Conformal upper bounds for the volume spectrum

Abstract

In this paper, we prove upper bounds for the volume spectrum of a Riemannian manifold that depend only on the volume, dimension and a conformal invariant.

Appendix A. Proof of Theorem 2.2

Proof of Theorem 2.2

We follows the steps given by Glynn-Adey and Liokumovich in [GL17], where they proved this theorem for $N=M$. Here we give the outline and point out some necessary modifications.

Suppose that N has smooth boundary. For any $\epsilon _0\in (0,1)$, take ${\bar{r}}(M,N,\epsilon _0)$ such that:

• for every $x\in \partial N$, we have that $B_r(x)$ is $(1 + \epsilon _0)$-bilipschitz diffeomorphic to the Euclidean ball of radius r and $B_r(x)\cap N$ is mapped onto a half-ball under the difformorphism. Denote by $B_{r}^+(x)=B_r(x)\cap N$;

• the monotonicity formula [GLZ20, Theorem 3.4] holds.

From now on, we fix some $\epsilon _0<1$.

Step 1: Suppose that N has smooth boundary. There exists $\epsilon =\epsilon (M,N,{\bar{r}})$ satisfying the following: for any domain $D\subset N$ with $|D|<\epsilon$, there exists a collection of domains $D(=:D_0)\supset D_1\supset D_2\supset \cdots \supset D_m$ satisfying

• $D_m\subset \mathrm {Int}N$;

• $|\partial D_j\cap \mathrm {Int}N|\ge |\partial D_{j+1}\cap \mathrm {Int}N|$ for $0\le j\le m-1$;

• for $0\le j\le m-1$, $D_j\setminus D_{j+1}$ is contained in some ball of radius ${\bar{r}}$ and center $x\in \partial N$;

Proof of Step 1

Suppose that $x\in \partial D_j\cap \partial N$, now we construct $D_{j+1}\subset D_j$. By the co-area formula, we can find $r'\in (3{\bar{r}}/4,{\bar{r}})$ such that $\partial D_j\cap \mathrm {Int}N$ is transverse to $\partial B_{r'}(x)$ and

$$\begin{aligned} |D_j\cap \partial B_{r'}(x)|\le (8/{\bar{r}})\cdot |D_j\cap B_{r'}(x)|. \end{aligned}$$

Denote by $S=\llbracket D_j\cap \partial B_{r'}(x)\rrbracket$. Let T be the minimizing current among all

$$\begin{aligned} T'\in \mathcal Z_{n-1}( B^+_{r'}(x),\partial B^+_{r'}(x);\mathbb Z_2) \text { with } {{\,\mathrm{spt}\,}}(\partial T'-\partial S)\subset \partial N. \end{aligned}$$

Then by the regularity theory [MOR03, Theorem 4.7] (see also [GLWZ19, Theorem 4.7]), T is induced by a free boundary hypersurface $\Sigma$ with $(n-8)$-dimensional singular set. By taking $\epsilon$ small enough, from the monotonicity formula [GLZ20, Theorem 3.4], $\Sigma \cap \partial N\cap B_{{\bar{r}}/2}(x)=\emptyset$. Using the monotonicity formula again, $\Sigma \cap B_{{\bar{r}}/4}(x)=\emptyset$. Note that by the isoperimetric choice [LZ], there exists $V\subset B_{{\bar{r}}}^+(x)$ such that $\partial \llbracket V\rrbracket =T-S$ and the volume of V is small. Hence V does not contain $B^+_{{\bar{r}}/4}(x)$. Together with the fact of $\partial V$ does not intersect $B^+_{{\bar{r}}/4}(x)$, we conclude that $V\cap B_{{\bar{r}}/4}^+(x)=\emptyset$. Now we define

$$\begin{aligned} D_{j+1}=D_j\cap (N\setminus (B^+_{{\bar{r}}}(x)\setminus V) ). \end{aligned}$$

Clearly, $D_j\setminus D_{j+1}$ is contained in $B_{{\bar{r}}}^+(x)$. Note that T is minimizing in $B_{{\bar{r}}}^+(x)$. Then it is minimizing in $B_{{\bar{r}}}^+(x)\setminus V$, i.e.

