Min–max theory for constant mean curvature hypersurfaces

Abstract

In this paper, we develop a min–max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature c. Moreover, if c is nonzero then our min–max solution always has multiplicity one.

Appendices

Appendix A: an interpolation lemma

The first lemma below was essentially due to Pitts [44, Lemma 3.8], but the modification to find the interpolation sequence using boundaries of Caccioppoli sets was completed by the first author [62, Proposition 5.3].

Lemma A.1

Suppose $L>0$, $\eta >0$, W is a compact subset of U, and $\Omega \in {\mathcal {C}}(M)$. Then there exists $\delta =\delta (L, \eta , U, W, \Omega )>0$, such that for any $\Omega _1, \Omega _2\in {\mathcal {C}}(M)$ satisfying

(a):

${\text {spt}}(\Omega _i-\Omega )\subset W$, $i=1, 2$,

(b):

$\mathbf {M}(\partial \Omega _i)\le L$, $i=1, 2$,

(c):

${\mathcal {F}}(\partial \Omega _1-\partial \Omega _2)\le \delta$,

there exist a sequence $\Omega _1=\Lambda _0, \Lambda _1, \ldots , \Lambda _m=\Omega _2\in {\mathcal {C}}(M)$ such that for each $j=0, \ldots , m-1$,

• (i) ${\text {spt}}(\Lambda _j-\Omega )\subset U$,

• (ii) ${\mathcal {A}}^c(\Lambda _j)\le \max \{{\mathcal {A}}^c(\Omega _1), {\mathcal {A}}^c(\Omega _2)\}+\eta$,

• (iii) $\mathbf {M}(\partial \Lambda _j-\partial \Lambda _{j+1})\le \eta$.

• (iv)
  $$\begin{aligned} \mathbf {M}(\partial \Lambda _j)\le \max \{\mathbf {M}(\partial \Omega _1), \mathbf {M}(\partial \Omega _2)\}+\frac{\eta }{2}. \end{aligned}$$
• (v)
  $$\begin{aligned} \mathbf {M}(\Lambda _j-\Omega _i)\le \frac{\eta }{2c}, \text { for } i =1, 2. \end{aligned}$$

Proof

Note that by a covering argument, one only needs to prove the case when $\partial \Omega _2$ is fixed, and in this case $\delta =\delta (L, \eta , U, W, \Omega , \Omega _2)$.

Under our assumptions on $\Omega _1, \Omega _2$, there are two issues for us that differ from the conclusions of Pitts [44, Lemma 3.8]:

• (1) we require the interpolating sequence $\{\partial \Lambda _j\}$ to consist of boundaries of Caccioppoli sets, while in [44] the interpolating sequence consists of integral currents $\{T_j\in {\mathcal {Z}}_n(M)\}$ (using notations therein);

• (2) for point (ii), we require that ${\mathcal {A}}^c(\Lambda _j)$ does not increase much from ${\mathcal {A}}^c(\Omega _1), {\mathcal {A}}^c(\Omega _2)$, while in [44] it was only proven that $\mathbf {M}(T_j)\le \max \{\mathbf {M}(\partial \Omega _1), \mathbf {M}(\partial \Omega _2)\}+\eta$.

These two minor points can be easily deduced from the mentioned work of Zhou [62], and now we point out the necessary details. In fact, with regards to point (1), the proof of [62, Proposition 5.3] already proceeds using boundaries, and we will first justify properties (i, iii, iv).

Specifically, given $L, \eta , \Omega , \Omega _2$ satisfying the assumptions in the Lemma, [62, Proposition 5.3] (applied to the case when $m=1, l=0$ therein) gives the desired $\delta >0$, such that if $\Omega _1$ satisfies the assumptions (a–c), then there exists a sequence $\Omega _1=\Lambda _0, \Lambda _1, \ldots , \Lambda _m=\Omega _2\in {\mathcal {C}}(M)$ such that for each $j=0, \ldots , m-1$, properties (iii) and (iv) are satisfied.

Although in [62, Proposition 5.3] $\Omega _i$ were not assumed to satisfy the additional condition ${\text {spt}}(\Omega _i-\Omega )\subset W$. It can be seen that the construction in [62, Proposition 5.3] indeed satisfies the required property (i) under this additional condition.

