1 Introduction

In the well-known paper [34], Loewner and Nirenberg studied the blowing up boundary value problem

$$\begin{aligned}&\Delta u=\frac{1}{4}n(n-2)u^{\frac{n+2}{n-2}},\,\,\text {in}\,\,\Omega ,\\&u(x)\rightarrow \infty ,\,\,\,\text {as}\,\,x\rightarrow \partial \Omega , \end{aligned}$$

with \(\Omega \) a bounded domain of \(C^2\) in \({\mathbb {R}}^n\). They proved that there exists a unique positive solution u to this problem, and there exists a constant \(C>0\) depending on the domain \(\Omega \) such that

$$\begin{aligned} |\text {dist}(x,\partial \Omega )^{\frac{n-2}{2}}u-1|\le C \text {dist}(x,\partial \Omega ) \end{aligned}$$

near the boundary, where \(\text {dist}(x,\partial \Omega )\) is the distance of x to \(\partial \Omega \). This is equivalent to seeking the conformal metric \(g=u^{\frac{4}{n-2}}\delta \) with \(\delta \) the Euclidean metric on \(\Omega \) has constant scalar curvature \(R_g=-n(n-1)\).

In [3] and [4], Aviles and McOwen generalized the Loewner–Nirenberg problem to compact Riemannian manifolds (M, g) with boundary. Denote \(M^{\circ }\) to be the interior of M. In particular, they considered the blowing up Dirichlet boundary value problem

$$\begin{aligned}&\frac{4(n-1)}{n-2}\Delta u-R_gu-n(n-1)u^{\frac{n+2}{n-2}}=0,\,\,\,\text {in}\,\,M^{\circ }, \end{aligned}$$
(1.1)
$$\begin{aligned}&u(p)\rightarrow \infty ,\,\,\,\text {as}\,\,p\rightarrow \partial M. \end{aligned}$$
(1.2)

We call (1.1)–(1.2) the Loewner–Nirenberg problem on (M, g). Using classical variational method they obtained a sequence of solutions to (1.1) with enlarging Dirichlet boundary data that go to infinity, and using maximum principle and an integral type weak Harnack inequality, they obtained the existence of the unique solution u to (1.1)–(1.2) and analyzed the asymptotic behavior of u near the boundary. For regularity of the Loewner–Nirenberg metric in the conformal class of a smooth compact manifold \((M^n,g)\) with boundary, an nice expansion of the solution near the boundary is given in [1, 38]. Recently, Xumin Jiang and Qing Han developed a type of weighted Schauder estimates near the boundary, see [25] (see also [30]), which fits in this expansion well, and for expansion of the solution near the boundary for manifolds with corners on the boundary see [27] and for more references on this topic one is referred to [26].

In this article, we derive two flow approaches to the Loewner–Nirenberg problem. Indeed, we introduce the Cauchy–Dirichlet problems to a direct scalar curvature flow (see (1.3)–(1.5)) and the Yamabe flow (1.6) on a compact Riemannian manifold (M, g) with boundary.

Let \((M^n,g)\) be a compact Riemannian manifold with boundary. Let \(M^{\circ }\) be the interior of M. Define the function spaces

$$\begin{aligned} C_{loc}^{k+\alpha ,m+\beta }(\partial M\times [0,+\infty ))&=\{u\in C^{k+\alpha ,m+\beta }(\partial M\times [0,T]\,\,\text {for any}\,\,T>0\},\,\,\text {and}\\ C_{loc}^{k+\alpha ,m+\beta }(M\times [0,+\infty ))&=\{u\in C^{k+\alpha , m+\beta }(M\times [0,T])\,\,\text {for any}\,\,T>0\}. \end{aligned}$$

Consider the Cauchy–Dirichlet problem

$$\begin{aligned}&u_t=\frac{4(n-1)}{n-2}\Delta u-R_gu-n(n-1)u^{\frac{n+2}{n-2}},\,\,\,\text {in}\,\,M\times [0,+\infty ), \end{aligned}$$
(1.3)
$$\begin{aligned}&u(p,0)=u_0(p),\,\,p\in M, \end{aligned}$$
(1.4)
$$\begin{aligned}&u(q,t)=\phi (q,t),\,\,q\in \,\partial M, \end{aligned}$$
(1.5)

with \(\phi \rightarrow \infty \) as \(t\rightarrow \infty \), which is called the direct flow in this paper. For the direct flow, we first derive the long time existence of the flow for general initial data, see Lemma 2.4. If the boundary data \(\phi \rightarrow \infty \) uniformly of certain speed (2.4) as \(t\rightarrow \infty \), we obtain an asymptotic blowing up lower bound estimates near the boundary, see Lemma 2.3. Together with the interior upper bound estimates (see Lemma 2.2, in comparison with the local upper bound estimates in [4]) and the Harnack inequality, we obtain the uniform upper and lower bound of u on any given compact subset of \(M^{\circ }\). The convergence of the flow is the main part of the discussion. We have the following theorem for the case \(R_g\ge 0\).

Theorem 1.1

Let (M, g) be a smooth compact manifold with boundary such that \(R_g\ge 0\). Assume two positive functions \(u_0\in C^{2,\alpha }(M)\) and \(\phi \in C_{loc}^{2,\alpha }(\partial M \times [0,+\infty ))\) satisfy the compatible condition (2.1) on \(\partial M\times \{0\}\). Moreover, assume \(\phi _t\ge 0\) for \(t\ge 0\), \(\phi \) satisfies (2.4) as \(t\rightarrow \infty \) and \(\liminf _{t\rightarrow \infty }\inf _{\partial M}\phi =\infty \). Then there exists a unique solution u to the Cauchy–Dirichlet problem (1.3)–(1.5), converging in \(C_{loc}^2(M^{\circ })\) to a solution \(u_{\infty }\) to the Loewner–Nirenberg problem (1.1)–(1.2) as \(t\rightarrow +\infty \). Moreover, there exists a constant \(C>0\) such that

$$\begin{aligned} \frac{1}{C}\big (\,x+\,(\inf _{p\in \partial M}\phi (p,t)\,)^{\frac{2}{2-n}}\,\big )^{\frac{2-n}{2}}-C\le u\le Cx^{\frac{2-n}{2}} \end{aligned}$$

near the boundary \(\partial M\), where x is the distance function to the boundary.

To achieve that, when the initial data is a subsolution to (1.1) and \(\phi \) is increasing in a certain speed to infinity as \(t\rightarrow \infty \), we first show that the solution is increasing and converges to the solution of the Loewner–Nirenberg problem, see Lemma 2.6 to Proposition 2.5. In particular, \(u(\cdot ,t)\) is a sub-solution to (1.1) for each \(t\ge 0\). Then for general positive initial data, we can first solve a Cauchy–Dirichlet problem with smaller initial data which is a subsolution to (1.1), and hence the solution \(u_1\) is increasing and converges to the solution of the Loewner–Nirenberg problem; and then we use \(u_1\) to give the lower bound of the flow with general initial data by maximum principle, and we use Hamilton’s technique in [24] to derive the convergence of the flow. We provide examples of the compact manifolds with boundary with positive scalar curvature in Sect. 3, and based on the positive mass theorem, we obtain that on a smooth compact manifold (M, g) with boundary, there exists a positive scalar curvature metric in the conformal class if and only if \(\inf _{u\in C^1(M)-\{0\}}Q(u)>-\infty \), where Q (see (3.2)) is the energy functional of the second type Escobar–Yamabe problem (3.1), see Theorem 3.2.

For a general conformal class (M, [g]), we have

Theorem 1.2

Let (M, g) be a compact Riemannian manifold with boundary of \(C^{4,\alpha }\). Let \(u_0\in C^{2,\alpha }(M)\) be any positive function such that \(u_0>v_0\) on M, where \(v_0\) is the largest solution to he homogeneous Dirichlet boundary value problem of the Yamabe equation (A.1)–(A.2). Then there exists a direct flow g(t) starting from \(g_0=u_0^{\frac{4}{n-2}}g\) and converges in \(C_{loc}^2(M^{\circ })\) to \(g_{\infty }=u_{LN}^{\frac{4}{n-2}}g\), where \(u_{LN}\) the solution to the Loewner–Nirenberg problem (1.1)–(1.2).

Notice that \(v_0>0\) in \(M^{\circ }\) when \(\lambda _1(L_g)<0\) (see Appendix A), where \(\lambda _1(L_g)\) is the first eigenvalue of the Dirichlet boundary value problem of the conformal Laplacian operator \(L_g=-\frac{4(n-1)}{n-2}\Delta _g +R_g\); while \(v_0=0\) by maximum principle when \(R_h\ge 0\) for some conformal metric \(h\in [g]\). When (M, g) is smooth and the positive mass theorem holds, if \(\lambda _1(L_g)>0\), there exists \(h\in [g]\) such that \(R_h>0\), see Sect. 3. To prove Theorem 1.2, we first take a conformal metric g with \(R_g=-n(n-1)\) as the background metric and show that the flow is monotone and converges to the Loewner–Nirenberg metric when we choose the initial data \(u_0=1\) and increasing boundary data in a certain speed; while for \(u_0>v_0\), we construct a compatible boundary data increasing in \(t\ge 0\) ina certain speed, and then use a solution to the Cauchy–Dirichlet problem (1.3)–(1.5) with smaller boundary data and smaller initial data which is a solution to (1.1) to give a lower bound, and we use Hamilton’s technique in [24] again for the upper bound control to conclude the convergence of the flow.

Yamabe flow is studied as an alternative approach to the Yamabe problem on closed manifolds, see [8, 9, 15, 49, 55], etc. It is also used in the study of general prescribed scalar curvature equation, see [12, 53], etc. For Yamabe flow on manifolds with boundary on the Neumann type boundary problems posed by Escobar, see [10, 11], etc. On complete non-compact Riemannian manifolds, there are also many works on the long time existence and convergence of the Yamabe flow, see [14, 37], etc. On compact manifolds with incomplete edge singularities Yamabe flow is studied in [5], etc. see also [48, 50] and the references there for Yamabe flow on incomplete manifolds. In [47], an instantaneously complete Yamabe flow was derived in the conformal class of the hyperbolic space. In [48], the author studied the relationship between the dimension of the singular sub-manifold and instantaneous completeness and incompleteness of the flow.

Notice that the Yamabe flow is conformally covariant and it makes no difference which metric in the conformal class is chosen as the background metric of the flow. We take a conformal metric g with \(R_g=-n(n-1)\) in the conformal class (M, [g]) as the background metric. Now we introduce the Cauchy–Dirichlet problem of the Yamabe flow

$$\begin{aligned} \begin{aligned}&(u^{\frac{n+2}{n-2}})_t=\frac{(n-1)(n+2)}{n-2}\left( \Delta _gu +\frac{n(n-2)}{4}(u-u^{\frac{n+2}{n-2}})\right) ,\\&u(q,0)=u_0(q),\,\,q\in M,\\&u(q,t)=\phi (q,t),\,\,(q,t)\in \partial M\times [0,+\infty ), \end{aligned} \end{aligned}$$
(1.6)

where \(\phi \rightarrow \infty \) as \(t\rightarrow \infty \), to get a natural connection between a compact metric g on M with the complete Loewner–Nirenberg metric in the conformal class. For the convergence, we have that

Theorem 1.3

Let \((M^n,g)\) be a compact Riemannian manifold with boundary of \(C^{4,\alpha }\) and \(R_g=-n(n-1)\). Let \(u_0\in C^{4,\alpha }(M)\) with \(u_0\ge 1\) be a subsolution of the equation (1.1) and satisfies

$$\begin{aligned} L_g(\mu )\ge 0 \end{aligned}$$
(1.7)

at the points \(q\in \partial M\) such that \(\mu =0\), where \(\mu \) and \(L_g(\mu )\) are defined in (4.5) and (4.4). Assume \(\phi \in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,\infty ))\) satisfies the compatible condition (4.3)–(4.4) and \(\phi _t\ge 0\) on \(\partial M\times [0,\infty )\). Moreover, assume \(\liminf _{t\rightarrow \infty }\inf _{\partial M}\phi =\infty \) and \(\phi \) satisfies (4.7)–(4.8) as \(t\rightarrow \infty \). Then there exists a unique positive solution u to (1.6) on \(M\times [0,+\infty )\) with \(u\in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,\infty ))\), and \(u\rightarrow u_{LN}\) in \(C_{loc}^4(M^{\circ })\) as \(t\rightarrow \infty \), where \(u_{LN}\) is the solution to the Loewner–Nirenberg problem (1.1)–(1.2).

