Editorial to the 'Special Issue—Energy transition in a circular economy perspective' of AIMS Energy

1.   Introduction

Let $(M, g)$ be a $n$-dimensional Riemannian manifolds. In the present paper, we focus on the following Monge-Ampère equation on compact Riemannian manifolds $(M, g)$ without boundary:

for any $z\in M$ where $\rho^2 = g^{ij}S_iS_j+S^2$ and $S_{ij}+g_{ij}S$ is a $(0, 2)$ type Codazzi tensor.

An equivalent form of (1.1) is

where $\sigma_n = \lambda_1\lambda_2\cdots\lambda_n$ and $\{\lambda_l\}^n_{l = 1}$ is a solution sequence to the following algebraic equation,

Noting that the volume element of Riemannian manifold is $dV = \sqrt{\det g_{ij}}dz$, we get

The left-side of (1.2) is called the density of $p$-integral Gaussian curvature measure for the log-convex measure $e^{-\varphi(\rho^2)}dx$ for Codazzi tensor in the present paper.

In particular, if $M = \mathbb{S}^2$, $p = 1$, $\varphi\equiv 0$ and $S$ is the support function of hypersurface $N\subseteq \mathbb{R}^3$, the second fundamental form $(h_{ij})_{2\times 2}$ of the hypersurface $N$ is given as

for any fixed $i, j\in\{1, 2\}$ where $S_{ij}$ is the covariant derivation of $S$ of second order. Then the left hand side of (1.4) becomes

which is associated with the well-known Gauss-Bonnet formula for $2$-dimensional Riemannian submanifold without boundary, see pp 358 of Kobayashi and Nomizu ^[1].

Lemma A.1(Gauss-Bonnet formula(^[1])). Let $N\subseteq \mathbb{R}^3$ be an orientable, compact, smooth $2$-dimensional submanifold without boundary. Then,

where $\chi(N)\in \mathbb{Z}$ denotes the Euler characteristic number of $N$.

For any fixed $n\ge 3$, one of the geometric interests for the measure $\sigma_ndV$ is the generalization of classical Gauss-Bonnet formula in higher dimensional space, see Fenchel ^[2], Allendoerfer ^[3], Allendoerfer and Weil ^[4], Chern ^[5,6], Chern and Lasf ^[7] and so on.

Another geometric interest is to the measure $\sigma_ndV$ in the well-known Steiner-Weyl formula, see Weyl ^[8], Federer ^[9], Chern ^[6], Schneider ^[10] and so on.

Since the measure $\sigma_ndV$ has its geometric origin, and therefore it is natural to get an intrinsic construction of the measure $\sigma_ndV$. In polar coordinates, one can formulate the measure $\sigma_ndV$ as follows:

for any Borel set $\omega\subseteq \mathbb{S}^n$, where $\nu_N$ and $r_N$ are the normal mapping and radial mapping of the hypersurface $N$,

and

see Oliker ^[11,12,13] or Schneider ^[10]. This observation led Aleksandrov to pose the following classical Aleksandrov problem, see Aleksandrov ^[14,15], Bakelman ^[16] or Guan, Li and Li ^[17].

Problem A.2 (The classical Aleksandrov problem). For any fixed $n\ge 1$, given a Borel measures $\mu$ which is supported on the unit sphere $\mathbb{S}^n$, finds a convex hypersurface $N\subseteq \mathbb{R}^{n+1}$ such that

for any Borel set $\omega\subseteq \mathbb{S}^n$ where $d\xi$, $\nu_N$ and $r_{N}$ are the standard $n$-dimensional spherical Lebesgue measure, normal mapping and radial mapping of the hypersurface $N$.

Aleksandrov ^[14,15] solved Problem A.2. via the mapping argument which is a kind of method of continuity, see also Bakelman ^[16] or Pogorelov ^[18]. Later, from the point of view of nonlinear analysis or PDEs theory, Oliker ^[11,12,13] resolved Problem A.2. Moreover, concerning the regularity or curvature bounds of the hypersurface, Treibergs ^[19] and Guan and Li ^[20] also analyzed the Problem A.2. Recently, Huang, Lutwak, Yang and Zhang ^[21] introduced the so-called $p$-integral Gaussian curvature measure and posed the $L_p$ Aleksandrov problem which can be stated as follows.