$$\begin{aligned} |\Sigma \cap D_j|\le |\partial D_j\cap \mathrm {Int}( B^+_{{\bar{r}}}(x)\setminus V)|. \end{aligned}$$

This implies

$$\begin{aligned} |\partial D_{j}\cap \mathrm {Int}N|-|\partial D_{j+1}\cap \mathrm {Int}N|=|\partial D_j\cap \mathrm {Int}( B^+_{{\bar{r}}}(x)\setminus V)|- |\Sigma \cap D_j|\ge 0. \end{aligned}$$

Thus Step 1 is completed. $\square$

Step 2: Suppose that N has smooth boundary. There exist constants $\beta _1=\beta _1(n)$ and $\epsilon =\epsilon (M,N,{\bar{r}})$ such that for any domain $D\subset N$ with $|D|\le \epsilon$, the following bound holds:

$$\begin{aligned} \omega _1(D,g)\le \beta _1|D|^{\frac{n-1}{n}}+|\partial D\cap \mathrm {Int}N|. \end{aligned}$$

(6.1)

Proof of Step 2

Let $\{ D_j\}_{j=1}^m$ be the domains constructed in Step 1. Then repeating the process inside N (see also [GL17, Proposition 4.3]), there exists $D_m\supset D_{m+1}\supset \cdots \supset D_L$ such that

• $|\partial D_j\cap \mathrm {Int}N|\ge |\partial D_{j+1}\cap \mathrm {Int}N|$ for $m\le j\le L-1$;

• for $m\le j\le L$, $D_j\setminus D_{j+1}$ is contained in some ball of radius ${\bar{r}}$ and center $x\in N$, where $D_{L+1}=\emptyset$;

By [GUT07], there exists $\beta _1=\beta _1(n)$ such that for $0\le j\le L$,

$$\begin{aligned} \omega _1(D_j\setminus D_{j+1},g)\le \beta _1|D_j\setminus D_{j+1}|^{\frac{n-1}{n}}. \end{aligned}$$

(6.2)

Now let $\Phi _j$ be a sweepout of $D_j\setminus D_{j+1}$ having no concentration of mass. Then there exist lifting maps $\widetilde{\Phi }_j:[0,1]\rightarrow \mathcal C(D_j\setminus D_{j+1})$ such that

$$\begin{aligned} \partial \circ \widetilde{\Phi }_j=\Phi _j \ \ \ \text { for } \ \ \ 0\le j\le L. \end{aligned}$$

Without loss of generality, we assume that $\widetilde{\Phi }_j(0)=0$, $\widetilde{\Phi }_j(1)=\llbracket D_j\setminus D_{j+1}\rrbracket$. By [GL17, Proposition 2.3], we can construct a sweepout of D as follows: we first define $\widetilde{\Phi }:[0,1]\rightarrow \mathcal C(D)$ by

$$\begin{aligned} \widetilde{\Phi }(t)=\widetilde{\Phi }_{L-j}\Big ((L+1)(t-\frac{j}{L+1})\Big ) +\llbracket D_{L+1-j}\rrbracket \ \ \text { for } \ \ \frac{j}{L+1}\le t\le \frac{j+1}{L+1}. \end{aligned}$$

Then $\Phi =\partial \circ \Phi$ is the desired sweepout, which has no concentration of mass. Such a construction gives that

$$\begin{aligned} \omega _1(D,g)\le \max _{0\le j\le L}\{\omega _1(D_j\setminus D_{j+1},g) +|\partial D_j\setminus \partial D| \} . \end{aligned}$$

Together with (6.2), we have

$$\begin{aligned} \omega _1(D,g)\le \beta _1|D|^{\frac{n-1}{n}} +|\partial D\cap N|. \end{aligned}$$

$\square$

Step 3: Suppose that N has smooth boundary. There exists $\beta _2=\beta _2(n)$ such that for any domain $D\subset N$, the following bound holds

$$\begin{aligned} \omega _1(D,g)\le \beta _2\cdot (1+|D|_{g_0}^{\frac{1}{n}})|D|^{\frac{n-1}{n}}+2|\partial D\cap \mathrm {Int}N|. \end{aligned}$$

(6.3)