Notice that the condition (1.7) is to guarantee the \(C^{4+\alpha ,2+\frac{\alpha }{2}}\) regularity of the solution u at \(\partial M\times \{0\}\) and the condition \(\phi _t\ge 0\) for \(t\ge 0\), and it holds automatically if \(u_0\) is a strict subsolution to (1.1) in a neighborhood of \(\partial M\), or \(u_0\) is a solution to (1.1) in a neighborhood of \(\partial M\). It is clear that \(\phi =\log (t),\,t,\,t^2,\,e^{t}\) and \(te^t\) satisfy (4.7). The strategy of the proof of this theorem is similar as that of Theorem 1.2, but more is involved because of the nonlinear term on the left hand side of the Yamabe flow equation. Notice that we do not show the convergence of the Yamabe flow for general initial data, because when using Hamilton’s technique in [24] to get the upper bound of u, we are not able to show that \(\limsup _{t\rightarrow \infty }\sup _{M^{\circ }}(u-u_{LN})\le 0\), although \(\sup _{M^{\circ }}(u(\cdot ,t)-u_{LN}(\cdot ))\) is decreasing in t, where \(u_{LN}\) is the solution to the Loewner–Nirenberg problem (1.1)–(1.2).

The Cauchy-Dirchlet problem of the Yamabe flow is generalized to the Cauchy-Dirchlet problem of \(\sigma _k\)-Ricci flow in [31], i.e., a flow approach to the generalized Loewner–Nirenberg problem of the fully nonlinear equations in [22] and [23], where the fact the sub-solution property is preserving along the flow also plays an important role for the convergence of the flow.

It would be interesting to know whether the direct flow converges to the solution to Loewner–Nirenberg problem when \(u_0<v_0\) somewhere in \(M^{\circ }\) when \(\lambda _1(L_g)<0\).

2 A Direct Flow

Let \((M^n,g)\) be a smooth compact Riemannian manifold with boundary \(\partial M\), with \(g\in C^{4,\alpha }\). We consider the direct flow, i.e. the Cauchy–Dirichlet problem (1.3)–(1.5) with \(\displaystyle \lim _{t\rightarrow +\infty }\phi (q,t)=+\infty \), uniformly. To guarantee the solution u is in \(C^{2+\alpha ,1+\frac{\alpha }{2}}(M\times [0,T_0])\) for some \(T_0>0\) and \(0<\alpha <1\), we need the compatible condition

$$\begin{aligned} u_0(p)&=\phi (p,0),\nonumber \\ \phi _t(p,0)&=\frac{4(n\!-\!1)}{n\!-\!2}\Delta u_0(p)\!-\!R_gu_0(p)\!-\!n(n\!-\!1)u_0(p)^{\frac{n+2}{n-2}} \,\,\,\,\,\text {for}\,\,p\in \,\partial M, \end{aligned}$$
(2.1)

with \(u_0\in \,C^{2,\alpha }(M)\) and \(\phi \in C^{2,\alpha }(\partial M\times [0,T])\) for all \(T>0\). Moreover, in order that \(u\in C^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,T_0])\), we need the additional condition

$$\begin{aligned} \phi _{tt}(p,0)=L(v)(p)\,\,\,\,\,\text {for}\,\,p\in \,\partial M, \end{aligned}$$
(2.2)

with \(u_0\in \,C^{4,\alpha }(M)\) and \(\phi \in C^{4,\alpha }(\partial M\times [0,T])\) for all \(T>0\), where

$$\begin{aligned} v=\frac{4(n-1)}{n-2}\Delta u_0-R_gu_0-n(n-1)u_0^{\frac{n+2}{n-2}} \end{aligned}$$

on M and L is a linear operator such that

$$\begin{aligned} L(\varphi )=\frac{4(n-1)}{n-2}\Delta \varphi -R_g\varphi -\frac{n(n-1)(n+2)}{n-2}u_0^{\frac{4}{n-2}}\varphi \end{aligned}$$

for any \(\varphi \in C^2(M)\).

For later use, we now present a well-known weak Harnack inequality for parabolic inequalities, see in [32] or Peter Li’s online lecture notes on geometric analysis for instance.

Lemma 2.1

Assume \({\bar{B}}_{r}(p)\) is a closed geodesic r-ball in a complete Riemannian manifold (M, g) with \(r\le 1\). Let \(u\in C^{2,1}(M\times [0,T_1])\) be a positive function such that

$$\begin{aligned} u_t \le \frac{4(n-1)}{(n-2)}\Delta u+C_0u, \end{aligned}$$

for some constant \(C_0>0\). Assume that there exists a constant \(C_S>0\), such that we have the Sobolev inequality

$$\begin{aligned} \frac{1}{|B_r(p)|}\int _{B_r(p)}|\nabla \phi |^2 \ge C_S r^{-2}\big (\frac{1}{|B_r(p)|}\int _{B_r(p)}\phi ^{\frac{2n}{n-2}}\big )^{\frac{n-2}{n}}, \end{aligned}$$

for each \(\phi \in C_0^1(B_r(p))\). Then for \(\theta \in (0,1)\) and \(T\in (T_0,T_1)\), there exists a constant \(C_1>0\) depending on \(C_0\), such that

$$\begin{aligned}&\displaystyle \sup _{q\in B_{\theta r}(p), t\in [T,T_1]}|u(q)|\\&\quad \le C_1C_S^{-\frac{n-2}{4}}\left( (1-\theta )^{-2}r^{-2}+(T-T_0)^{-1} \right) ^{\frac{n^2-4}{4n}}r^{\frac{n-2}{2}}(T_1-T_0)^{\frac{n-2}{2n}}\\&\qquad \times \big (\frac{1}{|T_1-T_0|\times |B_r(p)|}\int _{T_0}^{T_1} \int _{B_r(p)}|u|^{\frac{2n}{n-2}}\big )^{\frac{n-2}{2n}}. \end{aligned}$$

Remark. For a complete Riemannian manifold \((M^n, g)\), if \(Ric_g\ge -k(n-1)g\) with some constant \(k>0\), and assume that there exists a constant \(C_v>0\) such that for a geodesic ball \(B_r\subseteq M\), the volume ratio satisfies

$$\begin{aligned} \frac{|B_r|}{|B_r^0|}\ge C_v, \end{aligned}$$
(2.3)

where \(|B_r^0|\) is the volume of a geodesic ball of radius r in the space form of sectional curvature \(-k\), then the Sobolev inequality holds on \(B_r\) with \(C_S\) depending on k and \(C_v\). For a compact manifold (M, g) with boundary of \(C^{2,\alpha }\), such constants are uniform for each geodesic ball in the interior. Denote \(M^{\circ }\) the interior of M, for each \(p\in M^{\circ }\), we can also use classical parabolic theory in Eucliean domain to get the weak Harnack inequality. Indeed, let \(r=\frac{1}{2}\min \{r_0,\,\text {dist}(p,\,\partial M)\}\) with \(r_0\) the injectivity radius at p. In \(B_{2r}(p)\), using the geodesic normal coordinates, we can write the equality as an parabolic inequality in Euclidean domain \(B_r(0)\), and apply the weak Harnack theorem in [33] to get the same control.

We start with a parabolic analog to Aviles-McOwen’s local upper bound estimates in [4] on the solutions to (1.3).

Lemma 2.2

Assume \(u>0\) is a solution to (1.3) on \(M\times [0,T_1)\) with \(T_1>\epsilon _0\) for some \(2>\epsilon _0>0\). There exists a uniform constant \(C_3>0\) depending on \(\epsilon _0>0\) but independent of \(T_1\) and r, such that for each closed geodesic ball \({\bar{B}}_{2r}(p)\subseteq M^{\circ }\), we have

$$\begin{aligned} u(q,t)\le C_3r^{-\frac{(n-2)}{2}}, \end{aligned}$$

for \(q\in {\bar{B}}_r(p)\) with \(r\le \min \{1, \sqrt{\epsilon _0}\}\), and \(T_1> t\ge \frac{\epsilon _0}{2}\).

Proof

Pick up a function \(\varphi (q,t)=\xi (q)\eta (t)\in C^2( M\times [0,T_1))\) to be determined later. Let \(\xi \ge 0\) be a cut off function with compact support \(B_{2r}(p)\), so that \(\xi (q)=1\) for \(q\in B_r(p)\), \(0\le \xi (q)\le 1\) for \(q\in B_{2r}(p)\) and there exists a constant \(C>0\) independent of \(p\in M^{\circ }\) and r such that \(|\nabla \xi |\le Cr^{-1}\) in \(B_{2r}(p)\). Multiply \(u\varphi ^{\alpha }\) on both sides of (1.3) and do integration on M, we have

$$\begin{aligned} \int _M u u_t\varphi ^{\alpha }dV_g&=\int _M\left[ \frac{4(n-1)}{n-2}u\Delta u \varphi ^{\alpha }-R_gu^2\varphi ^{\alpha }-n(n-1)u^{\frac{2n}{n-2}}\varphi ^{\alpha }\right] dV_g \end{aligned}$$

Integration by parts, we obtain

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\big (\int _M u^2\varphi ^{\alpha }dV_g\big )-\frac{\alpha }{2} \int _M u^2\varphi ^{\alpha -1}\varphi _t dV_g\\&\quad =\int _M\left[ -\frac{4(n-1)}{n-2}|\nabla u|^2 \varphi ^{\alpha }-\frac{4(n-1)}{n-2} \alpha u\varphi ^{\alpha -1}\,\nabla u\cdot \nabla \varphi \right. \\&\qquad \left. -R_gu^2 \varphi ^{\alpha }-n(n-1)u^{\frac{2n}{n-2}}\varphi ^{\alpha } \right] dV_g\\&\quad \le \int _M\left[ u^2\big (\frac{(n-1)}{n-2}\alpha ^2\varphi ^{\alpha -2}\,|\nabla \varphi |^2-R_g\varphi ^{\alpha }\big )-n(n-1)u^{\frac{2n}{n-2}}\varphi ^{\alpha } \right] dV_g, \end{aligned}$$

where the last inequality is by Cauchy inequality. Therefore,

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\big (\int _M u^2\varphi ^{\alpha }dV_g\big )&\le \int _M\left[ \big (\frac{\alpha }{2}\varphi ^{\alpha -1}\varphi _t +\frac{(n-1)}{n-2}\alpha ^2\varphi ^{\alpha -2}\,|\nabla \varphi |^2-R_g\varphi ^{\alpha }\big )\,u^2\right. \\&\quad \left. -n(n-1)u^{\frac{2n}{n-2}}\varphi ^{\alpha }\right] dV_g. \end{aligned}$$

Now for any T such that \(T_1>T\,\ge \frac{\epsilon _0}{2}\), we assume \(1\ge \eta \ge 0\) with \(\eta (t)=0\) for \(t\le T-\frac{r^2}{4}\), \(\eta =1\) for \(t\ge T-\frac{r^2}{8}\) and \(|\eta '(t)|\le \frac{12}{r^2}\) for \(t>0\). Integrate the above inequality on \(t\in [T-\frac{r^2}{4},T]\) to have

$$\begin{aligned} 0&\le \frac{1}{2}\big (\int _M u^2\varphi ^{\alpha }dV_g\big )\big |_{t=T}\\&\le \int _{T-\frac{r^2}{4}}^{T}\int _M\left[ \big (\frac{\alpha }{2} \varphi ^{\alpha -1}\varphi _t+\frac{(n-1)}{n-2}\alpha ^2\varphi ^{\alpha -2}\,|\nabla \varphi |^2\!-\!R_g\varphi ^{\alpha }\big )\,u^2 \!-\! n(n\!-\!1)u^{\frac{2n}{n-2}}\varphi ^{\alpha } \right] dV_gdt. \end{aligned}$$

