October 16, 2025
In this paper, we show that on a compact Kähler manifold the Calabi flow can be extended as long as some space-time \(L^p\) integrals of the scalar curvature are bounded.
This paper is the continuation of the study on the extension of Calabi flow in . In , based on Chen-Cheng’s estimates in , we showed that the Calabi flow can be extended as long as the \(L^p\) norm of the scalar curvature is bounded. The estimates in are essentially elliptic. In this paper, we want to use the parabolic structure of the Calabi flow equation to study the extension of Calabi flow under some space-time integrals of the scalar curvature as in other second order geometric flows, such as Ricci flow and mean curvature flow etc.
Let \((M^n, g)\) be a compact Kähler manifold of complex dimension \(n\). To study the constant scalar curvature metrics in a Kähler class, E. Calabi in introduced the Calabi flow, which is the gradient flow of the Calabi energy. We call a family of Kähler metrics \(\omega_{\varphi(t)}(t\in [0, T])\) in the same Kähler class \([\omega_g]\) a solution of Calabi flow, if the Kähler potential \(\varphi(t)\) satisfies the equation \[\frac{\partial \varphi(t)}{\partial t}=R(\omega_{\varphi(t)})-\underline R, \label{eq000}\tag{1}\] where \(R(\omega_{\varphi(t)})\) denotes the scalar curvature of the metric \(\omega_{\varphi(t)}\) and \(\underline R\) denotes the average of the scalar curvature. The Calabi flow is expected to be an effective tool to find constant scalar curvature metrics in a Kähler class. However, since the Calabi flow a fully nonlinear fourth order partial differential equation, it is difficult to study its behavior by standard parabolic estimates. In this paper, we continue to study the extension problem of Calabi flow under some conditions on the scalar curvature.
There are many literatures on Calabi flow. The long time existence and convergence of Calabi flow on Riemann surfaces is completely solved by Chrusciel , Chen and Struwe independently by using different methods. In , Chen-He showed the short time existence and stability results of Calabi flow in general Kähler manifolds of higher dimensions. In a series of papers , Chen and He studied the long time existence and convergence under some curvature conditions. Moreover, Tosatti-Weinkove proved the long time existence and convergence under the assumption that the Calabi energy is small. Szekelyhidi in studied the Calabi flow on ruled surfaces, and in studied the Calabi flow under the assumption that the curvature tensor is uniformly bounded and the \(K\)-energy is proper. Streets showed the long time existence of a weak solution to the Calabi flow and Berman-Darvas-Lu showed the convergence of weak Calabi flow on general Kähler manifolds.
A conjecture of X. X. Chen in says that the Calabi flow always exists for all time for any initial Kähler metrics. Chen-He’s result in showed the extension result of Calabi flow under the assumption that the Ricci curvature stays bounded, and Huang in
proved the extension results of the Calabi flow on toric manifolds. In Li-Zheng showed the long time existence under the assumptions on the lower boundedness of Ricci curvature, the properness of the \(K\)-energy, and the
\(L^p(p>n)\) bound of scalar curvature. In , Li-Wang-Zheng used the ideas from Ricci flow in and to study the convergence of Calabi flow. A breakthrough was made by Chen-Cheng in and they showed that the Calabi flow
always exists as long as the scalar curvature is bounded.
In the previous paper , Li-Zhang-Zheng proved that the Calabi flow can be extended as long as the \(L^p\) scalar curvature is bounded. In this paper, based on Chen-Cheng’s estimates in we show that Calabi flow can be
extended as long as some space-time \(L^p\) integrals of the scalar curvature are bounded. The main theorem in this paper is the following result.
Theorem 1. Let \((M, \omega_g)\) be a compact Kähler manifold of complex dimension \(n\geq 2\), and \(\{\varphi(t), t\in [0, T)\}\) the solution to the Calabi flow (1 ) with \(T<\infty.\) If the scalar curvature satisfies \[\int_0^T\int_M\;\Big((\Delta_{\varphi}R)^{p+1}+|R|^{2p}\Big)\,\omega_{\varphi}^ndt\leq C, \label{eq:theo1}\qquad{(1)}\] for \(p>n\), the Calabi flow can be extended past time \(T\).
In Theorem 1, we need to assume a technical condition on the space-time \(L^p\) bound of \(\Delta_\varphi R\), which seems inevitable if we calculate the time derivative of the evolving metrics. It is possible that the condition on \(\Delta_\varphi R\) can be replaced by some other geometric conditions, and we will discuss this problem in future papers.
Theorem 1 is similar to the results in other geometric flows such as Ricci flow and mean curvature flow. For Ricci flow, B. Wang proved that on a compact Riemannian manifold
of real dimension \(m\) the Ricci flow can be extended if \[\int_0^T\int_M\;|Rm|^{p}\,\omega_{\varphi}^ndt\leq C, \quad p\geq \frac{m+2}{2}. {\nonumber}\] G. Di Matteo extends Wang’s result
to some mixed integral norms of the curvature tensor. For mean curvature flow, Xu-Ye-Zhao proved that the mean curvature flow \(\Sigma^m_t\subset {\mathbb{R}}^{m+1}\) can be extended if \[\int_0^T\int_M\;|A|^{p}\,d\mu dt\leq C, \quad p\geq m+2. {\nonumber}\] Le-Seum also showed some extension results of mean curvature flow under some mixed integral norms of the second fundamental form. Since Ricci flow and mean
curvature flow are second-order geometric flows, the usual parabolic Moser iteration argument applies once the Sobolev inequality holds. However, since Calabi flow is a fourth-order flow, we need to overcome new difficulties.
We outline the proof of Theorem 1. The proof is divided into several steps:
The \(C^0\) estimates of \(F\) and \(\varphi\). Lu-Seyyedali proved the \(C^0\) estimates of \(F\) and \(\varphi\) under the assumption that the \(L^p(p>n)\) norm of the scalar curvature is bounded. In the proof of Theorem 1 we use the parabolic version of Lu-Seyyedali’s argument to show that \(\|F\|_{C^0}\) and \(\|\varphi\|_{C^0}\) are bounded under the condition (?? ) of Theorem 1. Recall that using the method of Guo-Phong-Tong , Chen-Cheng in proved the \(L^{\infty}\) estimate of the parabolic complex Monge-Ampere flow: \[-\frac{\partial \psi}{\partial t}(\omega_g+\sqrt{-1}\partial\bar\partial\psi)^n=e^G\omega_{\varphi}^n.\] Based on Chen-Cheng’s estimates, we show that \(\|F\|_{L^{1+\delta}(M\times [0, T), \omega_{\varphi})}\) is uniformly bounded along the flow. This together with the assumption of Theorem 1 implies that \(\|\varphi\|_{C^0}\) is bounded along the Calabi flow. Thus using the parabolic maximum principles we show that \(\|F\|_{C^0}\) is bounded.
Higher order estimates of \(F\) and \(\varphi.\) We follow the argument of Chen-Cheng , Li-Zhang-Zheng and the parabolic Moser iteration to show that the space-time quantities \[\int_0^T\int_M\;(n+\Delta_g\varphi)^q\,\omega_g^ndt,\quad \int_0^T\int_M\;|\nabla F|_{\varphi}^{2{\kappa}}\,\omega_{\varphi}^ndt\]are bounded for some \({\kappa}>2n\) and any \(q\geq 1\). Using these estimates and the parabolic Moser iteration argument, we show that \(\|n+\Delta_g\varphi\|_{C^0}\) is bounded. Thus, using similar argument as in Chen-Cheng the higher order estimates of \(F\) and \(\varphi\) can be obtained. The argument is based on the Sobolev inequality of Guo-Phong-Song-Sturm or Guedj-Tô .
The organization of this paper is as follows. In Sec. 2 we recall some basic notations and show the parabolic Sobolev inequality on Kähler manifolds. In Sec. 3 we first show the \(L^{\infty}\) norm of \(F\) and \(\varphi\), and then we show the space-time \(L^p\) estimates of \(n+\Delta_{g}\varphi\) and \(|\nabla F|_{\varphi}\), which implies the \(L^{\infty}\) norm of \(n+\Delta_{g}\varphi\). Finally, in Sec. 4 we show the higher-order estimates along the Calabi flow.
In this section, we recall some basic notations and results on Kähler manifolds. Let \((M, \omega_g)\) be a compact Kähler manifold with complex dimension \(n\). We define the space of Kähler potentials \[{\mathcal{H}}(\omega_g)=\{\varphi\in C^{\infty}(M, {\mathbb{R}})\;|\;\omega_g+\sqrt{-1}\partial\bar\partial\varphi>0\},\] and we define the subset \({\mathcal{H}}_0\) of \({\mathcal{H}}(\omega_g)\) by \[\begin{align} {\mathcal{H}}_0:=\{\varphi\in{\mathcal{H}}(\omega_g)\;|\; I_{\omega_g}(\varphi)=0\}, \end{align}\] where the functional \(I_{\omega_g}\) is defined by \[\begin{align} I_{\omega_g}(\varphi)=\frac{1}{(n+1)!}\sum_{k=0}^{n}\int_M\varphi \omega^k\wedge\omega_\varphi^{n-k}. \end{align}\] It is clear that for any path \(\varphi(t)\in {\mathcal{H}}\), we have \[\frac{d}{dt}I_{\omega_g}(\varphi(t))=\frac{1}{n!}\int_M\; \frac{\partial \varphi(t)}{\partial t}\omega_{\varphi(t)}^n. \label{eq:I}\tag{2}\] The \(K\)-energy is defined by \[{\mathcal{K}}(\varphi)=-\int_0^1\,\int_M\;\frac{\partial \varphi_t}{\partial t}(R(\omega_{\varphi_t})-\underline R)\,\frac{\omega_{\varphi_t}^n}{n!}.\]Note that along the Calabi flow we have \[\frac{d}{dt}{\mathcal{K}}(\varphi(t))=-\int_M\;(R(\omega_{\varphi(t)})-\underline R)^2\,\frac{\omega_{\varphi_t}^n}{n!}\leq 0.\]Therefore, the \(K\)-energy is non-increasing along the Calabi flow. It is known that the \(K\)-energy can be written as \[{\mathcal{K}}(\varphi)=\int_M\; \log \frac{\omega_{\varphi}^n}{\omega_g^n}\,\frac{\omega_{\varphi}^n}{n!}+J_{-Ric(\omega_g)}(\varphi),\]where for a \((1, 1)\) form \(\chi\), we define \[J_{\chi}(\varphi)=\int_0^1\,\int_M\;\frac{\partial \varphi_t}{\partial t}\Big(\chi\wedge \frac{\omega_{\varphi_t}^{n-1}}{(n-1)!}-\underline\chi \frac{\omega_{\varphi_t}^n}{n!}\Big)\omega_{\varphi_t}^n\wedge dt,\] where \(\varphi_t\in {\mathcal{H}}\) is a path connecting \(0\) and \(\varphi.\) Here \[\underline\chi=\frac{\int_M\;\chi\wedge \frac{\omega_g^{n-1}}{(n-1)!}}{\int_M\;\frac{\omega_g^n}{n!}}.\] For any function \(\varphi\in {\mathcal{H}}(\omega_g)\), we define the function \(F\) by \[(\omega_g+\sqrt{-1}\partial\bar\partial\varphi)^n =e^F\omega_g^n.\] Let \(\varphi(x, t)\) be a family of Kähler potentials. We denote by \(R\) the scalar curvature of the metric \(\omega_{\varphi(x, t)}\), and \(R_g\) to denote the scalar curvature of the metric \(\omega_g\). For simplicity, we write \[\begin{align} \|f\|_s&=&\Big(\int_0^T\,\int_M\;|f(x, t)|^s\,\omega_{\varphi(x, t)}^n\,dt\Big)^\frac{1}{s},{\nonumber}\\ \|f\|_{s, t}&=&\Big(\int_M\;|f(x, t)|^{s}\,\omega_{\varphi(x, t)}^n\Big)^{\frac{1}{s}}.{\nonumber} \end{align}\] We denote by \(|\nabla f|_{\varphi}\) (resp. \(|\nabla f|_g\)) the norm of the gradient of \(f\) with respect to the metric \(\omega_{\varphi}\) (resp. \(\omega_g\)). Moreover, we denote by \(\Delta_{\varphi}\) (resp. \(\Delta_g\)) the Laplace operator with respect to the metric \(\omega_{\varphi}\) (resp. \(\omega_g\)).
Now we recall the following interpolation inequality.