Note that properties (iv, v) immediately imply (ii), so it remains only to show (v). Indeed, from the construction in [62, Proposition 5.3], one knows that the symmetric difference $\Lambda _j\triangle \Omega _i$ is a subset of the union of the symmetric difference $\Omega _1\triangle \Omega _2$ together with finitely many (depending only on L and $\eta$) balls of arbitrarily small radii in U. Since ${\text {Vol}}(\Omega _1\triangle \Omega _2)=\mathbf {M}(\Omega _1-\Omega _2)={\mathcal {F}}(\partial \Omega _1, \partial \Omega _2)$ by the isoperimetric lemma [62, Lemma 7.3], one thus obtains (v) by taking $\delta$ small enough. $\square$

Next we explain how to use Lemma A.1 to finish the proof of $(c)\Longrightarrow (d)$ in Proposition 5.3.

Proof of Proposition 5.3$(c)\Longrightarrow (d)$—continued It suffices to prove that for any given $\epsilon , \delta$ and $\Omega \in {\mathscr {A}}^c_n(U; \epsilon , \delta ; \mathbf {M})$, there exists $\delta '=\delta '(\epsilon , \delta , U, W, \Omega )>0$, such that $\Omega \in {\mathscr {A}}^c_n(W; \epsilon , \delta '; {\mathcal {F}})$. Indeed, suppose so, then by (c), we can find a sequence $\Omega _i\in {\mathscr {A}}^c_n(U; \epsilon _i, \delta _i; \mathbf {M})$ with $\lim _{i\rightarrow \infty }|\partial \Omega _i|=V$ as varifods. As $\Omega _i\in {\mathscr {A}}^c_n(W; \epsilon _i, \delta _i'; {\mathcal {F}})$ for some $\delta _i'$, this immediately implies that V is c-almost minimizing in W.

Given some $\delta '>0$, to show that $\Omega \in {\mathscr {A}}^c_n(W; \epsilon , \delta '; {\mathcal {F}})$, consider an arbitrary sequence $\Omega =\Omega _0, \Omega _1, \Omega _2, \ldots , \Omega _m\in {\mathcal {C}}(M)$ satisfying that:

• (i) ${\text {spt}}(\Omega _i-\Omega )\subset W$;

• (ii) ${\mathcal {F}}(\partial \Omega _{i+1}, \partial \Omega _i)\le \delta '$;

• (iii) ${\mathcal {A}}^c(\Omega _i)\le {\mathcal {A}}^c(\Omega )+\delta '$, for $i=1, \ldots , m$.

Note that the mass $\mathbf {M}(\partial \Omega _i)\le \mathbf {M}(\partial \Omega )+c{\text {Vol}}(M)+1$; in Lemma A.1 we let $L=\mathbf {M}(\partial \Omega )+c{\text {Vol}}(M)+1$, $\eta =\delta /2$, and hence there exists $\delta '=\delta (L, \eta , U, W, \Omega )$, and for each i there exist a sequence $\Omega _i=\Omega _{i, 1}, \ldots , \Omega _{i, l_i}=\Omega _{i+1}$, such that

• (i) ${\text {spt}}(\Omega _{i, l}-\Omega )\subset U$,

• (ii) ${\mathcal {A}}^c(\Omega _{i, l})\le \max \{{\mathcal {A}}^c(\Omega _i), {\mathcal {A}}^c(\Omega _{i+1})\}+\delta /2\le {\mathcal {A}}^c(\Omega )+\delta /2+\delta '$,

• (iii) $\mathbf {M}(\partial \Omega _{i, l}-\partial \Omega _{i, l+1})\le \delta /2$.