Therefore, by Hölder inequality,

$$\begin{aligned}&n(n-1)\int _{T-\frac{r^2}{4}}^T \int _Mu^{\frac{2n}{n-2}}\varphi ^{\alpha } dV_gdt\\&\quad \le \int _{T-\frac{r^2}{4}}^{T}\int _M\left[ \big (\frac{\alpha }{2} \varphi ^{\alpha -1}\varphi _t+\frac{(n-1)}{n-2}\alpha ^2\varphi ^{\alpha -2 }\,|\nabla \varphi |^2-R_g\varphi ^{\alpha }\big )\,u^2\right] dV_gdt\\&\quad \le \left[ \int _{T-\frac{r^2}{4}}^T\int _M u^{\frac{2n}{n-2}} \varphi ^{\alpha } dV_g dt\right] ^{\frac{n-2}{n}}\\&\qquad \left[ \int _{T-\frac{r^2}{4}}^T \int _M[\big (\frac{\alpha }{2}\varphi \varphi _t+\frac{(n-1)}{n-2} \alpha ^2\,|\nabla \varphi |^2-R_g\varphi ^2\big )\varphi ^{\alpha -2- \frac{n-2}{n}\alpha }]^{\frac{n}{2}}dV_gdt\right] ^{\frac{2}{n}}\\&\quad =\left[ \int _{T-\frac{r^2}{4}}^T\int _M u^{\frac{2n}{n-2}} \varphi ^{\alpha } dV_g dt\right] ^{\frac{n-2}{n}}\\&\qquad \left[ \int _{T-\frac{r^2}{4}}^T \int _M\left[ \big (\frac{\alpha }{2}\varphi \varphi _t+\frac{(n-1)}{n-2} \alpha ^2\,|\nabla \varphi |^2-R_g\varphi ^2\big )\varphi ^{-2+\frac{2}{n} \alpha }\right] ^{\frac{n}{2}}dV_gdt\right] ^{\frac{2}{n}}. \end{aligned}$$

Taking \(\alpha =n\), we have

$$\begin{aligned}&\big (\int _{T-\frac{r^2}{8}}^T \int _{B_r(p)}u^{\frac{2n}{n-2}}dV_gdt \big )^{\frac{2}{n}}\\ {}&\quad \le \left( \int _{T-\frac{r^2}{4}}^T \int _Mu^{\frac{2n}{n-2}} \varphi ^n dV_gdt\right) ^{\frac{2}{n}}\\&\quad \le \frac{1}{n(n-1)}\left[ \int _{T-\frac{r^2}{4}}^T\int _M\big (\frac{n}{2} \varphi \varphi _t+\frac{(n-1)}{n-2}n^2\,|\nabla \varphi |^2-R_g \varphi ^2\big )^{\frac{n}{2}}dV_gdt\right] ^{\frac{2}{n}}, \end{aligned}$$

where the right hand side is bounded from above independent of u, p, T and r. Recall that

$$\begin{aligned} u_t\le \frac{4(n-1)}{(n-2)}\Delta _gu-R_gu \end{aligned}$$

on \(M\times [0,T_1)\). Hence on \(B_r(p)\times [T-\frac{r^2}{4},T]\), combining with the weak Harnack type inequality on \(B_{2r}(q)\times [T-\frac{r^2}{4},T]\) with \(\theta =\frac{1}{2}\) and by the choice of \(\varphi \), we have

$$\begin{aligned} \displaystyle \sup _{q\in B_r(p),T-\frac{r^2}{8}\le t\le T}u(q,t)\le C_3r^{-\frac{n-2}{2}}, \end{aligned}$$

for \(T_1>T\ge \frac{\epsilon _0}{2}\), with a constant \(C_3=C_3(\epsilon _0)>0\) independent of \(p\in M^{\circ }\), u, \(T_1\) and r. In particular, for any compact subset in \(M^{\circ }\) and \(t\ge \frac{\epsilon _0}{2}\), u has a uniform upper bound independent of t. \(\square \)

For the Cauchy–Dirichlet problem (1.3)–(1.5) with \(\phi \rightarrow \infty \) uniformly on \(\partial M\), we now estimate the asymptotic behavior of the solution u near the boundary as \(t\rightarrow \infty \).

Lemma 2.3

Assume that the positive function \(u\in C^{2,1}(M\times [0,\infty ))\) is a solution to the Cauchy–Dirichlet problem (1.3)–(1.5) with \(\liminf _{t\rightarrow \infty }\inf _{\partial M}\phi =\infty \). Moreover, assume \(\phi \) satisfies that

$$\begin{aligned} \begin{aligned}&\phi ^{\frac{n}{2-n}}\frac{d\phi }{dt}\rightarrow 0,\,\,\,\phi ^{\frac{n-1}{2-n}}|\nabla _{g}\phi |\rightarrow 0,\\&\phi ^{\frac{n}{2-n}}|\nabla _{g}^2\phi |\rightarrow 0,\,\,\text {uniformly on}\,\,\,\partial M, \,\,\text {as}\,\,t\rightarrow +\infty . \end{aligned} \end{aligned}$$
(2.4)

Then there exist constants \(C>0\), \(t_1>0\) large and \(x_1>0\) small such that

$$\begin{aligned} u\ge \frac{1}{C}\big (\,x+\,(\inf _{p\in \partial M}\phi (p,t)\,)^{\frac{2}{2-n}}\,\big )^{\frac{2-n}{2}}-C, \end{aligned}$$

for \(x\le x_1\) and \(t\ge t_1\), where \(x=x(p)\) is the distance function to the boundary.

Proof

Recall that there exists a conformal metric h with \(g=f^{\frac{4}{n-2}}h\) so that \(R_h>0\) in a neighborhood V of the boundary \(\partial M\), where \(f\in C^{2,\alpha }(M)\) is a positive function, see in Lemma 3.1 in [1]. Indeed, for that one can use the formula of the scalar curvature under conformal change and do Taylor expansion of f in the direction of x in a small neighborhood of the boundary and just take f to be a quadratic polynomial of x with coefficients functions on \(\partial M\). Let x be the distance function to the boundary \(\partial M\) under h. Take \(x_1>0\) small so that the boundary neighborhood \(U=\{0\le x\le x_1\}\) lies in V and hence \(R_h>0\), and U is diffeomorphic to \(\partial M\times [0,x_1]\) under the exponential map \(F:\,\partial M\times [0,x_1]\rightarrow U\), where \(F(q, x)=\text {Exp}_q^h(x)\in U\) is on the geodesic starting from \(q\in \partial M\) in (V, h) in the inner normal direction of \(\partial M\) of distance x to q. We define a function \(\varphi \in C_{loc}^{2,\alpha }(U\times [0,+\infty ))\) such that

$$\begin{aligned} \varphi (\text {Exp}^h_q(x),t)=c\left[ (x+(f(q)\phi (q,t) )^{\frac{2}{2-n}})^{\frac{2-n}{2} }-(x_1+(f(q)\phi (q,t))^{\frac{2}{2-n}})^{\frac{2-n}{2}}\right] \end{aligned}$$
(2.5)

for \((q,x,t)\in \partial M\times [0,x_1]\times [0,+\infty )\), where \(c>0\) is a small constant to be determined. Let \({\tilde{u}}=fu\). Then we have

$$\begin{aligned}&{\tilde{u}}_t=f^{-\frac{4}{n-2}}\left[ \frac{4(n-1)}{n-2}\Delta _h {\tilde{u}} -R_h{\tilde{u}}-n(n-1){\tilde{u}}^{\frac{n+2}{n-2}}\right] , \,\,\,\text {in}\,\,M\times [0,+\infty ),\\&{\tilde{u}}(p,0)=f(p)u_0(p),\,\,p\in M,\\&{\tilde{u}}(q,t)=f(q)\phi (q,t),\,\,q\in \,\partial M. \end{aligned}$$

Note that h has the orthogonal decomposition \(h=dx^2+h_x\) on U with \(h_0=h\big |_{\partial M}\). Then by direct calculation, one has

$$\begin{aligned} \Delta _h\varphi&=\partial _x^2\varphi +\frac{1}{2}H_{x}\partial _x\varphi +\Delta _{h_x}\varphi ,\nonumber \\ \varphi _t&=cf^{\frac{2}{n-2}}[(x+(f(q)\phi (q,t))^{\frac{2}{2-n}} )^{\frac{-n}{2}}-(x_1+(f(q)\phi (q,t))^{\frac{2}{2-n}})^{\frac{-n}{2}} ]\phi ^{\frac{n}{2-n}}\frac{d\phi }{dt}\nonumber \\ \partial _x\varphi&=\frac{2-n}{2}c(x+(f(q)\phi (q,t))^{\frac{2}{2-n}} )^{\frac{-n}{2}},\,\,\,\,\partial _x^2\varphi =\frac{n(n-2)}{4}c (x+(f(q)\phi (q,t))^{\frac{2}{2-n}})^{\frac{-n-2}{2}},\nonumber \\ |\nabla _{h_x}\varphi |&\le c(x+(f(q)\phi (q,t))^{\frac{2}{2-n}} )^{\frac{-n}{2}}(f\phi )^{\frac{n}{2-n}}|\nabla _{h_x}(f\phi )|,\nonumber \\ |\nabla _{h_x}^2\varphi |&\le c(x+(f(q)\phi (q,t))^{\frac{2}{2-n}} )^{\frac{-n}{2}}\,\,\big [(f\phi )^{\frac{n}{2-n}}|\nabla _{h_x }^2(f\phi )|+\frac{n}{n-2}(f\phi )^{\frac{2n-2}{2-n}}|\nabla _{h_x}(f\phi )|^2\nonumber \\&\quad +\frac{n}{n-2}(x+(f(q)\phi (q,t))^{\frac{2}{2-n}})^{-1} (f\phi )^{\frac{2n}{2-n}}|\nabla _{h_x}(f\phi )|^2\big ], \end{aligned}$$
(2.6)

where \(H_{s}\) is the mean curvature of the hypersurface \(\Sigma _s=\{x=s\}\). Recall that \(\phi \rightarrow \infty \) uniformly on \(\partial M\) as \(t\rightarrow \infty \). By the assumption (2.4), we have that there exists \(t_1\ge 0\) such that for \(t\ge t_1\),

$$\begin{aligned} \varphi _t\le f^{-\frac{4}{n-2}}\left[ \frac{4(n-1)}{(n-2)}\Delta _h \varphi -R_h\varphi -n(n-1)\varphi ^{\frac{n+2}{n-2}}\right] \end{aligned}$$

on U, since all other terms are lower order terms as \(t\rightarrow \infty \) in comparison with the terms

$$\begin{aligned} \begin{aligned}&\frac{4(n-1)}{(n-2)}\Delta _h\varphi -n(n-1)\varphi ^{\frac{n+2}{n-2}}\\&\quad =n(n-1)c(x+(f(q)\phi (q,t))^{\frac{2}{2-n}})^{\frac{-n-2}{2}}(1+o(1))\\&\qquad -n(n-1)c^{\frac{n+2}{n-2}}[(x+(f(q)\phi (q,t))^{\frac{2}{2-n}} )^{\frac{2-n}{2}}\!-\!(x_1+(f(q)\phi (q,t))^{\frac{2}{2-n}} )^{\frac{2-n}{2}}]^{\frac{n+2}{n-2}}, \end{aligned} \end{aligned}$$
(2.7)

with \(c>0\) small and the term \(o(1)\rightarrow 0\) uniformly on U as \(t\rightarrow +\infty \). Let \(v={\tilde{u}}-\varphi \). Therefore,

$$\begin{aligned} v_t\ge f^{\frac{-4}{n-2}}\left[ \frac{4(n-1)}{(n-2)}\Delta _h v-\,\big (R_h+n(n-1)\xi (p,t)\,\big )\,v\,\right] \end{aligned}$$

on \(U\times [t_1,+\infty )\), where \(\xi (p,t)=\frac{u^{\frac{n+2}{n-2}}-\varphi ^{\frac{n+2}{n-2}}}{u-\varphi }\) when \(u\ne \varphi \), and \(\xi =\frac{n+2}{n-2}u^{\frac{4}{n-2}}\) otherwise, and hence, \(\xi (p,t)>0\) on \(U\times [t_1,+\infty )\). Since \({\tilde{u}}\) is continuous on \(U\times \{t=t_1\}\), we choose \(c>0\) small so that \({\tilde{u}}>\varphi \) on \(U\times \{t=t_1\}\). On the other hand, \(\varphi =0<{\tilde{u}}\) on \(\{x=x_1\}\times [t_1,\infty )\) and by definition \(\varphi \le {\tilde{u}}\) on \(\partial M\) and hence, \(v\ge 0\) on \(\partial U\times [t_1,+\infty )\bigcup U\times \{t_1\}\). Recall that \(R_h>0\) on U, and hence by maximum principle,

$$\begin{aligned} v\ge 0 \end{aligned}$$

on \(U\times [t_1,+\infty )\). Therefore,

$$\begin{aligned} u\ge f^{-1}\varphi \end{aligned}$$

on \(U\times [t_1,+\infty )\). Combining this lower bound estimate of u near the boundary with the Harnack inequality (2.9) based on the uniform interior upper bound estimates in Lemma 2.2, and also using finite cover of geodesic balls, on any compact sub-domain of \(M^{\circ }\) we obtain uniform positive lower bound of u for \(t\in [t_1,+\infty )\). \(\square \)

Now we give the long time existence of the flow.

