Sobolev inequalities in manifolds with asymptotically nonnegative curvature

Abstract

Using the ABP-method as in a recent work by Brendle (Commun Pure Appl Math 76:2192–2218, 2022), we establish some sharp Sobolev and isoperimetric inequalities for compact domains and submanifolds in a complete Riemannian manifold with asymptotically nonnegative Ricci/sectional curvature. These inequalities generalize those given by Brendle in the case of complete Riemannian manifolds with nonnegative curvature.

1 Introduction

It is known that Sobolev inequalities, as an important analytic tool in geometric analysis, have close connections with isoperimetric inequalities. The classical isoperimetric inequality for a bounded domain D in ${\mathbb {R}}^n$ says that

$$\begin{aligned} n^n|B^n||D|^{n-1}\le |\partial D|^n \end{aligned}$$

where $B^n$ denotes the unit ball in ${\mathbb {R}}^n$, and the equality holds if and only if D is a ball. There have been numerous works generalizing this inequality to different settings (cf. [14, 15, 33]).

The isoperimetric inequalities on minimal surfaces or minimal submanifolds have a long history. For example, [13, 14, 22, 29, 35,36,37] investigated the isoperimetric inequality on minimal surfaces under various conditions, while the famous Michael-Simon Sobolev inequality for general dimensions [5, 32] implies an isoperimetric inequality for minimal submanifolds, but with a non-sharp constant. It is conjectured that any n-dimensional minimal submanifold $\Omega$ of ${\mathbb {R}}^N$ satisfies the classical isoperimetric inequality: $n^n|B^n||\Omega |^{n-1}\le |\partial \Omega |^n$ with equality holds if and only if $\Omega$ is a ball in an n-plane of ${\mathbb {R}}^N$. Recently, S. Brendle [9], inspired by the ABP method as in [11] and [38], established a Michael-Simon-Sobolev type inequality on submanifolds of arbitrary dimension and codimension, which is sharp if the codimension is at most 2. In particular, his result implies a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2. Later, Brendle [10] also generalized his results in [9] to the case that the ambient space is a Riemannian manifold with nonnegative curvature. In [23], F. Johne gave a sharp Sobolev inequality for manifolds with nonnegative Bakry-Émery Ricci curvature, which generalizes Brendle’s results in [10]. In [7], Balogh and Krisály proved a sharp isoperimetric inequality in metric measure spaces satisfying ${{\textsf{C}}}{{\textsf{D}}}(0,N)$ condition which implies the sharp isoperimetric inequalities in [10] and [23]. Moreover, they also obtained a sharp $L^p$-Sobolev inequality for $p\in (1,n)$ on manifolds with nonnegative Ricci curvature and Euclidean volume growth. In a recent preprint [6], the authors also investigated sharp and rigid isoperimetric comparison theorems in $\textsf{RCD}(K,N)$ metric measure spaces.

In this paper, we generalize Brendle’s results in [10] to the case that the ambient space has asymptotically nonnegative curvature. The notion of asymptotically nonnegative curvature was first introduced by U. Abresch [1]. Some important geometric, topological and analysis problems have been investigated for this kind of manifolds (cf. [2, 3, 8, 21, 24, 25, 30, 31, 40, 41], etc). Now we recall its definition as follows. Let $\lambda :[0,+\infty )\rightarrow [0,+\infty )$ be a nonnegative and nonincreasing continuous function satisfying

$$\begin{aligned} b_0:=\int _0^{+\infty }s\lambda (s)ds<+\infty , \end{aligned}$$

(1.1)

which implies

$$\begin{aligned} b_1:=\int _0^{+\infty }\lambda (s)ds<+\infty . \end{aligned}$$

(1.2)

A complete noncompact Riemannian manifold (M, g) of dimension n is said to have asymptotically nonnegative Ricci curvature (resp. sectional curvature) if there is a base point $o\in M$ such that

$$\begin{aligned} \textrm{Ric}_q(\cdot ,\cdot )\ge -(n-1)\lambda (d(o,q))g \quad (resp.\ \textrm{Sec}_q\ge -\lambda (d(o,q))), \end{aligned}$$

(1.3)

where d(o, q) is the distance function of M relative to o. Clearly, this notion includes the manifolds whose Ricci (resp. sectional) curvature is either nonnegative outside a compact set or asymptotically flat at infinity. In particular, if $\lambda \equiv 0$ in (1.3), then this becomes the case treated in [10].

Let h(t) be the unique solution of

$$\begin{aligned} \left\{ \begin{aligned}&h''(t)=\lambda (t)h(t),\\&h(0)=0, h'(0)=1. \end{aligned} \right. \end{aligned}$$

(1.4)

By ODE theory, the solution h(t) of (1.4) exists for all $t\in [0,+\infty )$. According to [41] (see also Theorem 2.14 in [34]), the function

$$\begin{aligned} \frac{|\{q\in M:d(o,q)<r\}|}{n|B^n|\int _0^rh^{n-1}(t)dt} \end{aligned}$$

is a non-increasing function on $[0,+\infty )$ and thus we may introduce the asymptotic volume ratio of M by

$$\begin{aligned} \theta :=\lim _{r\rightarrow +\infty }\frac{|\{q\in M:d(o,q)<r\}|}{n|B^n|\int _0^rh^{n-1}(t)dt}, \end{aligned}$$

(1.5)

with $\theta \le 1$. In particular, we have $|\{q\in M:d(o,q)<r\}|\le |B^n|e^{(n-1)b_0}r^n$.

First, by combining the method in [10] with some comparison theorems, we establish a Sobolev type inequality for a compact domain in a Riemannian manifold with asymptotically nonnegative Ricci curvature as follows.

Theorem 1.1

Let M be a complete noncompact n-dimensional manifold of asymptotically nonnegative Ricci curvature with respect to a base point $o\in M$. Let $\Omega$ be a compact domain in M with boundary $\partial \Omega$, and let f be a positive smooth function on $\Omega$. Then

$$\begin{aligned} \int _{\partial \Omega } f+\int _\Omega |D f|+ 2(n-1)b_1\int _\Omega f \ge n|B^n|^{\frac{1}{n}}\theta ^{\frac{1}{n}} \Big (\frac{1+b_0}{e^{2r_0b_1+b_0}}\Big )^\frac{n-1}{n} \Big (\int _\Omega f^\frac{n}{n-1}\Big )^\frac{n-1}{n}, \end{aligned}$$

where $r_0=\max \{{d}(o,x)|x\in \Omega \}$, $\theta$ is the asymptotic volume ratio of M given by (1.5) and $b_0,b_1$ are defined in (1.1) and (1.2).

The following result characterizes the case of equality in Theorem 1.1:

Theorem 1.2

Let M be a complete noncompact n-dimensional manifold of asymptotically nonnegative Ricci curvature with respect to a base point $o\in M$. Let $\Omega$ be a compact domain in M with boundary $\partial \Omega$, and let f be a positive smooth function on $\Omega$. If

$$\begin{aligned} \int _{\partial \Omega } f+\int _\Omega |D f|+ 2(n-1)b_1\int _\Omega f = n|B^n|^{\frac{1}{n}}\theta ^{\frac{1}{n}} \Big (\frac{1+b_0}{e^{2r_0b_1+b_0}}\Big )^\frac{n-1}{n} \Big (\int _\Omega f^\frac{n}{n-1}\Big )^\frac{n-1}{n}, \end{aligned}$$

where $r_0=\max \{{d}(o,x)|x\in \Omega \}$, $\theta$ is the asymptotic volume ratio of M given by (1.5) and $b_0,b_1$ are defined in (1.1) and (1.2). Then $b_0=b_1=0$, M is isometric to Euclidean space, $\Omega$ is a ball, and f is constant.

Taking $f=1$ in Theorem 1.1, we obtain a sharp isoperimetric inequality:

Corollary 1.3

Let $M,\Omega , r_0,\theta ,b_0,b_1$ be as in Theorem 1.1. Then

$$\begin{aligned} |\partial \Omega |\ge \Big (n|B^n|^{\frac{1}{n}} \theta ^{\frac{1}{n}} \Big (\frac{1+b_0}{e^{2r_0b_1+b_0}}\Big )^\frac{n-1}{n} - 2(n-1)b_1|\Omega |^{\frac{1}{n}}\Big )|\Omega |^{\frac{n-1}{n}}. \end{aligned}$$

Furthermore, the equality holds if and only if M is isometric to Euclidean space and $\Omega$ is a ball.