Problem A.3 ($L_p$ Aleksandrov problem (^[21]). For any fixed $n\ge 1$ and $p\in \mathbb{R}$, given a Borel measures $\mu$ which is supported on the unit sphere $\mathbb{S}^n$, find a convex hypersurface $N\subseteq \mathbb{R}^{n+1}$ such that

for any Borel set $\omega\subseteq \mathbb{S}^n$ where $d\xi$, $\nu_N$, $u$ and $r_{N}$ are the standard spherical Lebesgue measure, normal mapping, support function and radial mapping of the hypersurface $N$.

Recently, more and more interesting geometric analysis has been focused on the weighted measure $e^{-\varphi(|x|^2)}dx$, see ^{[22,23,24,25,26,27]} and so on ^{[28,29,30,31,32,33,34]}. It may be interesting to mention that the convexity of $\varphi$ can deduce some interesting geometric inequalities for the measure $e^{-\varphi(|x|^2)}dx$, such as Brunn-Minkowski inequality, Prékopa-Leindler inequalities or Blaschke-Santaló inequalities, see ^{[10,35,36,37,38]}.

It is interesting to focus on the geometry of weighted measure $e^{-\varphi(|x|^2)}dx$ without the assumption of convexity of $\varphi$.

If $\varphi$ is concave, we call the measure $e^{-\varphi(|x|^2)}dx$ a log-convex measure.

One interest in the geometry of log-convex measure $e^{-\varphi(|x|^2)}dx$ is the so-called log-convex density conjecture in geometric measure theory which can be stated as follows.

Problem A.4 (Log-convex density conjecture). In $\mathbb{R}^{n+1}$ with a smooth, radial, log-convex density, balls about the origin provide isoperimetric regions of any given volume.

The Log-convex density conjecture was posed by Brakke and solved by Chambers ^[39]. More interesting comments on this topic can be referred to ^{[40,41,42,43]}.

Motivated by these beautiful results mentioned above, the main focus of the present paper is on $L_p$ Aleksandrov problem for log-convex measure $e^{-\varphi(|x|^2)}dx$ in the frame of Riemannian Geometry.

It is well-known that the main language of Riemannian Geometry is the so-called tensor, see Bishop and Goldberg ^[44] or Gerretsen ^[45]. This leads to our consideration on the problem in tensor spaces. By the analysis mentioned above, the core concept is the concept of Gaussian curvature. It is easy to see that the Gaussian curvature of a hypersurface can be calculated by means of the metric and second fundamental form of the hypersurface, see Kobayashi and Nomizu ^[1]. Therefore, to formulate a natural generalization of the classical $L_p$ Aleksandrov problem in tensor spaces, we need to replace the second fundamental form by some interesting symmetric tensors and pose a natural generalization of Gauss curvature. Since the second fundamental form of any hypersurface in a space of constant curvature satisfies the Codazzi equation, we may say a natural generalization of the second fundamental form of the hypersurface is the so-called Codazzi tensor of Riemannian manifolds in higher dimensional tensor space which is defined as follows.

For any connected smooth $n$-dimensional Riemannian manifolds $(M, g)$, we let $ST_2$ be the bundle of smooth symmetric $(0, 2)$ type tensor field over $M$, the covariant differential in the metric $g$ is denoted by $\nabla_X$ where $X$ is a vector field from the tangle bundle $TM$. The so-called Codazzi tensor is defined as follows:

Definition A.5 (Cadazzi tensor ^[46]). Let $A:M\rightarrow ST_2$ be a smooth section. It is called a Codazzi tensor if $A$ satisfies Codazzi equation,

for any $X, Y, Z\in TM$. The set of Codazzi tensors on $M$ is denoted by $Cod(M, g)$.

In particular, the second fundamental form of any hypersurface in a space of constant curvature is a Codazzi tensor, see pp. 26 of Kobayashi and Nomizu ^[1].

Some basic differential geometric theories about the Codazzi tensor are listed as follows, see ^[46].

Let $x:M\mapsto \mathbb{R}^{n+1}$ be an isometric immersion and assume also that rank $A = n$ at $z$. Then rank $A = n$ in some neighborhood $U$ of z. Suppose that $\xi$ is the unit normal vector field over $x(U)$.

Lemma A.6 (^[46]).