Lemma 2. (cf. [1], ) If \(0<p<r<q\), for any \(\epsilon>0\) we have \[\|f\|_{r, t}\leq\|f\|_{q, t}^{\theta}\|f\|_{p, t}^{1-\theta},\]where \(\theta=\frac{(r-p)q}{(q-p)r}\in (0, 1)\).
Following Guo-Phong-Song-Sturm or Guedj-Tô , the Sobolev constant of the metric \(\omega_{\varphi}\) is bounded under some conditions.
Theorem 3. (cf. , For any \(\gamma\in(1,\frac{n}{n-1})\) and \(u\in W^{1, 2}(M, \omega_{\varphi})\), we have the Sobolev inequality with respect to the metric \(\omega_\varphi\) \[\begin{align} \Big(\int_M\,|u|^{2\gamma}\;\omega_\varphi^n\Big)^{\frac{1}{\gamma}}\leq C(n,\omega_g,\gamma,\|F\|_\infty)\int_M \,(|u|^2+|\nabla u|_{\varphi}^2)\;\omega_\varphi^n. \end{align}\]
It is known that the following parabolic Sobolev inequality follows from Theorem 3, and we collect the proof for the readers’ convenience.
Lemma 4. For any \(0<\kappa<2<\beta<\gamma<\frac{2n}{n-1}\) and \(u\in W^{1, 2}(M\times [0, T), \omega_{\varphi})\), we have \[\int_{0}^{T}\;dt\int_M|u|^\beta\;\omega_{\varphi}^n\leq C\sup_{t\in [0,T)}\|u\|_{\kappa,t}^{(1-\frac{2}{\gamma})\kappa}\int_{0}^{T}\;dt\int_M\;\Big(|\nabla u|_\varphi^2+|u|^2\Big)\;\omega_{\varphi}^n.\label{eq:0461}\qquad{(2)}\] where \(C\) depends on \(\omega_g, n,\|F\|_\infty\) and \(\gamma\). Moreover, the constants \(\theta\in(0,1)\), \(\kappa, \beta, \gamma>0\) satisfy the conditions \[\frac{1}{\beta}=\frac{\theta}{\kappa}+\frac{1-\theta}{\gamma},\quad\quad (1-\theta)\beta=2.\label{eq:0460}\qquad{(3)}\]
Proof. Let \(\theta\), \(\kappa, \beta, \gamma>0\) be the constants satisfying (?? ). By Lemma 2, for any \(t\in[0,T)\), we have \[\|u\|_{\beta,t}\leq\|u\|_{\kappa,t}^{\theta}\|u\|_{\gamma,t}^{1-\theta}.{\nonumber}\] Now taking \(\beta\)-power and integrating with respect to \(t\), we get \[\begin{align} \int_{0}^{T}\,dt\int_M|u|^{\beta}\,\omega_\varphi^n&\leq& \sup_{[0,T)}\|u\|_{\kappa,t}^{\theta\beta} \int_{0}^{T}\|u\|_{\gamma,t}^{(1-\theta)\beta}\,dt{\nonumber}\\ &=&\sup_{[0,T)}\|u\|_{\kappa,t}^{\theta\beta} \int_{0}^{T}\|u\|_{\gamma,t}^{2}\,dt\label{eq:2467} \end{align}\tag{3}\] By Theorem 3, we have \[\|u\|_{\gamma,t}^2\leq C(\omega_g, n,\gamma,\|F\|_\infty)\int_M\Big(|\nabla u|^2+|u|^2\Big)\omega_\varphi^n.\] Substituting this result into (3 ) and using the assumption (?? ), we have the inequality (?? ). The lemma is proved. ◻
In this subsection, we use the parabolic version of Lu-Seyyedali to show that \(\|\varphi\|_{\infty}\) and \(\|F\|_{\infty}\) are bounded along the Calabi flow. To simplify the notations, we define the function \(\Phi(s)=\sqrt{1+s^2}\) and we introduce \(Q_F\), \(A_{R,p}\) and \(B_{R,p}\) as follows: \[\begin{align} Q_F&=&\Big(\int_{0}^{T}dt\int_M\Phi(F)\,\omega_\varphi^n\Big)^\frac{1}{n},\\ A_{R,p}&=&\Big(\int_{0}^{T}dt\int_M\Phi(R)^{p}\,\omega_{\varphi}^n\Big)^\frac{1}{n},\\ B_{R,p}&=&\Big(\int_{0}^{T}dt\int_M\Phi(\Delta_\varphi R)^{p}\,\omega_{\varphi}^n\Big)^\frac{1}{n}. \end{align}\]
The main result of this subsection is the following theorem.
Theorem 5. Let \(\varphi(x, t)(t\in [0, T))\) be the solution of Calabi flow (1 ) with \(T<\infty.\) If \(A_{R,p_1}\) and \(B_{R,p_2}\) are bounded with \(p_1> n+1\) and \(p_2> n+1\), and \(Q_F\) is also bounded. Then we have \[\|\varphi\|_{L^{\infty}(M\times [0, T)) }+ \|F\|_{L^{\infty}(M\times [0, T))}\leq C(n,\omega_g,Q_F, A_{R,p_1},B_{R,p_2}, \varphi(0), T). \label{eq:0465}\qquad{(4)}\]
First, we recall Chen-Cheng’s result.
Theorem 6. (cf. )Let \(T>0\). Consider the parabolic complex Monge-Ampere equation \[\begin{align} (-\partial_t\varphi)\;\omega_\varphi^n&=&e^H\omega_g^n,\label{eq:Z1}\\ \varphi(\cdot, 0)&=&\varphi_0. \label{eq:Z} \end{align}\] {#eq: sublabel=eq:eq:Z1,eq:eq:Z} We have the following results.
Assume that \(\varphi_0\in {\mathcal{H}}(\omega_g)\) and \(H(x, t)\) is smooth on \(M\times [0, T]\). Then there exists a unique smooth solution \(\varphi(x, t)\) to (?? )-(?? ) on \(M\times [0, T]\) starting from \(\varphi_0\) such that \(-\frac{\partial \varphi}{\partial t}>0\) and \(\omega_g+\sqrt{-1}\partial\bar\partial\varphi(x, t)>0.\)
If \(H\) satisfies the condition \[Ent_p(H):=\int_0^T\int_M\;e^H(|H|^p+1)\,\omega_g^n\,dt<\infty,\quad p>n+1,\]then we have \[\|\varphi\|_{L^{\infty}}\leq C\Big(\omega_g, p, n, \|\varphi_0\|_{L^{\infty}}, T, Ent_p(H)\Big).\]
The following result is proved by Lu-Seyyedali , and we conclude the proof for completeness.
Lemma 7. (cf. ) Let \(h:X\to\mathbf{R}\) be a positive smooth function and \(\varphi\) and \(v\) be Kähler potentials such that \[\begin{align} (\omega_g+\sqrt{-1}\partial\bar\partial\varphi)^n&=&e^F\omega_g^n,{\nonumber}\\ (\omega_g+\sqrt{-1}\partial\bar\partial v)^n&=&e^Fh^n\omega_g^n. \end{align}\] Then \(\Delta_\varphi v\geq nh-{\rm tr}_\varphi \omega_g\).
Proof. We compute \[\begin{align} \Delta_\varphi v&=&{\rm tr}_\varphi(\sqrt{-1}\partial\bar\partial v)={\rm tr}_\varphi(\omega_v-\omega_g) \geq n\Big(\frac{\omega_v}{\omega_\varphi}\Big)^{\frac{1}{n}}-{\rm tr}_\varphi\omega_g{\nonumber}\\ &\geq& n(e^{\frac{F}{n}} h)e^{-\frac{F}{n}}-{\rm tr}_\varphi\omega_g=nh-{\rm tr}_\varphi\omega_g .{\nonumber} \end{align}\] ◻
The next result shows that \(|\sup_M\varphi|\) is uniformly bounded along the Calabi flow.
Lemma 8. (cf. ) Let \(\varphi(t)(t\in [0, T))\) be a solution of Calabi flow (1 ) with \(T<\infty.\) Then \(|\sup_M\varphi|\) is bounded by \(\varphi(0)\) and \(T\).
Proof. The proof is divided into several steps.
(1). Let \(\psi(t)=\varphi(t+\frac{T}{2})\). Then \(\psi(t)(t\in[0,\frac{T}{2})\) is the solution to the Calabi flow. According to the distance \(d_2(\varphi(t),\psi(t))\) is non-increasing for \(t\in[0,\frac{T}{2})\). Therefore, \[\begin{align} d_2(\varphi(t),\psi(t))\leq d_2(\varphi(0),\psi(0))=d_2(\varphi(0),\varphi(\frac{T}{2})). \end{align}\] This implies that for any \(t\in[\frac{T}{2},T)\), we have \[\begin{align} d_2(\varphi(0),\varphi(t))&\leq& d_2(\varphi(0),\varphi(t-\frac{T}{2}))+d_2(\varphi(t-\frac{T}{2}),\varphi(t)){\nonumber}\\ &\leq&\max_{s\in[0,\frac{T}{2}]}d_2(\varphi(0),\varphi(s))+d_2(\varphi(0),\varphi(\frac{T}{2})).\label{eq:3461} \end{align}\tag{4}\]
(2). We show that \(d_1(\varphi(0),\varphi(t))\) is bounded. Indeed, for any two Kähler potentials \(\phi_0\), \(\phi_1\) and any smooth path \(\phi_s(s\in[0,1])\) connecting \(\phi_0\) and \(\phi_1\), we have \[\begin{align} L_1(\phi_0,\phi_1):=\int_{0}^{1}\|\phi_s\|_{L^1(\omega_{\phi_s)}}ds\leq {\rm vol}(\omega_g)^{\frac{1}{2}}\int_{0}^{1}\|\phi_s\|_{L^2(\omega_{\phi_s)}}ds:={\rm vol}(\omega_g)^{\frac{1}{2}}L_2(\phi_0,\phi_1).\label{eq:3467} \end{align}\tag{5}\] Taking the infimum with respect to all smooth path connecting \(\phi_0\) and \(\phi_1\), we have \[d_1(\phi_0, \phi_1)\leq {\rm vol}(\omega_{g})^{\frac{1}{2}} d_2(\phi_0, \phi_1). \label{eq:B13}\tag{6}\] Therefore, (4 ) and (6 ) imply that \(d_1(\varphi(0),\varphi(t))\) is bounded for \(t\in[0,T).\)
(3). We show that \(|\sup_M\varphi|\) is uniformly bounded for \(t\in[0,T)\). Without loss of generality, we may assume that \(\varphi(0)\in\mathcal{H}_0\). Then by the equality (2 ) we have \[\begin{align} \frac{d}{dt}I_\omega(\varphi(t))=\frac{1}{n!}\int_M\frac{\partial \varphi}{\partial t}\;\omega_{\varphi(t)}^n=\frac{1}{n!}\int_M(R-\underline{R})\omega_{\varphi(t)}^n=0.\label{eq:34610} \end{align}\tag{7}\] Thus (7 ) shows that \(\varphi(t)\in\mathcal{H}_0\) for all \(t\in[0,T).\) According to the Lemma 4.4 in Chen-Cheng, we have \[\begin{align} |\sup_M\varphi|\leq C\Big(d_1(0,\varphi)+1\Big)\leq C\Big(d_1(0,\varphi(0))+d_1(\varphi(0),\varphi(t))\Big)\label{eq:3469} \end{align}\tag{8}\] for some constant \(C\). Combining with (5 ) and (8 ), we conclude this lemma. ◻
Combining the above results, we show that the space-time integral of \(e^F\) is bounded for some \(q>1.\)
Lemma 9. Let \(\varphi(t)(t\in [0, T))\) be a solution of Calabi flow (1 ) with \(T<\infty.\) If \(A_{R,p_1}\) and \(B_{R,p_2}\) are bounded with \(\min\{p_1,p_2\}> n+1\) and \(Q_F\) is uniformly bounded, then there exist \(\delta_0>0\) and \(C\) depending on \(n,\omega_g,Q_F, A_{R,p_1}\), \(B_{R,p_2}\), \(\varphi(0)\) and \(T\) such that \[\int_{0}^{T}\,dt\int_{M}\;e^{(1+\delta_0)F}\,\omega_g^n\leq C.\label{eq:0462}\qquad{(5)}\]