Remark 1.4

If M has nonnegative Ricci curvature, then $b_0=b_1=0$ and Corollary 1.3 becomes

$$\begin{aligned} |\partial \Omega |\ge n|B^n|^{\frac{1}{n}}\theta ^{\frac{1}{n}}, \end{aligned}$$

which was first given by V. Agostiniani, M. Fogagnolo, and L. Mazziari [4] in dimension 3 and obtained by S. Brendle [10] for any dimension, see also [18] for related results in $3\le n\le 7$.

Similarly, we may establish a Sobolev type inequality for a compact submanifold (possibly with boundary) in a Riemannian manifold with asymptotically nonnegative sectional curvature as follows.

Theorem 1.5

Let M be a complete noncompact $(n+p)$-dimensional manifold of asymptotically nonnegative sectional curvature with respect to a base point $o\in M$. Let $\Sigma$ be a compact n-dimensional submanifold of M (possibly with boundary $\partial \Sigma$), and let f be a positive smooth function on $\Sigma$. If $p\ge 2$, then

$$\begin{aligned} \begin{aligned}&\int _{\partial \Sigma }f+\int _\Sigma \sqrt{|D^\Sigma f|^2+f^2|H|^2} +2nb_1\int _\Sigma f\\&\quad \ge n\Big (\frac{(n+p)|B^{n+p}|}{p|B^p|}\Big )^{\frac{1}{n}}\theta ^{\frac{1}{n}} \Big (\frac{1+b_0}{e^{2r_0b_1+b_0}}\Big )^{\frac{n+p-1}{n}} \Big (\int _\Sigma f^{\frac{n}{n-1}}\Big )^{\frac{n-1}{n}}, \end{aligned} \end{aligned}$$

where $r_0=\max \{d(o,x)|x\in \Sigma \}$, H is the mean curvature vector of $\Sigma$, $\theta$ is the asymptotic volume ratio of M given by (1.5) and $b_0,b_1$ are defined in (1.1) and (1.2).

Note that $(n+2)|B^{n+2}|=2|B^2||B^n|$. Hence, we obtain the following Sobolev type inequality for codimension 2:

Corollary 1.6

Let M be a complete noncompact $(n+2)$-dimensional manifold of asymptotically nonnegative sectional curvature with respect to a base point $o\in M$. Let $\Sigma$ be a compact n-dimensional submanifold of M (possibly with boundary $\partial \Sigma$), and let f be a positive smooth function on $\Sigma$. Then

$$\begin{aligned} \begin{aligned}&\int _{\partial \Sigma }f+\int _\Sigma \sqrt{|D^\Sigma f|^2+f^2|H|^2} +2nb_1\int _\Sigma f\\&\quad \ge n|B^n|^{\frac{1}{n}}\theta ^{\frac{1}{n}} \Big (\frac{1+b_0}{e^{2r_0b_1+b_0}}\Big )^{\frac{n+1}{n}} \Big (\int _\Sigma f^{\frac{n}{n-1}}\Big )^{\frac{n-1}{n}}, \end{aligned} \end{aligned}$$

where $r_0=\max \{d(o,x)|x\in \Sigma \}$, H is the mean curvature vector of $\Sigma$, $\theta$ is the asymptotic volume ratio of M given by (1.5) and $b_0,b_1$ are defined in (1.1) and (1.2).

The following result characterizes the case of equality in Corollary 1.6:

Theorem 1.7

Let $M, \Sigma , f, r_0, H, \theta , b_0, b_1$ as in Corollary 1.6. If

$$\begin{aligned} \begin{aligned}&\int _{\partial \Sigma }f+\int _\Sigma \sqrt{|D f|^2+f^2|H|^2} +2nb_1\int _\Sigma f\\&\quad = n|B^n|^{\frac{1}{n}}\theta ^{\frac{1}{n}} \Big (\frac{1+b_0}{e^{2r_0b_1+b_0}}\Big )^{\frac{n+1}{n}} \Big (\int _\Sigma f^{\frac{n}{n-1}}\Big )^{\frac{n-1}{n}}. \end{aligned} \end{aligned}$$

Then $b_0=b_1=0$ and M is isometric to Euclidean space, $\Sigma$ is a flat ball, and f is constant.

Letting $f=1$ in Corollary 1.6, we obtain a sharp isoperimetric inequality for minimal submanifolds of codimension 2 as follows.

Corollary 1.8

Let M be a complete noncompact $(n+2)$-dimensional manifold of asymptotically nonnegative sectional curvature with respect to a base point $o\in M$. Let $\Sigma$ be a compact n-dimensional mininal submanifold of M (possibly with boundary $\partial \Sigma$). Then

$$\begin{aligned} |\partial \Sigma |\ge n\Big (|B^n|^{\frac{1}{n}}\theta ^{\frac{1}{n}} \big (\frac{1+b_0}{e^{2r_0b_1+b_0}}\big )^{\frac{n+1}{n}}-2b_1|\Sigma |^{\frac{1}{n}}\Big )|\Sigma |^{\frac{n-1}{n}}, \end{aligned}$$

where $r_0=\max \{d(o,x)|x\in \Sigma \}$, $\theta$ is the asymptotic volume ratio of M given by (1.5) and $b_0,b_1$ are defined in (1.1) and (1.2). Furthermore, the equality holds if and only if M is isometric to Euclidean space and $\Sigma$ is a flat ball.

It is obvious that the above inequalities are nontrivial only when $\theta >0$. We say that a complete Riemannian manifold with asymptotically nonnegative (Ricci) curvature has maximal volume growth if $\theta >0$. Examples of such manifolds may be found in [1, 12, 19, 26, 27], and the first case of Theorem 1.2 in [39], etc.

2 The case of domains

Let (M, g) be a complete noncompact n-dimensional Riemannian manifold of asymptotically nonnegative Ricci curvature with respect to a base point $o\in M$. Let $\Omega$ be a compact domain in M with smooth boundary $\partial \Omega$ and f be a smooth positive function on $\Omega$. Without loss of generality, we assume hereafter that $\Omega$ is connected.

By scaling, we may assume that

$$\begin{aligned} \int _{\partial \Omega } f+\int _\Omega |D f|+\int _\Omega 2(n-1)b_1f=n\int _\Omega f^\frac{n}{n-1}. \end{aligned}$$

(2.1)

Due to (2.1) and the connectedness of $\Omega$, we can find a solution of the following Neumann boundary problem

$$\begin{aligned} \left\{ \begin{aligned}&\textrm{div}(fD u)=nf^\frac{n}{n-1}-2(n-1)b_1f-|D f|,&\text { in }\Omega ,\\&\langle Du,\nu \rangle =1,&\text { on }\partial \Omega , \end{aligned} \right. \end{aligned}$$

(2.2)

where $\nu$ is the outward unit normal vector field along $\partial \Omega$. By standard elliptic regularity theory (see Theorem 6.31 in [20]), we know that $u\in C^{2,\gamma }$ for each $0<\gamma <1$.

As in [10], we set

$$\begin{aligned} U:=\{x\in \Omega \setminus \partial \Omega :|Du(x)|<1\}. \end{aligned}$$

For any $r>0$, let

$$\begin{aligned} A_r=\{{\bar{x}}\in U: ru(x)+\frac{1}{2}{d} (x,\exp _{{\bar{x}}}(rD u({\bar{x}})))^2\ge ru({\bar{x}})+\frac{1}{2}r^2|D u({\bar{x}})|^2,\ \forall x\in \Omega \}. \end{aligned}$$

Define a transport map $\Phi _r:\Omega \rightarrow M$ for each $r>0$ by

$$\begin{aligned} \Phi _r(x)=\exp _x(rD u(x)), \quad \forall x\in \Omega . \end{aligned}$$

Since $\exp :TM\rightarrow M$ is smooth on any complete Riemannian manifold (see Proposition 5.7 in [28]), we known that the map $\Phi _r$ is of class $C^{1,\gamma },0<\gamma <1$.

