# Relating diameter and mean curvature for
Riemannian submanifolds

Jia-Yong Wu  Department of Mathematics, Shanghai Maritime University, Haigang
Avenue 1550, Shanghai 201306, P. R. China<EMAIL_ADDRESS>and  Yu Zheng
Department of Mathematics, East China Normal University, Dong Chuan Road 500,
Shanghai 200241, P. R. China<EMAIL_ADDRESS>
(Date: December 20, 2009.)

###### Abstract.

Given an $m$-dimensional closed connected Riemannian manifold $M$ smoothly
isometrically immersed in an $n$-dimensional Riemannian manifold $N$, we
estimate the diameter of $M$ in terms of its mean curvature field integral
under some geometric restrictions, and therefore generalize a recent work of
Topping in the Euclidean case (Comment. Math. Helv., 83 (2008), 539–546).

###### Key words and phrases:

Mean curvature; Riemannian submanifolds; Geometric inequalities; Diameter
estimate.

###### 2000 Mathematics Subject Classification:

Primary 53C40; Secondly 57R42.

This work is partially supported by the NSFC10871069.

##  1\. Introduction

Let $M\rightarrow N$ be an isometric immersion of Riemannian manifolds of
dimension $m$ and $n$, respectively. In this paper, we estimate the intrinsic
diameter of the closed submanifold $M$ in terms of its mean curvature vector
integral, under some geometric restrictions involving the volume of $M$, the
sectional curvatures of $N$ and the injectivity radius of $N$. In particular,
we can estimate the intrinsic diameter of the closed submanifold $M$ in terms
of its mean curvature vector integral without any geometric restriction,
provided the sectional curvatures of the ambient manifold $N$ is non-positive.
Our work was inspired by the following result of P. Topping [11] who treated
the case $N=\mathbb{R}^{n}$.

Theorem A. (P. Topping [11]) _For $m\geq 1$, suppose that $M$ is an
$m$-dimensional closed (compact, no boundary) connected manifold smoothly
immersed in $\mathbb{R}^{n}$. Then there exists a constant $C(m)$ dependent
only on $m$ such that its intrinsic diameter $d_{int}$ and mean curvature $H$
are related by_

(1.1) $d_{int}\leq C(m)\int_{M}|H|^{m-1}d\mu,$

_where $d_{int}:=\max_{x,y\in M}dist_{M}(x,y)$ and $\mu$ is the measure on $M$
induced by the ambient space. In particular, we can take
$C(2)=\frac{32}{\pi}$._

Prior to the Topping’s work, L. Simon in [6] (see also [8]) derived an
interesting estimate of the external diameter $d_{ext}:=\max_{x,y\in
M^{2}\hookrightarrow\mathbb{R}^{3}}|x-y|_{\mathbb{R}^{3}}$ of a closed
connected surface $M^{2}$ immersed in $\mathbb{R}^{3}$ in terms of its area
and Willmore energy:

(1.2)
$d_{ext}<\frac{2}{\pi}\left(Area(M^{2})\cdot\int_{M^{2}}|H|^{2}d\mu\right)^{\frac{1}{2}}.$

At the core of the proof of (1.2) is the following assertion that one cannot
simultaneously have small area and small mean curvature in a ball within the
surface. In other words, for all $r>0$, we have

(1.3)
$\pi\leq\frac{A_{ext}(x,r)}{r^{2}}+\frac{1}{4}\int_{B_{ext}(x,r)}|H|^{2}d\mu,$

where $B_{ext}(x,r)$ and $A_{ext}(x,r)$ denote the subset of $M^{2}$ immersed
inside the open extrinsic ball in $\mathbb{R}^{3}$ centred at $x$ of radius
$r>0$ and its area, respectively. This type of estimate is from [6], and with
these sharp constants from [8]. Combining this fact with a simple covering
argument, one can derive (1.2). Note that if $M^{2}$ is a surface of constant
mean curvature $H$ immersed in $\mathbb{R}^{3}$, P. Topping in [7] used a
different method and established the following inequality

$d_{ext}\leq\frac{A|H|}{2\pi}.$

Equality is achieved when $M^{2}$ is a sphere.