Lemma 2.4

There exists a unique solution \(u \in C_{loc}^{2+\alpha ,1+\frac{\alpha }{2}}(M\times [0,\infty ))\) to the Cauchy–Dirichlet boundary (1.3)–(1.5) with positive functions \(u_0\in C^{2,\alpha }(M)\) and \(\phi \in C_{loc}^{2+\alpha ,1+\frac{\alpha }{2}}(\partial M\times [0,\infty ))\) satisfying (2.1). Moreover, if in addition, \(u_0\in C^{4,\alpha }(M)\) and \(\phi \in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(\partial M\times [0,\infty ))\) satisfy (2.2), then \(u\in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,\infty ))\). Moreover, if \(\liminf _{t\rightarrow \infty }\inf _{\partial M}\phi =\infty \) and \(\phi \) satisfies (2.4) as \(t\rightarrow \infty \), then there exists constants \(C>0\), \(t_2>0\) large and \(x_1>0\) small such that

$$\begin{aligned} u\ge \frac{1}{C}\big (\,x+\,(\inf _{p\in \partial M}\phi (p,t) \,)^{\frac{2}{2-n}}\,\big )^{\frac{2-n}{2}}-C, \end{aligned}$$
(2.8)

for \(x\le x_1\) and \(t\ge t_2\), where \(x=x(p)\) is the distance function to the boundary, and hence, for each compact subset \(F\subseteq M^{\circ }\), there exists \(C=C(F)>0\) such that

$$\begin{aligned} u(q,t)\ge C \end{aligned}$$

for each \((q,t)\in F\times [0,\infty )\). Moreover, for each compact subset \(F\subseteq M^{\circ }\), there exists \(m=m(F)>0\) independent of \(u_0\) and \(\phi \) such that

$$\begin{aligned} \liminf _{t\rightarrow \infty }\inf _{q\in F}u(q,t)\ge m, \end{aligned}$$

provided that \(\liminf _{t\rightarrow \infty }\inf _{\partial M}\phi =\infty \).

Proof

Since (1.3) is parabolic, by the compatible condition and regularity of the positive functions \(u_0\) and \(\phi \), standard implicit function theorem yields the existence of a unique positive solution u to (1.3)–(1.5) on \(M\times [0,T_0)\) for some \(T_0>0\) with u satisfying \(u\in C^{2+\alpha ,1+\frac{\alpha }{2}}(M\times [0,T])\) for each \(0<T<T_0\) when \(u_0\in C^{2,\alpha }(M)\) and \(\phi \in C_{loc}^{2+\alpha ,1+\frac{\alpha }{2}}(M\times [0,\infty ))\) satisfying (2.1); moreover, \(u \in C^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,T])\) for each \(0<T<T_0\) when \(u_0\in C^{4,\alpha }(M)\) and \(\phi \in C^{4+\alpha ,2+\frac{\alpha }{2}}\) satisfy (2.2) in addition. We assume \(T_0\) is the maximum time for the existence of the positive solution u on \([0,T_0)\). By maximum principle, we have the upper bound

$$\begin{aligned} \sup _{q\in M,\,t\in [0,T)}u \le \max \left\{ \sup _{q\in \partial M,\,t\in [0,T)}\phi ,\,\sup _{q\in M}u_0(q),\,\big (\frac{1}{n(n-1)}\max \{-\inf _{q\in M}R_g(q),0\}\big )^{\frac{n-2}{4}}\right\} , \end{aligned}$$

for any \(0<T<T_0\), which is uniformly bounded on \(M\times [0,T_0)\). Then standard Schauder theory derives that there exists \(C=C(T_0)>0\) such that

$$\begin{aligned} \Vert u\Vert _{C^{2+\alpha ,1+\frac{\alpha }{2}}(M\times [0,T])}\le C \end{aligned}$$

in the \(C^{2+\alpha ,1+\frac{\alpha }{2}}\) setting Cauchy–Dirichlet problem, for each \(0<T<T_0\); while

$$\begin{aligned} \Vert u\Vert _{C^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,T])}\le C \end{aligned}$$

in the \(C^{4+\alpha ,2+\frac{\alpha }{2}}\) setting Cauchy–Dirichlet problem, for each \(0<T<T_0\). To establish the lower bound estimates, for any \(q\in M^{\circ }\), assume \(0<r<\frac{\text {dist}(q,\partial M)}{2}\) and \(r<1\). Then by the Harnack inequality, there exists a constant \(C>0\) depending on \(B_{2r}(q)\) but independent of T such that

$$\begin{aligned} \sup _{ B_r(q)\times [T-\frac{3r^2}{4}, T-\frac{r^2}{2}]}u\le C \inf _{B_r(q) \times [T-\frac{r^2}{4}, T]}u \end{aligned}$$
(2.9)

for each \(r^2<T<T_0\). By Lemma 2.2, \(|R_g+n(n-1)u^{\frac{4}{n-2}}|\) is uniformly bounded in \(B_{2r}(q)\) (independent of \(T_0\)) and hence, the constant C in (2.9) is independent of \(T_0\). Recall that \(\phi >0\) on \(\partial M \times [0,\infty )\). Therefore, u can be extended to a positive solution of \(C_{loc}^{2+\alpha ,1+\frac{\alpha }{2}}(M\times [0,\infty ))\) (resp. of \(C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,\infty ))\)).

Assume that \(\liminf _{t\rightarrow \infty }\inf _{\partial M}\phi =\infty \) and \(\phi \) satisfies (2.4) as \(t\rightarrow \infty \). Then by Lemma 2.3 we have that there exists \(x_1>0\), \(C>0\) and \(t_2>0\) such that (2.8) holds in \(\{0\le x\le x_1\}\) for \(t\ge t_2\). Using this uniform lower bound estimate near the boundary, combining with the above Harnack inequality and a finite geodesic ball cover of the path connecting to \(\{0\le x\le x_1\}\), we have that for each compact subset \(F\subseteq M^{\circ }\), there exists \(C=C(F)>0\) such that

$$\begin{aligned} u(q,t)\ge C \end{aligned}$$

for \((q,t)\in F\times [0,\infty )\), and hence by Lemma 2.2, u is uniformly bounded from above and below by positive constants in \(F\times [0,\infty )\).

On any compact subset \(F\subseteq M^{\circ }\), since the upper bound of u obtained in Lemma 2.2 is independent of \(u_0\) and \(\phi \), and so is \(|R_g+n(n-1)u^{\frac{4}{n-2}}|\) and hence the constant C in the Harnack inequality is independent of \(u_0\) and \(\phi \). Therefore, by Lemma 2.3 there exists \(m>0\) depending on F but independent of \(u_0\) and \(\phi \) such that

$$\begin{aligned} \liminf _{t\rightarrow \infty }\inf _{q\in F}u(q,t)\ge m. \end{aligned}$$

\(\square \)

Now we consider the convergence of the flow. To warm up, we first consider the case \(R_{g}\ge 0\).

Proposition 2.5

Let \((M^n,g)\) be a compact manifold with \(g\in C^{4,\alpha }\) up to the boundary such that \(R_{g}\ge 0\). Assume \(u_0\in C^{4,\alpha }(M)\) and \(\phi \in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,\infty ))\) satisfy (2.1) and (2.2), and also \(\phi _t\ge 0\) on \(M\times [0,\infty )\), \(\liminf _{t\rightarrow \infty }\inf _{\partial M}\phi =\infty \) and \(\phi \) satisfies (2.4) as \(t\rightarrow \infty \). Moreover, assume \(u_0\) is a positive subsolution to (1.1) and

$$\begin{aligned} L(v)(p)\ge 0 \end{aligned}$$
(2.10)

for any \(p\in \partial M\) such that \(v(p)=0\), where L and v are defined in (2.2). Then we have that the Cauchy–Dirichlet problem (1.3)–(1.5) has a unique global solution u, which converges in \(C_{loc}^4(M^{\circ })\) to the unique solution \(u_{\infty }\) to the Loewner–Nirenberg problem.

Remark that for a sub-solution \(u_0\) of (1.1), which is a strict sub-solution in a neighborhood of \(\partial M\), the condition (2.10) disappears automatically. For instance, if \(\varphi >0\) is a subsolution of (1.1), then \(\epsilon \varphi \) is a strict sub-solution for any constant \(0<\epsilon <1\). A solution \(\varphi >0\) to (1.1) automatically satisfies (2.10). Notice that the functions \(\log (t),\,\,t\) and \(e^t\) all satisfy (2.4). Also, if \(u_0\) is a solution to (1.1) in a neighborhood of \(\partial M\), (2.10) holds automatically. To prove this proposition, we need two lemmas.

Lemma 2.6

Let (M, g) be a smooth compact manifold with boundary such that \(R_g\ge 0\). Assume \(\phi \in C^{4}(\partial M \times [0,+\infty ))\) is a positive function with \(\phi _t\ge 0\), and \(u_0\in C^4(M)\) is a positive function which is a subsolution to (1.1) on M. Assume that a positive function \(u\in C^{4,2}(M\times [0,\infty ))\) is a solution to the Cauchy–Dirichlet problem (1.3)–(1.5). Then u satisfies that \(u_t\ge 0\) in \(M^{\circ }\times (0,+\infty )\). That’s to say, for any \(t>0\), \(u(\cdot ,t)\) is a subsolution to the Yamabe equation (1.1).

Proof

The proof is an application of the maximum principle to the equation satisfied by \(u_t\). Indeed, denote \(v=u_t\). It is clear that \(v\in C^{2,1}(M\times [0,+\infty ))\). We take derivative of t on both sides of (1.3) and (1.5) to obtain the Cauchy–Dirichlet problem

$$\begin{aligned}&v_t=\frac{4(n-1)}{n-2}\Delta v-R_gv-\frac{n(n-1)(n+2)}{n-2} u^{\frac{4}{n-2}}v,\,\,\,\text {in}\,\,M\times [0,+\infty ),\\&v(p,0)=u_t(p,0)=\frac{4(n-1)}{n-2}\Delta u_0(p)\!-\!R_gu_0(p)-n(n\!-\!1) u_0(p)^{\frac{n+2}{n-2}}\ge 0,\,\,p\!\in \! M,\nonumber \\&v(q,t)=\phi _t(q,t)\ge 0,\,\,q\in \,\partial M.\nonumber \end{aligned}$$
(2.11)

Therefore, by maximum principle, \(v=u_t\ge 0\) on \(M\times [0,+\infty )\). Moreover, if there exists a constant \(t_1>0\) such that \(\phi _t>0\) for \(t>t_1\), then by strong maximum principle, we have \(v=u_t>0\) in \(M^{\circ }\times (t_1,+\infty )\). \(\square \)

Proposition 2.5 is a direct consequence of the following lemma.