Now consider the new sequence by putting all these interpolations together: $\Omega _0=\Omega , \Omega _{0, 1}, \ldots , \Omega _{0, l_0}=\Omega _1, \Omega _{1, 1}, \ldots , \Omega _{1, l_1}=\Omega _2, \ldots , \Omega _3, \ldots , \cdots , \Omega _m$. It is evident that this new sequence satisfies assumptions (i), (ii), (iii) in Definition 5.1 for ${\mathscr {A}}^c_n(U; \epsilon , \delta ; \mathbf {M})$ if we further shrink $\delta '$ to make $\delta '\le \delta /2$, and hence we get ${\mathcal {A}}^c(\Omega _m)\ge {\mathcal {A}}^c(\Omega )-\epsilon$. This proves that $\Omega \in {\mathscr {A}}^c_n(W; \epsilon , \delta '; {\mathcal {F}})$. $\square$

Appendix B: interpolation process

Proof of Claim 2 in Proposition 4.4

Here we describe the construction of $\{\phi _i\}$ by interpolating $\{\phi ^1_i\}$.

For fixed $i\in {{\mathbb {N}}}$ and consider a 1-cell $\alpha \in I(1, k_i)$. We only need to show how to interpolate $\phi ^1_i$ when restricted to $\alpha _0$. For notational simplicity we write $\alpha =[0, 1]$. For $x\in \alpha$ let $\tilde{X}_i(x)$ be the linear interpolation between $\tilde{X}_i(0)=\tilde{X}(|\partial \phi ^*_i(0)|)$ and $\tilde{X}_i(1)=\tilde{X}(|\partial \phi ^*_i(1)|)$, where $\tilde{X}$ is defined in (4.6)). That is, $\tilde{X}_i(x)=(1-x)\tilde{X}_i(0)+x\tilde{X}_i(1)$.

The continuity of the map $V\rightarrow \tilde{X}(V)$ implies that $\Vert \tilde{X}_i(x)-\tilde{X}_i(0)\Vert _{C^1(M)}\rightarrow 0$ uniformly as $i\rightarrow \infty$, (since under the varifold $\mathbf {F}$-metric, $\mathbf {F}(|\partial \phi ^*_i(0)|, |\partial \phi ^*_i(1)|)\le \mathbf {f}(\phi ^*_i)\rightarrow 0$ uniformly in i). Define ${\bar{Q}}_i(x)$ to be the push-forward of $\phi ^*_i(0)$ by the flow of $\tilde{X}_i(x)$ up to time 1; this gives a map ${\bar{Q}}_i: \alpha \rightarrow \mathcal {C}(M)$. Note that $\partial {\bar{Q}}_i: \alpha \rightarrow {\mathcal {Z}}_n(M)$ is continuous under the $\mathbf {F}$-metric.

Since ${\bar{Q}}_i(x)$ and $\phi ^1_i(0)$ are the push-forwards of the same initial set $\phi ^*_i(0)$ under the flows of $\tilde{X}_i(x)$ and $\tilde{X}_i(0)$ respectively, we have

$$\begin{aligned} \left. \begin{array}{cl} \mathbf {F}(\partial {\bar{Q}}_i(x), \partial \phi ^1_i(0))\rightarrow 0 \\ \mathbf {M}({\bar{Q}}_i(x)- \phi ^1_i(0))\rightarrow 0 \end{array} \right. , \text { uniformly in }x,\alpha \hbox { as }i\rightarrow \infty . \end{aligned}$$

(B.1)

Note that the uniformity follows from a simple contradiction argument, using the uniform smallness between $\tilde{X}_i(x)$ and $\tilde{X}_i(0)$ together with the fact that the set of images of all the $\{\phi ^*_i\}$ is compact.

As ${\bar{Q}}_i(1)$ and $\phi ^1_i(1)$ are the respective push-forwards of $\phi ^*_i(0)$ and $\phi ^*_i(1)$ under the same flow of $\tilde{X}_i(1)$ (and since these flows have uniformly bounded Lipschitz constant, independent of i), we have

$$\begin{aligned}&\mathbf {M}(\partial {\bar{Q}}_i(1)-\partial \phi ^1_i(1)) \le C\, \mathbf {M}(\partial \phi ^*_i(1)-\partial \phi ^*_i(0))\nonumber \\&\quad \rightarrow 0, \text { uniformly in }\alpha \text { as }i\rightarrow \infty . \end{aligned}$$

(B.2)