Lemma 2.1

Assume that $x\in U$. Then we have

$$\begin{aligned} \frac{1}{n}\Delta u\le f^\frac{1}{n-1}-2\Big (\frac{n-1}{n}\Big )b_1. \end{aligned}$$

Proof

Using the Cauchy-Schwarz inequality and the property that $|Du|<1$ for $x\in U$, we get

$$\begin{aligned} -\langle Df,Du\rangle \le |Df|. \end{aligned}$$

In terms of (2.2), we derive that

$$\begin{aligned} \begin{aligned} f\Delta u&=nf^\frac{n}{n-1}-2(n-1)b_1f-|Df| -\langle Df,Du\rangle \\&\le nf^\frac{n}{n-1}-2(n-1)b_1f. \end{aligned} \end{aligned}$$

This proves the assertion. $\square$

The proofs of the following three lemmas are identical to those for Lemmas 2.2$-$2.4 in [10] without any change for the case of asymptotically nonnegative Ricci curvature. So we omit them here.

Lemma 2.2

The set

$$\begin{aligned} \{q\in M: d(x,q)<r,\ \forall x\in \Omega \} \end{aligned}$$

is contained in $\Phi _r(A_r)$.

Lemma 2.3

Assume that ${\bar{x}}\in A_r$, and let ${\bar{\gamma }}(t):=\exp _{{\bar{x}}}(tDu({\bar{x}}))$ for all $t\in [0,r]$. If Z is a smooth vector field along ${\bar{\gamma }}$ satisfying $Z(r)=0$, then

$$\begin{aligned} (D^2u)(Z(0),Z(0))+\int _0^r\big (|D_tZ(t)|^2-R({\bar{\gamma }}'(t),Z(t),{\bar{\gamma }}'(t),Z(t))\big )\ dt\ge 0. \end{aligned}$$

Lemma 2.4

Assume that ${\bar{x}}\in A_r$, and let ${\bar{\gamma }}(t):=\exp _{{\bar{x}}}(tD u({\bar{x}}))$ for all $t\in [0,r]$. Moreover, let $\{e_1,\dots ,e_n\}$ be an orthonormal basis of $T_{{\bar{x}}}M$. Suppose that W is a Jacobi field along ${\bar{\gamma }}$ satisfying

$$\begin{aligned} \langle D_tW(0),e_j\rangle =(D^2u)(W(0),e_j),\quad 1\le j\le n. \end{aligned}$$

If $W(\tau )=0$ for some $\tau \in (0,r)$, then W vanishes identically.

Now, we give two comparison results for later use. The proofs of the following two lemmas are inspired by the proofs of Lemma 2.1 and Corollary 2.2 in [34].

Lemma 2.5

Let G be a continuous function on $[0,+\infty )$ and let $\phi ,\psi \in C^2([0,+\infty ))$ be solutions of the following problems

$$\begin{aligned} \left\{ \begin{aligned}&\phi ''\le {G}\phi ,\quad t\in (0,+\infty ),\\&\phi (0)=1,\phi '(0)=b, \end{aligned} \right. \quad \left\{ \begin{aligned}&\psi ''\ge {G}\psi , \quad t\in (0,+\infty ),\\&\psi (0)=1,\psi '(0)={\tilde{b}}, \end{aligned} \right. \end{aligned}$$

where $b,{\tilde{b}}$ are constants and ${\tilde{b}}\ge b$. If $\phi (t)>0$ for $t\in (0,T)$, then $\psi (t)>0$ in (0, T) and

$$\begin{aligned} \frac{\phi '}{\phi }\le \frac{\psi '}{\psi }\quad \text {and}\quad \psi \ge \phi \quad \text {on }(0,T). \end{aligned}$$

Proof

Set $\beta =\sup \{t:\psi (t)>0\text { in }(0,t)\}$ and $\tau = \min \{\beta ,T\}$, so that $\phi$ and $\psi$ are both positive in $(0,\tau )$. The function $\psi '\phi -\psi \phi '$ is continuous on $[0,+\infty )$, nonnegative at $t=0$, and satisfies

$$\begin{aligned} (\psi '\phi -\psi \phi ')'=\psi ''\phi -\psi \phi ''\ge {G}(t)\psi \phi - {G}(t)\psi \phi =0, \end{aligned}$$

in $(0,\tau )$. Thus $\psi '\phi -\psi \phi '\ge 0$ on $[0,\tau )$, which implies

$$\begin{aligned} \frac{\psi '}{\psi }\ge \frac{\phi '}{\phi }\quad \text { in }[0,\tau ). \end{aligned}$$

(2.3)

Integrating (2.3) between 0 and t $(0<t<\tau )$ yields

$$\begin{aligned} \phi (t)\le \psi (t),\quad \text { in }[0,\tau ). \end{aligned}$$

Since $\phi >0\text { in }[0,\tau )$ by assumption, this forces $\tau =T$. $\square$

Lemma 2.6

Let G be a nonnegative continuous function on $[0,+\infty )$ satisfying

$\int _0^{+\infty }G\ dt<+\infty$. Let $h_1,h_2\in C^2([0,+\infty ))$ be solutions of the following problems

$$\begin{aligned} \left\{ \begin{aligned}&h_1''= {G}h_1,\quad t\in (0,+\infty ),\\&h_1(0)=0,h_1'(0)=1, \end{aligned} \right. \quad \left\{ \begin{aligned}&h_2''= {G}h_2, \quad t\in (0,+\infty ),\\&h_2(0)=1,h_2'(0)=0. \end{aligned} \right. \end{aligned}$$

(2.4)

Then we have

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{h_2}{h_1}=\lim _{t\rightarrow \infty }\frac{h'_2}{h'_1} \le \int _0^{+\infty }G\ dt<\infty . \end{aligned}$$

Proof

From (2.4), we derive

$$\begin{aligned} (h_2h_1'-h_1h_2')'(t)\equiv 0, \end{aligned}$$

and thus

$$\begin{aligned} (h_2h_1'-h_1h_2')(t)\equiv 1 \end{aligned}$$

(2.5)

in view of the initial values for $h_1$ and $h_2$. By derivation, one can find

$$\begin{aligned} \Big (\frac{h_2}{h_1}\Big )'=\frac{h_2'h_1-h_1'h_2}{h_1^2}=\frac{-1}{h_1^2}<0, \end{aligned}$$

which implies that $\lim _{t\rightarrow +\infty }\frac{h_2(t)}{h_1(t)}$ exists. It is easy to show that

$$\begin{aligned} 0\le \Big (\frac{h'_2}{h'_1}\Big )'= \frac{G(h_2h_1'-h_1h_2')}{(h'_1)^2} \le \frac{G}{(1+\int _0^tsG(s)ds)^2} \le G, \end{aligned}$$

so we get

$$\begin{aligned} \frac{h'_2(t)}{h'_1(t)}\le \int _0^{+\infty }G\ dt. \end{aligned}$$

By Lemma 2.13 in [34], we have $h_1(t)\ge t$. Consequently, using (2.5) and $h'_1=1+\int _0^tGh_1 ds$, we obtain

$$\begin{aligned} \frac{h_2}{h_1}=\frac{h'_2}{h'_1}+\frac{1}{h_1h'_1}\le \int _0^{+\infty }G\ dt+\frac{1}{t},\quad t\in (0,\infty ). \end{aligned}$$

(2.6)

Letting $t\rightarrow \infty$, we have

$$\begin{aligned} \begin{aligned} \lim _{t\rightarrow \infty }\frac{h_2}{h_1}=\lim _{t\rightarrow \infty }\frac{h'_2}{h'_1} \le \int _0^{+\infty }G\ dt. \end{aligned} \end{aligned}$$

$\square$

The next result is useful to study the growth of various balls on M when their radii approach to infinity.

Lemma 2.7

Let h be the solution of (1.4). Then

$$\begin{aligned} \lim _{t\rightarrow +\infty }\frac{h(t-C)}{h(t)}=1 \text { and } \lim _{t\rightarrow +\infty }\frac{h(tC)}{h(t)}=C, \end{aligned}$$

where C is any positive constant.