Proof of Step 3

We use the argument in [GL17, Theorem 5.1]. Let $\epsilon _1=25^{-n}\cdot \epsilon$. Take $\beta _2(n)=\beta _1(n)+3c(n)\cdot \Big [1-(1-25^{-n})^{\frac{n-1}{n}}\Big ]$. Here c(n) is the constant in [GL17, Lemma 3.4]. It follows that

$$\begin{aligned} \Big [1-(1-25^{-n})^{\frac{n-1}{n}}\Big ]\beta _2(n)\ge 3c(n). \end{aligned}$$

(6.4)

By Step 2, for $k\le 25^n$, (6.3) holds for D with $|D|\le k\epsilon _1$. We proceed by induction on k.

Suppose the inequality holds for compact domains with volume at most $k\epsilon$. Then for any $D\subset N$ with $k\epsilon _1<|D|\le (k+1)\epsilon _1$. By Theorem 4.1, there exists a hypersurface $\Sigma$ subdividing D into $D_0$ and $D_1$ such that $|D_j|\le (1-25^{-n})|D|$ (for $j=0,1$) and

$$\begin{aligned} |\Sigma |\le c(n)|D|^{\frac{n-1}{n}}(1+|D|_{g_0}^{\frac{1}{n}}). \end{aligned}$$

(6.5)

Then using the construction of sweepouts in Step 2, we have

$$\begin{aligned} \omega _1(D,g)\le \max _{j\in \{0,1\}}\{ \omega _1(D_j,g)+|\partial D_j\setminus \partial D| \}. \end{aligned}$$

(6.6)

Note that for $j=0,1$,

$$\begin{aligned} |D_j|\le (1-25^{-n})|D|\le |D|-\epsilon _1<(k+1)\epsilon _1-\epsilon _1<k\epsilon _1. \end{aligned}$$

Hence by the assumption,

$$\begin{aligned} \omega _1(D_j,g)&\le \beta _2\cdot (1+|D_j|_{g_0}^{\frac{1}{n}})|D_j|^{\frac{n-1}{n}}+2|\partial D_j\cap \mathrm {Int}N|\\&\le \beta _2\cdot (1+|D|_{g_0}^{\frac{1}{n}})|D|^{\frac{n-1}{n}} \cdot (1-25^{-n})^{\frac{n-1}{n}}+2|\partial D\cap \mathrm {Int}N|+2|\Sigma |\\&\le (\beta _2-3c(n))(1+|D|_{g_0}^{\frac{1}{n}})|D|^{\frac{n-1}{n}} +2|\partial D\cap \mathrm {Int}N|+2|\Sigma |\\&\le \beta _2\cdot (1+|D|_{g_0}^{\frac{1}{n}})|D|^{\frac{n-1}{n}}+2|\partial D\cap \mathrm {Int}N|-|\Sigma |. \end{aligned}$$

Here the third inequality is from (6.4) and we used (6.5) in the last one. Then together with (6.6), we conclude that

$$\begin{aligned} \omega _1(D,g)\le \beta _2\cdot (1+|D|_{g_0}^{\frac{1}{n}}) |D|^{\frac{n-1}{n}}+2|\partial D\cap \mathrm {Int}N|. \end{aligned}$$

This finishes Step 3. $\square$

Step 4: We prove the theorem for general compact domain N (having piecewise smooth boundary).

Proof of Step 4

Now let N be a compact domain with piecewise smooth boundary. Then we have a tubular neighborhood U of N such that U has smooth boundary and $|U|_{g_0}\le 2|N|_{g_0}$ and $|U|\le 2|N|$. Then by Step 3,

$$\begin{aligned} \omega _1(U,g)\le \beta _2\cdot (1+|U|_{g_0}^{\frac{1}{n}})|U|^{\frac{n-1}{n}}\le 2\beta _2\cdot (1+|N|_{g_0}^{\frac{1}{n}})|N|^{\frac{n-1}{n}} . \end{aligned}$$

Then the desired inequality follows from

$$\begin{aligned} \omega _1(N,g)\le \omega _1(U,g) \end{aligned}$$

if we take $K=2\beta _2(n)$. $\square$

So far, Theorem 2.2 is proved. $\square$