Proof. We construct auxiliary functions \(\psi\), \(\rho\) and \(v\) as the solutions to the following equations: \[\begin{align} &&(-\partial_t\psi)\omega_{\psi}^n=Q_F^{-n}\Phi(F)e^F\omega_g^n; \;\; \psi\Big|_{t=0}=0,{\nonumber}\\ &&(-\partial_t\rho)\omega_{\rho}^n=A_{R,p_1}^{-n}\Phi(R)^{p_1}e^F\omega_g^n;\;\; \rho\Big|_{t=0}=0,{\nonumber}\\ &&(-\partial_t v)\omega_{v}^n=B_{R,p_2}^{-n}\Phi(\Delta_\varphi R)^{p_2}e^F\omega_g^n,\;\; v\Big|_{t=0}=0. \end{align}\] Note that the existence of \(\psi,\rho,v\) is guaranteed by Theorem 6. For \(0<\epsilon\leq 1\), we define \[u=F+\epsilon\psi+\epsilon\rho +\epsilon v-\lambda\varphi.\] Using Lemma 7, we can compute \[\begin{align} &&e^{-\delta u}(\Delta_\varphi-\partial_t)(e^{\delta u})\geq \delta\Delta_\varphi u-\delta \dot{u}{\nonumber}\\ &&\geq\delta(-R+{\rm tr}_{\varphi}{Ric(\omega_g)})+\epsilon\delta\Big(nQ_F^{-1}(-\dot{\psi})^{-\frac{1}{n}}\Phi(F)^{\frac{1}{n}}-{\rm tr}_\varphi \omega_g\Big){\nonumber}\\ &&\quad+\epsilon\delta\Big(nA_{R,p_1}^{-1}(-\dot{\rho})^{-\frac{1}{n}}\Phi(R)^{\frac{p_1}{n}}-{\rm tr}_\varphi\omega_g\Big)+\epsilon\delta\Big(nB_{R,p_2}^{-1}(-\dot{v})^{-\frac{1}{n}}\Phi(\Delta_\varphi R)^{\frac{p_2}{n}}-{\rm tr}_\varphi \omega_g\Big){\nonumber}\\ &&\quad-n\lambda\delta+\delta\lambda {\rm tr}_\varphi\omega_g+\delta\Big(-\Delta_\varphi R-\epsilon\dot{\psi}-\epsilon\dot{\rho}-\epsilon\dot{v}+\lambda\dot{\varphi}\Big),\label{eq:34614} \end{align}\tag{9}\] where we write \(\dot{u}=\partial_t u\) for short. Choosing \(\lambda=3+|{Ric(\omega_g)}|_g\) in (9 ), we have \[\begin{align} &&e^{-\delta u}(\Delta_\varphi-\partial_t)(e^{\delta u}){\nonumber}\\&&\geq \delta\Big(-R+\epsilon nA_{R,p_1}^{-1}(-\dot{\rho})^{-\frac{1}{n}}\Phi(R)^{\frac{p_1}{n}}-\epsilon\dot{\rho}\Big)+\delta\Big(-\Delta_\varphi R{\nonumber}\\ &&\quad+\epsilon nB_{R,p_2}^{-1}(-\dot{v})^{-\frac{1}{n}}\Phi(\Delta_\varphi R)^{\frac{p_2}{n}}-\epsilon\dot{v}\Big)+\delta\epsilon\Big(nQ_F^{-1}(-\dot{\psi})^{-\frac{1}{n}}\Phi(F)^{\frac{1}{n}}-\dot{\psi}\Big){\nonumber}\\ &&\quad-n\lambda\delta+\delta\lambda(R-\underline{R}){\nonumber}\\ &&\geq\delta\Big((\lambda-1)R+\epsilon A_{R,p_1}^{-1}(-\dot{\rho})^{-\frac{1}{n}}\Phi(R)^{\frac{p_1}{n}}-\epsilon\dot{\rho}\Big)+\delta\Big(-\Delta_\varphi R{\nonumber}\\ &&\quad+\epsilon B_{R,p_2}^{-1}(-\dot{v})^{-\frac{1}{n}}\Phi(\Delta_\varphi R)^{\frac{p_2}{n}}-\epsilon\dot{v}\Big){\nonumber}\\ &&\quad+\delta\epsilon\Big(nQ_F^{-1}(-\dot{\psi})^{-\frac{1}{n}}\Phi(F)^{\frac{1}{n}}-\dot{\psi}\Big)-C, \end{align}\] where \(C=n\lambda\delta+\delta\lambda\underline{R}\). Since \(Bx^{-\frac{1}{n}}+x\geq C(n)B^{\frac{n}{n+1}}\) for all \(x> 0\), we get \[\begin{align} e^{-\delta u}(\Delta_\varphi-\partial_t)(e^{\delta u})&\geq&\delta\Big((\lambda-1)R+C\epsilon\Phi(R)^{\frac{p_1}{n+1}}\Big)+\delta\Big(-\Delta_\varphi R+C\epsilon\Phi(\Delta_\varphi R)^{\frac{p_2}{n+1}}\Big){\nonumber}\\ &\quad&+\delta\epsilon C\Phi(F)^{\frac{1}{n+1}}-C \end{align}\] where \(C\) depends on \(n,A_{R,p_1}\), \(B_{R,p_2}\) and \(Q_F\). Let \(\hat{\Phi}(F)=\delta\epsilon C\Phi(F)^{\frac{1}{n+1}}\). As a result, we have \[\begin{align} \int_{0}^{T}\;dt\int_{M}\;(\Delta_\varphi-\partial_t)(e^{\delta u})\omega_{\varphi}^n&\geq& \int_{0}^{T}\;dt\int_{M}\;e^{\delta u}\Big(\delta\Big((\lambda-1)R+C\epsilon\Phi(R)^{\frac{p_1}{n+1}}\Big){\nonumber}\\ &\quad&+\delta\Big(-\Delta_\varphi R+C\epsilon\Phi(\Delta_\varphi R)^{\frac{p_2}{n+1}}\Big) +\hat{\Phi}(F)-C\Big)\omega_{\varphi}^n.{\nonumber}\\ \quad\label{eq:34617} \end{align}\tag{10}\] Using the equation of Calabi flow, we have \[\begin{align} \int_{0}^{T}\;dt\int_{M}\;(\Delta_\varphi-\partial_t)(e^{\delta u})\omega_{\varphi}^n&=&\int_{0}^{T}\;dt\int_{M}-\partial_te^{\delta u}\;\omega_{\varphi}^n{\nonumber}\\ &&=\int_{0}^{T}-\partial_t\Big(\int_Me^{\delta u}\omega_{\varphi}^n\Big)\;dt+\int_{0}^{T}\;dt\int_M\;e^{\delta u}\partial_t(\omega_{\varphi}^n){\nonumber}\\ &&\leq\int_Me^{\delta u}\omega_{\varphi}^n\Big|_{t=0}+\int_{0}^{T}\;dt\int_M\;e^{\delta u}\dot{F}\;\omega_{\varphi}^n,\label{eq:34618} \end{align}\tag{11}\] Combining (10 ) and (11 ), we get \[\begin{align} \int_Me^{\delta u}\omega_{\varphi}^n\Big|_{t=0}&\geq&\int_{0}^{T}\;dt\int_{M}e^{\delta u}\Big(\delta\Big((\lambda-1)R+C\epsilon\Phi(R)^{\frac{p_1}{n+1}}\Big){\nonumber}\\ &\quad&+\delta\Big(-(1+\frac{1}{\delta})\Delta_\varphi R+C\epsilon\Phi(\Delta_\varphi R)^{\frac{p_2}{n+1}}\Big) +\hat{\Phi}(F)-C\Big)\,\omega_{\varphi}^n. \end{align}\] Since \(Cx^{\beta}-x\) has lower bound which is independent of \(x\) for all \(\beta> 1\) and \(\min\{p_1,p_2\}> n+1\), we get \[\begin{align} \int_Me^{\delta u}\omega_{\varphi}^n\Big|_{t=0}\geq\int_{0}^{T}\;dt\int_{M}\;e^{\delta u}\Big(\hat{\Phi}(F)-C(\lambda,\delta,\epsilon,n,A_{R,p_1},B_{R,p_2},Q_F,\omega_g)\Big)\omega_{\varphi}^n.\label{eq:34619} \end{align}\tag{12}\] Choosing \(\delta=\lambda^{-1}\alpha(M,\omega_g)\), where \(\alpha(M,\omega_g)\) is the \(\alpha\) invariant of \(\omega_g\), we have that \[\begin{align} \int_{0}^{T}\;dt\int_{M}\;e^{\delta u}\Big(\hat{\Phi}(F)-C\Big)\;\omega_{\varphi}^n\leq C(\delta,\lambda,\varphi(0)).\label{eq:34620} \end{align}\tag{13}\] Next we define \[\begin{align} E_1&=&\{(x,t)\in M\times[0,T):\hat{\Phi}(F)-C\geq1\},{\nonumber}\\ E_2&=&\{(x,t)\in M\times[0,T):\hat{\Phi}(F)-C< 1\}. \end{align}\] By definition, \(F\) is bounded on \(E_2\). Thus by (12 ) and (13 ) we have \[\begin{align} \int_{E_1}e^{\delta u}\;\omega_{\varphi}^n\,dt&\leq& \int_{E_1}e^{\delta u}\Big(\hat{\Phi}(F)-C\Big)\;\omega_{\varphi}^n\,dt{\nonumber}\\ &\leq&C -\int_{E_2}e^{\delta u}\Big(\hat{\Phi}(F)-C\Big)\;\omega_{\varphi}^n\,dt{\nonumber}\\ &\leq&C+C\int_{E_2}e^{\delta u}\,\omega_{\varphi}^n\,dt{\nonumber}\\ &\leq& C+C\int_{E_2}e^{\delta F-\lambda\delta \varphi}\;\omega_{\varphi}^n\,dt{\nonumber}\\ &\leq& C(n,\delta,\epsilon,\lambda,A_{R,p_1},B_{R,p_2},Q_F,\omega_g,\varphi(0),T).\label{eq:A1} \end{align}\tag{14}\] By definition of \(u\), we have \[\begin{align} \int_{E_1}e^{(1+\delta)F+\epsilon\delta(\psi+\rho+v)}\,\omega_g^n\,dt\leq e^{\delta\lambda|\sup_{M}\varphi|}\int_{E_1}e^{\delta u+F}\,\omega_g^n\,dt. \end{align}\] Since \(|\sup_M\varphi|\) is bounded by Lemma 8, we conclude that \(\int_{E_1}e^{(1+\delta)F+\epsilon\delta(\psi+\rho+v)}\omega_g^ndt\) is bounded. Using Hölder inequality, we get \[\begin{align} &&\int_{E_1}e^{(1+\frac{\delta}{2})F}\omega_g^ndt=\int_{E_1}e^{(1+\frac{\delta}{2})F+\frac{1+\frac{\delta}{2}}{1+\delta}\epsilon\delta(\psi+\rho+v)}e^{-\frac{1+\frac{\delta}{2}}{1+\delta}\epsilon\delta(\psi+\rho+v)}\omega_g^ndt{\nonumber}\\ &\leq&\Big(\int_{E_1}e^{(1+\delta)F+\epsilon\delta(\psi+\rho+v)}\omega_g^ndt\Big)^{\frac{1+\frac{\delta}{2}}{1+\delta}}\Big(\int_{E_1}e^{-\frac{1+\frac{\delta}{2}}{\frac{\delta}{2}}\epsilon\delta(\psi+\rho+v)}\omega_g^ndt\Big)^{\frac{\frac{\delta}{2}}{1+\delta}}. \end{align}\] Choosing \(\epsilon\) small enough such that \((2+\delta)\epsilon<\frac{\alpha(M,\omega_g)}{3}\), we conclude that \[\begin{align} \int_{E_1}e^{(1+\frac{\delta}{2})F}\omega_g^ndt\leq C(n,\lambda,\delta,\epsilon,A_{R,p_1},B_{R,p_2},Q_F,\omega_g,\varphi(0),T). \label{eq:0463} \end{align}\tag{15}\] Combining (15 ) with the fact that \(F\) is bounded on \(E_2\), we have (?? ). The lemma is proved. ◻
Using Lemma 9 and the Calabi flow equation, we show that the \(L^{q}(M, \omega_g)\) norm of \(F\) is bounded for some \(q>1.\)
Lemma 10. Under the assumption of Theorem 5, there exist \(\delta_1\) and \(C\) depending on \(n,\omega_g,Q_F, A_{R,p_1}\), \(B_{R,p_2}\), \(\varphi(0)\) and \(T\) such that \[\begin{align} \int_M e^{\delta_1 F}\,\omega_\varphi^n\leq C.\label{eq:0464} \end{align}\qquad{(6)}\]
Proof. Let \(\delta> 0\). Taking the derivative with respect to \(t\), we find that \[\begin{align} \frac{\partial}{\partial t}\Big(\int_M\;e^{\delta F}\omega_{\varphi}^n\Big)=\int_M(1+\delta)\dot{F}e^{\delta F}\;\omega_{\varphi}^n. {\nonumber} \end{align}\] Hence we have \[\begin{align} \int_M\;e^{\delta F}\;\omega_{\varphi}^n\Big|_t-\int_M\;e^{\delta F}\;\omega_{\varphi}^n\Big|_0\leq\int_{0}^{T}\;dt\int_{M}(1+\delta)|\dot{F}|e^{\delta F}\omega_{\varphi}^n. {\nonumber} \end{align}\] Using the Hölder inequality we have \[\begin{align} \int_{0}^{T}\;dt\int_{M}\;(1+\delta)\dot{F}e^{\delta F}\;\omega_{\varphi}^n\leq(1+\delta)\Big(\int_{0}^{T}\;dt\int_{M}\;e^{l\delta F}\;\omega_{\varphi}^n\Big)^{\frac{1}{l}}\Big(\int_{0}^{T}\;dt\int_{M}\;|\Delta_\varphi R|^{p_2}\;\omega_{\varphi}^n\Big)^{\frac{1}{p_2}},{\nonumber} \end{align}\] where \(\frac{1}{l}+\frac{1}{p_2}=1\). Choosing \(\delta\) small and using Lemma 9, we have (?? ). The lemma is proved. ◻
Combining the above estimates and using the maximum principles, we show Theorem 5.