$(i)$ Let $x:M\mapsto \mathbb{R}^{n+1}$ be an isometric immersion and $\xi$ be the unit normal vector field over $x(U)$. Then "support" function $f(\xi) = -(x, \xi)$ where $(\cdot, \cdot)$ is the inner product in $\mathbb{R}^{n+1}$.

$(ii)$ the second order covariant differential of $f$ and the coefficient of the second fundamental form $(b_{ij})_{n\times n}$ satisfies

Lemma A.7 (^[46]). Let $(M, g)$ be a Riemannian manifold of constant sectional curvature $K_{sec}$ (possibly zero) and $A\in Cod(M, g)$. Then for every point on $M$, there exists a neighborhood $V$ and a smooth function $f: M\rightarrow \mathbb{R}$ such that in $V$

where $(A)_{ij}(f)$ is the coefficient of $A = A(f)$. In addition, if $M$ is simply connected then such representation is available on the entire $M$. Conversely, on a manifold of constant curvature $K_{sec}$, any smooth function $f$ generates a Codazzi tensor $A(f)$ via Eq (1.15).

For any $f\in C^\infty(M)$, we let $A = (A_{ij})_{n\times n}$ be the Codazzi tensor generated by $f$, that is, $A_{ij}$ is given by (1.15). Let

Remark A.8. It is easy to see that $P^0_n$ satisfies the following equation

where $\lambda_1, \lambda_2, \cdots, \lambda_n$ are $n$ solutions to the following equation

This means that the geometric meaning of $P^0_n$ is that $P^0_n$ is a concept of "Gaussian" curvature for the quadratic form $A$.

Oliker and Simon ^[46] proved the following interesting prescribed Gaussian curvature problem for Codazzi tensor:

Theorem A.9 (^[46]). Let $(M, g)$ be a closed Riemannian manifold with constant sectional curvature $K_{sec}\neq 0$, and $\phi:M\rightarrow (0, \infty)$ is a strictly positive $C^\infty$ function. If $K_{sec} > 0$ suppose also that $M$ is not isometrically diffeomorphic to the sphere $\mathbb{S}^n$ of $\mathbb{R}^{n+1}$. Then there exists a (unique) function $f\in C^\infty(M)$ such that is a positive definite Codazzi tensor on $M$ and

Suppose the sectional curvature $K_{sec}$ of $M$ is positive constant, with loss of generality, we may assume that $K_{sec} = 1$. In (1.5), if we replace the standard Lebesgue $dx$ with $e^{-\varphi(|x|^2)}dx$, the $p$-integral Gaussian curvature function of Codazzi tensor with log-convex measure $e^{-\varphi(|x|^2)}dx$ can be defined as follows:

Therefore, we let

and call $P_{n, p}(f)$ as $p$-integral Gaussian curvature function for Codazzi tensor with log-convex measure $e^{-\varphi(|x|^2)}dx$.

In the present paper, we focus on $L_p$ Aleksandrov problem for Codazzi tensor with log-convex measures $e^{-\varphi(|x|^2)}dx$ which is stated as follows:

Problem A.10 ($L_p$ Minkowski problem for Codazzi tensor with log-convex measure). For any fixed $n\ge 1$ and $p\in \mathbb{R}$, does there exist a Codazzi tensor $A$ whose sectional curvature is 1 and is generated by $f$ such that

The main result of the present paper can be stated as follows.

Theorem 1.1. For any fixed $n\ge 1$ and $p > n+1$, there exist positive constants $c, \tau$ and a positive solution $S\in C^{2, \tau}(M)$ to the Eq (1.1) satisfying

where $\tau\in (0, 1)$, $c$ is independent of $S$ provided the following conditions hold.

$(A.1.)$ $0 < \phi\in C^4(M)$, $\varphi$ is a non-negative, radially symmetric, increasing, smooth and concave function in $\mathbb{R}$, $0 < \phi\in C^4(M)$ and

$(A.2.)$

$(A.3.)$ There exists $\delta_4 > 0$ such that

for any compact $[a, b]\subseteq(0, \infty)$.

$(A.4.)$

for any compact $[a, b]\subseteq(0, \infty)$.