Following the idea of L. Simon’s proof, P. Topping in [11] proved Theorem A by
considering a refined version of (1.3) for any dimensional manifold immersed
in $\mathbb{R}^{n}$. Roughly speaking, P. Topping asserted that the maximal
function and volume ratio (see their definitions in Section 2) cannot be
simultaneously smaller than a fixed dimensional constant. This assertion can
be confirmed by means of the Michael-Simon Sobolev inequality for submanifolds
of Euclidean space [4]. Then using this assertion and a covering lemma, one
can derive (1.1) immediately. As an application, H.-Z. Li in a recent paper
[3] used Theorem A to discuss the convergence of the volume-preserving mean
curvature flow in Euclidean space under some initial integral pinching
conditions.

On the other hand, as we all known, D. Hoffman and J. Spruck in [2] extended
the Michael-Simon result [4] to a general Sobolev inequality for submanifolds
of a Riemannian manifold under some geometric restrictions. To formulate their
result, we need some notations in [2]. Let $M\rightarrow N$ be an isometric
immersion of Riemannian manifolds of dimension $m$ and $n$, respectively. We
denote the sectional curvatures of $N$ by $K_{N}$. The mean curvature vector
field of the immersion is given by $H$. We write $\bar{R}(M)$ for the
injectivity radius of $N$ restricted to $M$ (or minimum distance to the cut
locus in $N$ for all points in $M$). Let us denote by $\omega_{m}$ the volume
of the unit ball in $\mathbb{R}^{m}$ and let $b$ be a positive real number or
a pure imaginary one.

Theorem B. (D. Hoffman and J. Spruck [2]) _Let $M\rightarrow N$ be an
isometric immersion of Riemannian manifolds of dimension $m$ and $n$,
respectively. Some notations are adopted as above. Assume $K_{N}\leq b^{2}$
and let $h$ be a non-negative $C^{1}$ function on $M$ vanishing on $\partial
M$. Then_

(1.4) $\left(\int_{M}h^{m/(m-1)}d\mu\right)^{(m-1)/m}\leq
c(m)\int_{M}\left[|\nabla h|+h|H|\right]d\mu,$

_provided_

(1.5)
$b^{2}(1-\beta)^{-2/m}\left(\omega_{m}^{-1}Vol(\mathrm{supp}h)\right)^{2/m}\leq
1$

_and_

(1.6) $2\rho_{0}\leq\bar{R}(M),$

_where_

$\rho_{0}=\left\\{\begin{aligned}
b^{-1}\sin^{-1}\left[b(1-\beta)^{-1/m}\left(\omega_{m}^{-1}Vol(\mathrm{supp}h)\right)^{1/m}\right]\quad\quad&\mathrm{for}\,\,\,b\,\,\,\mathrm{real},\\\
(1-\beta)^{-1/m}\left(\omega_{m}^{-1}Vol(\mathrm{supp}h)\right)^{1/m}\quad\quad&\mathrm{for}\,\,\,b\,\,\,\mathrm{imaginary}.\end{aligned}\right.$

_Here $\beta$ is a free parameter, $0<\beta<1$, and_

(1.7) $c(m):=c(m,\beta)=\pi\cdot
2^{m-1}\beta^{-1}(1-\beta)^{-1/m}\frac{m}{m-1}\omega_{m}^{-1/m}.$

######  Remark 1.1.

In Theorem B, we may replace the assumption $h\in C^{1}(M)$ by $h\in
W^{1,1}(M)$. As the mentioned remark in [2], the optimal choice of $\beta$ to
minimize $c$ is $\beta=m/(m+1)$. When $b$ is real we may replace condition
(1.6) by the stronger condition $\bar{R}\geq\pi b^{-1}$. When $b$ is a pure
imaginary number and the Riemannian manifold $N$ is simply connected and
complete, $\bar{R}(M)=+\infty$. Hence conditions (1.5) and (1.6) are
automatically satisfied.