Proof of Theorem 5. By Theorem 6 and Lemma 9, we conclude that \(\psi\) is bounded. Moreover by Lemma 10 we conclude that \(\varphi\) is also uniformly bounded. We define new auxiliary functions as the solutions of the following equations: \[\begin{align} (-\partial_t\rho)\omega_{\rho}^n&=&A_{R,q}^{-n}\Phi(R)^qe^F\omega_g^n,\;\; \rho\Big|_{t=0}=0,{\nonumber}\\ (-\partial_t v)\omega_{v}^n&=&B_{R,q}^{-n}\Phi(\Delta_\varphi R)^qe^F\omega_g^n,\;\; v\Big|_{t=0}=0, \end{align}\] where \(n+1< q< \min\{p_1,p_2\}\). For \(0<\sigma< \delta_0\), we have \[\begin{align} &&\int_{0}^{T}\;dt\int_{M}\;|\Phi(R)|^{(1+\sigma)q}e^{(1+\sigma)F}\;\omega_g^n=\int_{0}^{T}\;dt\int_{M}\;|\Phi(R)|^{(1+\sigma)q}e^{\sigma F}\;\omega_{\varphi}^n{\nonumber}\\ &\leq&\Big(\int_{0}^{T}\;dt\int_{M}\;|\Phi(R)|^{(1+\sigma)q\frac{\delta_0}{\delta_0-\sigma}}\;\omega_{\varphi}^n\Big)^{\frac{\delta_0-\sigma}{\delta_0}}\Big(\int_{0}^{T}\;dt\int_{M}\;e^{\delta_0F}\;\omega_\varphi^n\Big)^{\frac{\sigma}{\delta_0}}{\nonumber}\\ &\leq& C(n,\omega_g,Q_F,A_{R,p_1},B_{R,p_2},\varphi(0),T) \Big(\int_{0}^{T}\;dt\int_{M}\;|\Phi(R)|^{(1+\sigma)q \frac{\delta_0}{\delta_0-\sigma}}\omega_{\varphi}^n\Big)^{\frac{\delta_0-\sigma}{\delta_0}},{\nonumber} \end{align}\] where we used Lemma 9 in the last inequality. Now we can choose \(\sigma\) small enough such that \((1+\sigma)\frac{\delta_0}{\delta_0-\sigma}q< p_1\). Therefore, we conclude that \(\rho\) is bounded by Theorem 6. Similarly, we have that \(v\) is also bounded. Let \(u=F+\psi+\rho+v-\lambda \varphi\), we have \[\begin{align} (\Delta_\varphi-\partial_t)u\geq e^{\delta u}\Big(\hat{\Phi}(F)-C\Big), \end{align}\] where we use the same argument as in the proof of Theorem 9 and \(C\) depends on \(n,\omega_g,A_{R,p_1},B_{R,p_2}\) and \(Q_F\). Fixing \(\epsilon> 0\), we denote \((x_0,t_0)\) the maximum point of \(u\) on \(M\times[0,T-\epsilon]\). We have \[0\geq (\Delta_\varphi-\partial_t)u\geq e^{\delta u}\Big(\hat{\Phi}(F)-C\Big).{\nonumber}\] This implies that \(|F(x_0,t_0)|\) is bounded. As a result, for any \((x,t)\in M\times [0,T-\epsilon]\) \[\begin{align} u(x,t)\leq u(x_0,t_0)&=&F(x_0,t_0)+(\psi+\rho+v-\lambda\varphi)((x_0,t_0){\nonumber}\\&\leq& C(n,\omega_g,Q_F,A_{R,p_1},B_{R,p_2},\varphi(0),T).{\nonumber} \end{align}\] This implies \(F\leq C\). Replacing \(u\) by \(u'=-F+\psi+\rho+v-\lambda\varphi\), the same argument shows that \(F\geq -C\). Therefore we conclude that on \(M\times[0,T-\epsilon]\), \[|F|\leq C(n,\omega_g,Q_F, A_{R,p_1},B_{R,p_2}, \varphi(0), T).{\nonumber}\] Taking \(\epsilon\rightarrow 0\), we have (?? ). The theorem is proved. ◻
In this subsection, we follow similar method as in Chen-Cheng and Li-Zhang-Zheng to prove that \(\|n+\Delta\varphi\|_{s}\) is bounded if \(Q_F,A_{R,2p}^n,B_{R,p+1}^n\) are bounded with \(p> n\). We recall the following Chen-Cheng’s estimates in , see also Li-Zhang-Zheng .
Lemma 11. (cf. , ) We define \[v=e^{-\alpha(F+\lambda\varphi)}(n+\Delta_g\varphi). \label{eq:v3}\qquad{(7)}\] Let \(q>1\) and \(\alpha\geq q\). There exists a constant \(C(\omega_g)\) such that for \(\lambda>C(\omega_g)\), we have \[\begin{align} && \frac{3(q-1)}{q^2}\int_M\; |\nabla v^{\frac{q}{2}}|_{\varphi}^2\;\omega_{\varphi}^n+\frac{\lambda\alpha}{4}\int_M\; e^{\frac{\alpha{n-1}}{(}F+\lambda\varphi)-\frac{F}{n-1}} v^{q+\frac{1}{n-1}} \,\omega_{\varphi}^n \nonumber\\ &\leq & \int_M\; \tilde{R}v^{q}\omega_{\varphi}^n,\label{eq:v12} \end{align}\qquad{(8)}\] where \(\tilde{R}=\alpha(\lambda n-R)+\frac{\alpha\lambda}{\alpha-1}+\frac{1}{n} e^{-\frac{F}{n}}R_g\) .
Using the equation (1 ) of Calabi flow, we have
Lemma 12. Let \(v=e^{-\alpha(F+\lambda\varphi)}(n+\Delta_g\varphi)\) as in Lemma 11. For any \(q> 0\), we have \[\begin{align} \int_M\; v^{q}\;\omega_{\varphi}^n\Big|_t-\int_M\; v^{q}\;\omega_{\varphi}^n\Big|_0&\leq& Cq\Big(\int_{0}^{T}\;dt\int_{M}\;v^{qr}\;\omega_{\varphi}^n\Big)^{\frac{1}{r}}+Cq\Big(\int_{0}^{T}\;dt\int_{M}\;v^{qb}\;\omega_{\varphi}^n\Big)^{\frac{1}{b}}{\nonumber}\\\ &&+Cq\Big(\int_{0}^{T}\;dt\int_{M}\;v^{2q}\;\omega_{\varphi}^n\Big)^{\frac{1}{2}}, \label{eq:0466} \end{align}\qquad{(9)}\] where C depends on \(\alpha, A_{R,2p}^n, B_{R,p+1}^n, \|\varphi\|_\infty\) and \(\|F\|_\infty\). Moreover, \(p,r\) and \(b\) satisfy the following conditions: \[\frac{1}{2p}+\frac{1}{r}=1,\quad\quad \frac{1}{p+1}+\frac{1}{b}=1.\label{eq:2464}\qquad{(10)}\]
Proof. Taking the derivative with respect to \(t\), we get \[\begin{align} \frac{\partial}{\partial t}\Big(\int_M\; v^{q}\;\omega_\varphi^n\Big)=\int_M \;\Big(q v^{q-1}\dot{v}+v^{q}\Delta_{\varphi} R\Big)\;\omega_{\varphi}^n\label{eq:Q}. \end{align}\tag{16}\] Putting \(\dot{v}=-\alpha(\dot{F}+\lambda\dot{\varphi})v+e^{-\alpha(F+\lambda\varphi)}\Delta_g R\) into (16 ), we have \[\begin{align} \frac{\partial}{\partial t}\Big(\int_M \;v^{q}\;\omega_\varphi^n\Big)&=&\int_M\;\Big( -\alpha qv^q(\dot{F}+\lambda\dot{\varphi})+qv^{q-1}e^{-\alpha(F+\lambda\varphi)}\Delta_g R+v^q\Delta_{\varphi}R\Big)\;\omega_{\varphi}^n{\nonumber}\\ &=&\int_M\;\Big( (1-\alpha q)v^q\Delta_{\varphi}R-\alpha qv^q\lambda(R-\underline R)+qv^{q-1}e^{-\alpha(F+\lambda\varphi)}\Delta_g R\Big)\;\omega_{\varphi}^n.{\nonumber}\\\label{eq:2465} \end{align}\tag{17}\] Let \(p,r\) and \(b\) be the constants satisfying (?? ). Integrating both sides of (17 ) with respect to \(t\) and using the Hölder inequality, we have \[\begin{align} \int_M \;v^{q}\omega_{\varphi}^n\Big|_t-\int_M v^{q}\;\omega_{\varphi}^n\Big|_0&\leq& Cq\Big(\int_{0}^{T}\;dt\int_{M}\;v^{qr}\;\omega_{\varphi}^n\Big)^{\frac{1}{r}}+Cq\Big(\int_{0}^{T}\;dt\int_{M}\;v^{qb}\;\omega_{\varphi}^n\Big)^{\frac{1}{b}}{\nonumber}\\&+&Cq\int_{0}^{T}\;dt\int_{M}\;v^{q-1}|\Delta_g R|\;\omega_{\varphi}^n, \label{eq:A3} \end{align}\tag{18}\] where \(C\) depends on \(\alpha, \|\varphi\|_\infty,\|F\|_\infty, A_{R,2p}\) and \(B_{R,p+1}\). Using the inequality \(|\Delta_g R|\leq |\nabla^2R|_{\varphi}(n+\Delta_g\varphi)\), we have \[\begin{align} \int_{0}^{T}dt\int_{M}v^{q-1}|\Delta_g R|\,\omega_{\varphi}^n\leq C(\|\varphi\|_\infty,\|F\|_\infty) \Big(\int_{0}^{T}dt\int_{M}v^{2q}\;\omega_{\varphi}^n\Big)^{\frac{1}{2}}\Big(\int_{0}^{T}dt\int_{M}|\nabla^2R|_\varphi^2\,\omega_{\varphi}^n\Big)^{\frac{1}{2}}.{\nonumber}\\ \label{eq:34} \end{align}\tag{19}\] Note that \[\int_{0}^{T}\;dt\int_{M}\;|\nabla^2R|_{\varphi}^2\;\omega_{\varphi}^n= \int_{0}^{T}\;dt\int_{M}\;|\Delta_{\varphi}R|^2\;\omega_{\varphi}^n\label{eq:A4}\tag{20}\] and \(B_{R,p+1}^n\) is bounded with \(p> n+1\geq 2\). Combining (18 )-(20 ), we have the inequality (?? ). ◻
Combining Lemma 11, Lemma 12 with Lemma 4, we have the result.
