}u(x^1,\ldots ,x^n)&{}\text { if }x^n\ge 0\\ u(x^1,\ldots ,x^{n-1},-x^n)&{}\text { if }x^n<0. \end{array}\right. } \end{aligned}$$

To show $\Sigma _\mathcal {D}$ is smooth at $x_0$, it suffices to show that $u_\mathcal {D}$ is smooth along $\{x\in B_{r^\prime }^n(0):x^n=0\}$.

From the free boundary condition, we have $\frac{\partial u}{\partial {x^n}}\equiv 0$ on $\{x^n=0\}$ and so $u_\mathcal {D}$ has a continuous derivative on all of $B_{r^\prime }^n(0)$. Since ${\hat{\Sigma }}$ is smooth and minimal, $u_\mathcal {D}$ is smooth and solves the minimal graph equation (A.2) with respect to the metric ${\hat{g}}_\mathcal {D}$ in the strong sense on $\{x\in B_{r^\prime }^n(0):x^n\ne 0\}$. Moreover, it follows from $\frac{\partial u}{\partial {x^n}}\equiv 0$ on $\{x^n=0\}$ and the $\iota$-invariance of ${\bar{g}}_\mathcal {D}$ that $u_\mathcal {D}$ solves the minimal graph equation weakly on the entire ball $B^n_{r'}(0)$.

From this point, the smoothness of $u_\mathcal {D}$ is a standard application of tools from nonlinear elliptic PDE theory, so we will be brief (see [6, Lemma 7.2]). Standard estimates for minimizers implies $u_\mathcal {D}\in H^2(B^n_{r^\prime }(0))$ (see [13, Section 8.3.1]). Writing the equation (A.2) in divergence form, we have

$$\begin{aligned} \frac{\partial }{\partial x^i}\left( a^{ij}\frac{\partial u_\mathcal {D}}{\partial x^j}+b^i u_\mathcal {D}\right) =0, \end{aligned}$$

(A.6)

where the coefficients $a^{ij}$ and $b^i$ depend on $u_\mathcal {D}$ and are only differentiable. Since $u_\mathcal {D}$ weakly solves equation (A.6),

$$\begin{aligned} \int _{B^n_{r^\prime }(0)}\left( a^{ij}\frac{\partial u_\mathcal {D}}{\partial x^j}+b^i u_\mathcal {D}\right) \frac{\partial \psi }{\partial x^i} dx=0 \end{aligned}$$

for any test function $\psi \in C_0^\infty (B^n_{r^\prime }(0))$. Taking $\psi$ to be of the form $-\frac{\partial w}{\partial x^k}$ for some function w and integrating by parts, one finds $\frac{\partial u_\mathcal {D}}{\partial x^k}$ is a weak solution of a uniformly elliptic linear equation with $L^\infty$ coefficients for each $k=1,\ldots ,n$.

Now we may apply the DeGiorgi-Nash theorem (see [16, Theorem 8.24]) to conclude that, for each $r^{\prime \prime }<r^\prime$ there is an $\alpha \in (0,1)$ such that $\frac{\partial u_\mathcal {D}}{\partial x^k}\in C^{0,\alpha }(B^n_{r^{\prime \prime }}(0))$ for each $k=1,\ldots ,n$. Now $u_{\mathcal {D}}\in C^{1,\alpha }(B^n_{r^{\prime \prime }}(0))$ and the functions $\frac{\partial u_\mathcal {D}}{\partial x^k}$ solve a uniformly elliptic linear equation with Hölder coefficients. The Schauder estimates from Theorem 7 allow us to conclude that $\frac{\partial u_\mathcal {D}}{\partial x^k}\in C^{2,\alpha }(B_{r^\prime }(0))$. This argument may be iterated, see [16, Section 8], to conclude $u_\mathcal {D}\in C^{k,\alpha }(B^n_{r^{\prime \prime }}(0))$ for any k. This finishes the proof that $u_\mathcal {D}$ is a smooth solution to the mean curvature equation across the doubling locus $\{x^n=0\}$ and hence $\Sigma _\mathcal {D}$ is a smooth minimal hypersurface.