Motivated by the work of P. Topping, it is natural to expect that there exists
a general geometric inequality for submanifolds of a Riemannian manifold,
which is similar to Theorem A. Fortunately, following closely the lines of the
Topping’s proof of Theorem A in [11], we can employ the Hoffman-Spruck Sobolev
inequality for submanifolds of a Riemannian manifold together with a covering
lemma to derive the desired results.

######  Theorem 1.2.

For $m\geq 1$, suppose that $M$ is an $m$-dimensional closed connected
Riemannian manifold smoothly isometrically immersed in an $n$-dimensional
complete Riemannian manifold $N$ with $K_{N}\leq b^{2}$. For any $0<\alpha<1$,
if

(1.8) $b^{2}(1-\alpha)^{-2/m}\left(\omega_{m}^{-1}Vol(M)\right)^{2/m}\leq 1$

and

(1.9) $2\rho_{0}\leq\bar{R}(M),$

where

$\rho_{0}=\left\\{\begin{aligned}
b^{-1}\sin^{-1}\left[b(1-\alpha)^{-1/m}\left(\omega_{m}^{-1}Vol(M)\right)^{1/m}\right]\quad\quad&\mathrm{for}\,\,\,b\,\,\,\mathrm{real},\\\
(1-\alpha)^{-1/m}\left(\omega_{m}^{-1}Vol(M)\right)^{1/m}\quad\quad&\mathrm{for}\,\,\,b\,\,\,\mathrm{imaginary},\end{aligned}\right.$

then there exists a constant $C(m,\alpha)$ dependent only on $m$ and $\alpha$
such that

$d_{int}\leq C(m,\alpha)\int_{M}|H|^{m-1}d\mu.$

In particular, we can take $C(2,\alpha)=\frac{576\pi}{\alpha^{2}(1-\alpha)}$.

######  Remark 1.3.

In Theorem 1.2, the coefficients $C(m,\alpha)$ are not identical to (but
strongly dependent on) the coefficients $c(m)$ in Theorem B. From (2.7) and
(3.3) we can find that $C(m,\alpha)$ can still arrive at the minimum, when
$\alpha=\frac{m}{m+1}$. The conditions of (1.8) and (1.9) are similar to the
restrictions of (1.5) and (1.6) in Theorem B, and they guarantee that the
Hoffman-spruck Sobolev for submanifolds of a Riemannian manifold can be
applied in the proof of our theorem. When $b$ is real we may replace condition
(1.9) by the stronger condition $\bar{R}\geq\pi b^{-1}$. When $b$ is a pure
imaginary number and the Riemannian manifold $N$ is simply connected and
complete, $\bar{R}(M)=+\infty$, and hence conditions (1.8) and (1.9) are
automatically satisfied.

In particular, when $N=\mathbb{R}^{n}$, $K_{N}\equiv 0$ and
$\bar{R}(M)=+\infty$, and hence there are also no volume restrictions on $M$.
Combining this with Remark 1.3, if $b$ is pure imaginary or zero, then we see
that conditions (1.8) and (1.9) are automatically satisfied, and hence we
conclude that

######  Corollary 1.4.

For $m\geq 1$, suppose that $M$ is an $m$-dimensional closed connected
Riemannian manifold smoothly isometrically immersed in an $n$-dimensional
simply connected, complete, nonpositively curved Riemannian manifold $N$
($K_{N}\leq 0$). For any $0<\alpha<1$, then there exists a constant
$C(m,\alpha)$ dependent only on $m$ and $\alpha$ such that

$d_{int}\leq C(m,\alpha)\int_{M}|H|^{m-1}d\mu,$

where $\min_{0<\alpha<1}C(m,\alpha)=C(m,\frac{m}{m+1})$. In particular, we can
take $\min_{0<\alpha<1}C(2,\alpha)=C(2,\frac{2}{3})=3888\pi$.

