Existence of singular isoperimetric regions in 8-dimensional manifolds

Abstract

It is well known that isoperimetric regions in a smooth compact $(n+1)$-manifold are themselves smooth, up to a closed set of codimension at most 8. In this note, we construct an 8-dimensional compact smooth manifold whose unique isoperimetric region with half volume that of the manifold exhibits two isolated singularities. This stands in contrast with the situation in which a manifold is a space form, where isoperimetric regions are smooth in every dimension.

1 Introduction

Given an $(n+1)$-dimensional smooth compact Riemannian manifold (M, g), we will consider the isoperimetric problem, that is, given any positive number $0<t<|M|_g$, we look for a solution to the following constrained variational problem

$$\begin{aligned} \mathcal {I}(M, g, t):=\inf \{\textbf{P}_g(\Omega )\, :\,\Omega \in \mathcal {C}(M,t)\} \,, \end{aligned}$$

(1.1)

where $\mathcal {C}(M,t)$ is the class of sets of perimeters with enclosed volume t (for a more precise definition see Sect. 2.) To understand the structure of the boundary of $\Omega$, we say that a point on the boundary is regular if it is locally a smooth hypersurface. We denote it as $Reg(\partial \Omega )\subset \partial \Omega$, and for the complement, we call it the set of singularities in $\partial \Omega$, and denote it by $Sing(\partial \Omega )=\partial \Omega \setminus Reg(\partial \Omega )$. It is well known that minimizers exist and that they are regular outside of a closed set of codimension 8, which is discrete when $(n+1)=8$ (see Gonzalez, Massari, and Tamanini [10]).

A natural question is whether the bound is optimal or not. In the case of area-minimizing integral currents, the question is answered in the positive by the Simons cone in $\mathbb {R}^8$: $\mathbf{{C}}:=\{ (x,y): |x|=|y| \ \ \text {for} \ \ x,y\in \mathbb {R}^4 \}.$ However, minimizers of the problem above in $\mathbb {R}^n$ are euclidean balls and hence smooth for every $n\in \mathbb {N}$. Therefore, to construct a singular minimizer in dimension 8, we need to construct a manifold that is not a space form (see more explanation in Sect. 2.) On the other hand, Smale constructed in [22] a compact 8-manifold admitting a unique homological area-minimizing current with two singular points. Following his construction of local minimizing neighborhood (see Sect. 3.1), we can prove the following theorem for the isoperimetric region problem:

Theorem 1.1

(Singular isoperimetric region in 8-manifold) There exists a smooth closed Riemannian 8-manifold (M, g) whose unique isoperimetric region with volume $|M|_g/2$ has two isolated singularities. The unique tangent cone at each singular point is a Simons cone.

Remark 1.2

• In [22], we can prescribe the singularity for the homological area minimizers to be any strictly stable, strictly minimizing (tangent) cone with an isolated singularity (see Sect. 2 for the definitions), but in our construction, for the technical reason of the setting, we need in addition to assume that the unique (up to scaling) smooth area minimizing hypersurface on one side of the cone (see [12, Theorem 2.1]) is diffeomorphic to the one on the other side (e.g., Simons cone). However, it is promising that we can eliminate this requirement by modifying the construction. As far as the author’s knowledge, these are the first examples of isoperimetric regions with singularities.

• The Riemannian metric in the above theorem is only $C^\infty$ and not real analytic. In Theorem 3.1 (same as [22, Lemma 4]), Smale made a conformal change of the round metric on $\S ^8$ so that the new metric is not real analytic. It is an open question whether a singular example would exist for a real analytic metric.

1.1 Idea of the construction for isolated singularities

As in [22], the starting point is constructing a singular minimal surface in ${\mathbb {S}}^8$. We denote $\mathbf{{C}}$ as a 7-dimensional Simons cone. Then the product space ${\textbf {C}} \times \mathbb {R}$ will be an area-minimizing cone in $\mathbb {R}^9$. Now consider $\Sigma =(\mathbf{{C}}\times \mathbb {R})\cap {\mathbb {S}}^8$ in ${\mathbb {S}}^8$. Clearly, $\Sigma$ is a minimal hypersurface in $({\mathbb {S}}^8, g_S)$ with two isolated singularities, where $g_{S}$ is the round metric. It can be demonstrated that $\Sigma$ represents a suspension of $\mathbb {S}^3 \times \mathbb {S}^3$.

The important part of Smale’s work is to prove Theorem 3.1 [cf. [22, Lemma 4]]: under a conformal change of the standard metric of ${\mathbb {S}}^8$, there exists a smooth neighborhood V of $\Sigma$ such that $\Sigma$ is the unique homological area-minimizing current in V. Moreover, $\Sigma$ splits ${\overline{V}}$ into two parts, $V_+$ and $V_-$, and $\partial V$ has exactly two components (denoted by $\Gamma _+,\Gamma _-$) such that they lie in ${\overline{V}}_+, {\overline{V}}_-$ respectively. Each part of $\{\Gamma _\pm \}$ is homologous to $\Sigma$ and $\Gamma _+$ is diffeomorphic to $\Gamma _-$ (Theorem 3.2 below).

Next, we will construct a manifold with a singular isoperimetric region. Consider $(\Gamma ,h)$ the 7 dimensional Riemannian manifold which is diffeomorphic to $\Gamma _+$ and $\Gamma _-$, endowed with a “larger” metric (see Theorem 3.3 for details); denote $T_R:=\Gamma \times [0,R]$ a tube with length $R>0$, with the product metric $g=h + dr^2$. We glue the $\Gamma _+,\Gamma _-$ with the boundary of the tube, $\widetilde{\Gamma } _-:=\Gamma \times \{0\},\widetilde{\Gamma } _+:=\Gamma \times \{R\}$ respectively to form a torus, calling it $M_R$. Finally, we prove that (in Sect. 4) for sufficiently large R, the unique isoperimetric region with half the volume of $M_R$ has boundary $(\Gamma \times \{t_0\}) \cup \Sigma$ for some $t_0\in (0,R)$, where $\Sigma$ is the one described in the previous paragraph, with two isolated singularities (Fig. 1).

Fig. 1

$M(R):= \Gamma \times [0,R]\cup V/\Gamma _{\pm } \sim {\tilde{\Gamma }}_{\pm }$

2 Preliminaries

2.1 Sets of finite perimeters

Let $n\ge 1$ and (M, g) be an $(n+1)$-dimensional closed oriented Riemannian manifold. In most cases, we will consider $n=7$. In order to have an explicit definition of our variational problem, we first review the sets with well-defined measure theoretical perimeters.

Definition 2.1

(Caccioppoli sets/sets of finite perimeter; see e.g. [9]) Suppose E is a Lebesgue measurable subset of M, we define the perimeter of E by

$$\begin{aligned} \textbf{P}_g(E)=\sup \left\{ \int _E \div _g X \ d\mathcal {H}^{n+1} (x)\, \ X\in \Gamma ^1 (M), \ \Vert X\Vert _{g}\le 1 \right\} . \end{aligned}$$

We define the collection of sets of finite perimeters in (M, g) by

$$\begin{aligned} \mathcal {C}(M,g):= \{\Omega \subset M:\, \textbf{P}_g(\Omega )<\infty \}. \end{aligned}$$

In addition, for the subset of $\mathcal {C}(M)$ with a fixed volume, we denote

$$\begin{aligned} \mathcal {C}(M,g,t):= \{\Omega \subset M:\, \textbf{P}_g(\Omega )<\infty ,\,\, |\Omega |_g=t\}. \end{aligned}$$

We usually omit the subscript g above if the defining metric is understood.

By the Riesz Representation Theorem (see e.g. [20]), there is a TM-valued Radon measure $\mu _\Omega$ such that for any $X\in \Gamma ^1 (M)$, we have

$$\begin{aligned} \int _\Omega {\text {div}}_g X= \int _{M} X\cdot _g d\mu _\Omega . \end{aligned}$$

The total variation is denoted by $\Vert \mu _\Omega \Vert$. For an open set U, we denote

$$\begin{aligned} \textbf{P}(\Omega ;U)=\Vert \mu _\Omega \Vert (U), \end{aligned}$$

the relative perimeter in U.

By de Giorgi’s structure theorem, there is a n-rectifiable set $\partial ^* \Omega$ and $\nu _\Omega$ a Borel function on $\partial ^* \Omega$ with unit sphere such that

For simplicity, we will denote $\partial \Omega = {{\,\textrm{spt}\,}}\mu _\Omega$.

2.2 Isoperimetric problems and minimal surfaces

In this paper, we will study the following minimizing problem:

$$\begin{aligned} \mathcal {I}(M, g, t):=\inf \{\textbf{P}_g(\Omega )\, :\,\Omega \in \mathcal {C}(M,g,t) \}, \end{aligned}$$

where $(M^8,g)$ is a closed Riemannian manifold and $t\in (0, |\Omega |_g)$. We will call a Cacciopoppli set $\Omega$ a t-isoperimetric region in (M, g) if

$$\begin{aligned} \textbf{P}_g(\Omega )= \mathcal {I}(M,g,t)\ \ \text { and } \ \ \Omega \in \mathcal {C}(M,g,t). \end{aligned}$$

Intuitively, the boundary of $\Omega$ is a n-dimensional hypersurface.