The last step is to show that $\Sigma _\mathcal {D}$ is stable. Let $\phi \in C^\infty (\Sigma _\mathcal {D})$ define a normal variation and write $\phi =\phi _0+\phi _1$ where $\phi _0$ is invariant under the involution and $\phi _1$ is anti-invariant under the involution. Now we will consider the second variation of the volume of $\Sigma _\mathcal {D}$ with respect to $\phi$.

$$\begin{aligned} \delta _\phi ^2(\Sigma _\mathcal {D})&=\int _{\Sigma _\mathcal {D}}|\nabla \phi |^2-\phi ^2(\mathrm {Ric}(\nu ,\nu )+|A|^2)d\mu \\&=\int _{\Sigma _\mathcal {D}}|\nabla \phi _0|^2+2g(\nabla \phi _0,\nabla \phi _1)+|\nabla \phi _1|^2- (\phi _0^2+2\phi _0\phi _1+\phi _1^2)(\mathrm {Ric}(\nu ,\nu )+|A|^2)d\mu \\ {}&=\delta _{\phi _0}^2(\Sigma _\mathcal {D})+\delta _{\phi _1}^2(\Sigma _\mathcal {D})+ \int _{\Sigma _\mathcal {D}}2g(\nabla \phi _0,\nabla \phi _1)-2\phi _0\phi _1(\mathrm {Ric}(\nu ,\nu )+|A|^2)d\mu \\&=2\delta _{\phi _0|_\Sigma }^2(\Sigma )+2\delta _{\phi _1|_\Sigma }^2(\Sigma ) \ge 0 \end{aligned}$$

where the last equality follows from the fact that $g(\nabla \phi _0,\nabla \phi _1)$ and $\phi _0\phi _1$ are anti-invariant under the involution. This completes the proof of Lemma 8. $\square$

1.4 Second fundamental form bounds

In this section, we will prove Step 2 in Sect. 3.6. Let $(M_i,{\bar{g}}_i)$ and $W_i$ be as in Main Lemma. The uniform second fundamental form bounds for the stable minimal hypersurfaces $W_i\subset M_i$ can be deduced from a classical estimate due to Schoen–Simon [27] for stable minimal hypersurfaces in Riemannian manifolds. In the following, $(M,{\bar{g}})$ is a complete $(n+1)$-dimensional Riemannian manifold, $x_0\in M$, $\rho _0\in (0,\mathrm {inj}_{{\bar{g}}}(x_0))$, and $\mu _1$ is a constant satisfying

$$\begin{aligned} \begin{array}{c} \sup _{B_{\rho }(0)}\left| \frac{\partial {\bar{g}}_{ij}}{\partial x^k}\right| \le \mu _1,\quad \sup _{B_{\rho }(0)}\left| \frac{\partial ^2 {\bar{g}}_{ij}}{\partial x^k\partial x^l}\right| \le \mu _1^2, \end{array} \end{aligned}$$

(A.7)

on the metric ball $B_{\rho _0}(x_0)$ in geodesic normal coordinates $(x^1,\ldots ,x^{n+1})$ centered at $x_0$.

Theorem 9

(Corollary 1 [27]) Suppose $\Sigma$ is an oriented embedded $C^2$-hypersurface in an $(n+1)$-dimensional Riemannian manifold $(M,{\bar{g}})$ with $x_0\in \overline{\Sigma }$, $\mu _1$ satisfies (A.7), and $\mu$ satisfies the bound $\rho _0^{-n}\mathcal {H}^n(\Sigma \cap B_{\rho _0}(x_0))\le \mu$. Assume that

$$\begin{aligned}\mathcal {H}^n(\Sigma \cap B_{\rho _0}(x_0))<\infty \quad \mathrm{and}\quad \mathcal {H}^{n-2}(\mathrm {sing}(\Sigma )\cap B_{\rho _0}(x_0))=0. \end{aligned}$$

If $n\le 6$ and $\Sigma$ is stable in $B_{\rho _0}(x_0)$, then

$$\begin{aligned} \sup _{B_{\rho _0/2}(x_0)}|A^\Sigma |\le \frac{C}{\rho _0}, \end{aligned}$$

where C depends only on n, $\mu$, and $\mu _1\rho _0$.