We remark that the constants $C(2,\alpha)$ in Theorem 1.2 and Corollary 1.4
are not optimal in general. The proof of Theorem 1.2 follows the proof in the
Euclidean case [11]. Theorem 1.2 and Corollary 1.4 may have many interesting
applications which we have not discussed here. For example, we may borrow Li’s
idea of [3] and apply our Theorem 1.2 to study the convergence problem of the
volume-preserving mean curvature flow in Riemannian manifolds. We will explore
this aspect in the future.

Besides the above works, the closest precedent for our theorem is another P.
Topping’s work on diameter estimates for intrinsic manifolds evolving under
the Ricci flow [9]. In the Ricci flow case, P. Topping explored a log-Sobolev
inequality of the Ricci flow (see Theorem 3.4 in [9]), which can be derived by
the monotonicity of Perelman’s $\mathcal{W}$-functional (see [1], [5], [10]).
However a core tool of proving Theorem 1.2 is the Hoffman-Spruck Sobolev
inequality.

The rest of this paper is organized as follows. In Section 2, we will prove
Lemma 2.1. The proof needs the key Hoffman-Spruck Sobolev inequality. In
Section 3, we will finish the proof of Theorem 1.2 using Lemma 2.1 of Section
2 and a covering lemma.

##  2\. Estimates for maximal function and volume ratio

In this section we first introduce two useful geometric quantities: the
maximal function and the volume ratio. Then we apply the Hoffman-Spruck
Sobolev inequality to prove the following important Lemma 2.1, which is
essential in the proof of Theorem 1.2.

Given $x\in M^{m}$, with respect to a given metric, we denote the open
geodesic ball in $M^{m}$ centred at $x$ and of intrinsic radius $r>0$ by
$B(x,r)$, and its volume by

$V(x,r):=Vol(B(x,r)).$

Following Topping’s definitions in [11], when $m\geq 2$, we introduce the
_maximal function_

(2.1)
$M(x,R):=\sup_{r\in(0,R]}r^{-\frac{1}{m-1}}[V(x,r)]^{-\frac{m-2}{m-1}}\int_{B(x,r)}|H|d\mu$

and the _volume ratio_

(2.2) $\kappa(x,R):=\inf_{r\in(0,R]}\frac{V(x,r)}{r^{m}}$

for any $R>0$.

Similar to Lemma 1.2 in [11], we have the following general result.

######  Lemma 2.1.

For $m\geq 2$, suppose that $M$ is an $m$-dimensional Riemannian manifold
smoothly isometrically immersed in an $n$-dimensional Riemannian manifold $N$
with $K_{N}\leq b^{2}$, which is complete with respect to the induced metric.
For any $0<\alpha<1$, if conditions (1.8) and (1.9) are satisfied, then there
exists a constant $\delta>0$ dependent only on $m$ and $\alpha$ such that for
any $x\in M$ and $R>0$, at least one of the following is true:

  1. (1)

$M(x,R)\geq\delta$;

  2. (2)

$\kappa(x,R)>\delta$.

In the case of closed surfaces ($m=2$) in $N$, we can choose
$\delta=\frac{\alpha^{2}(1-\alpha)}{144\pi}$.

######  Remark 2.2.

In Lemma 2.1, when $b$ is a pure imaginary number and the Riemannian manifold
$N$ is simply connected and complete, $\bar{R}(M)=+\infty$. Hence conditions
(1.8) and (1.9) are automatically satisfied.

Now we will finish the proof of Lemma 2.1.