Lemma 2.7

Let (M, g), \(u_0\) and \(\phi \) be as in Proposition 2.5. Then the unique solution u to the Cauchy–Dirichlet problem (1.3)–(1.5) converges in \(C_{loc}^{4}(M^{\circ })\) to the solution \(u_{\infty }\) of the Loewner–Nirenberg problem (1.1)–(1.2) as \(t\rightarrow +\infty \). Moreover, there exists a constant \(C>0\) such that \(\frac{1}{C}x^{\frac{2-n}{2}}\le u_{\infty }\le Cx^{\frac{2-n}{2}}\) near the boundary \(\partial M\), where x is the distance function to the boundary.

Proof

By Lemma 2.4, the exists a unique positive solution \(u\in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,\infty ))\) to (1.3)–(1.5). By Lemma 2.6, \(u_t\ge 0\) on \(M\times [0,\infty )\). Since we have the upper and lower bound estimates on u on any compact sub-domain of \(M^{\circ }\) and u is increasing pointwisely on \(M^{\circ }\) as t increases, we have that u converges pointwisely in \(M^{\circ }\) as \(t\rightarrow +\infty \). By the Harnack inequality for \(v=u_t\) in the parabolic equation (2.11), we have that \(u_t\rightarrow 0\) locally uniformly and hence u converges locally uniformly in \(M^{\circ }\) to a positive function \(u_{\infty }\). Using classical parabolic estimates, we have that \(u(\cdot , t)\rightarrow u_{\infty }\) in \(C_{loc}^4(M)\) as \(t\rightarrow +\infty \). Therefore, \(u_{\infty }\) is a solution to (1.1) in \(M^{\circ }\). By the upper bound and lower bound estimates of u near the boundary \(\partial M\) in Lemma 2.2 and Lemma 2.3, we obtain the estimate of \(u_{\infty }\) near the boundary as stated in the lemma, and hence \(u_{\infty }\) is the unique solution to the Loewner–Nirenberg problem. This completes the proof of the lemma and Proposition 2.5. \(\square \)

Proof of Theorem 1.1

By Lemma 2.4, there exists a unique positive solution \(u\in C_{loc}^{2+\alpha ,1+\frac{\alpha }{2}}(M\times [0,\infty ))\) to (1.3)–(1.5).

By Lemma A.1, for any \(\epsilon >0\), there exists a unique positive solution \(w\in C^{4,\alpha }(M)\) to the Dirichlet boundary value problem

$$\begin{aligned}&\frac{4(n-1)}{n-2}\Delta w-R_gw-n(n-1)w^{\frac{n+2}{n-2}}=0,\\&w\big |_{\partial M}=\epsilon . \end{aligned}$$

By maximum principle, \(w\le \epsilon \).

Let \(\epsilon >0\) be a small constant so that \(\epsilon < \frac{1}{10n} \inf _{M}u_0\). It is easy to choose a boundary data \({\tilde{\phi }}(q,t)\in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(\partial M \times [0,+\infty ))\) such that \({\tilde{\phi }}=\epsilon \) and \({\tilde{\phi }}_t={\tilde{\phi }}_{tt}=0\) on \(\partial M\times \{0\}\), moreover, \({\tilde{\phi }}_t\ge 0\) and \({\tilde{\phi }}\le \phi \) on \(\partial M\times [0,+\infty )\). Also, we require that \({\tilde{\phi }}\rightarrow \infty \) and \({\tilde{\phi }}\) satisfies (2.4) as \(t\rightarrow \infty \). Therefore, by Proposition 2.5 there exists a unique solution \({\tilde{u}}\) to the problem

$$\begin{aligned}&{\tilde{u}}_t=\frac{4(n-1)}{n-2}\Delta {\tilde{u}} -R_g{\tilde{u}}-n(n-1){\tilde{u}}^{\frac{n+2}{n-2}}, \,\,\,\text {in}\,\,M\times [0,+\infty ),\\&{\tilde{u}}(p,0)=w,\,\,p\in M,\\&{\tilde{u}}(q,t)={\tilde{\phi }}(q,t),\,\,q\in \,\partial M, \end{aligned}$$

such that \({\tilde{u}}\) is increasing in t, and converges to the Loewner–Nirenberg solution \(u_{\infty }\). Let \(v=u-{\tilde{u}}\). Then v satisfies

$$\begin{aligned}&v_t=\frac{4(n-1)}{n-2}\Delta v-(R_g+n(n-1)\zeta )v, \,\,\,\text {in}\,\,M\times [0,+\infty ),\\&v(p,0)=u_0(p)-w(p)>0,\,\,p\in M,\\&v(q,t)=\phi (q,t)-{\tilde{\phi }}(q,t)\ge 0,\,\,q\in \,\partial M, \end{aligned}$$

where \(\zeta =\frac{u^{\frac{n+2}{n-2}}-{\tilde{u}}^{\frac{n+2}{n-2}}}{u-{\tilde{u}}}\) for \(u\ne {\tilde{u}}\), and \(\zeta =\frac{n+2}{n-2}u^{\frac{4}{n-2}}\) otherwise. In particular, \(\zeta >0\). By maximum principle, \(v\ge 0\) on \(M\times [0,+\infty )\).

Now we establish the upper bound of u. Let \(\xi (p,t)=u(p,t)-u_{\infty }(p)\), with \(u_{\infty }\) the Loewner–Nirenberg solution. Therefore, \(\xi (p,t)\rightarrow -\infty \) as \(p\rightarrow \partial M\) for each \(t>0\). Hence, for any \(t>0\), there exists \(p_t\in M^{\circ }\) such that \(\xi (p_t,t)=\sup _{p\in M^{\circ }}\xi (p,t)\), and hence \(\Delta _g\xi (p_t,t)\le 0\). Therefore, we have

$$\begin{aligned} \xi _t&=\frac{4(n-1)}{n-2}\Delta \xi -(R_g+n(n-1)\mu (x,t))\xi \\&\le -(R_g+n(n-1)\mu )\xi ,\nonumber \end{aligned}$$
(2.12)

at the point \((p_t,t)\) for \(t>0\), where \(\mu (x,t)=\frac{u^{\frac{n+2}{n-2}}-u_{\infty }^{\frac{n+2}{n-2}}}{u-u_{\infty }}\) for \(u\ne u_{\infty }\), and \(\mu =\frac{n+2}{n-2}u_{\infty }^{\frac{4}{n-2}}\) otherwise. In particular, \(\mu >0\). By maximum principle for the equation (2.12) satisfied by \(\xi \), if \(\xi (p_{t_1},t_1)\le 0\) for some \(t_1>0\), then \(\xi (p,t)\le 0\) for all \(t>t_1\). Therefore, there are two possibilities: one is that there exists \(t_2>0\) so that for \(t>t_2\) we have \(\xi \le 0\) on \(M^{\circ }\); the second case is that \(\xi (p_t,t)>0\) for \(t\in [0,+\infty )\). For the second case, let \(\eta (t)=\sup _{p\in M^{\circ }}\xi (p,t)\). By the inequality in (2.12), \(\eta (t)\) is decreasing since \(\eta (t)>0\). Now we use the discussion as in [24] to show that \(\displaystyle \limsup _{t\rightarrow +\infty }\eta (t)\le 0\). Denote \((\frac{d\eta }{dt})_+(t)=\displaystyle \limsup _{\tau \searrow 0}\frac{\eta (t+\tau )-\eta (t)}{\tau }\) and define the set

$$\begin{aligned} S(t)=\{p\in M^{\circ }\big |\, p\,\,\text {is a maximum point of }\,\xi \,\,\text {on}\,\,M^{\circ }\times \{t\}\}. \end{aligned}$$

Then S(t) is compact for any \(t\ge 0\). By Lemma 3.5 in [24],

$$\begin{aligned} \left( \frac{d\eta }{dt}\right) _+(t)\le \sup \{\xi _t(p_t,t)\big |\,p_t\in \, S(t)\}, \end{aligned}$$

since \(\eta (t)>0\). By the mean value theorem we have that \(\frac{u^{\frac{n+2}{n-2}}-u_{\infty }^{\frac{n+2}{n-2}}}{u-u_{\infty }}\ge \frac{n+2}{n-2}(\displaystyle \inf _{q\in M}u_{\infty })^{\frac{4}{n-2}}\). Therefore, for the second case,

$$\begin{aligned} \left( \frac{d\eta }{dt}\right) _+(t)\le -\left( R_g+\frac{n(n-1)(n+2)}{n-2}\left( \displaystyle \inf _{q\in M}u_{\infty }\right) ^{\frac{4}{n-2}}\right) \eta (t), \end{aligned}$$

for \(t\ge 0\). By integration or a contradiction argument we can easily obtain that \(\displaystyle \limsup _{t\rightarrow +\infty }\eta (t)\!\le \! 0\). In summary, for both cases we obtain the upper bound control that

$$\begin{aligned} \displaystyle \limsup _{t\rightarrow +\infty }\sup _{q\in M^{\circ }}(u(q,t)-u_{\infty }(q))\le 0. \end{aligned}$$

Based on the upper bound and lower bound control on u, we have that \(\displaystyle \lim _{t\rightarrow +\infty }u(q,t)\rightarrow u_{\infty }(q)\) locally uniformly on \(M^{\circ }\) as \(t\rightarrow +\infty \). By the standard interior parabolic estimates, we obtain that the convergence is in \(C_{loc}^2(M^{\circ })\) sense. \(\square \)

Let (M, g) be a general compact Riemannian manifold of \(C^{4,\alpha }\) with boundary. By the discussion in Appendix A, there exists a conformal metric \(h\in [g]\) of \(C^{4,\alpha }\) such that \(R_h=-n(n-1)\). We still denote it as g and choose it as the background metric of our flow, and hence the Cauchy–Dirichlet problem (1.3)–(1.5) becomes

$$\begin{aligned} \begin{aligned}&u_t=\frac{4(n-1)}{n-2}\Delta _{g} u+n(n-1)\big (u-u^{\frac{n+2}{n-2}}\big ),\,\,\,\text {in}\,\,M\times [0,+\infty ),\\&u(p,0)=u_0(p),\,\,p\in M,\\&u(q,t)=\phi (q,t),\,\,q\in \,\partial M, \end{aligned} \end{aligned}$$
(2.13)

with \(\phi > 0\) and \(u_0>0\) compatible, and \(\phi \rightarrow +\infty \) uniformly as \(t\rightarrow +\infty \). Recall that we have a unique positive solution \(u\in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,+\infty ))\) to the Cauchy–Dirichlet boundary value problem with uniform upper and lower bound estimates on any compact sub-domain in \(M^{\circ }\) for \(t\in [0,+\infty )\), and the upper bound and lower bound asymptotic behavior estimates near the boundary as \(t\rightarrow +\infty \). We now consider the convergence. To insure the asymptotic behavior near the boundary as \(t\rightarrow +\infty \), we always assume the boundary data \(\phi \) satisfies (2.4) for t large.

We start with the special initial data \(u_0=1\).

Lemma 2.8

Assume (M, g) is a compact Riemannian manifold with boundary of \(C^{4,\alpha }\) and \(R_g=-n(n-1)\). Let \(u>0\) be the solution to the Cauchy–Dirichlet problem

$$\begin{aligned} \begin{aligned}&u_t=\frac{4(n-1)}{n-2}\Delta _{g} u+n(n-1)\big (u-u^{\frac{n+2}{n-2}}\big ),\,\,\,\text {in}\,\,M\times [0,+\infty ),\\&u(p,0)=1,\,\,p\in M,\\&u(q,t)=\phi (q,t),\,\,q\in \,\partial M, \end{aligned} \end{aligned}$$
(2.14)

with a positive function \(\phi \in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(\partial M\times [0,+\infty ))\) such that \(\phi (q,0)=1\), \(\phi _t(q,0)=\phi _{tt}(q,0)=0\) for \(q\in \partial M\), \(\phi _t\ge 0\) on \(\partial M\times [0,+\infty )\) and \(\liminf _{t\rightarrow \infty }\inf _{\partial M}\phi =\infty \). Assume \(\phi \) satisfies (2.4). Then \(u\rightarrow u_{\infty }\) in \(C_{loc}^{4}(M^{\circ })\) as \(t\rightarrow +\infty \), where \(u_{\infty }\) is the solution to the Loewner–Nirenberg problem (1.1)–(1.2).