Proof of Step 2

By Lemma 8, the doubling $(W_i)_\mathcal {D}$ is a smooth stable minimal hypersurface of $(M_i)_\mathcal {D}$. In particular, the singular set of $(W_i)_\mathcal {D}$ is empty. Moreover, the manifolds $(M_i)_\mathcal {D}$ have uniformly bounded geometry so that the injectivity radius is uniformly bounded from below by some $\rho _0>0$, and there is a constant $\mu _1$ so that the bounds (A.7) hold in normal coordinates about any $x\in (M_i)_\mathcal {D}$, any $\rho \in (0,\rho _0)$, and all $i=1,2.\ldots$. According to Step 1, there is a constant $\mu$ such that

$$\begin{aligned} \rho _0^{-n}\mathrm {Vol}(W_i\cap B_\rho (x))\le \mu \end{aligned}$$

for all $i=1,2,\ldots$. Hence, we may uniformly apply Theorem 9 on any ball $B_{\rho _0}(x_0)\subset (M_i)_\mathcal {D}$ intersecting $W_i$ to obtain the bound in Step 2. $\square$

1.5 Generic regularity in dimension 8

It is well known that codimension one volume minimizing currents, in general, have singularities if the ambient space is of dimension 8 or larger. However, in [26] N. Smale developed a method for removing these singularities in 8-dimensional Riemannian manifolds by making arbitrarily small conformal changes. In this section, we will describe the modifications necessary to adapt his method to the case of Theorem 5 with $n=7$.

First, we will describe the perturbation result we will use. Let M be a compact $(n+1)$-dimensional manifold. For $k=3,4,\ldots$, let $\mathcal {M}^k_0$ denote the class of $C^k$ metrics on M which split isometrically as a product on some neighborhood of $\partial M$. Fix a relative homology class $\alpha \in H_n(M,\partial M;\mathbb {Z})$. We will show the following.

Theorem 10

Let $g_0\in \mathcal {M}^k_0$ and $n=7$. For $\epsilon >0$, there exists a metric $g\in \mathcal {M}^k_0$ and a $g_0$-volume minimizing current T representing $\alpha$ such that $||g-g_0||_{C^k}<\epsilon$ and $\mathrm {spt}(T)$ is smooth.

The proof of Theorem 10 follows by showing the constructions in [26] can be performed on the doubled manifold $M_{\mathcal {D}}$ (see Appendix A.3) in an involution-invariant manner. We proceed in two lemmas. The first lemma holds in any dimension.

Lemma 9

Let $g_0\in \mathcal {M}^k_0$ and suppose T is a homologicly $g_0$-volume minimizing current representing $\alpha$. For $\epsilon >0$, there is a metric $g\in \mathcal {M}^k_0$ such that $||g-g_0||_{C^k}<\epsilon$ and T is the only g-volume minimizing current representative of $\alpha$.

Proof

Let A, $d\mu =\theta d\mathcal {H}^n$, and $\xi$ be the underlying rectifiable set, measure, and choice of orientation for the approximate tangent space of A associated to the current T (see Sect. A.2). We may write $A=\cup _{j=1}^N A_j$ where each $A_j$ are connected. Choose $p_j\in \mathrm {reg}(A_j){\setminus }\partial M$ and $\rho >0$ so that

$$\begin{aligned} \left( B_\rho (p_j)\cap A_j\right) \subset \left( \mathrm {reg}(A){\setminus }\partial M\right) , \quad j=1,\ldots ,N. \end{aligned}$$

Perhaps restricting to smaller $\rho$, let $x=(x^1,\ldots ,x^n)$ be geodesic normal coordinates for $B_\rho (p_j)\cap A_j$ and let t be the signed distance on $B_\rho (p_j)$ from $A_j$ determined by $\xi$. This gives Fermi coordinates (t, x) on $B_\rho (p_j)$.