Remark 1.2. The main result of the present paper can be seen as an attempt at some new results on the integral geometry of differential forms which are associated with some interesting invariants arising in geometry and topology. Apart from the beautiful Gauss-Bonnet Theorem, there are many famous theorems which link the analysis, topology and geometry, such as Euler-Poincaré characteristic formula, Riemann-Roch-Hirzebruch-Grothendieck Theorem, Atiyah-Singer index Theorem, Chern-Simons invariants, see Palais ^[47], Chern-Simons ^[48], Atiyah ^[49], Shanahan ^[50], Mukherjee ^[51], Moore ^[52], Gilkey ^[53], Freed ^[54] and so on. Following some classical ideas of Steiner, Federer and Chern, it may be interesting to focus on some kinematic formulas for these invariants and consider analogous geometric problems which prescribe differential forms for these invariants. In particular, in the frame of Kähler Manifolds or Symplectic Manifolds, some great results on this topic can be referred to the great works of Patodi ^[55], Duistermaat ^[56], Atiyah and Bott ^[57], Karshon and Tolman ^[58], Donaldson ^[59,60,61], Abreu ^[62], Grossberg and Karshonand ^[63], Boyer, Calderbank and Tønnesen-Friedman ^[64] and so on. In our forthcoming study, we will focus on these topics.

Our proof of Theorem 1.1 is based on the well-known continuous method. We let the set of the positive continuous function on $M$ be $C_+(M)$ and

The main ingredient is the a priori bounds of solutions to the following auxiliary problem for any $S\in \mathcal{C}$:

for $t\in [0, 1]$.

Remark 1.3. It is worth mentioning that without the assumption of convexity of $\varphi$, some necessary geometric inequalities have not been established which does not guarantee the validity of the classical variational framework for Problem A.10. Therefore, we adopt the well-known continuous method to solve this problem. Moreover, by the a priori bounds (1.23) of $S$, we get the compactness of the solution set and the curvature estimate of the Codazzi tensor which have their independent interests.

Remark 1.4. Just like the case of concavity, log-convexity may be defined in the form of Prékopa-Leindler inequality: a function (or functional) $f: I\mapsto \mathbb{R}$ is called log-convex if $f$ satisfies the following condition,

for any $t\in [0, 1]$ and $x, y\in I$. The former case can be referred to pp. 369–374 of Schneider ^[10]. Some inequalities for log-convex functions (or functionals) have been analyzed in ^{[65,66,67,68]} and so on. It may be worth mentioning that Klartag ^[22] and Rotem ^[28] introduced some geometric notions for log-concave or more general $\alpha$-concave functions and measures, such as the support function and mean width. Motivated by these interesting results, it may be interesting to introduce similar notions and get more geometric analysis for log-convex measures and this will also be a topic of our future study.

The remaining part of this paper is arranged as follows: In Section 2, we prove the a priori bounds of $S$. In Section 3, we prove Theorem 1.1.

2.   A priori bounds of $S$

In this section, we consider the a priori bounds of solutions to the following equation on Riemannian manifolds $(M, g)$:

where $\rho^2 = |\nabla S|^2+S^2$ and the following conditions hold.

$(A.1.)$ $0 < \phi\in C^4(M)$, $\varphi$ is a non-negative, radially symmetric, increasing, smooth and convex function in $\mathbb{R}$, $0 < \phi\in C^4(M)$ and

$(A.2.)$

$(A.3.)$ There exist $\delta_4 > 0$ such that

for any compact $[a, b]\subseteq(0, \infty)$.

We let the set of the positive continuous function on $M$ be $C_+(M)$,

and

This main result of this section can be stated as follows,

Theorem 2.0. For any fixed $n\ge 1$ and $p > n+1$, we let $S\in \mathcal{C}\cap C_+(M)$ be a solution to (2.1). Suppose that $(A.1)\sim(A.3)$ hold. Then there exists a positive constant $c$, independent of $S$, such that

where $\tau\in (0, 1)$.

Now, we divide the proof of Theorem 2.0 into following four steps.

Step $(a)$. For any fixed $n\ge 1$ and $p > n+1$, we let $S\in \mathcal{C}\cap C_+(M)$ be a solution to (2.1). Suppose that $(A.1)\sim(A.3)$ hold. Then there exists a positive constant $c$ such that

for any $z\in M$.