Proof

By Lemma 2.4, we have that the exists a positive solution \(u\in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,\infty ))\) to the Cauchy–Dirichlet problem (2.14). By maximum principle, we have that \(u\ge 1\) in \(M\times [0,+\infty )\) and \(u>1\) for t large. As in Lemma 2.6, the solution u to (2.14) is increasing. Indeed, let \(v=u_t\), and then v satisfies

$$\begin{aligned}&v_t=\frac{4(n-1)}{n-2}\Delta _{g} v+n(n-1)\big (1-\frac{n+2}{n-2}u^{\frac{4}{n-2}}\big )v,\,\,\,\text {in}\,\,M\times [0,+\infty ),\\&v=0,\,\,p\in M,\\&v(q,t)=\phi _t(q,t)\ge 0,\,\,q\in \,\partial M. \end{aligned}$$

By maximum principle, \(v\ge 0\) for \(t\ge 0\) and \(v>0\) for t large. Therefore, u is increasing in t. By the uniform upper bound estimates on any compact sub-domain of \(M^{\circ }\), we have that u converges pointwisely to \(u_{\infty }>0\) locally in \(M^{\circ }\). Using the Harnack inequality on v we have that u converges to \(u_{\infty }\) locally uniformly in \(M^{\circ }\) as \(t\rightarrow +\infty \). By the standard parabolic estimates, the convergence is in \(C_{loc}^{4}(M^{\circ })\). By Lemma 2.2 and Lemma 2.3, there exists \(C>0\) such that

$$\begin{aligned} \frac{1}{C}x^{-\frac{n-2}{2}}\le u_{\infty }\le C x^{-\frac{n-2}{2}}, \end{aligned}$$

in a neighborhood of \(\partial M\), where x is the distance function to \(\partial M\). \(\square \)

Lemma 2.9

Assume (M, g) is a compact Riemannian manifold with boundary of \(C^{4,\alpha }\) and \(R_g=-n(n-1)\). For any \(u_0\in C^{2,\alpha }(M)\) such that \(u_0>1\) and a positive function \(\phi \in C_{loc}^{2+\alpha ,1+\frac{\alpha }{2}}(M,[0,+\infty ))\) satisfying the compatible condition (2.1) with \(u_0\), \(\phi _t\ge 0\) for \(t\ge 0\) and satisfying (2.4) as \(t\rightarrow \infty \), there exists a unique positive solution \(u\in C_{loc}^{2+\alpha ,1+\frac{\alpha }{2}}(M\times [0,+\infty ))\) to (2.13), and u converges to \(u_{\infty }\) in \(C_{loc}^2(M^{\circ })\) as \(t\rightarrow +\infty \), where \(u_{\infty }\) is the solution to the Loewner–Nirenberg problem (1.1)–(1.2).

Proof

Remark that for any \(u_0\in C^{2,\alpha }(M)\) such that \(u_0>1\), there always exists such a function \(\phi \) in the lemma. By Lemma 2.4, we have that the exists a positive solution \(u\in C_{loc}^{2+\alpha ,1+\frac{\alpha }{2}}(M\times [0,\infty ))\) to the Cauchy–Dirichlet problem (2.14). By maximum principle, we have that \(u\ge 1\) on \(M\times [0,\infty )\).

Now we pick up a function \({\tilde{\phi }}\in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(\partial M\times [0,\infty ))\) such that \({\tilde{\phi }}(q,0)=1\), \({\tilde{\phi }}_t(q,0)={\tilde{\phi }}_{tt}(q,0)=0\) on \(\partial M\), moreover, \({\tilde{\phi }}_t\ge 0\) and \({\tilde{\phi }}\le \phi \) on \(\partial M\times [0,\infty )\), and \({\tilde{\phi }}\) satisfies (2.4) as \(t\rightarrow \infty \). By Lemma 2.8, for the initial data \({\tilde{u}}_0=1\) and the boundary data \({\tilde{\phi }}\), there exists a unique positive solution \({\tilde{u}}\in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,\infty ))\) to the Cauchy–Dirichlet problem (2.14), and \({\tilde{u}}\) converges to \(u_{\infty }\) in \(C_{loc}^{4}(M^{\circ })\) as \(t\rightarrow \infty \), where \(u_{\infty }\) is the solution of the Loewner–Nirenberg problem (1.1)–(1.2). By maximum principle, we have that \({\tilde{u}}\ge 1\) in \(M\times [0,+\infty )\).

Now let \(\xi =u-{\tilde{u}}\). Then \(\xi \) satisfies

$$\begin{aligned}&\xi _t=\frac{4(n-1)}{n-2}\Delta _{g} \xi +n(n-1)\zeta \xi ,\,\,\,\text {in}\,\,M\times [0,+\infty ),\\&\xi (p,0)=u_0(p)-1>0,\,\,p\in M,\\&\xi (q,t)=\phi (q,t)-{\tilde{\phi }}(q,t)\ge 0,\,\,q\in \,\partial M, \end{aligned}$$

where \(\zeta =\frac{\big (({\tilde{u}}-{\tilde{u}}^{\frac{n+2}{n-2}})-(u-u^{\frac{n+2}{n-2}})\big )}{{\tilde{u}}-u}\) when \({\tilde{u}}\ne u\), and \(\zeta =1-\frac{n+2}{n-2}u^{\frac{4}{n-2}}\) otherwise, and hence \(\zeta <0\). By maximum principle, \( u\ge {\tilde{u}}\) in \(M\times [0,+\infty )\).

The upper bound estimates is the same as the case \(R_g\ge 0\). Notice that \(u_{\infty }>1\) by maximum principle of the equation (A.4). Let \(\eta =u-u_{\infty }\). Then \(\eta (q,t)\rightarrow -\infty \) as \(q\rightarrow \partial M\) for any \(t>0\). Notice that \(\eta \) satisfies the equation

$$\begin{aligned} \eta _t=\frac{4(n-1)}{n-2}\Delta _{g} \eta +n(n-1)\beta \eta ,\,\,\,\text {in}\,\,M\times [0,+\infty ), \end{aligned}$$

where \(\beta =\frac{(u-u^{\frac{n+2}{n-2}})-(u_{\infty }-u_{\infty }^{\frac{n+2}{n-2}})}{u-u_{\infty }}\) when \(u\ne u_{\infty }\) and \(\beta =1-\frac{n+2}{n-2}u^{\frac{4}{n-2}}\) otherwise, and hence, \(\beta <0\). Therefore, by the same argument as the case \(R_g\ge 0\), we have that \(\displaystyle \limsup _{t\rightarrow +\infty }\sup _{q\in M^{\circ }}\eta (q,t)\le 0\). Therefore, \(u(\cdot ,t)\rightarrow u_{\infty }(\cdot )\) locally uniformly on \(M^{\circ }\) as \(t\rightarrow \infty \). By standard interior estimates for the parabolic equation, we have that \({\tilde{u}}\rightarrow u_{\infty }\) in \(C_{loc}^{2}(M^{\circ })\) as \(t\rightarrow +\infty \). \(\square \)

Proof of Theorem 1.2

By the argument in Appendix A, there exists a positive solution \({\bar{u}}\in C^{2,\alpha }(M)\) to (1.1) such that \(v_0<{\bar{u}}<u_0\) on M. Let \({\bar{g}}={\bar{u}}^{\frac{4}{n-2}}g\) and hence \(R_{{\bar{g}}}=-n(n-1)\). Let \(\eta =\frac{u_0}{{\bar{u}}}\). Then by Lemma 2.9, we have that there exists a positive function \(\phi \in C_{loc}^{2,\alpha }(\partial M\times [0,+\infty ))\) such that to the flow

$$\begin{aligned}&u_t=\frac{4(n-1)}{n-2}\Delta _{{\bar{g}}} u+n(n-1) \big (u-u^{\frac{n+2}{n-2}}\big ),\,\,\,\text {in}\,\,M\times [0,+\infty ),\\&u(p,0)=\eta (p),\,\,p\in M,\\&u(q,t)=\phi (q,t),\,\,q\in \,\partial M, \end{aligned}$$

there exists a unique positive solution u, which converges to the Loewner–Nirenberg solution \(u_{\infty }\) in the conformal class in \(C_{loc}^2(M^{\circ })\). That is to say, there exists a direct flow starting from \(g_0=u_0^{\frac{4}{n-2}}g\) and converging to the Loewner–Nirenberg metric. Notice that if the first Dirichlet eigenvalue satisfies \(\lambda _1(L_g)\le 0\), by Lemma 3.1, it is equivalent to say that \(\displaystyle \inf _{u\in C^1(M)}Q(u)=-\infty \) when g is smooth, where the energy Q(u) is defined in (3.2). \(\square \)

3 Positive Scalar Curvature Metrics on Compact Manifolds with Boundary

In this section, we present examples for conformal classes on compact manifold with boundary that admit positive scalar curvature metrics. Let M be a compact smooth manifold with boundary \(\partial M\). It is well-known that there is not topological obstruction for the existence of positive scalar curvature metrics on M. For any given smooth Riemannian metric g on M, we can extend (M, g) to a smooth closed manifold (N, h). Then by the well known theorems by Kazdan and Warner, Theorem 3.3 in [28] and Theorem 6.2 in [29], we have that there exists a diffeomorphism \(F:\,N\rightarrow N\) and a smooth positive function \(u\in N\), such that the scalar curvature of \(u^{\frac{4}{n-2}}F^*h\) is positive on M where \(F^*h\) is the pull back metric of h by the map F. Notice that \(F^*h\) is not necessarily in the conformal class [g] on M. Another interesting result obtained by Shi-Wang-Wei (Theorem 1.1 in [51]) recently, answering a question by Gromov, states that, any smooth Riemannian metric h on \(\partial M\) can be extended to a smooth Riemannian metric g with positive scalar curvature on M.

Now we consider the existence of positive scalar curvature metrics in a conformal class on a compact manifold with boundary.

Let \((M^n,h)\) be a smooth compact Riemannian manifold with boundary \(\partial M\) of dimension \(n\ge 4\). Let g be a complete metric equipped on the interior \(M^{\circ }\) such that \(x^2g\) extends in \(C^{k,\alpha }\) up to the boundary, where x is the distance function to the boundary under the metric h, then we call \((M^{\circ }, g)\) conformally compact of \(C^{k,\alpha }\). If moreover, g is Einstein, we call \((M^{\circ }, g)\) conformally compact Einstein (CCE). In the well-known paper [42], Qing shows that for a conformally compact Einstein manifold which has conformal infinity of positive Yamabe constant, there exists a conformal compactification \({\bar{g}}\) with positive scalar curvature. Definitely, the CCE metric g is the Loewner–Nirenberg metric in the conformal class of \({\bar{g}}\), and hence the flow here connects these two metrics with the starting metric \({\bar{g}}\) and end at g as \(t\rightarrow +\infty \).