###### Proof of Lemma 2.1.

We follow the ideas of the proof of Lemma 1.2 in [11]. Suppose that
$M(x,R)<\delta$ for some constant $\delta>0$, which will be chosen later.
According to the definition of the maximal function $M(x,R)$, we know that for
all $r\in(0,R]$

(2.3) $\int_{B(x,r)}|H|d\mu<\delta
r^{\frac{1}{m-1}}[V(x,r)]^{\frac{m-2}{m-1}}.$

Note that for fixed $x$, $V(r):=V(x,r)$ is differentiable for almost all
$r>0$. For such $r\in(0,R]$, and any $s>0$, we define a Lipschitz cut-off
function $h$ on $M$ by

(2.4) $h(y)=\left\\{\begin{aligned} 1\quad\quad&y\in B(x,r)\\\
1-\frac{1}{s}(dist_{M}(x,y)-r)\quad\quad&y\in B(x,r+s)\setminus B(x,r)\\\
0\quad\quad&y\not\in B(x,r+s).\end{aligned}\right.$

Since function $\sin^{-1}x$ is increasing on $[0,1]$ and
$Vol(\mathrm{supp}h)\leq Vol(M)$, we easily see that conditions (1.8) and
(1.9) guarantee the function $h$ of (2.4) to satisfy conditions (1.5) and
(1.6), where $\beta=\alpha$. Substituting this function to the Hoffman-Spruck
Sobolev inequality from Theorem B, we derive that

$\displaystyle V(r)^{(m-1)/m}$
$\displaystyle\leq\left(\int_{M}h^{m/(m-1)}d\mu\right)^{(m-1)/m}$
$\displaystyle\leq
c(m)\left[\frac{V(r+s)-V(r)}{s}+\int_{B(x,r+s)}|H|d\mu\right],$

where $c(m):=c(m,\alpha)=\pi\cdot
2^{m-1}\alpha^{-1}(1-\alpha)^{-1/m}\frac{m}{m-1}\omega_{m}^{-1/m}$. Letting
$s\downarrow 0$, we conclude that

$\displaystyle V(r)^{(m-1)/m}\leq
c(m)\left[\frac{dV}{dr}+\int_{B(x,r)}|H|d\mu\right].$

Combining this with (2.3), we have

(2.5) $\frac{dV}{dr}+\delta
r^{\frac{1}{m-1}}V(r)^{\frac{m-2}{m-1}}-c(m)^{-1}V(r)^{\frac{m-1}{m}}>0.$

Now we assume that $\delta>0$ is sufficiently small so that
$\delta<\omega_{m}$, and define another smooth function

$v(r):=\delta r^{m}.$

Then a straightforward computation yields

(2.6) $\frac{dv}{dr}+\delta
r^{\frac{1}{m-1}}v(r)^{\frac{m-2}{m-1}}-c(m)^{-1}v(r)^{\frac{m-1}{m}}=\left(m\delta+\delta^{\frac{2m-3}{m-1}}-c(m)^{-1}\delta^{\frac{m-1}{m}}\right)r^{m-1}.$

We can see that

(2.7) $\frac{dv}{dr}+\delta
r^{\frac{1}{m-1}}v(r)^{\frac{m-2}{m-1}}-c(m)^{-1}v(r)^{\frac{m-1}{m}}\leq 0$

as long as $\delta>0$ is sufficiently small, depending only on $m$ and
$\alpha$.

Notice the fact that $V(r)/r^{m}\rightarrow\omega_{m}$ as $r\downarrow 0$,
while $v(r)/r^{m}=\delta<\omega_{m}$. And combining inequalities (2.5) and
(2.7), we conclude that

$V(r)>v(r)$

for all $r\in(0,R]$. Otherwise, there exists a fixed $r_{0}$ such that
$V(r_{0})=v(r_{0})$ and $V(r)>v(r)$ for all $r\in(0,r_{0})$. Then from (2.5)
and (2.7), we can derive

$\frac{dV}{dr}\Big{|}_{r=r_{0}}>\frac{dv}{dr}\Big{|}_{r=r_{0}}.$

Namely,

$\frac{dV}{dr}>\frac{dv}{dr}$

in any sufficiently small neighborhood of $r_{0}$, which is impossible since
$V(r_{0})=v(r_{0})$ and $V(r)>v(r)$ for all $r\in(0,r_{0})$.