Definition 2.2

Let $\Sigma \subset M$ be a smooth open hypersurface. We denote

$$\begin{aligned} Reg(\Sigma )\ {}&:= \{x\in \overline{\Sigma }\cap U: \overline{\Sigma } \text { is smooth, embedded hypersurface near }x \}; \\ Sing(\Sigma )&:= \overline{\Sigma }\setminus Reg(\Sigma ). \end{aligned}$$

By adding points to $\Sigma$ if necessary, we may identify $\Sigma = Reg(\Sigma )$ for simplicity. We say $\Sigma$ is regular, if $Sing(\Sigma )=\emptyset$. In addition, we will always assume $\Sigma$ has optimal regularity: $\mathcal {H}^{n-2} (Sing(\Sigma )) = 0$ and is locally finite.

Similar to the (local) area-minimizing hypersurfaces, for $2\le n\le 7$, it is a well-known result that $Sing(\Omega )=\emptyset$ if $\Omega$ is an isoperimetric region [2, 10, 11]. So higher dimension is the only possible case that singularities may appear. On the other hand, Simons cone [5] manifests an example of an area-minimizing hypersurface in Euclidean space, but isoperimetric regions are smooth regardless of dimensions.

Example 2.3

(Barbosa, do Carmo, and Eschenburg [3]) Let $M^{n+1}(c)$ denote the simply connected complete Riemannian manifold with constant sectional curvature c. Let $X:\Sigma ^n \rightarrow M^{n+1}$ be an immersion of a differentiable manifold $\Sigma ^n$. Suppose $X(\Sigma )$ has constant mean curvature. Then the immersion X is volume-preserving stable (see Definition 2.5) if and only if $X(\Sigma ) \subset M(c)$ is a geodesic sphere.

Similar to the area functional for the minimal surfaces, we define the following functional for isoperimetric regions. An important corollary is that all isoperimetric regions have constant mean curvature. The above examples are isoperimetric regions in space forms (see also [14, 18]).

Fixing a metric g, we can consider the functional $\mathcal {F}: \mathbb {R}\times \mathcal {C}(M)\rightarrow \mathbb {R}$ by

$$\begin{aligned} \mathcal {F}(\lambda , \Omega )=\textbf{P}(\Omega )+\lambda |\Omega |. \end{aligned}$$

Suppose X the smooth compactly supported vector field, $\{\phi _t\}$ the corresponding diffeomorphism, H the generalized mean curvature vector, and $\upsilon$ the outer normal vector for $\partial ^* \Omega$. By the area formula and first variation for potential energy (e.g., [15] Chapter 17), we can get the first variation with respect to $\Omega$:

$$\begin{aligned} \delta _2\mathcal {F}(\lambda , \Omega )&={ \left. \hspace{0.0pt}\frac{d}{dt} \mathcal {F}(\lambda ,\phi _t(\Omega )) \right| _{t=0} }\\&=\int _{\partial ^* \Omega } \div X \, d\mathcal {H}^{n-1}+ \lambda \int _{\partial ^*\Omega } X\cdot \upsilon \, d\mathcal {H}^{n+1}. \end{aligned}$$

Remark 2.4

In addition, if $\Omega$ is the isoperimetric region (i.e. the minimizer of (1.1)), we have $\delta _2\mathcal {F}(\lambda , \Omega )=0$. As remarked in [13, 19], by using the estimates of Hausdorff dimension for the singular set, $\mathcal {H}^{n-2} (\partial \Omega )=0$, then using a cutoff function argument for the test functions, we can see that $\lambda =-H\cdot \upsilon =-h$ for any points in $Reg (\partial \Omega )$, which provides $Reg (\partial \Omega )$ a constant mean curvature hypersurface in M. We call $\Sigma$ a H-hypersurface if $H_{\Sigma } \equiv H$.

Similar to the area-minimizing hypersurfaces, isoperimetric regions have the following sense of “stability”.

Definition 2.5

For $\Omega \in \mathcal {C}(M)$, we say $\Omega$ is volume-preserving stable if $\partial ^* \Omega$ has constant mean curvature and for any diffeomorphism $\phi$ with $|\phi (\Omega )|_g=|\Omega |_g$, we have $\delta ^2\textbf{P}(\Omega )\ge 0$.

Moreover, suppose $\partial \Omega$ is smooth. By [3, Proposition 2.5], suppose

$$\begin{aligned} f(x)=\langle \frac{\partial \phi }{\partial t}(x),\nu (x) \rangle , \end{aligned}$$

we have

$$\begin{aligned} \delta ^2\textbf{P}(\Omega )&=\delta ^2_2\mathcal {F}(-H,\Omega ) \\&=\int _{\partial \Omega } -f\Delta f- (|A_{\Sigma }|^2 + Ric_M(\nu , \nu ))f^2 \, d\mathcal {H}^n(x)\ge 0, \end{aligned}$$

where $\int _{\partial \Omega } f =0.$

Remark 2.6

For the sake of disambiguation, we will say that a hypersurface $\Sigma$ is “strongly stable” if $\delta ^2 \mathcal {P}(\Sigma ) \ge 0$ for any diffeomorphisms.

2.3 Cones with isolated singularities

Because the tangent cones of isoperimetric regions are area-minimizing cones, we will exhibit some properties about cones with isolated singularities, i.e., singular minimizing cones $K\subset \mathbb {R}^{n+1}$ with $Sing(K)=\{0\}$ and $n\ge 7$.

Example 2.7

(Minimal cones) Given $\Sigma ^{n-2} \subset {\mathbb {S}}^{n-1}$ a smooth minimal hypersurface, the cone based on $\Sigma$ is defined as

$$\begin{aligned} \mathbf{{C}}(\Sigma )=\{ \lambda x: \, x\in \Sigma , \ \lambda > 0 \}. \end{aligned}$$

(2.1)

In addition, if a cone $\mathbf{{C}}$ is a minimal surface, then $\mathbf{{C}}\cap {\mathbb {S}}^{n-1}$ is a minimal surface in ${\mathbb {S}}^{n-1}$. We denote $\mathbf{{C}}_1:=\mathbf{{C}}\cap {\mathbb {S}}^{n-1}$.

Definition 2.8

( [12, Sect.3]) Let ${{\textbf {C}}}$ be a regular hypercone (i.e. $Sing (\textbf{C})\subset \{0\}$), we say ${\textbf {C}}$ is strictly minimizing if there is an $\theta > 0$ such that

$$\begin{aligned} \textbf{M}({{\textbf {C}}}_1)\le \textbf{M}(S) - \theta \epsilon ^n, \end{aligned}$$

whenever $\epsilon >0$ and S is an integer multiplicity current with ${{\,\textrm{spt}\,}}S \Subset \mathbb {R}^{n+1}{\setminus } B_\epsilon$ and $\partial S= \partial {{\textbf {C}}}_1$.

Hardt and Simon in [12, Theorem 3.2] exhibit several equivalent definitions of strictly minimizing. The critical property of strictly minimizing hypercone is the following property of graphical local minimizing.

Theorem 2.9

([12, Theorem 4.4]) Suppose $\textbf{C}$ be a regular strictly minimizing (multiplicity one) hypercone in $\mathbb {R}^{n+1}$. Let M be a smooth oriented embedded hypersurface in $\mathbb {R}^{n+1}$ with the representation in the form

$$\begin{aligned} M = graph_{{{\textbf {C}}}_1} h = \{x+h(x)\nu (x): \ x\in \textbf{C}\setminus \{0\}\}, \end{aligned}$$

where $\nu$ is the unit normal vector and h is some function in $C^2({{\textbf {C}}}_1)$ such that

$$\begin{aligned} \Big | \frac{h(r\omega )}{r} \Big | + |Dh(r\omega )| \le C r^q, \ \ \ r\omega \in {{\textbf {C}}}_1, \end{aligned}$$

for some $q>0$.

Assume that $\mathbb {R}^{n+1}$ equipped a $C^3$ Riemannian metric $\displaystyle g = \sum _{i,j =1}^{n+1} g_{ij} dx^idx^j$, which satisfies

$$\begin{aligned} g_{ij}(0)=\delta _{ij}, \frac{\partial g_{ij}}{\partial x_k} (0) = 0, \ \ i,j,k = 1,\dots ,n+1. \end{aligned}$$

Suppose M is a minimal surface (mean curvature zero) in $(\mathbb {R}^{n+1},g)$, then there is a $\rho >0$ such that M is area minimizing in $B_\rho (0)$ with respect to the metric g.