Now fix a bump function $\eta :A\rightarrow [0,1]$ satisfying

$$\begin{aligned} \eta (x)={\left\{ \begin{array}{ll} 1&{}\text { for }x\in B_{\rho /2}(p_j)\cap A_j\\ 0&{}\text { for }x\in B_\rho (p_j){\setminus } B_{3\rho /4}(p_j) \end{array}\right. } \end{aligned}$$

for each $j=1,\ldots N$. Also fix a smooth function $\phi :\mathbb {R}\rightarrow \mathbb {R}$ with $\mathrm {spt}(\phi )\subset [-3/4,3/4]$,

$$\begin{aligned} \phi (t)\ge 0\text { on }[-1,1], \phi (0)=1,\text { and } \phi (r)<1\text { if }r\ne 0. \end{aligned}$$

Consider the function $\phi _{{\bar{\epsilon }}}:M\rightarrow \mathbb {R}$ given by

$$\begin{aligned} \phi _{{\bar{\epsilon }}}(y)={\left\{ \begin{array}{ll} 1-{{\bar{\epsilon }}}^{k+1}\phi (t/{{\bar{\epsilon }}})\eta (x)&{}\text { if }y=(x,t)\in B_\rho (p_j)\text { for some }j\\ 1&{}\text { otherwise} \end{array}\right. } \end{aligned}$$

for ${\bar{\epsilon }}>0$ satisfying $\mathrm {spt}(\phi _{{\bar{\epsilon }}})\subset \cup _{j=1}^NB_{3\rho /4}(p_j)$. We have the perturbed metrics $g_{{\bar{\epsilon }}}=\phi _\epsilon ^{\frac{2}{n}}g_0\in \mathcal {M}^k_0$. It is straight-forward to show that there exists $\epsilon _1\in (0,\epsilon )$ such that, for any ${\bar{\epsilon }}\in (0,\epsilon _1]$, T is the only $g_{{\bar{\epsilon }}}$-volume minimizing representative of $\alpha$ (see [26]). Perhaps restricting to smaller values of ${\bar{\epsilon }},$ we may also arrange for $||g-g_{{\bar{\epsilon }}}||_{C^k}<\epsilon$. This completes the proof of Lemma 9. $\square$

Lemma 10

Let $n=7$, $k\ge 3$, $g_0\in \mathcal {M}^k$, and $\epsilon >0$. Suppose T is the only $g_0$-volume minimizing representative of $\alpha$, then there exists $g\in \mathcal {M}^k$ such that $||g-g_0||_{C^k}<\epsilon$ and $\alpha$ may be represented (up to multiplicity) by a smooth g-volume minimizing hypersurface.

Proof

Following [26], we construct a conformal factor which will slide the minimizing current off itself in one direction and appeal to a perturbation result for isolated singularities which allows us to conclude that this new current has no singularity. Write $(M_\mathcal {D},g_{0,\mathcal {D}})$ for the doubling of $(M,g_0)$ (see Sect. A.3) with involution $\iota :M_\mathcal {D}\rightarrow M_\mathcal {D}$. The current T may also be doubled to obtain an involution-invariant current $T_\mathcal {D}$ on $M_\mathcal {D}$. Similarly to Section A.3, $T_\mathcal {D}$ is locally $g_{0,\mathcal {D}}$-volume minimizing. Let $A=\cup _{j=1}^NA_j$, $d\mu =\theta d\mathcal {H}^7$, and $\xi$ be the underlying set, measure, and orientation associated to T, as in the proof of Lemma 9.