Therefore

$\kappa(x,R):=\inf_{r\in(0,R]}\frac{V(x,r)}{r^{m}}>\delta,$

which completes the proof of Lemma 2.1.

In the case of closed surfaces ($m=2$) in $N$, we can choose
$\delta=\frac{c(2,\alpha)^{-2}}{9}=\frac{\alpha^{2}(1-\alpha)}{144\pi}$ to
satisfy (2.7) and the constraint condition $\delta<\omega_{2}=\pi$. ∎

##  3\. Diameter Control

In this section we can follow the lines of [11] or [9], and easily prove
Theorem 1.2 by using Lemma 2.1 and a covering lemma. For the completeness of
this paper, here we still give the detailed proof of Theorem 1.2.

###### Proof of Theorem 1.2.

We may assume $m\geq 2$ since the case $m=1$ is trivial. Now we choose $R>0$
sufficiently large so that the total volume of the closed manifold $M$ is less
than $\delta R^{m}$, where $\delta$ is given by Lemma 2.1 (Notice that
$\delta$ does not depend on $R$). In particular, for all $z\in M$, we must
have

$\kappa(z,R)\leq\frac{V(z,R)}{R^{m}}\leq\delta.$

Hence by Lemma 2.1, as long as conditions (1.8) and (1.9) are satisfied, we
must have the maximal function $M(x,R)\geq\delta$. Namely, there exists
$r=r(z)$ such that

(3.1) $\displaystyle\delta$ $\displaystyle\leq
r^{-\frac{1}{m-1}}V(z,r)^{-\frac{m-2}{m-1}}\int_{B(z,r)}|H|d\mu$
$\displaystyle\leq
r^{-\frac{1}{m-1}}\left(\int_{B(z,r)}|H|^{m-1}d\mu\right)^{\frac{1}{m-1}},$

where we used the Hölder inequality for the second inequality above. Hence

(3.2) $r(z)\leq\delta^{1-m}\int_{B(z,r(z))}|H|^{m-1}d\mu.$

Now we have to pick appropriate points $z$ at which to apply (3.2). Let
$x_{1},x_{2}\in M$ be extremal points in $M$. This means that
$d_{int}=dist_{M}(x_{1},x_{2})$. Let $\Sigma$ be a shortest geodesic
connecting $x_{1}$ and $x_{2}$. Obviously, $\Sigma$ is covered by the balls
$\\{B(z,r(z)):z\in\Sigma\\}$. By a modification of the covering lemma (see
Lemma 5.2 in [9]), there exists a countable (possibly finite) set of points
$\\{z_{i}\in\Sigma\\}$ such that the balls $\\{B(z_{i},r(z_{i}))\\}$ are
disjoint, and cover at least a fraction $\rho$, where $\rho\in(0,\frac{1}{2})$
of $\Sigma$:

$\rho d_{int}\leq\sum_{i}2r(z_{i}).$

Combining this with (3.2), we have

$\displaystyle d_{int}$ $\displaystyle\leq\frac{2}{\rho}\sum_{i}r(z_{i})$
$\displaystyle\leq\frac{2}{\rho}\delta^{1-m}\sum_{i}\int_{B(z_{i},r(z_{i}))}|H|^{m-1}d\mu$
$\displaystyle\leq\frac{2}{\rho}\delta^{1-m}\int_{M}|H|^{m-1}d\mu,$

where $\delta>0$ is sufficiently small, depending only on $m$ and $\alpha$.
Letting $\rho\to\frac{1}{2}$, we arrived at

(3.3) $d_{int}\leq 4\delta^{1-m}\int_{M}|H|^{m-1}d\mu.$

Hence the desired theorem follows. If $m=2$, we can choose
$4\delta^{1-m}=\frac{576\pi}{\alpha^{2}(1-\alpha)}$, since
$\delta=\frac{\alpha^{2}(1-\alpha)}{144\pi}$. ∎

## Acknowledgment

The authors thank the referee for various comments and suggestions that helped
improve this paper.

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