Remark 2.10

• In [22], Smale generalized the above local minimizing property in the manifold setting (Theorem 3.1 below), with additionally assuming the cone ${\textbf {C}}$ is strictly stable. Moreover, Zhihan Wang in [23, Theorem 5.1] also presents a proof of local minimizing property by constructing a foliation over $\Sigma$.

• Simons cone ${\textbf {C}}$ is regular, strictly stable, and strictly minimizing. And it splits $\mathbb {R}^8$ into two parts $E_+$ and $E_-$. By [12, Theorem 2.1], each $E_+$, $E_-$ contains one smooth area minimizing hypersurface up to scaling, call them $R_+,R_-$ respectively. By the symmetry of ${\textbf {C}}$, $R_+,R_-$ are diffeomorphic to each other. So Simons cone satisfies the requirement in Remark 1.2.

3 Constructing the manifolds with Smale’s ideas

This section is dedicated to constructing a family of manifolds, a suitable choice of which will later give us the main Theorem 1.1. We divide it into two parts: first, using a result of Smale [22], we obtain the first piece of our manifold, then we suitably modify it to glue it to a cylinder to obtain the desired construction. The second part is more similar to an 8-dimensional torus example in [7, Lemma 4.1] by Chodosh, Engelstein, and Spolaor.

3.1 Smale’s main result

We recall here the main result from [22], which will be the starting point of our construction.

Theorem 3.1

([22, Lemma 4]) Let $\textbf{C}$ be any strictly stable and strictly minimizing cone, e.g., Simons cone. Let $\Sigma :=(\mathbf{{C}}\times \mathbb {R})\cap {\mathbb {S}}^8$. There exists a $C^\infty$ metric g on ${\mathbb {S}}^8$ and $\delta >0$ such that $\Sigma$ is uniquely homologically area-minimizing in the tubular neighborhood

$$\begin{aligned} U_\delta := \{ x\in {\mathbb {S}}^8: d_{{\mathbb {S}}^8}(x,\Sigma )\le \delta \}, \end{aligned}$$

with respect to the metric g.

Proof

The proof of this result can be found in [22, Lemma 4]. $\square$

In order to glue the $U_\delta$ along the boundaries with a manifold (with boundary), we need the boundary of $U_\delta$ to be smooth. Because of the singularities of $\Sigma$, we cannot expect the smoothness of the boundaries of $U_\delta$ for any small $\delta >0$. Fortunately, as remarked in the proof of [22, Lemma 4], we can find a smaller neighborhood $V\subset U_\delta$ such that $\Sigma$ is still homological area-minimizing in V, and the boundary $\partial V$ is smooth. The basic idea is to glue in pieces of foliations.

Theorem 3.2

Denote $\textbf{C}$ the Simons cone. Let $\Sigma :=(\mathbf{{C}}\times \mathbb {R})\cap {\mathbb {S}}^8$ and let $U_\delta$ and g be as in the previous theorem. There exists an open subset V such that ${{\overline{V}}}\Subset U_\delta$ and

• $\Sigma \subset V$ and it is homologically minimizing in V with respect to the metric g;

• $V\setminus \Sigma$ consist of 2 connected components, $V_\pm$ and $\partial V_\pm =\Sigma \cup \Gamma _\pm$, disjoint union, and $\Gamma _\pm$ are smooth and diffeomorphic to each other;

• $\Sigma$ is a deformation retract of V.

Proof

The construction of V is almost the same as the argument in [22].

For $\delta$ small, $\Sigma$ splits $U_\delta$ into two parts, call them $U_+,U_-$. Note that $Sing(\Sigma ) = \{ p_+, p_- \}$. For $\sigma >0$ smaller than $\delta /8$, we have $B_\sigma (p_\pm ) \subset U_\delta$. Consider the Fermi’s coordinate

$$\begin{aligned} \{(x,t): x\in \Sigma , \ \ \rho (x)> \sigma , \ \ |t|<\delta \}, \end{aligned}$$

around $\Sigma$, here we denote $\rho (x):=d_{{\mathbb {S}}^8}(x,p_+\cup p_-)$. We denote $\Sigma _\sigma = \{ q \in \Sigma : \rho (q) < \sigma \}$, then consider the constant graph on $\Sigma \setminus \Sigma _\sigma$:

$$\begin{aligned} \Gamma _t = graph_{\Sigma \setminus \Sigma _\sigma } t, \end{aligned}$$

with $|t|< \delta$.

Let $p: = p_+$ or $p_-$, and denote $S_t = \{(x,t): \rho (x,0) = 4\sigma \}$. By [12, Theorem 5.6], $S_t$ bounds a 7-dimensional smooth submanifold $R_t$ which is area-minimizing in $B_{5\sigma }(p)$, and as $t \rightarrow 0$, $R_t \rightarrow \Sigma _{4\sigma }$ in both the current and Hausdorff senses. Denote $\lambda _t:= d_{\S ^8}(p, R_t)$; as argued in [12, Theorem 2.1], we have $\eta _{p,\lambda _t \#} R_t \rightarrow S$, where S is the unique minimal hypersurface in E with $d(S,p)=1$, where E is one side of $\textbf{C}_p$ such that $S\subset E$. Because S is smooth, $\eta _{p,\lambda _t \#} R_t \rightarrow S$ in $C^2_{loc}$. By [12, Theorem 2.1], S has the following properties:

• (1) for any vector $\xi \in E$, the ray $\{\lambda \xi : \lambda > 0\}$ intersects S at a single point, and the intersection is transverse;

• (2) there exists a constant $C:=C(\textbf{C}_p)$ such that is a graph of a function on $\textbf{C}_p$.

On the other hand, for any small t, there exists a $C^2$ function $u_t$ on $\Sigma _{4\sigma } \setminus \Sigma _{C\lambda _t}$ such that $R_t \setminus B_{C\lambda _t}(p)$ can be represented as the graph of $u_t$. We denote $\epsilon _t:= C\lambda _t$.

We can find a smooth cutoff function $\chi$ on $\Sigma$ such that

$$\begin{aligned} \chi (x) = {\left\{ \begin{array}{ll} 1 &{} \text {for } \rho (x) \ge 3\sigma , \\ 0 &{} \text {for } \rho (x) \le 2\sigma . \end{array}\right. } \end{aligned}$$

We can get a smooth hypersurface by gluing the smooth submanifold $\Gamma _t$ with $R_t$ through a function w on $\Sigma {\setminus }\Sigma _{\sigma }$ by

$$\begin{aligned} w_t= \chi u_t + (1-\chi ) t. \end{aligned}$$

(3.1)

Therefore,according to our construction above, we can denote $V_{t_0}$ as a foliation in the following sense: there exists $t_0 > 0$ such that

$$\begin{aligned} V_{t_0}:= \bigcup _{t \in [-t_0, t_0]}W_t, \end{aligned}$$

where $W_0: = \Sigma$ and for each $t \ne 0$, $W_t$ is the smooth hypersurface formed by gluing the smooth submanifold $\Gamma _t$ with $R_t$ through the function $w_t$ defined in (3.1).

Then, we explore the topology of $V_{t_0}$. As mentioned above, for small enough t, for any $q \in {\text {spt}} \eta _{p,\lambda _t \#} R_t \cap B_{C}(p)$, there is a unique ray from p connecting p and q such that the ray intersects $\eta _{p,\lambda _t \#} R_t \cap B_{C}(p)$ only at q. So, after rescaling back, there exists ${\bar{t}} > 0$ (smaller than $t_0$) such that

$$\begin{aligned} \left( \bigcup _{t \in [-{\bar{t}}, {\bar{t}}]} W_t \right) \cap B_{\varepsilon _{{{\bar{t}}}}}(p) \end{aligned}$$

forms a cone centered at p, which could collapse to the point p. Denote

$$\begin{aligned} V:=\left( \bigcup _{t \in [-{\bar{t}}, {\bar{t}}]} W_t \right) \end{aligned}$$

After this deformation retraction, the new space is homotopy equivalent to V. Denote by $\Sigma '$ the $\Sigma$ under this deformation retract. Because $W_{{\bar{t}}}$ and $W_{-{\bar{t}}}$ can be represented by graphs on $\Sigma {\setminus } B_{\epsilon _{{\bar{t}}}}$, there consequently exists $u \in C(\overline{\Sigma '})$ with $u \ge 0$ on ${\text {Reg}}(\Sigma ')$ and $u = 0$ at $p:= p_{\pm }$, such that V is a deformation retract of a space X, where

$$\begin{aligned} X = \{(x,t): x \in \Sigma ' \text { and } -u(x) \le t \le u(x)\}. \end{aligned}$$

Therefore, $\Sigma$ is a deformation retract of X. And thus, $\Sigma$ is a deformation retract of V.

Denote by $\Gamma _+:= W_{{{\bar{t}}}}$ (resp. $\Gamma _-:=W_{-{{\bar{t}}}}$), the smooth hypersurface in $U_+$ (resp. $U_-$). Moreover, note that $\textbf{C}$ splits $\mathbb {R}^8$ into two parts, by the symmetry of the Simons cone, $R_t$ is diffeomorphic to $R_{-t}$ for all $|t|\le {{\bar{t}}}$. Therefore, $\Gamma _+$ and $\Gamma _-$ are diffeomorphic to each other.