To continue, we now introduce a well-known Neumann type boundary value problem on \((M^,g)\) introduced by Escobar ([17], see also [16]), which is sometimes called the second type Escobar–Yamabe problem:

$$\begin{aligned} \begin{aligned}&-\frac{4(n-1)}{(n-2)}\Delta _g u +R_gu=0,\,\,\text {in}\,\,M,\\&\frac{4(n-1)}{n-2}\frac{\partial }{\partial n_g}u +2(n-1)H_gu=2(n-1)u^{\frac{n}{n-2}},\,\,\text {on}\,\,\partial M, \end{aligned} \end{aligned}$$
(3.1)

where \(n_g\) is the unit out-normal vector field on \(\partial M\) and \(H_g\) is the mean curvature on the boundary. The solution of problem (3.1) is a critical point of the energy functional:

$$\begin{aligned} Q(u)=\frac{\int _M(|\nabla u|^2+\frac{n-2}{4(n-1)}R_gu^2)dV_g+\frac{n-2}{2}\int _{\partial M}H_g u^2 dS}{\big (\int _{\partial M}|u|^{\frac{2(n-1)}{n-2}}dS\big )^{\frac{n-2}{n-1}}}, \end{aligned}$$
(3.2)

for any \(u\in C^1(M)\), where \(H_g\) is the mean curvature of the boundary \(\partial M\) on (M, g), \(dV_g\) is the volume element of (M, g) and dS is the volume element of \((\partial M, g\big |_{\partial M})\). Notice that the functional does not always have a finite lower bound in a general conformal class, as pointed by Zhiren Jin, see [19]. For instance, \(\displaystyle \inf _{u\in C^1(M)-\{0\}}Q(u)=-\infty \) when (M, g) is obtained by deleting a small geodesic ball on a closed Riemannian manifold with negative scalar curvature. Under the assumption \(\displaystyle \inf _{u\in C^1(M)}Q(u)>-\infty \) and the positive mass theorem, the problem has been solved: it was studied in [13, 19, 20, 39, 40] assuming the positive mass theorem or the Weyl tensor \(W_g=0\) on M, and the remaining cases (\(n\ge 6\), \(\partial M\) is umbilic, \(\displaystyle \inf _{u\in C^1(M)-\{0\}}Q(u)>0\) and the subset on \(\partial M\) where the vanishing of the Weyl tensor of g is of certain order is the whole boundary \(\partial M\)) were recently solved in [41] without the positive mass theorem.

Recall that by an easy computation, Escobar observed in [19] the following lemma.

Lemma 3.1

(Escobar, [19]) Assume \((M^n,g)\) is a smooth compact Riemannian manifold with boundary. Let \(\lambda _1(L_g)\) be the first eigenvalue of the Dirichlet boundary value problem of the conformal Laplacian \(L_g=-\frac{4(n-1)}{n-2}\Delta _g+R_g\). Then \(\displaystyle \inf _{u\in C^1(M)-\{0\}}Q(u)>-\infty \) if and only if \(\lambda _1(L_g)>0\).

When \(R_g\ge 0\), we know that \(\lambda _1(L_g)>0\), and hence \(\displaystyle \inf _{u\in C^1(M)-\{0\}}Q(u)>-\infty \). Based on the final solution of the second type of Escobar–Yamabe problem (3.1) in [41], we will use a perturbation argument to show that the other direction holds.

Theorem 3.2

Let (M, g) be a smooth compact Riemannian manifold with boundary. Assuming either the condition in Theorem 1.1 in [41] holds on (M, g) or the positive mass theorem holds, there exists a positive scalar curvature metric in the conformal class of (M, g) if and only if \(\displaystyle \inf _{u\in C^1(M)-\{0\}}Q(u)>-\infty \).

Proof

Necessity is a trivial consequence of Lemma 3.1. Now we show the other direction. Claim. If there is a metric \(h\in [g]\) such that \(R_h=0\) on M, then there exists a metric \(h_1\in [g]\) such that \(R_{h_1}>0\) on M. This is an application of the implicit function theorem as proof of Theorem 6.11 in [2]. Consider the Dirichlet boundary value problem

$$\begin{aligned}&-\frac{4(n-1)}{n-2}\Delta _h u=f u^{\frac{n+2}{n-2}},\,\,\,\text {in}\,M,\\&u=1,\,\,\text {on}\,\partial M, \end{aligned}$$

for any given function \(f\in C^{\alpha }(M)\). We want to obtain a positive solution to the problem. Let \({\mathcal {C}}_0=\{u\in C^{2,\alpha }( M):\,\,u\big |_{\partial M}=0\}\). Define the map \(F:\,{\mathcal {C}}_0\times C^{\alpha }(M)\rightarrow C^{\alpha }(M)\) such that

$$\begin{aligned} F(u, f)=-\frac{4(n-1)}{n-2}\Delta _h (1+u)-f (1+u)^{\frac{n+2}{n-2}}. \end{aligned}$$

This is a \(C^1\) map. We take derivative

$$\begin{aligned} D_uF(v,f)=-\frac{4(n-1)}{n-2}\Delta _h v- \frac{n+2}{n-2}f (1+u)^{\frac{4}{n-2}}v. \end{aligned}$$

Recall that \(F(0,0)=0\) since \(R_h=0\). At (0, 0),

$$\begin{aligned} D_uF(v,0)=-\frac{4(n-1)}{n-2}\Delta _h v \end{aligned}$$

is invertible. By implicit function theorem, there exists a constant \(\epsilon >0\) small such that for any \(f\in C^{\alpha }(M)\) with \(\Vert f\Vert _{C^{\alpha }(M)}\le \epsilon \), we have that the Dirichlet problem has a unique solution \({\tilde{u}}=1+u\) in the neighborhood of 1 in \(C^{2,\alpha }(M)\). Take f to be a small positive constant. We have that \({\tilde{u}}>0\) on M. This completes the proof of Claim. The solution of Escobar–Yamabe problem (3.1), where positive mass theorem is used for certain cases, tells that when \(\displaystyle \inf _{u\in C^1(M)}Q(u)>-\infty \), there exists a metric in the conformal class with zero scalar curvature, and hence by Claim, we have that there exists a positive scalar curvature metric in the conformal class. This completes the proof of the Theorem. \(\square \)

In particular, for manifolds of dimension \(3\le n\le 7\) (see [43,44,45]), or spin manifolds [54], positive mass theorem holds. Recently, Schoen-Yau [46] presented a proof that positive mass theorem holds true in general dimension.

Remark. For positive mass theorem and the existence theory of solutions to the Escobar–Yamabe problem, for the regularity of the metric, it is enough that the metric is of \(C^k\) for some sufficiently large \(k>0\). The above argument shows that for a smooth compact Riemannian manifold (M, g), if \(\displaystyle \inf _{u\in C^1(M)-\{0\}}Q(u)=-\infty \), then there exists no conformal metric \(g_1=u^{\frac{4}{n-2}}g\) with \(u\in C^{2,\alpha }(M)\) such that \(R_{g_1}=0\).

4 The Yamabe Flow: A Conformally Invariant Flow

We consider the Yamabe flow

$$\begin{aligned} u_t=(n-1)u^{-\frac{4}{n-2}}\left( \Delta _gu-\frac{n-2}{4(n-1)} (R_gu+n(n-1)u^{\frac{n+2}{n-2}})\right) , \end{aligned}$$
(4.1)

which can also be written as

$$\begin{aligned} (u^{\frac{n+2}{n-2}})_t=\frac{(n-1)(n+2)}{n-2} \left( \Delta _gu-\frac{n-2}{4(n-1)}(R_gu+n(n-1)u^{\frac{n+2}{n-2}})\right) . \end{aligned}$$
(4.2)

The flow is conformally covariant in the sense that, under the conformal change \(g=\varphi ^{\frac{4}{n-2}}h\), the above equation (4.1) becomes

$$\begin{aligned} v_t=(n-1)v^{-\frac{4}{n-2}}\left( \Delta _hv-\frac{n-2}{4(n-1)} (R_hv+n(n-1)v^{\frac{n+2}{n-2}})\right) , \end{aligned}$$

where \(v=u\varphi \). So without loss of generality, let g be the metric on M in the conformal class such that \(R_g=-n(n-1)\). We consider the Cauchy–Dirichlet problem (1.6) where \(u_0\in C^{2k+2,\alpha }(M)\) and \(\phi (q,t)\in C_{loc}^{2k+2+\alpha ,k+1+\frac{\alpha }{2}}(\partial M\times [0,+\infty ))\) with \(k=0,\,1\) satisfying the \(C^{2+\alpha ,1+\frac{\alpha }{2}}\) compatible condition

$$\begin{aligned}&u_0(p)=\phi (p,0), \nonumber \\&\phi (p,0)^{\frac{4}{n-2}} \phi _t(p,0)=(n-1)\left[ \Delta _g u_0(p)+\frac{n(n-2)}{4}(u_0(p)-u_0(p)^{\frac{n+2}{n-2}})\right] , \end{aligned}$$
(4.3)

for \(p\in \,\partial M\), and in addition the \(C^{4+\alpha ,2+\frac{\alpha }{2}}\) compatible condition

$$\begin{aligned} \phi (p,0)^{\frac{4}{n-2}}\phi _{tt}(p,0)+\frac{4}{n-2} \phi (p,0)^{\frac{6-n}{n-2}}(\phi _t(p,0))^2=(n-1)L_g(\mu ), \end{aligned}$$
(4.4)

where the linear operator \(L_g\) is defined as

$$\begin{aligned} L_g(\mu )=\Delta _g\mu +\frac{n(n-2)}{4}\left( 1-\frac{n+2}{n-2}u_0(p)^{\frac{4}{n-2}}\right) \mu , \end{aligned}$$

and the function \(\mu \) is defined as

$$\begin{aligned} \mu (p)=(n-1)u_0(p)^{-\frac{4}{n-2}}\left[ \Delta _g u_0(p)+\frac{n(n-2)}{4}(u_0(p)-u_0(p)^{\frac{n+2}{n-2}}\right] . \end{aligned}$$
(4.5)

Lemma 4.1

Assume \((M^n,g)\) is a compact Riemannian manifold with boundary of \(C^{4,\alpha }\) such that \(R_g=-n(n-1)\). Let \(u_0\in C^{2k+2,\alpha }(M)\) be a positive function, and \(\phi \in C_{loc}^{2k+2+\alpha ,k+1+\frac{\alpha }{2}}(M\times [0,\infty ))\) be a positive function satisfying the compatible condition (4.3) for \(k=0\) and (4.4) in addition for \(k=1\). Then there exists a unique positive solution \(u\in C_{loc}^{2k+2+\alpha ,k+1+\frac{\alpha }{2}}(M\times [0,\infty ))\) to the Cauchy–Dirichlet problem (1.6). Moreover, if \(u_0\ge 1\) and \(\phi \ge 1\), then \(u\ge 1\).

Proof

Since \(u_0\in C^{2k+2,\alpha }(M)\) is positive, the equation (4.1) is uniform parabolic, and by the compatible condition on \(u_0\) and \(\phi \), there exists \(T>0\) such that a positive solution u on \(M\times [0,T)\) such that \(u\in C^{2k+2+\alpha ,k+1+\frac{\alpha }{2}}(M\times [0,T_1])\) for any \(T_1<T\) and \(k=0,\,1\) respectively. Now for any \(0<T_1<T\), by maximum principle,

$$\begin{aligned} u\ge \min \{1,\,\displaystyle \inf _{M}u_0,\,\inf _{M\times [0,T_1])}\phi \} \end{aligned}$$

on \(M\times [0,T_1]\). In fact, if there exists \((q,t)\in M^{\circ }\times (0,T_1]\), such that \(0<u(q,t)=\inf _{M\times [0,T_1]}u<1\), then we have \((u^{\frac{n+2}{n-2}})_t(q,t)\le 0,\,\,\Delta _gu(q,t)\ge 0, (u-u^{\frac{n+2}{n-2}})(q,t)>0\), contradicting with the equation. Also, by similar argument, we have that

$$\begin{aligned} u\le \max \{1,\,\sup _{M}u_0,\,\sup _{M\times [0,T_1]}\phi \}, \end{aligned}$$

on \(M\times [0,T_1]\). Therefore, by the standard a prior \(C^{2k+2+\alpha , 1+k+\frac{\alpha }{2}}\) estimates of parabolic equations, the solution \(u>0\) can be extended on \(M\times [0,+\infty )\) with \(u\in C^{2k+2+\alpha ,k+1+\frac{\alpha }{2}}(M\times [0,T])\) for any \(T>0\). \(\square \)

Lemma 4.2

Let (M, g), \(u_0\in C^{4,\alpha }(M)\) and \(\phi \in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,\infty ))\) be as in Lemma 4.1 with the compatible condition (4.3)–(4.4). In particular, \(R_g=-n(n-1)\). Moreover, let \(u_0\ge 1\) be a subsolution to the equation (1.1). And moreover, we assume \(u_0\in C^{4,\alpha }(M)\) and

$$\begin{aligned} L_g(\mu )\ge 0 \end{aligned}$$

at the points \(q\in \partial M\) such that \(\mu (q)=0\) where \(L_g\) and \(\mu \) are as in (4.4). Let a positive function \(\phi \in C_{loc}^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,\infty ))\) satisfy the compatible condition (4.3) and (4.4). Also we assume \(\phi _t\ge 0\) on \(\partial M\times (0,+\infty )\). Then the solution u to (1.6) satisfies \(u_t\ge 0\) on \(M\times [0,+\infty )\). In particular, \(u(\cdot ,t)\) is a sub-solution to (1.1) for each \(t\ge 0\).