Let $\rho _0>0$ and fix a smooth function $\phi :\mathbb {R}\rightarrow \mathbb {R}$ satisfying

• (1) $\phi (-t)=-\phi (t),$

• (2) $\phi (t)\ge 0\text { for }t\ge 0,$

• (3) $\phi (t)=t\text { for }t\in [0,\frac{\rho _0}{4}],$

• (4) $\phi (t)=\frac{\rho _0}{2}\text { for }t\in [\frac{\rho _0}{2},\frac{3\rho _0}{4}],$

• (5) $\phi (t)=0\text { for }t\ge \rho _0.$

Let $\{B_\rho (p_j)\}_{j=1}^N$ be a collection of disjoint metric balls in $\mathring{M}$ centered at regular points $p_j\in A_j$. Choose $\rho _0>0$ small enough to ensure that, in Fermi coordinates (t, x) for $A_j$ with $\xi$ pointing into the side corresponding to $t>0$, the function $(t,x)\mapsto \phi (t)\eta (x)$ is supported in $\cup _{j=1}^NB_\rho (p_j)$. For a fixed $s\in (0,1)$ and a parameter ${\bar{\epsilon }}\in (0,1)$, consider the functions $u_{{\bar{\epsilon }}}:\Sigma \rightarrow \mathbb {R}$ given by

$$\begin{aligned} u_{{\bar{\epsilon }}}(y)={\left\{ \begin{array}{ll} 1-\bar{\epsilon }^s\phi (t)\eta (x)&{}\text { if }y=(t,x)\in \cup _{j=1}^NB_\rho (p_j)\\ 1&{}\text { otherwise.} \end{array}\right. } \end{aligned}$$

The conformal metrics $g_{{\bar{\epsilon }}}=u_{{\bar{\epsilon }}}^{\frac{2}{n}}g_0$ will be used to find the desired smooth representative. Since $g_{{\bar{\epsilon }}}$ splits as a product near $\partial M$, we may consider the corresponding $\iota$-invariant metric $g_{{\bar{\epsilon }},\mathcal {D}}$ on $M_\mathcal {D}$.

For sake of contradiction, suppose that there is a sequence ${\bar{\epsilon }}_i\rightarrow 0$ and homologically $g_{{\bar{\epsilon }}_i}$-volume minimizing currents $T_i$ representing $\alpha$ with $\mathrm {sing}(T_i)\ne \emptyset$ for all $i=1,2,\ldots .$ Since $\mathbf {M}(T_i)$ is uniformly bounded in i, $T_i$ weakly converges to some homologically $g_0$-volume minimizing current $T_\infty$ which also represents $\alpha$. Since T is assumed to be the unique such current, we must have $T_\infty =T$. Write $P_i$, $d\mu _i$, and $\xi _i$ for the set, measure, and orientation corresponding to $T_i$ for $i=1,2,\ldots$. Let $Q_i$ be a connected component of $P_i$ with $\mathrm {sing}(Q_i)\ne \emptyset$ for each $i=1,2,\ldots$. Now $Q_i$ converges in the Hausdorff sense to some sheet Q of T. By the Allard regularity theorem [2], this convergence is smooth away from $\mathrm {sing}(Q)$. Hence, after passing to a subsequence, $y_i$ converges to some $y\in \mathrm {sing}(Q)$.

In terms of the doubled manifold, the $\iota$-invariant currents $T_{i,\mathcal {D}}$ are homologically $g_{{\bar{\epsilon }}_i,\mathcal {D}}$-volume minimizing, $T_{i,\mathcal {D}}$ weakly converge to $T_{0,\mathcal {D}}$, and the doubled sets $Q_{i,\mathcal {D}}$ converge to $Q_{\mathcal {D}}$ smoothly away from $\mathrm {sing}(Q_\mathcal {D})$. Now let $\mathcal {N}\subset M_\mathcal {D}$ be a small distance neighborhood of $Q_\mathcal {D}$ so that $\mathcal {N}{\setminus } Q_\mathcal {D}$ consists of two disjoint, open sets $\mathcal {N}_-$ and $\mathcal {N}_+$ on which the signed distance to $Q_\mathcal {D}$ is negative and positive, respectively. In the doubled manifold, we may directly apply the following results from [26].