In summary, $\Sigma$ splits V into two parts: $V_+,V_-$. And $\partial V_+$ has two components: the smooth part $\Gamma _+$, and $\Sigma$ (similarly for $\partial V_-$). Then we have $\partial V= \Gamma _+ \cup \Gamma _-$, where $\Gamma _+,\Gamma _-$ are both smooth and homologous to $\Sigma$. Moreover, $\Gamma _+$ is diffeomorphic to $\Gamma _-$. We will use the set V in the next subsection to construct the manifolds with singular isoperimetric regions. $\square$

3.2 Construction of the toric manifolds

Next, we will construct a collection of 8-dimensional closed Riemannian manifolds, which we will use in the next section. Again we denote V, the smooth neighborhood of $\Sigma$ from the last subsection.

Theorem 3.3

Denote $\Gamma$ the 7-manifold which is diffeomorphic to $\Gamma _+$ and $\Gamma _-$ by $F_\pm :\Gamma _\pm \rightarrow \Gamma$. Consider the smooth manifold defined by gluing the boundaries of V and $\Gamma \times [0,R]$:

$$\begin{aligned} M(R):=\Gamma \times [0,R]\cup V/\Gamma _{\pm }\times \{0,R\} \sim _{F_\pm } {\tilde{\Gamma }}_{\pm }, \end{aligned}$$

(3.2)

where

$$\begin{aligned} {\tilde{\Gamma }}_+:= \Gamma \times \{ 0\}, \qquad \qquad {\tilde{\Gamma }}_-:= \Gamma \times \{ R\}. \end{aligned}$$

There exist $R_0>0$ and a one parameter family of $C^\infty$-metrics $(g_R)_{R>0}$ on M(R) such that, if V is as in Theorem 3.2, then for every $R>R_0$, the followings hold:

• (1) V is isometrically embedded into $(M(R), g_R)$;

• (2) $H_7(M(R); \mathbb {Z}_2) \cong \mathbb {Z}_2 \cong \langle [\Gamma ] \rangle \cong \langle [\Sigma ] \rangle$, where $[\Gamma ], [\Sigma ]$ denote the homology classes represented by $\Gamma$ and $\Sigma$ respectively;

• (3) $\Sigma$ is the unique homological area minimizer in $(M(R),g_R)$.

The idea is the following: suppose h is some Riemannian metric of $\Gamma$ to be determined. At first, we conformally change (enlarge) the metric g of V in $U_\delta$ near $\Gamma _+,\Gamma _-$ such that the outside of a tubular neighborhood of $\Gamma _+$ (and $\Gamma _-$) is isometric to the cylindrical metric $(\Gamma \times [0,\epsilon ],h+dr^2)$ for sufficiently small $\epsilon$,. Additionally, we require that $\Sigma$ is still homological area-minimizing in $(V,\widetilde{g} )$, where $\widetilde{g}$ is the metric after the conformal change.

Finally, we glue $(V,\widetilde{g} )$ with $(\Gamma \times [0, R], h +dr^2)$ along the boundaries respectively. There is a Riemannian metric on M(R) depending on the length R. And we denote

$$\begin{aligned} (M(R),g_R):=\Gamma \times [0,R]\cup V/\Gamma _{\pm } \sim {\tilde{\Gamma }}_{\pm }. \end{aligned}$$

Proof of Theorem 3.3

For each $i=\pm$, consider $U_i(\supset \Gamma _i)$ the Fermi’s coordinate (x, t) for $x\in \Gamma _{i}$, here choosing t with the positive direction towards outside of $\Sigma$, then there exists an $\epsilon >0$ such that

• (1) for $|t|\le \epsilon$, we have $(x,t) \in U_{i}$ respectively;

• (2) there exists an $\epsilon _1>0$ such that

  $$\begin{aligned} \inf _{|t|\le \epsilon }\inf _{ x\in \Gamma _{\pm } } dist((x,t), \Sigma )>\epsilon _1. \end{aligned}$$

For each small t, we define

$$\begin{aligned} \Gamma _{i}(t):= \{ (x,t):\, x\in \Gamma _i \}, \end{aligned}$$

again here we define the positive signs for each $\Gamma _i$ the side opposite from $\Sigma$, so each $\Gamma _i(t)$ forms a layer in $U_i$.

Note that for any $|t|\le \epsilon$, $\Gamma _+(t),\Gamma _-(t)$ bound an open neighborhood of $\Sigma$ as well. For $t= -\epsilon$, we denote $V_0$ the neighborhood of $\Sigma$ with boundary $\Gamma _+(- \epsilon ) \cup \Gamma _- (-\epsilon )$. In another word, $\Gamma _+(-\epsilon ), \Gamma _- (- \epsilon )$ split the neighborhood $U_\epsilon$ into three parts, and $\Sigma$ lies in the middle part.

For $-\epsilon < \sigma \le \epsilon$, denote

$$\begin{aligned} V_+(\sigma )&:= \{(x,t):\, x\in \Gamma _+ , \ -\epsilon \le t<\sigma \};\\ V_-(\sigma )&:= \{(x,t):\, x\in \Gamma _- , \ -\epsilon \le t<\sigma \}. \end{aligned}$$

We can denote the neighborhood (depending on $\sigma$) by

$$\begin{aligned} V(\sigma ):= V_0 \cup V_+(\sigma ) \cup V_-(\sigma ). \end{aligned}$$

(3.3)

Note that for each small $\sigma$, $V(\sigma )$ is a neighborhood of $\Sigma$ with two smooth boundaries $\Gamma _i(t)$ such that each $\Gamma (t)$ is homologous to $\Sigma$. In addition, by Theorem 3.1, $\Sigma$ is uniquely homological area-minimizing in $V(\sigma )$ for any $-\epsilon \le \sigma \le \epsilon$.

In addition, we observe that $V: = V(0).$ Following the ideas at the beginning of this section, we will glue the smooth neighborhood $V(\epsilon )$ along the boundary with a manifold (with boundary) endowed with a cylindrical metric.

Now, we will keep the metric on $V_0$ and deform the metrics on $V_\pm (\epsilon )$ to the cylindrical metric (near $\Gamma _{\pm }(\epsilon )$). Note that for each $i=\pm$, under the Fermi coordinates in $V_i(\epsilon )$, the metric has the form:

$$\begin{aligned} g(x,t) = g_i (x,t) + \eta _i(x,t)dt^2, \end{aligned}$$

where for each small t, $g_i(\cdot , t)$ is a metric on $\Gamma _i$ respectively, and $\eta _i$ is a smooth positive function.

In order to construct a deformation of the metric on $V(\epsilon )$, we first define new metrics on $\Gamma _+(\epsilon ),\Gamma _-(\epsilon )$. Note that $\Gamma _+(\epsilon ),\Gamma _-(\epsilon )$ are both diffeomorphic to $\Gamma$. Consider h the Riemannian metric on $\Gamma$ and the diffeomorphisms $F_\pm$:

$$\begin{aligned} F_+: \Gamma _+(\epsilon ) \longrightarrow \Gamma (=(\Gamma ,h)), \ \ \ \ F_-: \Gamma _-(\epsilon ) \longrightarrow \Gamma , \end{aligned}$$

such that for any $x\in \Gamma _\pm (\epsilon )$ and any $v\in T_x \Gamma _\pm (\epsilon )$, we have

$$\begin{aligned} F_\pm ^* h(v,v) \ge 2 g(v,v), \end{aligned}$$

where g denotes the original metric in $V(\epsilon )$.

To construct a new metric on $V(\epsilon )$, for $i = \pm$, we define the metric ${\overline{g}}_i$ on $\Gamma _i$ and the number ${\overline{\eta }}_i$ by

$$\begin{aligned} {{\overline{g}}}_i (x)&:= F^*_i (h) (x),\end{aligned}$$

(3.4)

$$\begin{aligned} {\overline{\eta }}_i&:= \max _{{{\overline{V}}}_i(\epsilon )} \eta _i(x,t). \end{aligned}$$

(3.5)

Denote the cutoff function $\phi :\mathbb {R}\rightarrow \mathbb {R}$ a smooth non-negative function such that

$$\begin{aligned} \phi (x)= {\left\{ \begin{array}{ll} 0 &{} t\le 0;\\ 1 &{} t\ge \epsilon /2, \end{array}\right. } \end{aligned}$$

Then we can define new metrics $\widetilde{g} _i$ on $V_i(\epsilon )$, where $i=+$ or −, by the following:

$$\begin{aligned} \widetilde{g} _i(x,t)=&(1-\phi (t))g_i(x,t) +\phi (t) {{\overline{g}}}_i(x)\nonumber \\&+[(1-\phi (t))\eta _i(x,t) +\phi (t){\overline{\eta }}_i] dt^2. \end{aligned}$$

(3.6)

So clearly, each of  $\widetilde{g} _+$ and $\widetilde{g} _-$ is simply an interpolation such that, as t increases from 0 to $\epsilon /2$, the original metric deforms to the cylindrical metric. Moreover, we leave the metric unchanged when t is non-positive.