Lemma 13. Under the assumption that \(Q_F,A_{R,2p}\) and \(B_{R,p+1}\) are bounded with \(p> n\), for any \(s\geq 1\), there exists a constant \(C\) depending on \(n,s,\omega_g,Q_F,A_{R,2p},B_{R,p+1}\) and \(\varphi(0)\) and \(T\) such that \[\int_{0}^{T}\;dt\int_{M}\;(n+\Delta_g \varphi)^s\;\omega_{\varphi}^n\leq C. \label{eq:0467}\qquad{(11)}\]
Proof. By Lemma 4 and Lemma 11, for any \(q> 1\) we have \[\begin{align} &&\int_{0}^{T}dt\int_Mv^{\frac{\beta q}{2}}\;\omega_{\varphi}^n{\nonumber}\\&& \leq C(n,\omega_g,\|F\|_\infty,\gamma)\sup_{[0,T)}\|v^{\frac{q}{2}}\|_{\kappa,t}^{(1-\frac{2}{\gamma})\kappa}\int_{0}^{T}dt\int_M(|\nabla v^{\frac{q}{2}}|_\varphi^2+|v|^q)\;\omega_{\varphi}^n{\nonumber}\\ &&\leq C\frac{q^2}{3(q-1)}\sup_{t\in[0,T)}\|v^{\frac{q}{2}}\|_{\kappa,t}^{(1-\frac{2}{\gamma})\kappa}\int_{0}^{T}dt\int_M(\tilde{R}+1)v^q\,\omega_{\varphi}^n{\nonumber}\\ &&\leq C(n,\omega_g,\|F\|_\infty,\gamma,A_{R,2p})\frac{q^2}{3(q-1)}\sup_{t\in[0,T)}\|v^{\frac{q}{2}}\|_{\kappa,t}^{(1-\frac{2}{\gamma})\kappa}\Big(\int_{0}^{T}\;dt\int_M\;v^{qr}\;\omega_{\varphi}^n\Big)^{\frac{1}{r}}.{\nonumber}\\\label{eq:B1} \end{align}\tag{21}\] Note that \(\|v^{\frac{q}{2}}\|_{\kappa,t}^{(1-\frac{2}{\gamma})\kappa}=\|v\|_{\frac{q\kappa}{2},t}^{(\frac{1}{2}-\frac{1}{\gamma})\kappa q}\). By Lemma 12 we have \[\begin{align} \|v^{\frac{q}{2}}\|_{\kappa,t}^{(\frac{1}{2}-\frac{1}{\gamma})\kappa }&\leq& \Big(\int_M\;v^{\frac{q\kappa}{2}}\;\omega_{\varphi}^n\Big|_{t=0}+\frac{Cq\kappa}{2}\Big(\int_{0}^{T}\;dt\int_{M}\;v^{\frac{q\kappa r}{2}}\;\omega_{\varphi}^n\Big)^{\frac{1}{r}}{\nonumber}\\&\quad&+\frac{Cq\kappa}{2}\Big(\int_{0}^{T}\;dt\int_{M}\;v^{\frac{q\kappa b}{2}}\;\omega_{\varphi}^n\Big)^{\frac{1}{b}}+\frac{Cq\kappa}{2}\Big(\int_{0}^{T}\;dt\int_{M}\;v^{q\kappa}\;\omega_{\varphi}^n\Big)^{\frac{1}{2}}\Big)^{1-\frac{2}{\gamma}}.{\nonumber} \end{align}\] Taking the \(\frac{\beta q}{2}\)-root in (21 ), we have \[\begin{align} \|v\|_{\frac{\beta q}{2}}&\leq& C^{\frac{2}{\beta q}}\Big(\frac{q^2}{3(q-1)}\Big)^{\frac{2}{\beta q}}\Big(C +\frac{Cq\kappa}{2}\Big(\int_{0}^{T}\;dt\int_{M}\;v^{\frac{q\kappa r}{2}}\;\omega_{\varphi}^n\Big)^{\frac{1}{r}} {\nonumber}\\&\quad&+\frac{Cq\kappa}{2}\Big(\int_{0}^{T}\;dt\int_{M}\;v^{\frac{q\kappa b}{2}}\;\omega_{\varphi}^n\Big)^{\frac{1}{b}}+\frac{Cq\kappa}{2}\Big(\int_{0}^{T}\;dt\int_{M}\;v^{q\kappa}\;\omega_{\varphi}^n\Big)^{\frac{1}{2}}\Big)^{\frac{2\theta}{q\kappa}}\|v\|_{qr}^{\frac{2}{\beta}},{\nonumber}\\ \quad \end{align}\] where \(C\) depends on \(\alpha, n,\omega_g,\|\varphi\|_\infty,\|F\|_\infty,A_{R,2p},B_{R,p+1},\gamma\) and \(\varphi(0)\). Since \(p> n\), we have that \(r=\frac{2p}{2p-1}< \frac{2n}{2n-1}< 2\) and \(b=\frac{p+1}{p}<2\). We choose \(\beta\) and \(\kappa\) such that \[\begin{align} \frac{\beta}{2}> \max\{\kappa,r\},\label{eq:34641} \end{align}\tag{22}\] or equivalently, \[\begin{align} \frac{2r-2}{1-\frac{2}{\gamma}}<\kappa<\frac{2}{1+\frac{2}{\gamma}}.\label{eq:B8} \end{align}\tag{23}\] Since \(r< \frac{2n}{2n-1}\), we can choose \(\gamma\) close to \(\frac{2n}{n-1}\) such that \(\frac{2r-2}{1-\frac{2}{\gamma}}<\frac{2}{1+\frac{2}{\gamma}}\). For such \(\kappa\), \(\gamma\) and large \(q\) with \(q\kappa> 1\), we have \[\begin{align} \|v\|_{\frac{\beta q}{2}}&\leq&C^{\frac{2}{\beta q}}\Big(\frac{q^2}{3(q-1)}\Big)^{\frac{2}{\beta q}}\Big(C+Cq\kappa\|v\|_{{q\kappa}}^{\frac{q\kappa}{2}}\Big)^{\frac{2\theta}{q\kappa}}\|v\|_{{qr}}^{\frac{2}{\beta}}{\nonumber}\\ &\leq&C^{\frac{2}{\beta q}}\Big(\frac{q^2}{3(q-1)}\Big)^{\frac{2}{\beta q}}C^{\frac{2\theta}{q\kappa}}(q\kappa)^{\frac{2\theta}{q\kappa}} \|v\|_{{q\max\{r,\kappa\}}}, \end{align}\] where \(C\) depends on \(\alpha, \omega_g,\kappa,\gamma,\|\varphi\|_\infty,\|F\|_\infty, \varphi(0),A_{R,2p},B_{R,p+1}\) and in the last inequality we used the fact that \[\begin{align} v=e^{-\alpha(F+\lambda\varphi)}(n+\Delta_g\varphi)\geq C(\alpha,\|\varphi\|_\infty,\|F\|_\infty)\frac{1}{n}e^{\frac{F}{n}}\geq C(n,\alpha,\|\varphi\|_\infty,\|F\|_\infty). \end{align}\] Let \(\alpha=2p.\) By the iteration argument there exists \(q_0> 1\) such that for any \(q> q_0\) we have \[\begin{align} \|v\|_{q}\leq C(n,\omega_g,q,\kappa,\gamma,\|F\|_\infty,\|\varphi\|_\infty,A_{R,2p},B_{R,p+1},\varphi(0))\|v\|_{q_0}.\label{eq:0468} \end{align}\tag{24}\] Since \(\|v\|_{q_0}\leq\epsilon\|v\|_{q}+C(\epsilon)\|v\|_{1}\), we have \(\|v\|_{q}\leq C\|v\|_{1}\) for small \(\epsilon\). Now \[\begin{align} \|v\|_{1}&=&\int_{0}^{T}dt\int_M\;e^{-\alpha(F+\lambda\varphi)} (n+\Delta_g\varphi)\;\omega_{\varphi}^n\\&\leq & C(q,\|F\|_\infty,\|\varphi\|_\infty)\int_{0}^{T}\;dt\int_M(n+\Delta_g\varphi)\;\omega_g^n \\&\leq& C(n, q,\|\varphi\|_\infty,\|F\|_\infty,T).{\nonumber} \end{align}\]Combining this with (24 ), we have the inequality (?? ). The lemma is proved. ◻
In this subsection we show that \(\|\nabla F\|_{{2s}}\) is bounded for any \(s< 2p\). Note that we assumed the condition that \(p>n\) in the assumption of Theorem 1.
Lemma 14. Under the assumption of Lemma 13, for any \(s<2p\) there exists a constant \(C\) depending on \(n,s,\omega_g,Q_F,A_{R,2p},B_{R,p+1},\varphi(0)\) and \(T\) such that \[\begin{align} \int_{0}^{T}\;dt\int_{M}\;|\nabla F|_{\varphi}^{2s}\;\omega_{\varphi}^n \leq C. \end{align}\]
To show Lemma 14, we first show the following result by using the equation (1 ) of Calabi flow.