Proof

From Lemma 2.13 in [34], we know $t\le h(t)\le e^{b_0}t$, and thus

$$\begin{aligned} h'(t)=1+\int _0^t \lambda h\ dt\le 1+b_0e^{b_0}. \end{aligned}$$

(2.7)

Clearly (2.7) means that $h'$ is nondecreasing and bounded from above. Consequently we have

$$\begin{aligned} \lim _{t\rightarrow +\infty }\frac{h(t-C)}{h(t)}= \lim _{t\rightarrow +\infty }\frac{h'(t-C)}{h'(t)}=1 \end{aligned}$$

and

$$\begin{aligned} \lim _{t\rightarrow +\infty }\frac{h(tC)}{h(t)}=\lim _{t\rightarrow +\infty }\frac{Ch'(tC)}{h'(t)}=C. \end{aligned}$$

$\square$

We are now turning to the proof of Theorem 1.1.

Proof of Theorem 1.1

For any $r>0$ and ${\bar{x}}\in A_r$, let $\{e_1,\dots ,e_n\}$ be an orthonormal basis of the tangent space $T_{{\bar{x}}}M$. Choosing the geodesic normal coordinates $(x^1,\dots ,x^n)$ around ${\bar{x}}$, such that $\frac{\partial }{\partial x^i}=e_i$ at ${\bar{x}}$. Let ${\bar{\gamma }}(t):=\exp _{{\bar{x}}}(tD u({\bar{x}}))$ for all $t\in [0,r]$. For $1\le i\le n$, let $E_i(t)$ be the parallel transport of $e_i$ along ${\bar{\gamma }}$. For $1\le i\le n$, let $X_i(t)$ be the Jacobi field along ${\bar{\gamma }}$ with the initial conditions of $X_i(0)=e_i$ and

$$\begin{aligned} \langle D_tX_i(0),e_j\rangle =(D^2u)(e_i,e_j),\quad 1\le j\le n. \end{aligned}$$

Let $P(t)=(P_{ij}(t))$ be a matrix defined by

$$\begin{aligned} P_{ij}(t)=\langle X_i(t),E_j(t)\rangle , \quad 1\le i,j\le n. \end{aligned}$$

From Lemma 2.4, we known $\det P(t)>0,\forall t\in [0,r)$. Obviously, $|\det D\Phi _t({\bar{x}})|=\det P(t)>0$ for $t\in [0,r)$. Let $S(t)=(S_{ij}(t))$ be a matrix defined by

$$\begin{aligned} S_{ij}(t)=R({\bar{\gamma }}'(t),E_i(t),{\bar{\gamma }}'(t),E_j(t)), \quad 1\le i,j\le n, \end{aligned}$$

where R denotes the Riemannian curvature tensor of M. By the Jacobi equation, one can obtain

$$\begin{aligned} \left\{ \begin{aligned}&P''(t)=-P(t)S(t),\quad t\in [0,r],\\&P_{ij}(0)=\delta _{ij},P_{ij}'(0)=(D^2u)(e_i,e_j). \end{aligned} \right. \end{aligned}$$

(2.8)

Let $Q(t)=P(t)^{-1}P'(t),t\in (0,r)$. Using (2.8), a simple computation yields

$$\begin{aligned} \frac{d}{dt}Q(t)=-S(t)-Q^2(t), \end{aligned}$$

where Q(t) is symmetric. The assumption of asymptotically nonnegative Ricci curvature gives

$$\begin{aligned} \begin{aligned} \frac{d}{dt}[\textrm{tr} Q(t)]+\frac{1}{n}[\textrm{tr}Q(t)]^2&\le \frac{d}{dt}[\textrm{tr} Q(t)]+\textrm{tr}[Q^2(t)]\\&=-\textrm{tr}S(t)\\&\le (n-1)|D u({\bar{x}})|^2\lambda (d(o,{\bar{\gamma }}(t))), \end{aligned} \end{aligned}$$

(2.9)

where o is the base point. Using triangle inequality and the definition of $A_r$, it is easy to see that

$$\begin{aligned} d(o,{\bar{\gamma }}(t))\ge \big |d(o,{\bar{x}})-d({\bar{x}},{\bar{\gamma }}(t))\big | =\big |d(o,{\bar{x}})-t|Du({\bar{x}})|\big |. \end{aligned}$$

(2.10)

Set

$$\begin{aligned} \begin{aligned}&g=\frac{1}{n}\textrm{tr}Q,\\&\Lambda _{{\bar{x}}}(t)=\frac{(n-1)}{n} |D u({\bar{x}})|^2 \lambda (\big |d(o,{\bar{x}})-t|D u({\bar{x}})|\big |). \end{aligned} \end{aligned}$$

Noting that $\lambda$ is nonincreasing, it follows from (2.8), (2.9), (2.10) that

$$\begin{aligned} \left\{ \begin{aligned}&g'(t)+g(t)^2\le \Lambda _{{\bar{x}}}(t), \quad t\in (0,r),\\&g(0)=\frac{1}{n}\Delta u({\bar{x}}). \end{aligned} \right. \end{aligned}$$

If we take $\phi =e^{\int _0^tg(\tau )d\tau }$, then $\phi$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\phi ''\le \Lambda _{{\bar{x}}}(t)\phi , \quad t\in (0,r),\\&\phi (0)=1,\phi '(0)=\frac{1}{n}\Delta u({\bar{x}}). \end{aligned} \right. \end{aligned}$$

(2.11)

Next, we denote by $\psi _1,\psi _2$ the solutions of the following problems

$$\begin{aligned} \left\{ \begin{aligned}&\psi _1''= \Lambda _{{\bar{x}}}(t)\psi _1,\quad t\in (0,r),\\&\psi _1(0)=0,\psi _1'(0)=1, \end{aligned} \right. \quad \left\{ \begin{aligned}&\psi _2''= \Lambda _{{\bar{x}}}(t)\psi _2,\quad t\in (0,r),\\&\psi _2(0)=1,\psi _2'(0)=0. \end{aligned} \right. \end{aligned}$$

(2.12)

Similar to the proof of (2.6), it is easy to verify that

$$\begin{aligned} \frac{\psi _2}{\psi _1}(r)\le \int _0^{+\infty }\Lambda _{{\bar{x}}}(t)\ dt+\frac{1}{r} \le 2\Big (\frac{n-1}{n}\Big )b_1|Du({\bar{x}})|+\frac{1}{r}. \end{aligned}$$

Since $|Du({\bar{x}})|<1$, we obtain

$$\begin{aligned} \frac{\psi _2}{\psi _1}(r)\le 2\Big (\frac{n-1}{n}\Big )b_1+\frac{1}{r}. \end{aligned}$$

(2.13)

Using Lemma 2.13 in [34] and (2.12), we deduce that

$$\begin{aligned} \begin{aligned} \psi _1(t)&\le \int _0^te^{\int _0^s \tau \Lambda _{{\bar{x}}}(\tau )d\tau }ds \\&\le te^{\int _0^\infty \tau \Lambda _{{\bar{x}}}(\tau )d\tau } \\&=te^{\frac{n-1}{n}\int _0^\infty w\lambda (|d(o,{\bar{x}})-w|)dw} \\&\le te^{\frac{n-1}{n}(2r_0b_1+b_0)}, \end{aligned} \end{aligned}$$

(2.14)

where $r_0=\max \{d(o,x)|x\in \Omega \}$.