Lemma 11

[26, Proposition 1.6] For large i, we have

• (1) $Q_{i,\mathcal {D}}\cap \mathcal {N}_-=\emptyset ;$

• (2) $Q_{i,\mathcal {D}}\cap \mathcal {N}_+{\setminus } \mathrm {spt}(\phi _{\epsilon _i}\eta )_\mathcal {D}\ne \emptyset$.

In light of Lemma 11, the Simon maximum principle [30] shows

$$\begin{aligned} \left( Q_{i,\mathcal {D}}{\setminus } \mathrm {spt}(\phi _i\eta )_\mathcal {D}\right) \subset \left( \mathcal {N}_+{\setminus } \mathrm {spt}(\phi _i\eta )_\mathcal {D}\right) \end{aligned}$$

for each $i=1,2,\ldots$. Recalling that $Q_{i,\mathcal {D}}$ converges to $Q_\mathcal {D}$ in the Hausdorff distance, we may apply the perturbation result [19, Theorem 5.6] to conclude that $Q_{i,\mathcal {D}}$ is smooth for sufficiently large i. This contradiction finishes the proof of Lemma 10. $\square$

Theorem 10 follows by first applying Lemma 9 to approximate $g_0$ with a metric $g_1$ supporting a unique minimizing representative of $\alpha$ then applying Lemma 10 to approximate $g_1$ with a metric $g_2$ and obtain a $g_2$-volume minimizing representative of $\alpha$.

Proof of Theorem 5 for $n=7$. We will closely follow the argument presented in Sect. 4. Let $(Z,{\bar{g}},{\bar{\gamma }}): (Y_0,g_0,\gamma _0) \rightsquigarrow (Y_1,g_1,\gamma _1)$ be a psc-bordism and let $(Z_i,{\bar{g}}_i,{\bar{\gamma }}_i)$ be the corresponding i-collaring for $i=1,2,\ldots$. As usual, we denote by ${\bar{\alpha }}_i\in H_7(Z_i,\partial Z_i;\mathbb {Z})$ the Poincaré dual to ${\bar{\gamma }}_i$.

For each $i=1,2,\ldots$, we apply Theorem 10 to obtain a metric ${\hat{g}}_i$ on $Z_i$ so that

$$\begin{aligned} ||{\hat{g}}_i-{\bar{g}}_i||_{C^i_{{\bar{g}}_i}}\le \frac{1}{i} \end{aligned}$$

and ${\bar{\alpha }}_i$ can be represented by a smooth ${\hat{g}}_i$-volume minimizing weighted oriented hypersurface $(W_i,\theta _i)$. It follows from the proofs of Lemmas 9 and 10 that ${\hat{g}}_i$ can and will be chosen so that the set on which ${\hat{g}}_i\ne {\bar{g}}_i$ is contained within $M_1\subset M_i$ for $i=1,2,\ldots$. Indeed, the perturbations required to form ${\hat{g}}_i$ are supported on balls centered about chosen regular points of ${\bar{g}}_i$-volume minimizing currents and one can always find regular points of minimizers of ${\bar{\alpha }}_i$ in $M_1\subset M_i$. Evidently, ${\hat{g}}_i$ has positive scalar curvature for all sufficiently large i. Since ${\hat{g}}_i={\bar{g}}_i$ on $Y\times [-i,0]\subset Z_i$, the proof of the Main Lemma shows that there is a subconvergence

$$\begin{aligned} (Z_i,(W_i,\theta _i),{\hat{g}}_i,\mathsf {S}_i)\rightarrow (Y\times (-\infty ,0],(X\times (-\infty ,0],\theta _X),g+dt^2,\mathsf {S}), \end{aligned}$$

where $Y, (X,\theta _X^\prime ), g, \mathsf {S}_i$, and $\mathsf {S}_\infty$ are defined as in Sect. 4. One can now directly apply the argument from 4.2 to finish the proof of Theorem 5 for $n=7$. $\square$