Now we can apply the discretization-interpolation theorem [62, Theorem 5.1] (c.f. [33, Theorem 13.1]) to ${\bar{Q}}_i$. In fact, given any continuous map ${\bar{\phi }}$ from a 1-cell $\alpha$ to ${\mathcal {C}}(M)$, the discretization-interpolation theorem implies that for any $\eta >0$, there exists a large integer $l_\eta >0$, and a discrete map $\phi$ defined on the vertices of some refinement $\alpha (l_\eta )$, so that: (i) the masses of $\phi$ do not go up much, (ii) the fineness of $\phi$ is controlled by $\eta$, and (iii) $\phi$ stays close to ${\bar{\phi }}$ under the flat norm. In particular, the triple $(\eta , l_\eta , \phi )$ can be chosen to be any triple $(\delta _j, k_j, \phi _j)$ given by [62, Theorem 5.1] (here the latter triple uses the notation of that theorem) satisfying $\delta _j\le \eta$.

Applying this to ${\bar{Q}}_i$, for any $\eta >0$, we obtain a large integer $l_\eta >0$ and a map $Q_i: \alpha (l_\eta )_0\rightarrow {\mathcal {C}}(M)$, such that

• (i) given $x\in \alpha (l_\eta )_0$,

  $$\begin{aligned} \mathbf {M}(\partial Q_i(x)) \le \mathbf {M}(\partial {\bar{Q}}_i(x))+\eta /2, \end{aligned}$$

  and also by the same argument as in the proof of point (v) of Lemma A.1,

  $$\begin{aligned} \mathbf {M}(Q_i(x)-{\bar{Q}}_i(x)) \le \eta /(2c), \end{aligned}$$

  and hence

  $$\begin{aligned} {\mathcal {A}}^c(Q_i(x))\le {\mathcal {A}}^c({\bar{Q}}_i(x))+\eta ; \end{aligned}$$

• (ii) $\mathbf {f}(Q_i)\le \eta$;

• (iii) $\sup \{{\mathcal {F}}(\partial Q_i(x)-\partial {\bar{Q}}_i(x)): x\in \alpha (l_\eta )_0\}<\eta$.

When $\eta \rightarrow 0$, by (i, iii) and [44, 2.1(20)]¹ (see also [33, Lemma 4.1]), we have

$$\begin{aligned} \lim _{\eta \rightarrow 0}\sup \{\mathbf {F}(\partial Q_i(x), \partial {\bar{Q}}_i(x)): x\in \alpha (l_\eta )_0\}=0. \end{aligned}$$

To finish the interpolation, take a sequence $\eta _i\rightarrow 0$, and denote $l_i=k_i+l_{\eta _i}+1$, then we construct $\phi _i: I(1, k_i+l_{\eta _i}+1)\rightarrow {\mathcal {C}}(M)$ by defining $\phi _i$ on each $\alpha (l_{\eta _i}+1)_0$ by

$$\begin{aligned} \phi _i(x)= \left\{ \begin{array}{cl} Q_i(3x) &{} \text { for }x\in [0, 1/3]\cap \alpha (l_{\eta _i}+1)_0 \\ \phi _i^1(1) &{} \text { otherwise.} \end{array} \right. \end{aligned}$$

The desired properties (a, b, c, d) of $\phi _i$ can be read off from (B.1), (B.2) and the properties of $Q_j$. Since ${\bar{Q}}_i$ is obtained from a continuous deformation from $\phi ^*_i$, a further interpolation argument shows that S is homotopic to $S^*$, and hence we finish the proof of Claim 2.

Appendix C: good replacement property and regularity

Here we record the notions of good replacements and the good replacement property. Recall the following definitions by Colding–De Lellis [11]. Consider two open subsets $W\subset \subset U\subset M^{n+1}$.

Definition C.1

[11, Definition 6.1]. Let $V \in V_n(U)$ be stationary in U. A stationary varifold $V' \in V_n(U)$ is said to be a replacement for V in W if

Definition C.2

[11, Definition 6.2]. Let $V \in V_n(U)$ be stationary in U. V is said to have the good replacement property in W if

• (a) there is a positive function $r: W\rightarrow {{\mathbb {R}}}$ such that for every annulus $A_{s, t}(x)\cap M\subset W$ with $0<s<t<r(x)$, there is a replacement $V'$ for V in $A_{s, t}(x)\cap M$;

• (b) the replacement $V'$ has a replacement $V''$ in every annulus $A_{s, t}(y)\cap M\subset W$ with $0<s<t<r(y)$;

• (c) $V''$ has a replacement $V'''$ in every annulus $A_{s, t}(z)\cap M\subset W$ with $0<s<t<r(z)$.