Let $\psi (t)=\psi _2(t)+\frac{1}{n}\Delta u({\bar{x}})\psi _1(t)$. Using Lemma 2.5, one can get

$$\begin{aligned} \frac{1}{n}\textrm{tr}Q(t)=\frac{\phi '}{\phi }\le \frac{\psi '}{\psi },\quad \forall t\in (0,r). \end{aligned}$$

Thus,

$$\begin{aligned} \frac{d}{dt}\log \det P(t)=\textrm{tr}Q(t)\le n\frac{\psi '}{\psi }. \end{aligned}$$

(2.15)

Consequently, (2.15) implies

$$\begin{aligned} |\det D\Phi _t({\bar{x}})|=\det P(t)\le \psi ^n(t)=(\psi _2(t)+\frac{1}{n}\Delta u({\bar{x}})\psi _1(t))^n \end{aligned}$$

for all $t\in [0,r]$. This gives

$$\begin{aligned} |\det D\Phi _r({\bar{x}})|&\le \Big (\frac{\psi _2(r)}{\psi _1(r)}+\frac{1}{n}\Delta u({\bar{x}})\Big )^n\psi ^n_1(r) \end{aligned}$$

for any ${\bar{x}}\in A_r$. Note that $0\le \phi \le \psi$. Using (2.13), (2.14) and Lemma 2.1, we derive that

$$\begin{aligned} \begin{aligned} |\det D\Phi _r({\bar{x}})|&\le e^{(n-1)(2r_0b_1+b_0)}\Big (2\big (\frac{n-1}{n}\big )b_1+\frac{1}{r}+\frac{1}{n}\Delta u({\bar{x}})\Big )^n r^n\\&\le e^{(n-1)(2r_0b_1+b_0)}\Big (\frac{1}{r}+f^{\frac{1}{n-1}}({\bar{x}})\Big )^n r^n \end{aligned} \end{aligned}$$

(2.16)

for any ${\bar{x}}\in A_r$. Moreover, by (1.4), we obtain $h(t)\ge t$ and

$$\begin{aligned} \lim _{t\rightarrow \infty }h'(t)=1+\int _0^\infty h(s)\lambda (s)\ ds\ge 1+\int _0^\infty s\lambda (s)\ ds =1+b_0. \end{aligned}$$

(2.17)

Combining Lemma 2.2, (2.16) with the formula for change of variables in multiple integrals, we find that

$$\begin{aligned} \begin{aligned}&|\{q\in M:d(x,q)<r\text { for all }x\in \Omega \}|\\&\le \int _{A_r}|\det D\Phi _r|\\&\le \int _{\Omega }e^{(n-1)(2r_0b_1+b_0)}(\frac{1}{r}+f^{\frac{1}{n-1}})^n r^n. \end{aligned} \end{aligned}$$

(2.18)

For $r>r_0$, the triangle inequality implies that

$$\begin{aligned} B_{r-r_0}(o)\subset \{q\in M:d(x,q)<r\text { for all }x\in \Omega \} \subset B_{r+r_0}(o). \end{aligned}$$

(2.19)

From (1.5), (2.19) and Lemma 2.7, it is easy to show that

$$\begin{aligned} \begin{aligned} |B^n|\theta =&\lim _{r\rightarrow +\infty }\frac{B_{r-r_0}(o)}{n\int _0^{r-r_0}h(t)^{n-1}dt} \frac{\int _0^{r-r_0}h(t)^{n-1}dt}{\int _0^{r}h(t)^{n-1}dt} \\&\le \lim _{r\rightarrow +\infty }\frac{|\{q\in M:d(x,q)<r\text { for all }x\in \Omega \}|}{n\int _0^rh(t)^{n-1}dt}\\&\le \lim _{r\rightarrow +\infty }\frac{B_{r+r_0}(o)}{n\int _0^{r+r_0}h(t)^{n-1}dt} \frac{\int _0^{r+r_0}h(t)^{n-1}dt}{\int _0^{r}h(t)^{n-1}dt} \\ =&|B^n|\theta . \end{aligned} \end{aligned}$$

(2.20)

Dividing (2.18) by $n\int _0^rh(t)^{n-1}dt$ and sending $r\rightarrow \infty$, it follows from (2.17) and (2.20) that

$$\begin{aligned} \begin{aligned} |B^n|\theta&\le e^{(n-1)(2r_0b_1+b_0)}\int _\Omega f^{\frac{n}{n-1}} \lim _{r\rightarrow \infty }\frac{r^n}{n\int _0^rh(t)^{n-1}dt} \\&=e^{(n-1)(2r_0b_1+b_0)}\int _\Omega f^{\frac{n}{n-1}}\lim _{r\rightarrow \infty }\frac{1}{h'(t)^{n-1}} \\&\le \Big (\frac{e^{2r_0b_1+b_0}}{1+b_0}\Big )^{n-1}\int _\Omega f^{\frac{n}{n-1}}. \end{aligned} \end{aligned}$$

Hence we obtain

$$\begin{aligned} \int _{\partial \Omega } f+\int _\Omega |D f|+2(n-1)b_1\int _\Omega f \ge n|B^n|^{\frac{1}{n}} \theta ^{\frac{1}{n}} \Big (\frac{1+b_0}{e^{2r_0b_1+b_0}}\Big )^\frac{n-1}{n} \Big (\int _\Omega f^\frac{n}{n-1}\Big )^\frac{n-1}{n}. \end{aligned}$$

$\square$

Proof of Theorem 1.2

Suppose the equality of Theorem 1.1 holds. Then we have equalities in (2.13) and (2.17) which force $\lambda \equiv 0$. Thus M has nonnegative Ricci curvature. The assertion follows immediately from Theorem 1.2 in [10]. $\square$

3 The case of submanifolds

In this section, we assume that the ambient space M is a complete noncompact $(n+p)$-dimensional Riemannian manifold of asymptotically nonnegative sectional curvature with respect to a base point $o\in M$. Let $\Sigma \subset M$ be a compact submanifold of dimension n with or without boundary, and f be a positive smooth function on $\Sigma$. Let ${\bar{D}}$ denote the Levi-Civita connection of M and let $D^\Sigma$ denote the induced connection on $\Sigma$. The second fundamental form B of $\Sigma$ is given by

$$\begin{aligned} \langle B(X,Y),V\rangle =\langle {\bar{D}}_XY,V\rangle , \end{aligned}$$

where X, Y are the tangent vector fields on $\Sigma$, V is a normal vector field along $\Sigma$. The mean curvature vector of $\Sigma$ is defined by $H=\textrm{tr}B$.

We only need to treat the case that $\Sigma$ is connected. By scaling, we can assume that

$$\begin{aligned} \int _{\partial \Sigma }f+\int _\Sigma \sqrt{|D^\Sigma f|^2+f^2|H|^2} +2nb_1\int _\Sigma f=n\int _\Sigma f^{\frac{n}{n-1}}. \end{aligned}$$

(3.1)

By the connectedness of $\Sigma$ and (3.1), there exists a solution of the following Neumann boundary problem

$$\begin{aligned} \left\{ \begin{aligned}&\text {div}_\Sigma (fD^\Sigma u)=nf^\frac{n}{n-1}-2nb_1f-\sqrt{|D^\Sigma f|^2+f^2|H|^2},&\text { in }\Sigma ,\\&\langle D^\Sigma u,\nu \rangle =1,&\text { on }\partial \Sigma , \end{aligned} \right. \end{aligned}$$

(3.2)

where $\nu$ is the outward unit normal vector field of $\partial \Sigma$ with respect to $\Sigma$. Note that if $\partial \Sigma =\varnothing$, then the boundary condition in (3.2) is void. By standard elliptic regularity theory (see Theorem 6.31 in [20]), we know that $u\in C^{2,\gamma }$ for each $0<\gamma <1$.

As in [10], we define

$$\begin{aligned} \begin{aligned} U:&=\{x\in \Sigma \setminus \partial \Sigma :|D^\Sigma u(x)|<1\},\\ E:&=\{(x,y):x\in U,y\in T^\perp _x\Sigma ,|D^\Sigma u(x)|^2+|y|^2<1\}. \end{aligned} \end{aligned}$$

For each $r>0$, we denote by $A_r$ the set of all points $({\bar{x}},{\bar{y}})\in E$ satisfying

$$\begin{aligned} ru(x)+\frac{1}{2}{d} (x,\exp _{{\bar{x}}}(rD^\Sigma u({\bar{x}}))+r{\bar{y}})^2\ge ru({\bar{x}})+\frac{1}{2}r^2(|D^\Sigma u({\bar{x}})|^2+|{\bar{y}}|^2) \end{aligned}$$

for all $x\in \Sigma$. Define the transport map $\Phi _r:T^\perp \Sigma \rightarrow M$ for each $r>0$ by

$$\begin{aligned} \Phi _r(x,y)=\exp _x(rD^\Sigma u(x)+ry) \end{aligned}$$

for all $x\in \Sigma$ and $y\in T^\perp _x\Sigma$. The regularity of u implies that $\Phi _r$ is of class $C^{1,\gamma }$, $0<\gamma <1$.