Proof

Let \(v=u_t\), condition on \(u_0\) implies \(v(q,0)\ge 0\). By assumption, we have \(\phi \ge 1\) on \(\partial M\times [0,+\infty )\) and hence, by Lemma 4.1, \(u\ge 1\) in \(M\times [0,\infty )\). Take derivative of t on both sides of the equation in (1.6), and we have

$$\begin{aligned}&\frac{n+2}{n-2}u^{\frac{4}{n-2}}v_t+\frac{4(n+2)}{(n-2)^2} u^{\frac{6-n}{n-2}}v^2\nonumber \\&\quad =\frac{(n-1)(n+2)}{n-2}\left( \Delta _gv +\frac{n(n-2)}{4}(1-\frac{n+2}{n-2}u^{\frac{4}{n-2}})v\right) . \end{aligned}$$
(4.6)

By maximum principle, v can not obtain a negative minimum on \(M\times [0,T]\) at a point \((q,t)\in M^{\circ }\times (0,T]\), for any \(T>0\). Indeed, if otherwise, since \(v\ge 0\) on \(M\times \{0\}\,\bigcup \,\partial M\times [0,\infty )\) and recall that \(u\ge 1\), by continuity of v, there exists \(t_1>0\) and \(p\in M^{\circ }\) such that

$$\begin{aligned} v(p,t_1)=\inf _{M\times [0,t_1]}v<0, \end{aligned}$$

and \(|v(p,t_1)|\) is so small that at the point \((p,t_1)\),

$$\begin{aligned} -\frac{4(n+2)}{(n-2)^2}u^{\frac{6-n}{n-2}}v^2+\frac{n(n-1) (n+2)}{4}\left( 1-\frac{n+2}{n-2}u^{\frac{4}{n-2}}\right) v>0, \end{aligned}$$

contradicting with the equation (4.6) and the fact \(v_t(p,t_1)\le 0\) and \(\Delta _gv(p,t_1)\ge 0\). Therefore, \(v\ge 0\) in \(M\times [0,+\infty )\). \(\square \)

Lemma 4.3

Let (M, g), \(u_0\) and \(\phi \) be as in Lemma 4.2. Moreover, assume \(\phi \) satisfies that there exists a constant \(\beta >0\) such that

$$\begin{aligned} \phi ^{-1}\phi _t\le \beta \end{aligned}$$
(4.7)

on \(\partial M\times [0,\infty )\), and

$$\begin{aligned} \begin{aligned}&\phi ^{\frac{n-1}{2-n}}|\nabla _{g}\phi |\rightarrow 0,\\&\phi ^{\frac{n}{2-n}}|\nabla _{g}^2\phi |\rightarrow 0,\,\,\text {uniformly on} \,\,\,\partial M, \,\,\text {as}\,\,t\rightarrow +\infty . \end{aligned} \end{aligned}$$
(4.8)

Let x be the distance function to the boundary on (M, g). Let \(U=\{0\le x \le x_1\}\). Then there exist constants \(C>0\), \(x_1>0\) small and \(t_1>0\) large such that the solution \(u>0\) to (1.6) satisfies

$$\begin{aligned} u(q,t)\ge C(x+\phi ^{\frac{2}{2-n}})^{\frac{2-n}{2}}-C \end{aligned}$$

on \(U\times [t_1,+\infty )\).

For instance, take \(\phi (t)=e^{t},\,t^2e^{t},\,t,\) or any other monotone function of polynomial growth for t large.

Proof

The proof is similar to Lemma 2.3. Let \(f\equiv 1\) and \(x_1>0\) small be as in Lemma 2.3 such that the exponential map \(F=\text {Exp}:\,\partial M\times [0,x_1]\rightarrow \,U\) is a diffeomorphism, and let the barrier function \(\varphi \in C_{loc}^{2,\alpha }(U\times [0,\infty ))\) be defined in (2.5). For \(\varphi \) we have the estimates (2.6). In particular,

$$\begin{aligned} \varphi ^{\frac{4}{n-2}}\varphi _t&=\varphi ^{\frac{4}{n-2}} cf^{\frac{2}{n-2}}[(x+(f(q)\phi (q,t))^{\frac{2}{2-n}})^{\frac{-n}{2}}\\&\quad -(x_1+(f(q)\phi (q,t))^{\frac{2}{2-n}})^{\frac{-n}{2}}]\phi ^{\frac{n}{2-n}} \,\frac{\partial \phi }{\partial t}\\&\le c^{\frac{n+2}{n-2}}(x+\phi (q,t)^{\frac{2}{2-n}})^{-\frac{n+4}{2}} \phi ^{\frac{n}{2-n}}\frac{\partial \phi }{\partial t}\\&\le c^{\frac{n+2}{n-2}}(x+\phi (q,t)^{\frac{2}{2-n}})^{-\frac{n+2}{2} }\phi ^{-1}\frac{\partial \phi }{\partial t}, \end{aligned}$$

and hence by (2.7) and our condition on \(\phi \), for the constants \(c>0\) and \(x_1\) small enough and \(t_1>0\) large enough, we have

$$\begin{aligned} (\varphi ^{\frac{n+2}{n-2}})_t\le \frac{(n-1)(n+2)}{n-2} \left( \Delta _g\varphi +\frac{n(n-2)}{4}(\varphi -\varphi ^{\frac{n+2}{n-2}})\right) \end{aligned}$$

on \(U\times [t_1,\infty )\). Now we take \(c>0\) small enough so that \(u>\varphi \) on \(U\times \{t_1\}\). By the definition of \(\varphi \), we also know that \(u>\varphi \) on \(\partial U\times [0,+\infty )\). Let \(v=u-\varphi \). Take difference of this inequality and the equation satisfied by u in (1.6), we have

$$\begin{aligned}&u^{\frac{4}{n-2}}v_t+(u^{\frac{4}{n-2}}-\varphi ^{\frac{4}{n-2}}) \varphi _t\\&\quad \ge (n-1)(\Delta _gv+\frac{n(n-2)}{4}(u-u^{\frac{n+2}{n-2}}-\varphi +\varphi ^{\frac{n+2}{n-2}})) \end{aligned}$$

on \(U\times [t_1,\infty )\). Assume that there exists \((q,t)\in U^{\circ }\times (t_1,+\infty )\) such that \(v(q,t)=\displaystyle \inf _{M\times [t_1,t]}v<0\), and then since \(u\ge 1\), we have \(\varphi (q,t)>1\) and hence at the point (q, t), we have

$$\begin{aligned}&u^{\frac{4}{n-2}}v_t\le 0,\,\,(u^{\frac{4}{n-2}}-\varphi ^{\frac{4}{n-2}})\varphi _t\le 0,\\&\Delta _gv\ge 0,\,\,\,u-u^{\frac{n+2}{n-2}}-\varphi +\varphi ^{\frac{n+2}{n-2}}>0, \end{aligned}$$

where \(\varphi _t\ge 0\) by the assumption \(\phi _t\ge 0\), contradicting with the above inequality. Therefore, \(v\ge 0\) on \(U\times [t_1,+\infty )\). This completes the proof of the lemma. \(\square \)

Proof of Theorem 1.3

By Lemma 4.1 and Lemma 4.2, we have that there exists a positive solution u on \(M\times [0,+\infty )\) with \(u\in C^{4+\alpha ,2+\frac{\alpha }{2}}(M\times [0,T])\) for any \(T>0\), and \(u_t\ge 0\), i.e., \(u(\cdot ,t)\) is a sub-solution to (1.1) for any \(t\ge 0\) and also, \(u\ge 1\). By maximum principle, \(u(q,t)\le u_{LN}(q)\) for any \((q,t)\in M^{\circ }\times [0,+\infty )\), where \(u_{LN}\) is the solution to the Loewner–Nirenberg problem (1.1)–(1.2). Alternatively, one can use the local super-solution constructed in Lemma 5.2 in [23] to give the local upper bound estimates of u in \(M^{\circ }\). Indeed, let \(x=x(q)\) be the distance function of \(q\in M^{\circ }\) to \(\partial M\). There exists \(x_1>0\) small such that for \(q\in \{0<x(q)\le x_1\}\), the injectivity radius i(q) at q is larger than \(\frac{x(q)}{2}\). We then define a function \({\bar{u}}\) on \(B_{R}(q)\) by

$$\begin{aligned} {\bar{u}}(p)=\left( \frac{2R}{R^2-r(p)^2}\right) ^{\frac{n-2}{2}} e^{\frac{n-2}{2}(\sqrt{R^2-r(p)^2+\epsilon ^2}-\epsilon )} \end{aligned}$$

for \(p\in B_R(q)\) where \(R=\frac{x(q)}{2}\), \(\epsilon >0\) is some small constant and r(p) is the distance function from p to q, and hence \({\bar{u}}\in C^{2}(B_R(q))\) and \({\bar{u}}=\infty \) on \(\partial B_{R}(q)\). In fact \(x_1\) and \(\epsilon \) are chosen small enough as in [23] so that \({\bar{u}}\) is a super-solution to (1.1) on \(B_R(q)\). By maximum principle, \(u\le {\bar{u}}\) on \(B_R(q)\), and hence there exists a uniform constant \(C>0\), such that

$$\begin{aligned} u(q,t)\le Cx(q)^{\frac{2-n}{2}} \end{aligned}$$

on \(\{0<x(q)\le x_1\}\) for \(t\ge 0\). For any \(q\in M-\{0\le x\le x_1\}\), taking \(R<\min \{\frac{i(q)}{2},x_1\}\) and \(\epsilon >0\) small so that \({\bar{u}}\) is a super-solution to (1.1) on \(B_R(q)\), and hence we have that there exists a uniform constant \(C>0\) depending on \(B_R(q)\) such that \(u(p,t)\le C\) for \(p\in B_{\frac{R}{2}}(q)\) and \(t\ge 0\). Now by standard interior Schauder estimates of parabolic equations, we have that for any compact subset \(F\subseteq M^{\circ }\), there exists a uniform constant \(C=C(F)>0\) such that

$$\begin{aligned} \Vert u\Vert _{C^{4+\alpha ,2+\frac{\alpha }{2}}(F\times [T,T+1])}\le C \end{aligned}$$

for any \(T\ge 0\). Since u is locally uniformly bounded from above in \(M^{\circ }\) and \(u_t\ge 0\) on \(M\times [0,\infty )\), by Harnack inequality with respect to the equation (4.6) satisfied by \(u_t\), we have that u converges locally uniformly in \(M^{\circ }\) to a positive function \(u_{\infty }\) on \(M^{\circ }\) as \(t\rightarrow +\infty \). By interior Schauder estimates of the uniform parabolic equation in (1.6), we have that \(u\rightarrow u_{\infty }\) in \(C_{loc}^4(M^{\circ })\), and hence \(u_{\infty }\) is a solution to (1.1) in \(M^{\circ }\). By the lower bound estimates near the boundary in Lemma 4.3 and the above upper bound estimates, there exists a constant \(C>0\) such that

$$\begin{aligned} C x^{\frac{2-n}{2}}\ge u_{\infty }\ge \frac{1}{C} x^{\frac{2-n}{2}} \end{aligned}$$

in \(\{0<x\le x_1\}\) for some constant \(x_1>0\), where x is the distance function to the boundary on (M, g). Therefore, by uniqueness of the solution to (1.1)–(1.2), \(u_{\infty }=u_{LN}\). This completes the proof of the theorem. \(\square \)