Lemma 15. Let \(w=e^{\frac{F}{2}}|\nabla F|_{\varphi}^2+1\) and \(z=w^q\) with \(q> \frac{1}{\kappa}\). We have \[\begin{align} && \|z\|_{\kappa,t}^{\kappa}-\|z\|_{\kappa,0}^\kappa\leq Cq\kappa\|z\|_{b\kappa}^\kappa+Cq\kappa(q\kappa -\frac{1}{2})\Big(\frac{2q\kappa}{q\kappa -1}\Big)^{\frac{1}{2}}\|z\|_{\frac{d(2q\kappa -2)}{q}}^{\frac{q\kappa -1}{q}}{\nonumber}\\&&+Cq\kappa(q\kappa -\frac{1}{2})(q\kappa-1)^{-\frac{1}{2}}\|z\|_{\frac{r(2q\kappa-1)}{q}}^{\frac{2q\kappa-1}{2q}}+Cq\kappa\|z\|_{\frac{a(q\kappa -1)}{q}}^{\frac{q\kappa -1}{q}}+Cq\kappa\|z\|_{2\kappa}^\kappa, \end{align}\] where \(C\) only depends on \(n,p,\omega_g,Q_F,A_{R,2p}\), \(B_{R,p+1},\varphi(0)\) and \(T\). Here \(a\), \(p\) and \(d\) satisfy the following conditions: \[\frac{1}{a}+\frac{1}{2p}+\frac{1}{p+1}=1,\quad\quad \frac{1}{d}+\frac{1}{p}=1.\]
Proof. Taking the derivative with respect to \(t\) and using (1 ), we get \[\begin{align} \frac{\partial}{\partial t}\|z\|_{\kappa,t}^{\kappa}=\frac{\partial}{\partial t}\int_{M}\;w^{\kappa q}\;\omega_{\varphi}^n=\int_{M}\;\Big(\kappa qw^{\kappa q-1}\dot{w}+w^{\kappa q}\Delta_{\varphi}R\Big)\;\omega_{\varphi}^n.\label{eq:A2c} \end{align}\tag{25}\] Note that \[\begin{align} \dot{w}&=&\frac{\partial}{\partial t}(e^{\frac{1}{2} F}|\nabla F|_{\varphi}^2){\nonumber}\\&=&\frac{1}{2} \dot{F}(w-1)+2e^{\frac{1}{2} F}\text{Re}(\nabla\Delta_\varphi R\cdot_\varphi\nabla F)-e^{\frac{1}{2} F}\nabla^2R(\nabla F,\nabla F), \label{eq:12} \end{align}\tag{26}\] Therefore, (26 ) and (25 ) imply that \[\begin{align} \|z\|_{\kappa,t}^{\kappa}-\|z\|_{\kappa,0}^\kappa&=&\int_{0}^{t}\;dt\int_{M}\;\Big(\kappa qw^{\kappa q-1}\Big(\frac{1}{2} \dot{F}(w-1)+2e^{\frac{1}{2} F}\text{Re}(\nabla\Delta_{\varphi}R\cdot_{\varphi}\nabla F){\nonumber}\\&\quad&-e^{\frac{1}{2} F}\nabla^2R(\nabla F,\nabla F)\Big)+w^{\kappa q}\Delta_{\varphi}R\;\Big)\omega_{\varphi}^n{\nonumber}\\&:=&I_0+I_1+I_2+I_3+I_4. \end{align}\] We will estimate each term \(I_i\). By direct calculation, we have \[\begin{align} I_0&=&\int_{0}^{t}\;dt\int_{M}\;\frac{\kappa q}{2}z^\kappa\dot{F}\;\omega_{\varphi}^n\leq C q\kappa\|z\|_{b\kappa}^{\kappa},\\ I_1&=&-\int_{0}^{t}\;dt\int_M \;\frac{q\kappa}{2} w^{\kappa q-1}\dot{F}\;\omega_\varphi^n\leq C q\kappa\|z\|_{b\kappa}^{\kappa},\label{eq:34652} \end{align}\tag{27}\] where \(C\) depends on \(B_{R,p+1}\). Moreover, we have \[\begin{align} I_2&=&\int_{0}^{t}\;dt\int_{M}\;2\kappa qw^{\kappa q-1}e^{\frac{F}{2}}\text{Re}(\nabla\Delta_{\varphi }R\cdot_{\varphi}\nabla F)\,\omega_\varphi^n{\nonumber}\\&=&-2q\kappa\int_{0}^{t}\;dt\int_{M}\;\nabla(w^{\kappa q-1}e^{\frac{F}{2}}\nabla F)\Delta_{\varphi} R\;\omega_{\varphi}^n{\nonumber}\\ &=&-2q\kappa\int_{0}^{t}\;dt\int_{M}\;\Big((\kappa q-1)w^{\kappa q-2}\nabla w\cdot_{\varphi}\nabla F e^{\frac{F}{2}}\Delta_{\varphi }R{\nonumber}\\ &&+\frac{1}{2}w^{\kappa q-1}e^{\frac{1}{2}F}|\nabla F|_{\varphi}^2\Delta_{\varphi} R+w^{\kappa q-1}e^{\frac{F}{2}}\Delta_{\varphi}F\Delta_\varphi R\Big)\;\omega_{\varphi}^n.{\nonumber} \end{align}\] Therefore we have \[\begin{align} I_2&\leq& 2q\kappa\int_{0}^{T}\;dt\int_{M}\;\Big((\kappa q-1)w^{\kappa q-\frac{3}{2}}|\nabla w|_{\varphi}|\Delta_{\varphi}R|+\frac{1}{2}w^{\kappa q}|\Delta_{\varphi}R|{\nonumber}\\&&+w^{\kappa q-1}e^{\frac{F}{2}}|\Delta_{\varphi}F||\Delta_{\varphi}R|\Big)\;\omega_{\varphi}^n{\nonumber}\\ &\leq& 2q\kappa\int_{0}^{T}\;dt\int_{M}\frac{2q\kappa-2}{2q\kappa-1}|\nabla w^{q\kappa-\frac{1}{2}}|_{\varphi}|\Delta_{\varphi}R|\,\omega_{\varphi}^n+Cq\kappa||z||_{\kappa b}^{\kappa}+Cq\kappa||w^{\kappa q-1}||_{a}{\nonumber}\\ &\leq &Cq\kappa\Big(\int_{0}^{T}\;dt\int_{M}|\;\nabla w^{\kappa q-\frac{1}{2}}|_\varphi^2\;\omega_{\varphi}^n\Big)^{\frac{1}{2}}+Cq\kappa\|z\|_{\kappa b}^{\kappa}+Cq\kappa\|z\|_{\frac{a(\kappa q-1)}{q}}^{\frac{\kappa q-1}{q}}, \label{eq:1461} \end{align}\tag{28}\] where \(C\) depends on \(A_{R,2p},B_{R,p+1},\|F\|_\infty,\|n+\Delta_g\varphi\|_{2p(n-1)}\) and we used the fact that \(\Delta_\varphi F\in L^{2p}(M\times[0,T),\omega_\varphi^n\wedge dt)\) in the second inequality. In fact, we have \[\begin{align} \Delta_\varphi F&=&-R+{\rm tr}_\varphi{Ric(\omega_g)}\leq -R+C(g)\sum_{i=1}^{n}\frac{1}{1+\varphi_{i\overline{i}}}{\nonumber}\\ &\leq&-R+C(g)(n+\Delta\varphi)^{n-1}e^{-F}=-R+C(g)\tilde{v}^{n-1}e^{-F},\label{eq:34655} \end{align}\tag{29}\] where \(\tilde{v}=n+\Delta_g\varphi\). By Lemma 13 we have \(\tilde{v}\in L^{s_0}(M\times[0,T),\omega_\varphi^n\wedge dt)\) for any \(s_0>1\). Therefore, we have \(\Delta_\varphi F\in L^{2p}(M\times[0,T),\omega_\varphi^n\wedge dt)\).
Using (2.31) of Li-Zhang-Zheng , for any \(q>0\) we have \[\begin{align} && \int_{0}^{T}dt\int_M|\nabla(w^{q+\frac{1}{2}})|_\varphi^2\,\omega_\varphi^n\leq C(\omega_g,\|F\|_\infty)(q+\frac{1}{2})^2\int_{0}^{T}dt\int_M\,\Big(\frac{q+1}{q}w^{2q}R^2{\nonumber}\\&\quad&+\frac{q+1}{q}w^{2q}\tilde{v}^{2n-2}+\frac{1}{q} w^{2q+1}|R|+\frac{1}{q}w^{2q+1}\tilde{v}^{n-1}\Big)\,\omega_\varphi^n. \label{eq:1462} \end{align}\tag{30}\] Combining (30 ) with (28 ), we have \[\begin{align} I_2&\leq& Cq\kappa(q\kappa -\frac{1}{2})\Big(\int_{0}^{T}\;dt\int_{M}\;\Big(\frac{q\kappa }{q\kappa -1}w^{2\kappa q-2}R^2+\frac{q\kappa }{q\kappa -1}w^{2q\kappa -2}\tilde{v}^{2n-2}+\frac{1}{q\kappa-1}w^{2q\kappa -1}|R|{\nonumber}\\&\quad&+\frac{1}{q\kappa-1}\tilde{v}^{n-1}w^{2q\kappa -1}\Big)\;\omega_{\varphi}^n\Big)^{\frac{1}{2}} +Cq\kappa\|z\|_{\kappa b}^{\kappa}+Cq\kappa\|z\|_{\frac{a(\kappa q-1)}{q}}^{\frac{\kappa q-1}{q}}{\nonumber}\\ &\leq& Cq\kappa(q\kappa -\frac{1}{2})\Big(\frac{2q\kappa}{q\kappa -1}\Big)^{\frac{1}{2}}\|w^{2\kappa q-2}\|_d^{\frac{1}{2}}+Cq\kappa(q\kappa -\frac{1}{2})(q\kappa-1)^{-\frac{1}{2}}\|w^{2q\kappa-1}\|_r^{\frac{1}{2}}{\nonumber}\\&\quad&+Cq\kappa\|z\|_{\kappa b}^\kappa+Cq\kappa\|z\|_{\frac{a(\kappa q-1)}{q}}^{\frac{\kappa q-1}{q}},{\nonumber} \end{align}\] where \(C\) depends on \(n,p,\omega_g,Q_F,A_{R,2p},B_{R,p+1},\varphi(0)\) and \(T\). Hence, we have \[\begin{align} I_2&\leq &Cq\kappa(q\kappa -\frac{1}{2})\Big(\frac{2q\kappa}{q\kappa -1}\Big)^{\frac{1}{2}}\|z\|_{\frac{d(2q\kappa -2)}{q}}^{\frac{q\kappa -1}{q}}+Cq\kappa(q\kappa -\frac{1}{2})(q\kappa-1)^{-\frac{1}{2}}\|z\|_{\frac{r(2q\kappa-1)}{q}}^{\frac{2q\kappa-1}{2q}}{\nonumber}\\&\quad&+Cq\kappa\|z\|_{\kappa b}^\kappa+Cq\kappa\|z\|_{\frac{a(q\kappa -1)}{q}}^{\frac{q\kappa -1}{q}}.\label{eq:34656} \end{align}\tag{31}\] Moreover, we have \[\begin{align} I_3&=&-q\kappa\int_{0}^{t}\;dt\int_{M}\;w^{q\kappa-1}e^{\frac{F}{2}}\nabla^2R(\nabla F,\nabla F)\;\omega_{\varphi}^n{\nonumber}\\ &\leq& q\kappa\int_{0}^{T}\;dt\int_{M}\;w^{q\kappa-1}e^{\frac{F}{2}}|\nabla^2R|_{\varphi}|\nabla F|_{\varphi}^2\;\omega_{\varphi}^n{\nonumber}\\ &\leq&q\kappa\int_{0}^{T}\;dt\int_{M}\;w^{q\kappa}|\nabla^2R|_{\varphi}\;\omega_{\varphi}^n{\nonumber}\\ &\leq& q\kappa\Big(\int_{0}^{T}\;dt\int_{M}\;w^{2q\kappa}\;\omega_{\varphi}^n\Big)^{\frac{1}{2}}\Big(\int_{0}^{T}\;dt\int_{M}\;|\nabla^2R|_{\varphi}^2\;\omega_{\varphi}^n\Big)^{\frac{1}{2}}{\nonumber}\\ &=&C(B_{R,p+1})q\kappa\|z\|_{2\kappa}^\kappa,\label{eq:34657} \end{align}\tag{32}\] and \[\begin{align} I_4=\int_{0}^{t}\;dt\int_{M}\;w^{q\kappa}\Delta_{\varphi}R\;\omega_{\varphi}^ndt\leq C(B_{R,p+1})\|z\|_{b\kappa}^{\kappa}.\label{eq:34658} \end{align}\tag{33}\] Combining the inequalities (27 ), (31 ), (32 ) and (33 ), we have \[\begin{align} \|z\|_{\kappa,t}^{\kappa}-\|z\|_{\kappa,0}^\kappa&\leq& Cq\kappa\|z\|_{b\kappa}^\kappa+Cq\kappa(q\kappa -\frac{1}{2})\Big(\frac{2q\kappa}{q\kappa -1}\Big)^{\frac{1}{2}}\|z\|_{\frac{d(2q\kappa -2)}{q}}^{\frac{q\kappa -1}{q}}{\nonumber}\\&\quad&+Cq\kappa(q\kappa -\frac{1}{2})(q\kappa-1)^{-\frac{1}{2}}\|z\|_{\frac{r(2q\kappa-1)}{q}}^{\frac{2q\kappa-1}{2q}}+Cq\kappa\|z\|_{\frac{a(q\kappa -1)}{q}}^{\frac{q\kappa -1}{q}}+Cq\kappa\|z\|_{2\kappa}^\kappa,{\nonumber} \end{align}\] where \(C\) depends on \(n,p,\omega_g,Q_F,A_{R,2p},B_{R,p+1},\varphi(0)\) and \(T\). The lemma is proved. ◻
Using Lemma 15 and the parabolic Sobolev inequality Lemma 4 , we can show Lemma 14.