Lemma 3.1

Assume that $(x,y)\in E$. Then we have

$$\begin{aligned} \frac{1}{n} \left( \Delta _\Sigma u(x)-\langle H(x),y\rangle \right) \le f^\frac{1}{n-1}(x)-2b_1. \end{aligned}$$

Proof

Combining $|D^\Sigma u(x)|^2+|y|^2<1$ with Cauchy-Schwarz inequality, we obtain

$$\begin{aligned} \begin{aligned}&-\langle D^\Sigma f(x),D^\Sigma u(x)\rangle -f(x)\langle H(x),y\rangle \\&\le \sqrt{|D^\Sigma f(x)|^2+f(x)^2|H(x)|^2}\sqrt{|D^\Sigma u(x)|^2+|y|^2}\\&\le \sqrt{|D^\Sigma f(x)|^2+f(x)^2|H(x)|^2}. \end{aligned} \end{aligned}$$

(3.3)

In terms of (3.2) and (3.3), one derives that

$$\begin{aligned} \begin{aligned}&f(x)\Delta _\Sigma u(x)-f(x)\langle H(x),y\rangle \\&\quad =nf(x)^{\frac{n}{n-1}}-2nb_1f-\sqrt{|D^\Sigma f(x)|^2+f(x)^2|H(x)|^2}\\&\qquad -\langle D^\Sigma f(x),D^\Sigma u(x)\rangle -f(x)\langle H(x),y\rangle \\&\quad \le nf(x)^{\frac{n}{n-1}}-2nb_1f. \end{aligned} \end{aligned}$$

The proof is completed. $\square$

The following three lemmas are due to Brendle (Lemmas 4.2, 4.3, 4.5 in [10]). Their proofs are independent of the curvature condition of ambient space too.

Lemma 3.2

For each $0\le \sigma <1$, the set

$$\begin{aligned} \{q\in M:\sigma r<d(x,q)<r,\ \forall x\in \Sigma \} \end{aligned}$$

is contained in the set

$$\begin{aligned} \Phi _r(\{(x,y)\in A_r:|D^\Sigma u(x)|^2+|y|^2>\sigma ^2\}). \end{aligned}$$

Lemma 3.3

Assume that $({\bar{x}},{\bar{y}})\in A_r$, and let ${\bar{\gamma }}(t):=\exp _{{\bar{x}}}(tD^\Sigma u({\bar{x}})+t{\bar{y}})$ for all $t\in [0,r]$. If Z is a smooth vector field along ${\bar{\gamma }}$ satisfying $Z(0)\in T_{{\bar{x}}}\Sigma$ and $Z(r)=0$, then

$$\begin{aligned} \begin{aligned}&((D^\Sigma )^2u)(Z(0),Z(0))-\langle B(Z(0),Z(0)),{\bar{y}}\rangle \\&\quad +\int _0^r\big (|{\bar{D}}_tZ(t)|^2-{\bar{R}}({\bar{\gamma }}'(t),Z(t),{\bar{\gamma }}'(t),Z(t))\big )dt\ge 0. \end{aligned} \end{aligned}$$

Lemma 3.4

Assume that $({\bar{x}},{\bar{y}})\in A_r$, and let ${\bar{\gamma }}(t):=\exp _{{\bar{x}}}(tD^\Sigma u({\bar{x}})+t{\bar{y}})$ for all $t\in [0,r]$. Let $\{e_1,\dots ,e_n\}$ be an orthonormal basis of $T_{{\bar{x}}}\Sigma$. Suppose that W is a Jacobi field along ${\bar{\gamma }}$ satisfying $W(0)\in T_{{\bar{x}}}\Sigma$ and $\langle {\bar{D}}_tW(0),e_j\rangle =((D^\Sigma )^2u)(W(0),e_j)-\langle B(W(0),e_j),{\bar{y}}\rangle$ for each $1\le j\le n$. If $W(\tau )=0$ for some $\tau \in (0,r)$, then W vanishes identically.

Now we begin the proof of Theorem 1.5.

Proof of Theorem 1.4

For any $r>0$ and $({\bar{x}},{\bar{y}})\in A_r$, let $\{e_i\}_{1\le i\le n}$ be any given orthonormal basis in $T_{{\bar{x}}}\Sigma$. Choose a normal coordinate system $(x^1,\cdots ,x^n)$ on $\Sigma$ around ${\bar{x}}$ such that $\frac{\partial }{\partial x^i}=e_i$ at ${\bar{x}}\ (1\le i\le n)$. Let $\{e_\alpha \}_{n+1\le \alpha \le n+p}$ be an orthonormal frame field of $T^\perp \Sigma$ around ${\bar{x}}$ such that $\big ((D^\Sigma )^\perp e_\alpha \big )_{{\bar{x}}}=0$ for $n+1\le \alpha \le n+p$, where $(D^\Sigma )^\perp$ denotes the normal connection in the normal bundle $T^\perp \Sigma$ of $\Sigma$. Any normal vector y around ${\bar{x}}$ can be written as $y=\sum _{\alpha =n+1}^{n+p}y^\alpha e_\alpha$, and thus $(x^1,\cdots ,x^n,y^{n+1},\cdots ,y^{n+p})$ becomes a local coordinate system on the total space of the normal bundle $T^\perp \Sigma$.

Let ${\bar{\gamma }}(t):=\exp _{{\bar{x}}}(tD^\Sigma u({\bar{x}})+t{\bar{y}})$ for all $t\in [0,r]$. For each $1\le A\le n+p$, we denote by $E_A(t)$ the parallel transport of $e_A({\bar{x}})$ along ${\bar{\gamma }}$. For each $1\le i\le n$, let $X_i$ be the Jacobi field along ${\bar{\gamma }}$ with the following initial conditions

$$\begin{aligned} \begin{aligned} X_i(0)&=e_i,\\ \langle {\bar{D}}_tX_i(0),e_j\rangle&=((D^\Sigma )^2u)(e_i,e_j)-\langle B(e_i,e_j),{\bar{y}} \rangle ,\quad 1\le j\le n,\\ \langle {\bar{D}}_tX_i(0),e_\beta \rangle&= \langle B(e_i,D^\Sigma u({\bar{x}})),e_\beta \rangle ,\quad n+1\le \beta \le n+p. \end{aligned} \end{aligned}$$

(3.4)

For each $n+1\le \alpha \le n+p$, let $X_\alpha$ be the Jacobi field along ${\bar{\gamma }}$ satisfying

$$\begin{aligned} X_\alpha (0)=0,\quad {\bar{D}}_tX_\alpha (0)=e_\alpha . \end{aligned}$$

(3.5)

Using Lemma 3.4, we known that $\{X_A(t)\}_{1\le A\le n+p}$ are linearly independent for each $t\in (0,r)$.