Now we can define a new metric $\widetilde{g}$ on the whole neighborhood $V(\epsilon )$ by the following:

$$\begin{aligned} \widetilde{g} (p)= {\left\{ \begin{array}{ll} g(p), &{} p\in V_0, \\ \widetilde{g} _i(p), &{} p\in V_i(\epsilon ),\ i=+ \ \text {or}\ -. \end{array}\right. } \end{aligned}$$

Therefore, we can construct the collection of manifolds $\{(M(R),g_R)\}_R$. Note that $V(\epsilon )$ has cylindrical metric on $V(\epsilon ){\setminus } V(\epsilon /2)$, and $\Gamma _\pm (\epsilon )$ are both isometric to $(\Gamma ,h)$. Then consider the maps $F_\pm :\Gamma _\pm \rightarrow \Gamma$ defined above. Denote $W(R)= [0,R] \times \Gamma$ for $R>0$ a Riemannian manifold with the boundary equipped with the product metric. Along with the diffeomorphisms $F_\pm$, we glue $\Gamma _+(\epsilon )$ with $\{0\} \times \Gamma$ and glue $\Gamma _-(\epsilon )$ with $\{R\} \times \Gamma$. Then we get a connected smooth Riemannian manifold depending on the positive number R, denote it by $(M(R), g_R)$.

Note that under the Riemannian metric $\widetilde{g}$, (3.4)–(3.6) show that we leave the metric in V unchanged and enlarge the metric on $V_+(\epsilon ),V_-(\epsilon )$. So $\Sigma$ is still the unique homological area minimizer in $V(\epsilon )$. Moreover, by [22, Lemma 5], we see that for sufficiently large $R>0$, under the metric defined above, $\Sigma$ is the unique homological area minimizer in M(R).

Finally, we study the homology group $H_7(M(R), \mathbb {Z}_2)$. It is evident that $\Gamma \times [0, R]$ is homotopy equivalent to $\Gamma$. Note that, by our construction, topologically, $\Sigma$ is a suspension of $\mathbb {S}^3 \times \mathbb {S}^3$, and the smooth hypersurface $\Gamma$ is $\mathbb {S}^3 \times \mathbb {S}^3 \times [-1,1]$, with two boundaries filled in by two copies of $\mathbb {S}^3 \times \mathbb {B}^4$. Thus, $\Gamma$ is homotopy equivalent to the product of $\mathbb {S}^3$ with the suspension of $\mathbb {S}^3$. Therefore, $H_6(\Gamma , \mathbb {Z}_2) = 0$. On the other hand, by Theorem 3.2, $\Sigma$ is a deformation retract of V. Then, by computing the Mayer-Vietoris sequence for M,

$$\begin{aligned}&\dots \rightarrow H_7(\Gamma \sqcup \Gamma ) \rightarrow H_7(V, \mathbb {Z}_2) \oplus H_7(\Gamma \times [0, R],\mathbb {Z}_2) \\&\quad \rightarrow H_7(M(R), \mathbb {Z}_2) \rightarrow H_6(\Gamma \sqcup \Gamma ) \rightarrow \dots \end{aligned}$$

We have

$$\begin{aligned} \dots \rightarrow \mathbb {Z}_2 \oplus \mathbb {Z}_2&\rightarrow \mathbb {Z}_2 \oplus \mathbb {Z}_2 \rightarrow H_7(M, \mathbb {Z}_2) \rightarrow 0 \rightarrow \dots \\ (1,0)&\mapsto (1,1) \\ (0,1)&\mapsto (1,1) \end{aligned}$$

we see that $H_7(M, \mathbb {Z}_2) \cong \mathbb {Z}_2 \cong \langle [\Gamma ] \rangle$. $\square$

4 Proof of the main Theorem 1.1

In Sect. 3.2, we have constructed a collection of closed manifolds that we need later as ambient spaces. Then, in Sect. 4.1, we will construct the singular isoperimetric region in the corresponding manifold. In this section, we will redefine V for convenience by

$$\begin{aligned} V := V(\epsilon ) \end{aligned}$$

(4.1)

where $V(\epsilon )$ is defined in (3.3).

4.1 Construction of singular isoperimetric region

Theorem 4.1

Let M(R) be the family of manifolds constructed in Theorem 3.3. There exists $R_1>0$ such that for any $R>R_1$, there is $t_0\in (0,R)$ such that the boundary of the two unique isoperimetric regions (the one and its complement) with volume $|M(R)|_g/2$ is of the form

$$\begin{aligned} \Sigma \cup \left[ \Gamma \times \{t_0\}\right] . \end{aligned}$$

This theorem directly implies Theorem 1.1.

Example 4.2

Consider the torus ${\mathbb {S}}^7\times {\mathbb {S}}^1(R)$ with product metric for large length R. [7] shows that for large R, the boundary of an isoperimetric region with half volume is of the form

$$\begin{aligned} \left[ {\mathbb {S}}^7 \times \{0\} \right] \cup \left[ {\mathbb {S}}^7 \times \{\pi R\}\right] . \end{aligned}$$

To prove Theorem 4.1, we want to have a uniform bound on the mean curvature of isoperimetric regions with volume bounded away from zero and that of the manifold. For a closed manifold with dimension $2\le n\le 7$, this lemma is proved in [7]. In the higher dimensional cases, the boundary can have singularities; we want to have the mean curvature bounded in $Reg(\partial \Omega )$. Adapting the argument of [7, Lemma C.1] shows the following lemma:

Lemma 4.3

(cf. [7, Lemma C.1] and [6, 17]) For $n \ge 2$, fix $\delta >0$, and $(M^{n},g)$ a closed Riemannian manifold with $C^3$-metric, there is $C = C(M,g,\delta ) <\infty$ so that if $\Omega \in \mathcal {C}(M, t)$ is an isoperimetric region with volume $|\Omega |_g \in (\delta , |M|_g - \delta )$, then the mean curvature of $Reg(\partial \Omega )$ satisfies $|H|\le C$.

Proof

[7, Lemma C.1] proves the case $2\le n\le 7$. Next, we assume $n\ge 8$, the proof is similar to [7, Lemma C.1].

Assuming not, we would have a sequence of isoperimetric regions $\Omega _j \subset (M,g)$ with $|\Omega _j|_g \in (\delta , |M|_g -\delta )$ with divergent constant mean curvature $H_j$ such that $\lambda _j:=| H_j| \rightarrow \infty$.

Choosing any $x_j \in \partial \Omega _j$, we can rescale the metric in $\lambda _j$ at the point $x_j$, denote $\widetilde{g} _j = \lambda _j^2\,g$. So we have $(M, \widetilde{g} _j, x_j)$ converges in $C^3_{loc}$ to the flat metric on $(\mathbb {R}^n,0)$. Also, for each j, we have $\widetilde{\Omega } _j$ an isoperimetric region in $(M, \widetilde{g} _j)$. Passing to a subsequence, there is a locally isoperimetric region¹$\widetilde{\Omega }$ in $\mathbb {R}^n$ such that $\widetilde{\Omega } _j$ converges to $\widetilde{\Omega }$ in the local Hausdorff sense, and $|\partial ^* \widetilde{\Omega } _j| \rightarrow |\partial ^* \widetilde{\Omega } |$ locally in the varifold sense. Therefore, for any $p \in Reg(\partial \widetilde{\Omega } )$, there exists a neighborhood $B_r(p)$ around p for some $r>0$ such that $\partial \widetilde{\Omega } _j \cap B_r(p) \subset Reg(\partial \widetilde{\Omega } _j)$, and therefore $Reg(\partial \widetilde{\Omega } _j)\cap B_r(p)$ converges in $C^{2,\alpha }$ to $Reg(\partial \widetilde{\Omega } ) \cap B_r(p)$. Therefore, the mean curvature of $\widetilde{\Omega }$ is $\pm 1$. Moreover, because $\widetilde{\Omega }$ is a locally isoperimetric region, $\partial \widetilde{\Omega }$ is thus volume-preserving stable.

Next, we claim that there exists a sequence of choices of $x_j \in \partial \Omega _j$ such that the boundary of the limit space $\partial \widetilde{\Omega }$ is non-compact. Suppose not, and assume that $\partial \widetilde{\Omega }$ is compact for any choices of $x_j \in \partial \Omega _j$. Then, either $\widetilde{\Omega }$ or its complement is compact and must be an isoperimetric region in $\mathbb {R}^n$. By [15, Theorem 14.1], it must be a ball. Given the mean curvature $|{\widetilde{H}}| = 1$, $\widetilde{\Omega }$ or its complement $\widetilde{\Omega }^c$ is necessarily a unit ball.