Proof of Step $(a)$. We consider the following extremal problem,

It follows from the compactness of $M$ and the continuity of $S$ that there exists $z_1\in M$ such that

It follows from the Eq (2.1) that at the point $z = z_1$,

Combining this and condition $(A.1)$, we can see that there exists a positive constant $c > 0$ such that

We next consider the following extremal problem,

Adopting a similar argument mentoned above, we also can see that there exists a positive constant $c > 0$ such that

(2.7) and (2.9) yield the desired conclusion of Step $(a)$.

Step $(b)$. For any fixed $n\ge 1$ and $p > n+1$, we let $S\in \mathcal{C}\cap C_+(M)$ be a solution to (2.1). Suppose that $(A.1)\sim(A.3)$ hold. Then there exists a positive constant $c$ such that

Proof of Step $(b)$. The proof is based on Maximum Principle. We let

Suppose that there exists $z_0\in M$ such that

Then,

for any fixed $l\in \{1, 2, \cdots, n\}$ at the point $z_0$. It follows from Lemma 2.1 that there exists a positive constant $c$,

at the point $z = z_0$ where

Noting that $g = (g_{ij})_{n\times n}$ is strictly positive, it follows from (2.14) and (2.15) that the matric $(S_{il}+Sg_{il})_{n\times n}$ is reversible at the point $z = z_0$ and therefore,

at the point $z = z_0$ for any fixed $l\in \{1, 2, \cdots, n\}$ due to (2.13) and the positivity of $g$. From (2.16), we can see that

at the point $z = z_0$. Therefore, it follows from Lemma 2.1 that there exists a positive constant $c$, independent of $S$,

for any $z\in M$. This completes the proof of Step $(b)$.

Before getting the higher order estimates of $S$, we let

and

Then, Eq (2.1) becomes

Step $(c)$. For any fixed $n\ge 1$ and $p > n+1$, we let $S\in \mathcal{C}\cap C_+(M)$ be a solution to (2.1) and $u\in \mathcal{\widetilde{C}}\cap C_+(M)$ be a solution to (2.21). Suppose that $(A.1)\sim(A.3)$ hold. Then there exists a positive constant $c$, independent of $S$, such that

Proof of Step $(c)$. We let $H = \sum_{i}u_{ii}$. Suppose that $H$ achieves it maximum at the point $z = z_0$. Without loss of generality, we may $(H_{ij})_{n\times n}$ is diagonal at the point $z = z_0$. Therefore, at the point $z = z_0$,

and $(H_{ij})_{n\times n}$ is non-positive. We let

for any fixed $i, j, s, t\in \{1, 2, \cdots, n\}$. Therefore, at the point $z = z_0$,

By the commutator identity, we have,

Putting (2.25) into (2.26), we get

Taking the $\alpha$-th partial derivatives on both sides of (2.21) twice for any fixed $\alpha\in \{1, 2, \cdots, n\}$, we have

for any fixed $\alpha\in \{1, 2, \cdots, n\}$. By the concavity of $\mathcal{G}$, we have

This implies that

at the point $z = z_0$. Therefore,

at the point $z = z_0$. It follows from Newton-MacLaurin inequality that

see Guan and Ma ^[69]. Putting (2.31), (2.32) into (2.27), we have, at the point $z = z_0$,

We let

It follows from Lemmas 2.1 and 2.2 that

Now, we claim that at the point $z = z_0$,

where $\delta_4 > 0$ to be chosen. Indeed, it follows from the definition of $\Psi$ that

Noting $\rho^2 = |\nabla S|^2+S^2$, for any fixed $\alpha\in \{1, 2, \cdots, n\}$, taking $\alpha$-th partial derivatives on both sides of $(2.37)$ twice, we have

and

where

and

We now get some estimates of $I_2$. Since $\phi\in C^4(M)$, it follows from Lemmas 2.1 and 2.2 that

By the definition of $H$, we have,

Therefore, it follows from Lemma 2.1 and Hölder inequality that

for some $c$ which means that

Moreover, it follows from (2.43), (2.23) and Lemma 2.2 that

at the point $z = z_0$. Therefore, combining (2.42), (2.45) and (2.46), we have,

at the point $z = z_0$.

Since $\varphi\in C^2$ is concave, we have,

for any $z\in M$. Noting that

Therefore,

We let

It follows from Step $(a)$ and Step $(b)$ that

Therefore, it follows from $(A.3)$ that there exists $\delta_4 > 0$ such that

Therefore,

Therefore, (2.47) and (2.54) yield

at the point $z = z_0$. This is the desired inequality (2.36).