Let $P(t)=(P_{AB}(t))$ and $S(t)=(S_{AB}(t))$ be the matrices given by

$$\begin{aligned} \begin{aligned} P_{AB}(t)&=\langle X_A(t),E_B(t)\rangle ,\\ S_{AB}(t)&={\bar{R}}({\bar{\gamma }}'(t),E_A(t),{\bar{\gamma }}'(t),E_B(t)) \end{aligned} \end{aligned}$$

for $1\le A,B\le n+p$ and $t\in [0,r]$, where ${\bar{R}}$ denotes the Riemannian curvature tensor of M. Using the Jacobi equation and the initial conditions (3.4), (3.5), we have

$$\begin{aligned} \begin{aligned} P''(t)&=-P(t)S(t),\\ P_{AB}(0)&=\begin{bmatrix} \delta _{ij}&{}0\\ 0&{}0 \end{bmatrix}, \\ P'_{AB}(0)&=\begin{bmatrix} ((D^\Sigma )^2u)(e_i,e_j)-\langle B(e_i,e_j),{\bar{y}} \rangle &{}\langle B(e_i,D^\Sigma u({\bar{x}})),e_\beta \rangle \\ 0&{}\delta _{\alpha \beta } \end{bmatrix}. \end{aligned} \end{aligned}$$

(3.6)

Set $Q(t)=P(t)^{-1}P'(t),t\in (0,r)$. By (3.6), a simple computation yields

$$\begin{aligned} \frac{d}{dt}Q(t)=-S(t)-Q^2(t), \end{aligned}$$

(3.7)

where Q(t) is symmetric. For the matrices P(t), Q(t), it is easy to derive their following asymptotic expansions (cf. [10])

$$\begin{aligned} \begin{aligned} P(t)&=\begin{bmatrix} \delta _{ij}+O(t)&{}O(t)\\ O(t) &{} t\delta _{\alpha \beta }+O(t^2) \end{bmatrix},\\ Q(t)&=\begin{bmatrix} (D^\Sigma )^2u(e_i,e_j)-\langle B(e_i,e_j),{\bar{y}} \rangle +O(t)&{}O(1) \\ O(1)&{}\frac{1}{t}\delta _{\alpha \beta }+O(1) \end{bmatrix} \end{aligned} \end{aligned}$$

(3.8)

as $t\rightarrow 0^+$. In terms of (3.7) and the curvature assumption for M, we deduce

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}Q_{AA}(t)+Q_{AA}(t)^2\le \frac{d}{dt}Q_{AA}(t)+\sum _{B=1}^{n+p}Q_{AB}Q_{BA}(t) \\&\quad =-S_{AA}(t) \\&\quad \le (|D^\Sigma u({\bar{x}})|^2+|{\bar{y}}|^2-\langle D^\Sigma u({\bar{x}})+{\bar{y}},e_A\rangle ^2)\lambda (d(o,{\bar{\gamma }}(t)))\\&\quad \le (|D^\Sigma u({\bar{x}})|^2+|{\bar{y}}|^2-\langle D^\Sigma u({\bar{x}})+{\bar{y}},e_A\rangle ^2)\lambda (\big |d(o,{\bar{x}})-t|D^\Sigma u({\bar{x}})+{\bar{y}}|\big |) \end{aligned} \end{aligned}$$

(3.9)

for $1\le A\le n+p$, where the last inequality follows from the following triangle inequality

$$\begin{aligned} d(o,{\bar{\gamma }}(t))\ge \big |d(o,{\bar{x}})-d({\bar{x}},{\bar{\gamma }}(t))\big | =\big |d(o,{\bar{x}})-t|D^\Sigma u({\bar{x}})+{\bar{y}}|\big |. \end{aligned}$$

For $1\le A\le n+p$, we set

$$\begin{aligned} \Lambda _{{\bar{x}},A}(t)=(|D^\Sigma u({\bar{x}})|^2+|{\bar{y}}|^2-\langle D^\Sigma u({\bar{x}})+{\bar{y}},e_A\rangle ^2)\lambda (\big |d(o,{\bar{x}})-t|D^\Sigma u({\bar{x}})+{\bar{y}}|\big |). \end{aligned}$$

Then we have

$$\begin{aligned} \left\{ \begin{aligned}&Q'_{ii}(t)+Q_{ii}(t)^2\le \Lambda _{{\bar{x}},i}(t),\quad t\in (0,r), \\&\lim _{t\rightarrow 0^+}Q_{ii}(t)=\lambda _i, \end{aligned} \right. \end{aligned}$$

where $\lambda _i=P'_{ii}(0)$. Let $\phi _i$ be defined by

$$\begin{aligned} \phi _i(t)=e^{\int _0^t Q_{ii}(\tau )d\tau }. \end{aligned}$$

Then $\phi _i$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\phi _i''\le \Lambda _{{\bar{x}},i}\phi _i, \quad t\in (0,r), \\&\phi _i(0)=1,\phi _i'(0)=\lambda _i. \end{aligned} \right. \end{aligned}$$

(3.10)

Next, we denote by $\psi _{1,i},\psi _{2,i}$ solutions to the following problems

$$\begin{aligned} \left\{ \begin{aligned}&\psi _{1,i}''= \Lambda _{{\bar{x}},i}\psi _{1,i},\quad t\in (0,r), \\&\psi _{1,i}(0)=0,\psi _{1,i}'(0)=1, \end{aligned} \right. \quad \left\{ \begin{aligned}&\psi _{2,i}''= \Lambda _{{\bar{x}},i}\psi _{2,i},\quad t\in (0,r), \\&\psi _{2,i}(0)=1,\psi _{2,i}'(0)=0. \end{aligned} \right. \end{aligned}$$

(3.11)

Similar to the proof of (2.6), (2.13) and (2.14), we obtain

$$\begin{aligned} \begin{aligned} \frac{\psi _{2,i}}{\psi _{1,i}}(r)&\le \int _0^{+\infty }\Lambda _{{\bar{x}},i}(t)\ dt+\frac{1}{r}\\&\le 2b_1{\frac{|D^\Sigma u({\bar{x}})|^2+|{\bar{y}}|^2-\langle D^\Sigma u({\bar{x}})+{\bar{y}},e_i\rangle ^2}{\sqrt{|D^\Sigma u({\bar{x}})|^2+{\bar{y}}^2}}}+\frac{1}{r}\\&\le 2b_1{{\sqrt{|D^\Sigma u({\bar{x}})|^2+{\bar{y}}^2}}}+\frac{1}{r} \end{aligned} \end{aligned}$$

(3.12)

and

$$\begin{aligned} \psi _{1,i}(t)\le te^{\frac{|D^\Sigma u({\bar{x}})|^2+{\bar{y}}^2-\langle D^\Sigma u({\bar{x}})+{\bar{y}},e_i\rangle ^2}{|D^\Sigma u({\bar{x}})|^2+{\bar{y}}^2}(2r_0b_1+b_0)}, \quad t\in (0,r), \end{aligned}$$

(3.13)

where $r_0=\max \{d(o,x)|x\in \Sigma \}$. Using Lemma 2.5, one can find from (3.10) and (3.11) that

$$\begin{aligned} Q_{ii}(t)=\frac{\phi '_i}{\phi _i}(t)\le \frac{\psi '_{2,i}+\lambda _i\psi '_{1,i}}{\psi _{2,i}+\lambda _i\psi _{1,i}}(t). \end{aligned}$$

(3.14)

Similarly we obtain from (3.8) and (3.9) that

$$\begin{aligned} \left\{ \begin{aligned}&Q'_{\alpha \alpha }(t)+Q_{\alpha \alpha }(t)^2\le \Lambda _{{\bar{x}},\alpha }(t),\quad t\in (0,r),\\&Q_{\alpha \alpha }(t)=\frac{1}{t}+O(1),\quad \text {as }t\rightarrow 0^+ \end{aligned}\right. \end{aligned}$$

for $n+1\le \alpha \le n+p$. Set $\phi _\alpha (t)=te^{\int _0^t(Q_{\alpha \alpha } (\tau )-\frac{1}{\tau })d\tau }$. Then $\phi _\alpha$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\phi _\alpha ''\le \Lambda _{{\bar{x}},\alpha }\phi _\alpha , \quad t\in (0,r),\\&\phi _\alpha (0)=0,\phi _\alpha '(0)=1. \end{aligned} \right. \end{aligned}$$

Next, we denote by $\psi _{1,\alpha }$ the unique solution to the following problem

$$\begin{aligned} \left\{ \begin{aligned}&\psi _{1,\alpha }''= \Lambda _{{\bar{x}},\alpha }\psi _{1,\alpha }, \quad t\in (0,r),\\&\psi _{1,\alpha }(0)=0,\psi _{1,\alpha }'(0)=1. \end{aligned} \right. \end{aligned}$$

(3.15)

Similar to (2.14), we derive that

$$\begin{aligned} \psi _{1,\alpha }\le e^{\frac{|D^\Sigma u({\bar{x}})|^2+{\bar{y}}^2-\langle D^\Sigma u({\bar{x}})+{\bar{y}},e_\alpha \rangle ^2}{|D^\Sigma u({\bar{x}})|^2+{\bar{y}}^2}(2r_0b_1+b_0)}t, \end{aligned}$$