Therefore, $\partial \Omega _j$ will also be regular. Because the only possible $\widetilde{\Omega }$ is the unit ball or its complement, $(\partial \widetilde{\Omega }_j, {\widetilde{g}}_j)$ would be composed of a union of connected components, each approximating a unit geodesic sphere.

For $\lambda _j$ sufficiently large, $\frac{1}{2\lambda _j}$ will be the lower bound of the diameters of each connected component (almost a geodesic sphere) of $\partial \Omega _j$. Without loss of generality, we assume M is connected. Therefore, either $\widetilde{\Omega }_j$ consists of a union of balls, or $\widetilde{\Omega }_j^c$ consists of a union of balls. Consider the first case and denote by $V_1 > 0$ the upper bound of the volumes of $(\Omega _j)_j$. Thus, for any sufficiently large j, $\widetilde{\Omega } _j$ contains at most $V_1\lambda _j^{n}$ balls. Consequently, we have

$$\begin{aligned} \textbf{P}(\Omega _j) \ge C(n) V_1\lambda _j^n\textbf{P}(B_{\lambda _j}) \ge C(n)\frac{V_1}{\lambda _j^{n-1}} \lambda _j^{n}. \end{aligned}$$

Therefore, as $\lambda _j \rightarrow \infty$, $\textbf{P}(\Omega _j) \rightarrow \infty$, which is a contradiction. In the second case, where $\widetilde{\Omega }_j^c$ consists of a union of balls, we may consider $V_2 > 0$ as the uniform lower bound of the volumes of $(\Omega _j)_j$ and similarly obtain a contradiction. Therefore, we can assume that $\partial \widetilde{\Omega }$ is non-compact.

Denote $\widetilde{H} :=$ the mean curvature of $Reg(\partial \widetilde{\Omega } )$, and $\widetilde{A} :=$ the second fundamental form of $Reg(\partial \widetilde{\Omega } )$. Because $|\widetilde{H} | =1$, we have $|\widetilde{A} |^2 \ge \frac{1}{n}$. Because $\widetilde{\Omega }$ is a locally isoperimetric region, $\partial \widetilde{\Omega }$ is volume preserving stable (see Definition 2.5). Therefore, for any $\phi \in C^1_c(Reg(\partial \widetilde{\Omega } ) )$ with $\int _{Reg(\partial \widetilde{\Omega } )} \phi \, d\mathcal {H}^{n-1} = 0$, we have

$$\begin{aligned} \int _{Reg(\partial \widetilde{\Omega } )} \phi ^2 \, d\mathcal {H}^{n-1} \le \int _{Reg(\partial \widetilde{\Omega } )} n|\widetilde{A} |^2 \phi ^2 \, d\mathcal {H}^{n-1} \le n\int _{Reg(\partial \widetilde{\Omega } )} |\nabla \phi |^2 \, d\mathcal {H}^{n-1}. \end{aligned}$$

(4.2)

Following the remark from [8] and [4, Proposition 2.2], we claim that $\partial \widetilde{\Omega }$ is strongly stable outside of a compact set, i.e., there exists a large enough $R>0$ such that for any $\phi \in C^1_c(Reg(\partial \widetilde{\Omega } ) {\setminus } B_R )$, we have

$$\begin{aligned} \int _{Reg(\partial \widetilde{\Omega } )} \phi ^2 \, d\mathcal {H}^{n-1} \le n \int _{Reg(\partial \widetilde{\Omega } )} |\nabla \phi |^2 \, d\mathcal {H}^{n-1}. \end{aligned}$$

(4.3)

Assume not, there is $R_1>0$ and some $\phi _1\in C^1_c(Reg(\partial \widetilde{\Omega } ) {\setminus } B_{R_1} )$ such that (4.3) fails for $\phi := \phi _1$. Denote $R_2>0$ such that ${{\,\textrm{spt}\,}}\phi _1 \Subset B_{R_2}$, then there exists $\phi _2\in C^1_c(Reg(\partial \widetilde{\Omega } ) \setminus B_{R_2} )$ such that for $\phi := \phi _2$. Denote $\phi _3:=c_1\phi _1+c_2\phi _2$ for some $c_1,c_2\in \mathbb {R}$ such that $\int _{Reg(\partial \widetilde{\Omega } )} \phi _3^2 \, d\mathcal {H}^{n-1} =0.$ So we also have

$$\begin{aligned} \int _{Reg(\partial \widetilde{\Omega } )} \phi _3^2 \, d\mathcal {H}^{n-1} > n \int _{Reg(\partial \widetilde{\Omega } )} |\nabla \phi _3|^2 \, d\mathcal {H}^{n-1}, \end{aligned}$$

which contradicts (4.2). Thus we prove the claim.

Consider the $C^1$ radial cutoff function $\phi$ in $\mathbb {R}^n$ such that: fixing any $\rho >R+1$,

$$\begin{aligned} \phi (x)= {\left\{ \begin{array}{ll} 1 &{} |x|\in [R+1, \rho ],\\ 0 &{} |x|\in [0, R] \cup [2\rho , \infty ]. \end{array}\right. } \end{aligned}$$

with $|D\phi (x)| \le C(n) \rho ^{-1}$ for $|x|>\rho$ and $|D\phi (x)| \le C(R)$ where C(R) depending on R, and D is the Euclidean connection on $\mathbb {R}^n$. Because $\widetilde{\Omega }$ is a locally isoperimetric region, we have $\mathcal {H}^{n-3}(Sing (\partial \widetilde{\Omega } )) = 0$. Given any $\epsilon >0$, consider $\{B_{r_j}(p_j)\}_j$ a collection of geodesic balls which cover $Sing(\partial \widetilde{\Omega } )$, such that

$$\begin{aligned} \sum _j r_j^{n-3} <\epsilon . \end{aligned}$$

We define $\psi _j$ a smooth cutoff function by

$$\begin{aligned} \psi _j (x)= {\left\{ \begin{array}{ll} 1 &{} \text {if } x\notin B_{2r_j} (p_j), \\ 0 &{} \text {if } x\in B_{r_j} (p_j). \end{array}\right. } \end{aligned}$$

with $|D \psi _j|\le C(n) r_j^{-1}$. Now we define

$$\begin{aligned} \psi _\epsilon&:= \inf _j \psi _j \\ \Phi _\epsilon&:= (\phi )^{\frac{n-1}{2}} \cdot \psi _\epsilon . \end{aligned}$$

So we note that $\Phi _\epsilon$ is a Lipschitz compactly supported function on $Reg(\partial \widetilde{\Omega } )$. So by (4.3), we have

$$\begin{aligned}&\int _{Reg(\partial \widetilde{\Omega } )} \Phi _\epsilon ^{2} \, d\mathcal {H}^{n-1} \le n\int _{Reg(\partial \widetilde{\Omega } )} |\nabla \Phi _\epsilon |^{2} \, d\mathcal {H}^{n-1}\\&\quad \le 2n \int _{Reg(\partial \widetilde{\Omega } )} |D\phi ^{\frac{n-1}{2}} |^2 \cdot \psi _\epsilon ^2 + \phi ^{n-1} \cdot |D \psi _\epsilon |^{2} \, d\mathcal {H}^{n-1}\\&\quad = 2n \int _{Reg(\partial \widetilde{\Omega } )} \left( \frac{n-1}{2}\right) ^2 \phi ^{n-3} |D\phi |^2 \cdot \psi _\epsilon ^2 + \phi ^{n-1} \cdot |D \psi _\epsilon |^{2} \, d\mathcal {H}^{n-1}\\&\quad = \int _{Reg(\partial \widetilde{\Omega } )} \left( \frac{n-3}{n-1}\cdot \phi ^{n-1} +C(n) |D\phi |^{n-1} \right) \cdot \psi _\epsilon ^2 \, d\mathcal {H}^{n-1} \\&\qquad + 2n\int _{Reg(\partial \widetilde{\Omega } )} \phi ^{n-1} \cdot |D \psi _\epsilon |^{2} \, d\mathcal {H}^{n-1}. \end{aligned}$$

Therefore, abusing the notations of constants C(n), we have

$$\begin{aligned}&\int _{Reg(\partial \widetilde{\Omega } )} \Phi _\epsilon ^{2} \, d\mathcal {H}^{n-1} \le C(n) \int _{Reg(\partial \widetilde{\Omega } )} |D\phi |^{n-1} \cdot \psi _\epsilon ^2 \, d\mathcal {H}^{n-1}\nonumber \\&\qquad + C(n)\int _{Reg(\partial \widetilde{\Omega } )} \phi ^{n-1} \cdot |D \psi _\epsilon |^{2} \, d\mathcal {H}^{n-1}. \end{aligned}$$

(4.4)