It follows from (2.33) and (2.36) that

at the point $z = z_0$. It follows from Hölder inequality that

at the point $z = z_0$. Combining (2.57), the definition of $u$ and Step $(a)$, it is easy to get the inequality (2.22) which completes the proof of Step $(c)$.

Step $(d)$. It follows from (2.21) that Eq (2.1) becomes

provided $\mathcal{F}(u_{ij}) = \mathcal{G}(u_{ij})-\Psi$. We let $\mathcal{F}_{ij} = \frac{\partial \mathcal{F}}{\partial u_{ij}}$. It follows from Step $(a)$, Step $(b)$ and Step $(c)$ that there exist positive constants $\lambda$ and $\Lambda$, independent of $S$, such that

and

for any $\zeta = (\zeta_1, \zeta_2, \cdots, \zeta_n)\in \mathbb{R}^n$. That is,

(d.ⅰ) (2.58) is elliptic uniformly.

Now, we claim that

(d.ⅱ) $\mathcal{F}$ is concave with respect to $(u_{ij})_{n\times n}$.

It follows from the definition of $\mathcal{F}$ that it suffices to prove that $\mathcal{G} = \det^{\frac{1}{n}}$ is concave with respect to $(u_{ij})_{n\times n}$. Indeed, For any $u, v\in \mathcal{C}$ and $t\in [0, 1]$, we let $\{\delta^1_i\}^n_{i = 1}$ and $\{\delta^2_i\}^n_{i = 1}$ be the eigenvalue sequence of $(u_{ij})_{n\times n}$ and $(v_{ij})_{n\times n}$ respectively. Then $\{t\delta^1_i+(1-t)\delta^2_i\}^n_{i = 1}$ is a eigenvalue sequence of $(tu_{ij}+(1-t)v_{ij})_{n\times n}$. Moreover, since $(u_{ij})_{n\times n}$ and $(v_{ij})_{n\times n}$ are convex, we have

for any fixed $i\in \{1, 2, \cdots, n\}$ and $j\in \{1, 2\}$ and therefore,

for any $t\in [0, 1]$. Combining the definition of $\mathcal{G}$ and (2.62), we get

which proves that $\mathcal{G} = \det^{\frac{1}{n}}$ is concave with respect to $(u_{ij})_{n\times n}$ and this completes the proof of the claim.

Then, it follows from (d.ⅰ), (d.ⅱ) and Theorem 17.14 of Gilbarg and Trudinger ^[70] that there exist $\tau_1\in (0, 1)$ and positive constant $c$, independent of $S$, such that

(see pp. 457–461 of Gilbarg and Trudinger ^[70]). Therefore there exist $\tau\in (0, 1)$ and positive constant $c$, independent of $S$, such that

This is the desired conclusion of Theorem 2.0.

3.   Existence

This section is devoted to the proof of Theorem 1.1.

Motivated by ^[69,71] and so on, we consider the following auxiliary problem with a parameter $t\in [0, 1]$,

for any $z\in M$ where $\rho^2 = |\nabla S|^2+S^2 = g^{ij}S_iS_j+S^2$, $0 < \phi\in C^4(M)$ and the following conditions hold.

$(A.1.)$ $0 < \phi\in C^4(M)$, $\varphi$ is a non-negative, radially symmetric, increasing, smooth and convex function in $\mathbb{R}$, $0 < \phi\in C^4(M)$ and

$(A.2.)$

$(A.3.)$ There exist $\delta_4 > 0$ such that

for any compact $[a, b]\subseteq(0, \infty)$.

$(A.4.)$

for any compact $[a, b]\subseteq(0, \infty)$.

We let the set of the positive continuous function on the Riemannian manifolds $(M, g)$ be $C_+(M)$ and

Since $f_t$ is a contious function, independent of $z$, satisfying

for any $t\in [0, 1]$ and $z\in M$, adopting some similar arguments in Section 2, we get

Lemma 3.1. For any fixed $n\ge 1$, $p > n+1$ and $t\in [0, 1]$, we let $S_t\in \mathcal{C}\cap C_+(M)$ be a solution to (3.1). Suppose that $(A.1)\sim(A.3)$ hold. Then there exists a constant $c$, independent of $t$, such that

for any $t\in [0, 1]$ and some $\tau\in(0, 1)$.