Note that our formulations are local compared to those in [11]. Indeed, the proofs of [11, Proposition 6.3] and [14, Theorem 2.8] are purely local, so the following proposition still holds:

Proposition C.3

[11, Proposition 6.3], [14, Proposition 2.8] When $2\le n\le 6$, if $V\in V_n(U)$ has the good replacement property in W, then is an integer multiple of some smooth embedded minimal hypersurface $\Sigma$.

Appendix D: Almgren–Pitts deformation process

Here we sketch the deformation process of Almgren–Pitts [44, 4.10] that we used in the proof of Theorem 5.6. Recall that we assumed for the sake of contradiction that there is no $V\in C(S)$ that is c-almost minimizing on small annuli.

As the critical set C(S) (Definition 3.6) is compact under the varifold $\mathbf {F}$-norm, there exists some uniform $\epsilon >0$, such that for each $V\in C(S)$, there exist $p_V\in M$ and some $\tilde{r}>0$, so that given any $\delta >0$ and $\tilde{r}>r+2s>r-2s>0$, and $\Omega \in {\mathcal {C}}(M)$ with $\mathbf {F}(|\partial \Omega |, V)<\epsilon$, we must have $\Omega \notin {\mathscr {A}}^c_n(A_{r-2s, r+2s}(p_V)\cap M; \epsilon , \delta ; \mathbf {M})$. Moreover, we can pick finitely many $\{V_1, \ldots , V_m\}\subset C(S)$, such that the $\mathbf {F}$-balls $\{B^{\mathbf {F}}_{\epsilon /4}(V_j)\}$ form a cover of C(S). Given j, denote $p_j=p_{V_j}$.

Now consider $\phi _i\in S$ for i sufficiently large. Denote by $I_i$ the subset of $\text {domain}(\phi _i)$, such that if $x\in I_i$, then ${\mathcal {A}}^c(\phi _i(x))$ is close enough to $\mathbf {L}^c$. Then for each $x\in I_i$, $\phi _i(x)$ must be close to some $V_j$ under the $\mathbf {F}$-norm, and so for any $\delta >0$,

$$\begin{aligned} \phi _i(x)\notin {\mathscr {A}}^c_n(A_{r-2s, r+2s}(p_j)\cap M; \epsilon , \delta ; \mathbf {M}), \end{aligned}$$

where $r, s>0$ can be as small as we want. By definition of ${\mathscr {A}}^c_n$ there exists a discrete deformation sequence, supported in $A_{r-2s, r+2s}(p_j)\cap M$ and starting from $\phi _i(x)$, which is almost continuous at the scale of $\delta$ under the $\mathbf {M}$-norm and eventually deforms ${\mathcal {A}}^c(\phi _i(x))$ down by at least $\epsilon$.

We can then choose $\delta$ to be the fineness of $\phi _i$, and use the corresponding deformation sequence to deform $\phi _i$. The choice of $\delta$ will make this deformation a homotopy. Note that to ensure $\phi _i$ is a sweepout, we not only need to deform $\phi _i(x)$ but also its adjacent slices. The strategy is then to allow two deformations, taking place in disjoint annuli, for each $x\in I_i$. A covering argument then shows that any slice is indeed affected by at most two such deformations. Finally, the deformations are chosen so that at least in one small annulus, the ${\mathcal {A}}^c$-value of $\phi _i(\cdot )$ is deformed down by $\epsilon$, while the ${\mathcal {A}}^c$-value of $\phi _i(\cdot )$ in the other deformation annulus is increased no more than $\delta$. Choosing $\delta <\epsilon /2$, the ${\mathcal {A}}^c$-values of $\phi _i(\cdot )$ will therefore be deformed down by $\epsilon /2$ everywhere in $I_i$, and this implies $\mathbf {L}^c(\tilde{S})<\mathbf {L}^c(S)$.