For the first part of the right hand side of (4.4), by the definition of $\phi ,\psi _\epsilon$, we have

$$\begin{aligned}&\int _{Reg(\partial \widetilde{\Omega } )} |D\phi |^{n-1} \cdot \psi _\epsilon ^2 \, d\mathcal {H}^{n-1} \le \rho ^{1-n} \mathcal {H}^{n-1} (\partial \widetilde{\Omega } \cap B_{2\rho })+ C(R) \mathcal {H}^{n-1} (\partial \widetilde{\Omega } \cap A_{R,R+1}), \end{aligned}$$

where $A_{R,R+1}:= B_{R+1}\setminus {\overline{B}}_R$. For the second part of (4.4), we have

$$\begin{aligned} \int _{Reg(\partial \widetilde{\Omega } )} \phi ^{n-1} \cdot |D \psi _\epsilon |^{2} \, d\mathcal {H}^{n-1}&\le \sum _j\int _{Reg(\partial \widetilde{\Omega } )\cap B_{2r_j}(p_j) } r_j^{-2} \, d\mathcal {H}^{n-1}\\&\le \sum _j r_j^{-2} \cdot C r_j^{n-1}\\&\le C \epsilon . \end{aligned}$$

Here the volume bound comes from the monotonicity formula [20, 17.6], i.e., we have $\mathcal {H}^{n-1} (Reg(\partial \widetilde{\Omega } ) \cap B_{2r_j} (p_j)) \le Cr_j^{n-1}$ for the constant C depending on $|\widetilde{ H} |$, which is constantly 1 in our case.

Therefore, as $\epsilon \rightarrow 0$, the dominated convergence theorem implies that

$$\begin{aligned} \mathcal {H}^{n-1} (\partial \widetilde{\Omega } \cap (B_{\rho } \setminus B_{R+1} )) \le C (1+ \rho ^{1-n} \mathcal {H}^{n-1} (\partial \widetilde{\Omega } \cap B_{2\rho }) ), \end{aligned}$$

(4.5)

where the constant C is independent with $\rho$. Note that as $\rho \rightarrow \infty$, and $\partial \widetilde{\Omega }$ is not compact, we can cover $\partial \widetilde{\Omega }$ by a countable collection of geodesic balls $(B_j)_j$ such that the concentric balls in $(B_j)_j$ with the half radius are pairwise disjoint. Then because $|\widetilde{H} | = 1$, by the monotonicity formula (from below), we have that

$$\begin{aligned} \mathcal {H}^{n-1} (\partial \widetilde{\Omega } \cap (B_\rho \setminus B_R )) \rightarrow \infty . \end{aligned}$$

On the other hand, note that $\widetilde{\Omega }$ is a locally isoperimetric region. For any $\rho$ large, consider $0<r(\rho )\le \rho$ with the property that

$$\begin{aligned} |\widetilde{\Omega } \cap B_{\rho }| = | (\widetilde{\Omega } \setminus B_\rho ) \cup B_{r(\rho )}|, \end{aligned}$$

i.e., $[\widetilde{\Omega } \cap B_{\rho } ]\Delta [(\widetilde{\Omega } {\setminus } B_\rho ) \cup B_{r(\rho )} ]\Subset B_{\rho +\epsilon }$ and have the same volume in $B_{\rho +\epsilon }$ for any $\epsilon >0$. Because $\widetilde{\Omega }$ is a local isoperimetric region, so for any large $\rho$, we have

$$\begin{aligned} \textbf{P}(\widetilde{\Omega } ; B_{\rho +\epsilon })\le \textbf{P}((\widetilde{\Omega } \setminus B_\rho ) \cup B_{r(\rho )}; B_{\rho +\epsilon } ). \end{aligned}$$

Denote $E:=\widetilde{\Omega } \setminus B_\rho , F:=B_{r(\rho )}$, $G:=B_{\rho +\epsilon },$ and $\nu _E,\nu _F$ the outer normal vector fields on the reduced boundaries. By the set operations on Gauss-Green measures ( [15, Theorem 16.3]), we have

$$\begin{aligned} \textbf{P}(E\cup F;G)&=\textbf{P}((\widetilde{\Omega } \setminus B_\rho ) \cup B_{r(\rho )}; B_{\rho +\epsilon } )\\&= \textbf{P}( \widetilde{\Omega } \setminus B_\rho ; B_{r(\rho )}^{(0)}\cap B_{\rho +\epsilon }) + \textbf{P}(B_{r(\rho )}; (\widetilde{\Omega } \setminus B_\rho )^{(0)} \cap B_{\rho +\epsilon })\\&\quad +\mathcal {H}^{n-1}(\{\nu _E=\nu _F\} \cap B_{\rho +\epsilon })\\&\le \textbf{P}( \widetilde{\Omega } \setminus B_\rho ; B_{\rho +\epsilon }\setminus \overline{B_{r(\rho )}}) + C \rho ^{n-1} \end{aligned}$$

As $\epsilon \rightarrow 0^+$, we get

$$\begin{aligned} \textbf{P}(\widetilde{\Omega } ; B_{\rho }) \le C \rho ^{n-1}. \end{aligned}$$

So we have a contradiction with (4.5). $\square$

Lemma 4.4

Let $\Omega _R \in (M(R),g_R)$ be the isoperimetric region with volume |M(R)|/2, and let $\partial \Omega _R=\bigcup _{i=1}^LI_i$, where $\{I_i\}$ are the connected components of $\partial \Omega _R$. Then there exists a nonnegative constant $R_0=R_0(V)$ (V is defined in (4.1)), such that for any $R>R_0$, the following hold:

• (1) $\textbf{P}(\Omega _R) \le \textbf{M}(\Gamma ) +\textbf{M}(\Sigma );$

• (2) each $I_i$ has uniformly bounded diameter;

• (3) there exists an integer $L_0=L_0(V,R_0)>1$ such that the number of connected components satisfies $1<L<L_0$;

• (4) $I_1$ and $I_2$ are homologous to $[1]:=[\Gamma ] \in H_7(M(R),\mathbb {Z}_2)$.

Proof

(1) Note that

$$\begin{aligned} \textbf{P}(\Omega _R)\le \textbf{M}(\Gamma )+ \textbf{M}(\Sigma ), \end{aligned}$$

(4.6)

because there is clearly a $U_R\in C(M(R),g_R,|M(R)|_{g_R}/2)$ such that $U_R = V_+ \cup (\Gamma \times [0, t_0])$ for some $t_0$, which has half volume and with the boundary

$$\begin{aligned} \big [ \{t_0\}\times \Gamma \big ] \cup \Sigma . \end{aligned}$$

(2) Next suppose that there is a sequence $R_k \rightarrow \infty$, denote with $\Omega _k \subset (M(R_k),g_k)$ the corresponding isoperimetric regions with $|\Omega _k|_{g_k} =\frac{1}{2} |M(R_k)|_{g_k}$. Here we denote $g_k:=g_{R_k}$. Suppose $I_k\in \partial \Omega _{k}$ is a component such that ${{\,\textrm{diam}\,}}(I_k) \rightarrow \infty .$ We observe that all the metrics $g_R$ are locally isometric, so by Lemma 4.3, we have a uniform bound of mean curvature for $Reg(\partial \Omega _k)$ and all $k\in \mathbb {N}$. Then monotonicity formula [1, 6] shows that there exists $r_0>0$, for any $x\in \partial \Omega _k$, we have

$$\begin{aligned} f(r) := e^{7 (\Vert H_k\Vert _{\infty } +C) r} \frac{\textbf{P}( \Omega _k ;B_r(x))}{ \omega _7 r^7}, \end{aligned}$$

(4.7)

is non-decreasing for $r\in (0,r_0]$. Where C depends on the upper bound of sectional curvatures of $M(R_k)$ and $r_0$ depends on both the sectional curvatures bound and the injectivity radius (so they are independent with $R_k$), $H_k$ is the mean curvature of $Reg(\partial \Omega _k)$, and $\omega _7$ the volume of a unit ball of dimension 7.

If ${{\,\textrm{diam}\,}}(I_k) \rightarrow \infty$, there must exist $t_k\ge 0$ and $T_k\rightarrow \infty$ such that $I_k$ intersects with $\{t\}\times \Gamma$ for all $t\in [t_k,t_k+T_k]$. Thus we can cover $I_k\cap \left( [t_k,t_k+T_k] \times \Gamma \right)$ by balls with radius $r_0$ and half radius are pairwise disjoint, the cover contains at least $\frac{T_k}{2r_0}$ many balls. Then by the monotonicity formula (4.7), we have $\textbf{M}(I_k) \rightarrow \infty$. Hence, we obtain a contradiction with (1).

(3) Given any sequence $R_k\rightarrow \infty$, first, suppose that the isoperimetric regions are connected. Note that by (1) and (2):

$$\begin{aligned} \textbf{P}(\Omega _k)\le \textbf{M}(\Gamma )+ \textbf{P}(\Sigma ), \end{aligned}$$

(4.8)

and they have a uniform bound of diameters. So for each k large, $\partial \Omega _k\subset M(R_k){\setminus } \left( [\frac{1}{8} {R_k},\right. \left. \frac{7}{8} {R_k}] \times \Gamma \right)$, therefore, $|\Omega _k|$ cannot be equal to $\frac{1}{2} |M(R_k)|$. This leads to a contradiction.