Proof of Lemma 14. Let \(w=e^{\frac{1}{2} F}|\nabla F|_{\varphi}^2+1\) as above. By the inequality (4.4)-(4.6) of Chen-Cheng or (2.27) of Li-Zhang-Zheng , we have \[\begin{align} \Delta_{\varphi}w&\geq&2e^{\frac{F}{2}}\nabla_{\varphi} F\cdot_{\varphi} \nabla\Delta_{\varphi}F-C(g, \|F\|_\infty)\tilde{v}^{n-1} \,w-\frac{1}{2} Rw+\frac{1}{2}R.\label{eq:B2} \end{align}\tag{34}\] Multiplying both sides of (34 ) by \(w^{2q}\) and integrating by parts, for any \(q> 0\) we have \[\begin{align} &&\int_{0}^{T}\;dt\int_{M}\;2qw^{2q-1}|\nabla w|_{\varphi}^2\;\omega_{\varphi}^n= \int_{0}^{T}\;dt\int_{M}\;-w^{2q}\Delta_{\varphi}w\;\omega_{\varphi}^n{\nonumber}\\ &\leq& \int_{0}^{T}\;dt\int_{M}\;-2e^{\frac{F}{2}}\nabla_{\varphi}F\cdot_{\varphi}\nabla\Delta_{\varphi}Fw^{2q}+C\tilde{v}^{n-1}w^{2q}+ |R| w^{2q+1}\;\omega_{\varphi}^n{\nonumber}\\ &\leq&\int_{0}^{T}\;dt\int_{M}\; \Big(q w^{2q-1}|\nabla w|_{\varphi}^2+(4q+2)w^{2q}e^{\frac{1}{2} F}(\Delta_{\varphi}F)^2 +w^{2q+1}|\Delta_{\varphi}F|\Big)\,\omega_{\varphi}^n{\nonumber}\\&\quad&+\int_{0}^{T}\;dt\int_{M}\;\Big(C\tilde{v}^{n-1}w^{2q}+ |R|w^{2q+1}\Big)\;\omega_{\varphi}^n,\label{eq:B3} \end{align}\tag{35}\] where in the last equality we used the inequality (4.19) of Chen-Cheng . Note that \[|\Delta_{\varphi}F|\leq |R|+|{\rm tr}_{\varphi}{Ric(\omega_g)}|\leq |R|+C(g, \|F\|_\infty)\tilde{v}^{n-1}. \label{eq:B11}\tag{36}\] Combining (35 ) with (36 ), we have \[\begin{align} &&\int_{0}^{T}\;dt\int_{M}\; q w^{2q-1}|\nabla w|_{\varphi}^2\,\omega_{\varphi}^n \leq C(\omega_g, \|F\|_\infty)\int_{0}^{T}\;dt\int_{M}\; \Big((q+1) w^{2q}R^2{\nonumber}\\&&+q w^{2q}\tilde{v}^{2n-2} +w^{2q+1}|R|+\tilde{v}^{n-1}w^{2q+1}\Big)\,\omega_{\varphi}^n.\label{eq:A2b} \end{align}\tag{37}\] Set \(z=w^{q+\frac{1}{2}}\). By the Sobolev inequality Lemma 4, we have \[\begin{align} &&\int_{0}^{T}\;dt\int_{M}z^{\beta}\omega_{\varphi}^n\leq C(n,\omega_g,\gamma,\|F\|_\infty)\sup_{t\in[0,T)}\|z\|_{\kappa,t}^{\theta\beta}\int_{0}^{T}\;dt\int_{M}\;\Big(|\nabla z|_{\varphi}^2+z^2\Big)\;\omega_{\varphi}^n{\nonumber}\\ &&\leq C\sup_{t\in[0,T)}\|z\|_{\kappa,t}^{\theta\beta}(q+\frac{1}{2})^2\int_{0}^{T}\;dt\int_{M}\;\Big(\frac{q+1}{q}z^{\frac{4q}{2q+1}}R^2{\nonumber}\\&&\quad+z^{\frac{4q}{2q+1}}\tilde{v}^{2n-2}+\frac{1}{q}z^2|R|+\frac{1}{p} \tilde{v}^{n-1}z^2\Big)\,\omega_{\varphi}^n{\nonumber}\\ &&\leq C(q+\frac{1}{2})^2\sup_{t\in[0,T)}\|z\|_{\kappa,t}^{\theta\beta}\Big(\frac{2q+1}{q}\|z\|_{\frac{4qd}{2q+1}}^{\frac{4q}{2q+1}}+\frac{1}{q}\|z\|_{2r}^2\Big),\label{eq:34662} \end{align}\tag{38}\] where \(C\) depends on \(n,\omega_g,\gamma,\|F\|_\infty,A_{R,2p}\),\(B_{R,p+1}\) and \(\varphi(0)\). By Lemma 15, we have \[\begin{align} \|z\|_{\kappa,t}^{\kappa}-\|z\|_{\kappa,0}^\kappa&\leq& C(e+1)\|z\|_{b\kappa}^\kappa+C(e+1) (e+\frac{1}{2})\Big(\frac{2e+2}{e}\Big)^{\frac{1}{2}}\|z\|_{\frac{4de}{2q+1}}^{\frac{2e}{2q+1}}{\nonumber}\\&\quad&+C(e+1)(e+\frac{1}{2})e^{-\frac{1}{2}}\|z\|_{\frac{r(4e+2)}{2q+1}}^{\frac{2e+1}{2q+1}}+C(e+1)\|z\|_{\frac{2ae}{2q+1}}^{\frac{2e}{2q+1}}+C(e+1)\|z\|_{2\kappa}^\kappa,{\nonumber}\\ \quad\label{eq:34663} \end{align}\tag{39}\] where \(\kappa>\frac{2}{2q+1}\) by Lemma 15 and \(e:=q\kappa+\frac{1}{2}\kappa-1> 0\) . Combining (38 ) and (39 ), we get \[\begin{align} \|z\|_\beta&\leq& C^{\frac{1}{\beta}}(q+\frac{1}{2})^{\frac{2}{\beta}}\Big(\|z\|_{\kappa,0}^\kappa+C(e+1)\|z\|_{b\kappa}^\kappa+C(e+1) (e+\frac{1}{2})^2(\frac{2e+2}{e})^{\frac{1}{2}}\|z\|_{\frac{4de}{2q+1}}^{\frac{2e}{2q+1}}{\nonumber}\\&\quad&+C(e+1)(e+\frac{1}{2})^2e^{-\frac{1}{2}}\|z\|_{\frac{r(4e+2)}{2q+1}}^{\frac{2e+1}{2q+1}}+C(e+1)\|z\|_{\frac{2ae}{2q+1}}^{\frac{2e}{2q+1}}+C(e+1)\|z\|_{2\kappa}^\kappa\Big)^{\frac{\theta}{\kappa}}{\nonumber}\\&\quad&\cdot\Big(\frac{2q+1}{q}\|z\|_{\frac{4qd}{2q+1}}^{\frac{4q}{2q+1}}+\frac{1}{q}\|z\|_{2r}^2\Big)^{\frac{1}{\beta}}.\label{eq:B10} \end{align}\tag{40}\] In order to use the iteration argument, we need to choose the constants in (40 ) satisfying \[\begin{align} \beta=2+(1-\frac{2}{\gamma})\kappa>c:= \max\Big\{b\kappa,\frac{4de}{2q+1},\frac{r(4e+2)}{2q+1},2\kappa,\frac{4qd}{2q+1},2r\Big\},{\nonumber} \end{align}\] or equivalently, \[\begin{align} &&\max\Big\{\frac{2r-2}{D},\frac{2}{2q+1},\frac{4qd-4q-2}{(2q+1)D}\Big\}<\kappa<{\nonumber}\\&&\min\Big\{\frac{1}{2d-D}(2+\frac{4d}{2q+1}),\frac{1}{2r-D}(2+\frac{2r}{2q+1}),\frac{2}{1+\frac{2}{\gamma}}\Big\},\label{eq:B4}{\nonumber} \end{align}\tag{41}\] where \(D=1-\frac{2}{\gamma}> 0\). By Lemma 16 below, when \(\frac{1}{2}-\frac{dD}{2(d+1)}\leq q\leq\frac{r}{2(d-r)}\), such a pair\((q,\kappa)\) exists. Moreover, Lemma 16 implies that \[\begin{align} \max\Big\{\frac{2r-2}{D},\frac{2}{2q+1},\frac{4qd-4q-2}{(2q+1)D}\Big\}&=&\frac{2r-2}{D},{\nonumber}\\ \min\Big\{\frac{1}{2d-D}(2+\frac{4d}{2q+1}),\frac{1}{2r-D}(2+\frac{2r}{2q+1}),\frac{2}{1+\frac{2}{\gamma}}\Big\}&=&\frac{1}{2d-D}(2+\frac{4d}{2q+1}){\nonumber}\\ \quad{\nonumber} \end{align}\] in this case. Therefore (40 ) implies that \[\begin{align} \|z\|_\beta\leq C\|z\|_{c},\label{eq:34667} \end{align}\tag{42}\] where \(C\) depends on \(n,q,\omega_g,\kappa,\gamma,\|F\|_\infty,A_{R,2p},B_{R,p+1}\) and \(\varphi(0)\). Taking the \((q+\frac{1}{2})\)-root in (42 ), we get \[\begin{align} \|w\|_{\beta(q+\frac{1}{2})}\leq C\|w\|_{c(q+\frac{1}{2})}.\label{eq:34668} \end{align}\tag{43}\] Note that \(\beta(q+\frac{1}{2})\to 2p\) when \(\kappa\to \frac{2r-2}{D}\) and \(q\to\frac{r}{2(d-r)}\). Therefore (43 ) implies that for any \(s< 2p\), there exists \(k<s\) such that \[\begin{align} \|w\|_s\leq C(n,\omega_g,s,A_{R,2p},B_{R,p+1},\gamma,\kappa,\|F\|_\infty,\varphi(0))\|w\|_k.\label{eq:1463} \end{align}\tag{44}\] By the interpolation inequality, we have \[\begin{align} \|w\|_k\leq C(\epsilon)\|w\|_1+\epsilon\|w\|_s.\label{eq:1464} \end{align}\tag{45}\] Combining (44 ) with (45 ) and choosing \(\epsilon\) small enough, we get \[\begin{align} \|w\|_s\leq C\|w\|_1.\label{eq:34670} \end{align}\tag{46}\] Note that \[\begin{align} \|w\|_1&=&\int_{0}^{T}dt\int_{M}\;\Big(e^{\frac{F}{2}}|\nabla F|_{\varphi}^2+1\Big)\;\omega_{\varphi}^n{\nonumber}\\ &\leq& C(\|F\|_\infty)\int_{0}^{T}\;dt\int_{M}\;|\nabla F|_{\varphi}^2\;\omega_{\varphi}^n+C(\|F\|_\infty){\rm vol}_{\omega_g}(M)T{\nonumber}\\ &=&-C\int_{0}^{T}\;dt\int_{M}\;F\;\Delta_\varphi F\;\omega_{\varphi}^n+C{\rm vol}_{\omega_g}(M)T{\nonumber}\\ &\leq& C(\omega_g,\|F\|_\infty,A_{R,2p},\|n+\Delta_g\varphi\|_{2p(n-1)},T).{\nonumber} \end{align}\] Since \(n+\Delta_g \varphi\in L^{2p(n-1)}(M\times[0,T),\omega_\varphi^n\wedge dt)\) by Lemma 13, we finish this proof. ◻
The following result was used in the proof of Lemma 14.
Lemma 16. Given the constants \(n(n\geq2),p(p>n),\gamma\in(2,\frac{2n}{n-1})\) with \(\frac{2p}{2p-1}<\frac{2\gamma}{\gamma+2}\). We define \[\begin{align} r=\frac{2p}{2p-1} ,\quad d=\frac{p}{p-1},\quad D=1-\frac{2}{\gamma}.\label{eq:34671} \end{align}\qquad{(12)}\] Then there exists a pair \((q,\kappa)\) satisfying the following conditions: \[\begin{align} &&\max\Big\{\frac{2r-2}{D},\frac{2}{2q+1},\frac{4qd-4q-2}{(2q+1)D}\Big\}<\kappa{\nonumber}\\ &&<\min\Big\{\frac{1}{2d-D}(2+\frac{4d}{2q+1}),\frac{1}{2r-D}(2+\frac{2r}{2q+1}),\frac{2}{1+\frac{2}{\gamma}}\Big\}.\label{eq:34672} \end{align}\qquad{(13)}\] More precisely, we have \[\begin{align} &&\max\Big\{\frac{2r-2}{D},\frac{2}{2q+1},\frac{4qd-4q-2}{(2q+1)D}\Big\}=\frac{2r-2}{D},{\nonumber}\\ &&\min\Big\{\frac{1}{2d-D}(2+\frac{4d}{2q+1}),\frac{1}{2r-D}(2+\frac{2r}{2q+1}),\frac{2}{1+\frac{2}{\gamma}}\Big\}=\frac{1}{2d-D}(2+\frac{4d}{2q+1}){\nonumber} \end{align}\] when \(\frac{1}{2}-\frac{dD}{2(d+1)}\leq q\leq\frac{r}{2(d-r)}\). In this case, we have \[\begin{align} \frac{1}{2d-D}\Big(2+\frac{4d}{2q+1}\Big)>\frac{2r-2}{D}.{\nonumber} \end{align}\]
Proof. Firstly, we show that \[\begin{align} \max\Big\{\frac{2r-2}{D},\frac{2}{2q+1},\frac{4qd-4q-2}{(2q+1)D}\Big\}=\frac{2r-2}{D}\label{eq:34680a} \end{align}\tag{47}\] when \(\frac{D-r+1}{2(r-1)}\leq q\leq\frac{r}{2(d-r)}\). Let \(q_0\) and \(q_1\) be the solutions to the following equations respectively: \[\begin{align} \frac{2}{2q_0+1}&=&\frac{2r-2}{D},{\nonumber}\\ \frac{4q_1d-4q_1-2}{(2q_1+1)D}&=&\frac{2r-2}{D}.{\nonumber} \end{align}\] We get that \(2q_0+1=\frac{D}{r-1}\) and \(2q_1+1=\frac{d}{d-r}\). Since \(\frac{D}{r-1}<\frac{d}{d-r}\) by definition, we have that \[\frac{2}{2q+1}<\frac{2r-2}{D},\quad \frac{2r-2}{D}>\frac{4q_1d-4q_1-2}{(2q_1+1)D}{\nonumber}\]when \(\frac{D-r+1}{2(r-1)}\leq q\leq\frac{r}{2(d-r)}\). Therefore, (47 ) is proved.