As a corollary of Lemma 3.1, we have,

Corollary 3.2. For any fixed $n\ge 1$, $p > n+1$, we let $\mathcal{I}$ is the set defined in (3.2). Suppose that $(A.1)\sim(A.3)$ hold. Then $\mathcal{I}$ is closed.

Proof. It suffices to show that for any sequence $\{t_j\}_{j = 1}^\infty\subseteq\mathcal{I}$ satisfying

as $j\to \infty$ for some $t_0\in [0, 1]$, we need to prove $t_0\in\mathcal{I}$.

We let $S^j$ be a solutions of problem (3.1) at $t = t_j$. It follows from the conclusion of Lemma 3.1 that there exists a positive constant $c$, independent of $j$ such that

it follows from Ascoli-Arzela Theorem that up to a subsequence, there exists a $S^0\in C^2(M)$, such that

as $j\to \infty$. It is easy to see that

uniformly on $M$ as $j\to \infty$ where $(\rho^j)^2 = (S^j)^2+|\nabla S^j|^2$ for any $j\in \{0, \, \cdots \}$. Letting $j\to \infty$, we can see that $(t_0, S^0)$ is a solution to the following problem:

for any $z\in M$. (3.4) implies that $t_0\in\mathcal{I}$. This is the desired conclusion of Corollary 3.2.

Lemma 3.3. For any fixed $n\ge 1$, $p > n+1$, we let $\mathcal{I}$ is the set defined in $(3.2)$. Suppose that $(A.1)\sim(A.4)$ hold. Then $\mathcal{I}$ is open.

Proof. Suppose that there exists a $\bar{t}\in \mathcal{I}$, it suffices to prove $t\in \mathcal{I}$ for any $t\in B_\delta(\bar{t})\cap [0, 1]$. To achieve this goal, joint with Implicit Function Theorem, we need to analyze the kernel of linearized equation associated to $(3.1)$. We assume that $\bar{S}$ is a solution to $(3.1)$ at $t = \bar{t}$. For any $\zeta\in M$, we let

and

By the Eq (3.1), we have

Taking logarithm on both sides of (3.8), since $f_t$ is independent of $\bar{S}$, we get,

where $(\bar{P}_{ij})_{n\times n}$ is the inverse of the matrix $(\bar{S}_{ij}+\bar{S}g_{ij})_{n\times n}$ and

We let $\zeta = \bar{S}v$. Direct Calculation shows that

and

Therefore, we get

which implies that

It follows from (3.12) and (3.10) that

and thus,

due to the symmetry of $(\bar{P}_{ij})_{n\times n}$. Putting (3.14) and (3.16) into (3.9), we have,

where

and

Since $\bar{S}, M(\bar{S}) > 0$, $(\bar{P}_{ij})_{n\times n}$ is positive, we see that $(a_{ij})_{n\times n}$ is positive. It follows from Lemma 3.1 that $b_i$ is bounded. By the condition $(A.4.)$, we have

Since $M(\bar{S})$ is positive, we have,

Therefore, it follows from (3.20) and (3.21) we get $N < 0$. By Strong Maximum Principle for elliptic equations of second order, we see that

(see pp. 35 of Gilbarg and Trudinger ^[70]) and thus,

since $\bar{S} > 0$. Then by the standard Implicit Function Theorem, for any $t\in B_\delta(\bar{t})\cap [0, 1]$, there exists a $S\in C^{2, \tau}(M)$, such that $G_t(S) = 0$. This means that $t\in \mathcal{I}$ and completes the proof of Lemma 3.3.

Final proof of Theorem 1.1. It is easy to see that $S\equiv 1$ is a solution of (3.1) at $t = 0$. This means that $\mathcal{I}$ is not-empty. This, together with Corollary 3.2 and Lemma 3.3, implies that $\mathcal{I} = [0, 1]$. Taking $t = 1$, we get the proof of Theorem 1.1.

Acknowledgments

The work was supported by China Postdoctoral Science Foundation (Grant: No.2021M690773). The author would like to thank heartily to the anonymous referees for their invaluable comments which are helpful to improve this paper's quality and to editors' hard work on the publication of the paper and to Professor Daomin Cao and Professor Qiuyi Dai for their useful guidance on nonlinear PDE theory and helpful comments on this work.

Conflict of interest

There is no conflict of interest in this work.