(3.16)

for $t\in (0,r)$. By Lemma 2.1 in [34] we have

$$\begin{aligned} Q_{\alpha \alpha }(t)=\frac{\phi '_\alpha }{\phi _\alpha }(t)\le \frac{\psi '_{1,\alpha }}{\psi _{1,\alpha }}(t). \end{aligned}$$

(3.17)

From (3.14) and (3.17), it follows that

$$\begin{aligned} \frac{d}{dt}\log \det P(t)=\textrm{tr}(Q(t))\le \sum _i \frac{\psi '_{2,i}+\lambda _i\psi '_{1,i}}{\psi _{2,i}+\lambda _i\psi _{1,i}}(t)+\sum _\alpha \frac{\psi '_{1,\alpha }}{\psi _{1,\alpha }}(t). \end{aligned}$$

(3.18)

Combining (3.11), (3.15) with the asymptotic properties in (3.8), we conclude that

$$\begin{aligned} \lim _{t\rightarrow 0^+}\frac{\det P(t)}{\prod _i(\psi _{2,i}(t)+\lambda _i\psi _{1,i}(t)) \prod _\alpha \psi _{1,\alpha }(t)}=1. \end{aligned}$$

(3.19)

Integrating (3.18) over $[\varepsilon ,t]$ for $0<\varepsilon <t$ and using (3.19) by letting $\varepsilon \rightarrow 0^+$, it is easy to show that

$$\begin{aligned} |\det {\bar{D}}\Phi _t({\bar{x}},{\bar{y}})|=\det P(t)\le \prod _i(\psi _{2,i}(t)+\lambda _i\psi _{1,i}(t)) \prod _\alpha \psi _{1,\alpha }(t). \end{aligned}$$

Note that $0\le \phi _i\le (\psi _{2,i}+\lambda _i\psi _{1,i})$ and $\psi _{1,i}\ge 0\ (1\le i\le n)$. Combining (3.13), (3.16) with arithmetric-geometric mean inequality, we obtain

$$\begin{aligned} \begin{aligned} |\det {\bar{D}}\Phi _t({\bar{x}},{\bar{y}})|&\le \Big (\frac{1}{n}\sum _i\frac{\psi _{2,i}(t)}{\psi _{1,i}(t)}+\frac{1}{n} (\Delta _\Sigma u({\bar{x}})-\langle H({\bar{x}}),{\bar{y}}\rangle )\Big )^n\prod _A\psi _{1,A}(t)\\&\le \Big (\frac{1}{n}\sum _i\frac{\psi _{2,i}(t)}{\psi _{1,i}(t)}+ \frac{1}{n}(\Delta _\Sigma u({\bar{x}})-\langle H({\bar{x}}),{\bar{y}}\rangle )\Big )^nt^{n+p}e^{(n+p-1)(2r_0b_1+b_0)} \end{aligned} \end{aligned}$$

which yields by (3.12) that

$$\begin{aligned} \begin{aligned} |\det {\bar{D}}\Phi _r({\bar{x}},{\bar{y}})|&\le (2b_1{{\sqrt{|Du({\bar{x}})|^2+{\bar{y}}^2}}}+\frac{1}{r}+\frac{1}{n} (\Delta _\Sigma u({\bar{x}})\\&\quad \ -\langle H({\bar{x}}),{\bar{y}}\rangle ))^nr^{n+p}e^{(n+p-1)(2r_0b_1+b_0)} \end{aligned} \end{aligned}$$

(3.20)

for all $({\bar{x}},{\bar{y}})\in A_r$. Noting that ${\sqrt{|Du({\bar{x}})|^2+{\bar{y}}^2}}<1$, we derive by Lemma 3.1 and (3.20) that

$$\begin{aligned} |\det {\bar{D}}\Phi _r({\bar{x}},{\bar{y}})| \le (\frac{1}{r}+f^\frac{1}{n-1}({\bar{x}}))^n r^{n+p}e^{(n+p-1)(2r_0b_1+b_0)} \end{aligned}$$

(3.21)

for all $({\bar{x}},{\bar{y}})\in A_r$. Using Lemma 3.2 and (3.21), one may find in a similar way as the proof of Theorem 1.4 in [10] that

$$\begin{aligned} \begin{aligned}&|\{p\in M:\sigma r<d(x,p)<r,\forall x\in \Sigma \}|\\&\quad \le \frac{p}{2}|B^p|(1-\sigma ^2)e^{(n+p-1)(2r_0b_1+b_0)}\int _\Sigma (\frac{1}{r}+f^\frac{1}{n-1}({\bar{x}}))^nr^{n+p}, \end{aligned} \end{aligned}$$

(3.22)

for all $r>0$ and all $0\le \sigma <1$. Similar to the proof of (2.20), one can obtain by using Lemma 2.7 that

$$\begin{aligned} \begin{aligned}&\lim _{r\rightarrow +\infty }\frac{|\{p\in M:\sigma r<d(x,p)<r,\forall x\in \Sigma \}|}{(n+p)\int _0^rh^{n+p-1}dt}\\&\quad =|B^{n+p}|\theta \lim _{r\rightarrow +\infty } (1-\sigma \frac{h^{n+p-1}(\sigma r)}{h^{n+p-1}(r)})\\&\quad =|B^{n+p}|(1-\sigma ^{n+p})\theta . \end{aligned} \end{aligned}$$

(3.23)

Dividing (3.22) by $(n+p)\int _0^rh(t)^{n+p-1}dt$ and sending $r\rightarrow +\infty$, we deduce by using (2.17) and (3.23) that

$$\begin{aligned} \begin{aligned}&=|B^{n+p}|(1-\sigma ^{n+p})\theta \\&\quad \le \frac{p}{2}|B^p|(1-\sigma ^2)e^{(n+p-1)(2r_0b_1+b_0)}\int _\Sigma f ^{\frac{n}{n-1}} \lim _{r\rightarrow +\infty }\frac{r^{n+p}}{(n+p)\int _0^rh(t)^{n+p-1}dt} \\&\quad \le \frac{p}{2}|B^p|(1-\sigma ^2)\Big (\frac{e^{2r_0b_1+b_0}}{1+b_0}\Big )^{n+p-1} \int _\Sigma f ^{\frac{n}{n-1}}. \end{aligned}\qquad \end{aligned}$$

(3.24)

for all $0\le \sigma <1$. Now, if we divide (3.24) by $1-\sigma$ and let $\sigma \rightarrow 1$, we have

$$\begin{aligned} (n+p)|B^{n+p}|\theta \le p|B^p|\Big (\frac{e^{2r_0b_1+b_0}}{1+b_0}\Big )^{n+p-1} \int _\Sigma f ^{\frac{n}{n-1}}. \end{aligned}$$

(3.25)

Hence (3.1) and (3.25) imply that

$$\begin{aligned} \begin{aligned}&\int _{\partial \Sigma }f+\int _\Sigma \sqrt{|D^\Sigma f|^2+f^2|H|^2} +2nb_1\int _\Sigma f\\&\quad \ge n\Big (\frac{(n+p)|B^{n+p}|}{p|B^p|}\Big )^{\frac{1}{n}}\theta ^{\frac{1}{n}} \Big (\frac{1+b_0}{e^{2r_0b_1+b_0}}\Big )^{\frac{n+p-1}{n}} \Big (\int _\Sigma f^{\frac{n}{n-1}}\Big ) ^{\frac{n-1}{n}}. \end{aligned} \end{aligned}$$

$\square$

Proof of Theorem 1.6

Suppose the equality of Theorem 1.5 holds. Then we have equality in both (2.17) and (3.12) and either one forces $\lambda \equiv 0$. Thus M has nonnegative sectional curvature. The assertion follows immediately from Theorem 1.6 in [10]. $\square$

Finally we would like to mention that we have established a Sobolev type inequality for manifolds with density and asymptotically nonnegative Bakery-Émery Ricci curvature in [16] and a logarithmic Sobolev type inequality for closed submanifolds in manifolds with asymptotically nonnegative sectional curvature in [17].