In addition, by the uniform bound of the mean curvatures for $\Omega _k$ with all large R, the monotonicity formula (4.7) implies there exists a $c=c( V)$ such that the perimeter of each component of $\partial \Omega _R$ is bounded below by c. By (4.6), we get a uniform bound L about the number of components of $\partial \Omega$ for all large R.

(4) Denote $\partial \Omega _R= \bigcup _i I_i$. Suppose (4) fails. Note that by Theorem 3.3 (2), $H_7(M;\mathbb {Z}_2) \cong \langle [\Gamma ]\rangle$, then by the boundedness of diameter, all $\{I_i\}$ are boundaries. Reasoning as in the first part of (3), there exists $(t_i)_i$ and $(T_i)_i$ such that

$$\begin{aligned} I_i \setminus V\subset \mathcal {T}_i:=[t_i,t_i+T_i]\times \Gamma , \qquad \text {and}\qquad T_i<T_0<\infty , \end{aligned}$$

for every $i=1,\dots , L$, where $T_0$ is independent of i and exists by (2). In particular, since each $I_i$ is a boundary, we have

$$\begin{aligned} \Omega _R\subset V\cup \bigcup _{i=1}^L \mathcal {T}_i. \end{aligned}$$

Therefore we conclude

$$\begin{aligned} {{\,\textrm{Vol}\,}}(\Omega _R)\le {{\,\textrm{Vol}\,}}\left( V\cup \bigcup _{i=1}^L \mathcal {T}_i\right) \le {{\,\textrm{Vol}\,}}(V)+ \sum _{i=1}^L T_0\cdot \textbf{M}(\Gamma ). \end{aligned}$$

which for sufficiently large R cannot be half the volume of the whole manifold. $\square$

Proof of Theorem 4.1

We still denote $\Omega _R$ the isoperimetric region in M(R) with half volume. In order to prove the isoperimetric region is of the form we want, we will consider the connected components of $\partial \Omega _R$ into two types:

$$\begin{aligned} \{ \Lambda _i \}&:=\ \{\Lambda _i \subset \partial \Omega _R : \Lambda _i \cap {{\overline{V}}} = \emptyset \} ;\\ \{\Delta _i \}&:= \ \{ \Delta _i \subset \partial \Omega _R : \Delta _i \cap {{\overline{V}}} \ne \emptyset \}. \end{aligned}$$

Combining the results in Lemma 4.4, we claim the following result for isoperimetric regions $\partial \Omega _R$ with large R.

Claim: With $R_0$ from Lemma 4.4, there exists $R_1>R_0$, such that for any $R>R_1$, $\partial \Omega _R$ has exactly 2 components $\Lambda ,\Delta$, where $\Lambda \in \{ \Lambda _i \}$ and $\Delta \in \{\Delta _i \}$. Specifically, $[\Lambda ]=[\Delta ]=[1]$ where $[1]:=[\Sigma ] \in H_7(M(R),\mathbb {Z}_2)$.

Proof of the claim:

Step 1. $\{\Delta _i\} \ne \emptyset$ and at least one $\Delta _i$ is homologous to $\Sigma$.

Suppose not, then $I_1,I_2\subset [0,T] \times \Gamma$, where $I_1, I_2$ are as in Lemma 4.4(4). Since $[I_1]= [I_2] =[1] \in H_7(M(R),\mathbb {Z}_2)$ and $I_1,I_2\in \{\Lambda _i\}$, this implies that

$$\begin{aligned} \textbf{P}(\Omega _R)\ge \textbf{M}(I_1)+\textbf{M}(I_2) \ge 2\textbf{M}(\Gamma ) > \textbf{M}(\Sigma )+\textbf{M}(\Gamma ){\mathop {\ge }\limits ^{(1)}} \textbf{P}(\Omega _R). \end{aligned}$$

A contradiction.

Step 2. $\{\Lambda _i\} \ne \emptyset$ and at least one $\Lambda _i$ is homologous to $\Sigma$.

Suppose not, then we have that $\partial \Omega _R=\bigcup _{i=1}^{L_1}\Delta _i \cup \bigcup _{i=1}^{L_2}\Lambda _i$, with $L_2=0$ if $\{\Lambda \} = \emptyset$; and in the other case, each $\Lambda _i=\partial U_i$, for some open connected $U_i\subset \Omega _R$. Moreover, by Lemma 4.4(3), ${{\,\textrm{diam}\,}}(\Delta _i)<d_0,{{\,\textrm{diam}\,}}(\Lambda _i)<d_0$ and $L_1+L_2<L_0$, with $L_0, d_0$ independent of R. Now notice that since $\Delta _i\cap V\ne \emptyset$, there exist $T_0>0$, depending only on $d_0$, such that

$$\begin{aligned} \Omega _R\subset \Big ( V\cup [(0, T_0)\cup (R-T_0,R)] \times \Gamma \Big )\cup \bigcup _{i=1}^{L_2}U_i, \end{aligned}$$

with ${{\,\textrm{Vol}\,}}(U_i)\le T_0\cdot \Gamma$.This yields

$$\begin{aligned} {{\,\textrm{Vol}\,}}(\Omega _R)\le (L_2+2) \cdot T_0\cdot \textbf{M}(\Gamma ), \end{aligned}$$

which leads a contradiction for R sufficiently large.

Step 3. $\partial \Omega _R$ has no other components.

We have concluded that there is at least one $\Delta \in \{ \Delta _i \}$ and at least one $\Lambda \in \{ \Lambda _i \}$ such that

$$\begin{aligned}{}[\Delta ]&=[1]\in H_7(M(R), \mathbb {Z}_2) ,\\ [\Lambda ]&=[1]\in H_7(M(R), \mathbb {Z}_2). \end{aligned}$$

Clearly $\Lambda \subset [0,R]\times \Gamma$ where each slice $\{t\}\times \Gamma$ is a homological area-minimizing in the tube Tube(R). So we directly have

$$\begin{aligned} \textbf{M}(\Lambda )&\ge \textbf{M}(\Gamma ). \end{aligned}$$

On the other hand, we know that $\Sigma$ is the unique homological area minimizer in M(R). So clearly

$$\begin{aligned} \textbf{M}(\Delta ) \ge \textbf{M}(\Sigma ). \end{aligned}$$

So overall, by the perimeter upper bound (4.6), the only case that would happen is

$$\begin{aligned} \partial \Omega _R = \left[ \{t_0\} \times \Gamma \right] \cup \Sigma . \end{aligned}$$

We have two isolated singularities on $\Sigma$, so we prove Theorem 1.1. $\square$

5 Open problems

In this section, we will discuss two open questions about isoperimetric regions. The first is about more general types of examples. The second is about the generic regularity of isoperimetric regions.

1. About the existence of a singular example with non-zero mean curvature

There are two natural questions we may proceed with:

• 1. Prescribed regular tangent cones: Our construction in Theorem 4.1 requires the tangent cones to be Simons cones. Then we can guarantee that $\Gamma _+$ and $\Gamma _-$ are diffeomorphic to each other. If we generally choose the tangent cones as regular, strictly stable, strictly minimizing hypercones, $\Gamma _+$ may not be diffeomorphic to $\Gamma _-$.

• 2. Prescribed mean curvature: Note that $\partial \Omega$ is particularly composed of area-minimizing hypersurfaces. So this singular example comes from the existence of singular area minimizers. So now we come up with a new question: whether there exists a singular isoperimetric region with mean curvature constantly non-zero? In Morgan and Johnson [16, Theorem 2.2], we see that regardless of dimensions, if an isoperimetric region encloses a sufficiently small region, it will not have singularities (it will be a nearly round sphere.) However, for an isoperimetric with mean curvature small, a singular example is still unknown.

2. About generic regularity of isoperimetric regions

For the homological area-minimizing case, Smale in [21] shows that there exists a $C^\infty$-generic Riemannian metric such that for every nontrivial element in $H_7(M,\mathbb {Z})$, there exists a (unique) smooth homological area-minimizing hypersurface.

So we may ask whether we could generically “smooth" the singular isoperimetric regions. Let (M, g) be an 8-dimensional closed Riemannian manifold. For $k = 3,4,...$, denote $\mathcal {M}^k$ the class of $C^k$ metrics on M, endowed with the $C^k$ topology. For any $t\in (0,|M|_g)$, we define the subclass $\mathcal {F}^k_t \subset \mathcal {M}^k$ to be the set of metrics such that there is a smooth isoperimetric region with volume t (relative to the new ${{\overline{g}}}\in \mathcal {F}^k_t$).

Conjecture 5.1

$\mathcal {F}^k_t$ is dense in $\mathcal {M}^k$ for any $k\ge 3$ and $t\in (0, |M|_g))$.