Next, we show that if \(q\geq\frac{2-rD}{4(r-1)}\), then the inequality holds \[\begin{align} \min\Big\{\frac{1}{2d-D}\Big(2+\frac{4d}{2q+1}\Big),\frac{1}{2r-D}\Big(2+\frac{2r}{2q+1}\Big),\frac{2}{1+\frac{2}{\gamma}}\Big\}=\frac{1}{2d-D}\Big(2+\frac{4d}{2q+1}\Big).\label{eq:34682} \end{align}\tag{48}\] Let \(q_2\) and \(q_3\) be the solutions to the following equations: \[\begin{align} \frac{1}{2d-D}\Big(2+\frac{4d}{2q_2+1}\Big)&=&\frac{2}{1+\frac{2}{\gamma}},{\nonumber}\\ \frac{1}{2r-D}\Big(2+\frac{2r}{2q_3+1}\Big)&=&\frac{2}{1+\frac{2}{\gamma}}.{\nonumber} \end{align}\] Then we have \[2q_2+1=\frac{d(2-D)}{d-1},\quad 2q_3+1=\frac{r(2-D)}{2r-2}.{\nonumber}\]Since \(r=\frac{2p}{2p-1}=\frac{2d}{d+1}\) by (?? ), we have that \(q_2=q_3\). Note that \[\lim_{q\to\infty}\frac{1}{2d-D}\Big(2+\frac{4d}{2q+1}\Big)= \frac{2}{2d-D}<\lim_{q\to\infty}\frac{1}{2r-D}\Big(2+\frac{2r}{2q+1}\Big),{\nonumber}\]we have the equality (48 ). Moreover, we have \(\frac{d(2-D)}{d-1}>\frac{D}{r-1}\) since \(n\geq 2\).
Therefore, we have \[\begin{align} \max\Big\{\frac{2r-2}{D},\frac{2}{2q+1},\frac{4qd-4q-2}{(2q+1)D}\Big\}&=&\frac{2r-2}{D},{\nonumber}\\ \min\Big\{\frac{1}{2d-D}\Big(2+\frac{4d}{2q+1}\Big),\frac{1}{2r-D}\Big(2+\frac{2r}{2q+1}\Big),\frac{2}{1+\frac{2}{\gamma}}\Big\}&=&\frac{1}{2d-D}\Big(2+\frac{4d}{2q+1}\Big){\nonumber}. \end{align}\] when \(\frac{d(2-D)}{d-1}\leq2q+1\leq\frac{d}{d-r}\). Note that \[\frac{1}{2d-D}\Big(2+\frac{4d}{2q+1}\Big)\geq \frac{1}{2d-D}\Big(2+\frac{4d}{\frac{d}{d-r}}\Big)=\frac{1}{2d-D}(2+4d-4r)>\frac{2r-2}{D}, {\nonumber}\]we conclude that there exists \(\kappa\) satisfying (?? ). ◻
In this subsection, we show that \(\|\nabla\varphi\|_\infty\) is bounded. First, we recall the following result from Chen-Cheng , see also Lemma 2.5 in Li-Zhang-Zheng .
Lemma 17. (cf. , ) Let \[\begin{align} A(F,\varphi)&=&-(F+\lambda\varphi)+\frac{1}{2}\varphi^2,\\ u&=&e^A(|\nabla\varphi|_g^2+10), \end{align}\]where \(\lambda\) depends only on \(\|\varphi\|_\infty\) and \(\omega_g\). Then we have the inequality \[\begin{align} \Delta_{\varphi}u\geq \hat{R} u+\frac{1}{n-1}|\nabla\varphi|_g^{2+\frac{2}{n}}e^{-\frac{F}{n}}e^A, \end{align}\]where \(\hat{R}=R-\lambda n(n+2) +(n+2)\varphi\).
Using the equation (1 ) of Calabi flow, we have the result.
Lemma 18. Let \(z=u^q(q>1)\) where \(u\) is defined in Lemma 17. We have \[\begin{align} \|z\|_{\kappa,t}^{\kappa}- \|z\|_{\kappa, 0}^{\kappa}\leq C\|z\|_{2\kappa}^\kappa +C\|z\|_{b\kappa}^\kappa, \end{align}\] where C depends on \(n,\omega_g,Q_F,A_{R,2p}\), \(B_{R,p+1}\), \(\|\varphi\|_\infty, \|F\|_\infty,\varphi(0)\) and \(T\). Here \(b\) and \(p\) satisfy the equality \(\frac{1}{p+1}+\frac{1}{b}=1.\)
Proof. Taking the derivative with respect to \(t\), we have \[\begin{align} \frac{\partial}{\partial t}\|z\|_{\kappa,t}^{\kappa}=\frac{\partial}{\partial t}\int_M\;|z|^{\kappa}\;\omega_\varphi^n=\int_M\;\Big(\kappa z^{\kappa-1}\dot{z}+z^{\kappa}\dot{F}\Big)\;\omega_\varphi^n.{\nonumber} \end{align}\] Using \(\dot{z}=qu^{ q-1}\dot{u}\) and \[\begin{align} &&\dot{u}=\dot{A}u+2e^A\text{Re}(\nabla R\cdot\nabla\varphi ),{\nonumber}\\ &&\dot{A}=-(\dot{F}+\lambda\dot{\varphi})+\varphi\dot{\varphi},{\nonumber} \end{align}\] where \(\nabla R\cdot \nabla\varphi\) is taken with respect to \(\omega_g\), we have \[\begin{align} \|z\|_{\kappa,t}^{\kappa}- \|z\|_{\kappa, 0}^{\kappa}&\leq&\int_{0}^{T}\;dt\int_M\Big(\;\kappa z^{\kappa-1}qz^{\frac{q-1}{q}}\Big(\dot{A}u+2e^A\text{Re}(\nabla R\cdot\nabla\varphi)\Big)+\dot{F}z^{\kappa}\Big)\;\omega_{\varphi}^n{\nonumber}\\ &:=&J_1+J_2+J_3.{\nonumber} \end{align}\] We estimate \(J_1,J_2\) and \(J_3\) respectively. Note that \[\begin{align} J_1=\int_{0}^{T}\;dt\int_{M}\;\kappa q z^{\kappa}\dot{A}\;\omega_{\varphi}^n&\leq& C(\|\varphi\|_\infty,A_{R,2p},B_{R,p+1})q\kappa \Big(\int_{0}^{T}\;dt\int_{M}\;z^{b\kappa}\;\omega_{\varphi}^n\Big)^{\frac{1}{b}}{\nonumber}\\&=&Cq\kappa \|z\|_{b\kappa}^{\kappa},\label{eq:34679} \end{align}\tag{49}\] where \(b\) and \(p\) satisfy \(\frac{1}{p+1}+\frac{1}{b}=1,\) and \[\begin{align} J_2&=&2q\kappa \int_{0}^{T}\;dt\int_{M}\;z^{\kappa-\frac{1}{q}}e^A\text{Re}(\nabla R\cdot\nabla\varphi)\;\omega_{\varphi}^n{\nonumber}\\ &\leq& 2q\kappa \Big(\int_{0}^{T}\;dt\int_{M}\;z^{2\kappa-\frac{2}{q}}e^{2A}|\nabla\varphi|_g^2\;\omega_{\varphi}^n\Big)^{\frac{1}{2}}\Big(\int_{0}^{T}\;dt\int_{M}\;|\nabla R|_g^2\;\omega_{\varphi}^n\Big)^{\frac{1}{2}}.\label{eq:2466} \end{align}\tag{50}\] Note that \[\begin{align} \int_{0}^{T}\;dt\int_M|\nabla R|_g^2\,\omega_\varphi^n&\leq& C(\|F\|_\infty)\int_{0}^{T}\;dt\int_M |R\Delta_g R|\,\omega_g^n{\nonumber}\\ &\leq&C\int_{0}^{T}\;dt\int_M|R||\nabla^2R|_\varphi(n+\Delta_g\varphi)\,\omega_g^n{\nonumber}\\ &\leq&C\Big(\int_{0}^{T}\;dt\int_M|R|^{2p}\,\omega_\varphi^n\Big)^{\frac{1}{2p}} \Big(\int_{0}^{T}\;dt\int_M|\nabla^2R|_\varphi^2\,\omega_\varphi^n\Big)^{\frac{1}{2}}{\nonumber}\\&& \Big(\int_{0}^{T}\;dt\int_M\tilde{v}^{s}\,\omega_\varphi^n\Big)^{\frac{1}{s}},\label{eq:2468} \end{align}\tag{51}\] where \(s\) and \(p\) satisfy \(\frac{1}{2p}+\frac{1}{2}+\frac{1}{s}=1\). Combining (50 ) with (51 ), we have \[\begin{align} J_2 &\leq& C(\|F\|_\infty,A_{R,2p},B_{R,p+1},\|n+\Delta_g\varphi\|_s)q\kappa \Big(\int_{0}^{T}\;dt\int_{M}\;z^{2\kappa-\frac{2}{q}}u\;\omega_\varphi^n\Big)^{\frac{1}{2}}{\nonumber}\\&=&Cq\kappa \Big(\int_{0}^{T}\;dt\int_{M}\;z^{2\kappa-\frac{1}{q}}\;\omega_{\varphi}^n\Big)^{\frac{1}{2}}{\nonumber}\\ &\leq& C(\|\varphi\|_\infty,\|F\|_\infty,A_{R,2p},B_{R,p+1},\|n+\Delta_g\varphi\|_s)q\kappa \Big(\int_{0}^{T}\;dt\int_{M}\;z^{2\kappa}\;\omega_{\varphi}^n\Big)^{\frac{1}{2}}{\nonumber}\\&=&Cq\kappa \|z\|_{2\kappa}^{\kappa}.\label{eq:34680} \end{align}\tag{52}\] Moreover, we have \[\begin{align} J_3=\int_0^T\,dt\int_M\,\dot{F}z^{\kappa}\,\omega_{\varphi}^n\leq C(B_{R,p+1})\Big(\int_{0}^{T}\;dt\int_{M}\;z^{b\kappa}\;\omega_{\varphi}^n\Big)^{\frac{1}{b}}=C\|z\|_{\kappa b}^{\kappa}.\label{eq:34681} \end{align}\tag{53}\] Combining (49 ), (52 ) with (53 ), we get \[\begin{align} \|z\|_{\kappa,t}^{\kappa}- \|z\|_{\kappa, 0}^{\kappa}\leq Cq\kappa\|z\|_{b\kappa}^\kappa+Cq\kappa\|z\|_{2\kappa}^\kappa,{\nonumber} \end{align}\] where \(C\) depends on \(n,\omega_g,Q_F,A_{R,2p}\), \(B_{R,p+1}\), \(\|\varphi\|_\infty, \|F\|_\infty,\varphi(0)\) and \(T\). The lemma is proved. ◻
Using Lemma 18 and Lemma 4, we have the result.
Lemma 19. Under the assumption of Lemma 13, we have \[|\nabla\varphi(x, t)|_g\leq C, \label{eq:1466}\qquad{(14)}\] where C depends on \(n,\omega_g,\|\varphi\|_\infty,\|F\|_\infty,Q_F,A_{R,2p},B_{R,p+1},\varphi(0)\) and \(T\).
Proof. Let \(q> 1\). Since by Lemma 17 \(u=e^A(|\nabla\varphi|_g^2+10)\) satisfies \[\begin{align} \Delta_{\varphi}u\geq \hat{R} u+h, \label{eq:u}{\nonumber} \end{align}\tag{54}\] where \(h=\frac{1}{n-1}|\nabla\varphi|_g^{2+\frac{2}{n}}e^{-\frac{F}{n}}e^A,\) multiplying both sides by \(u^{q-1}\) and integrating by parts we have \[\begin{align} &&\frac{4(q-1)}{q^2}\int_{0}^{T}dt\int_M\; |\nabla(u^{\frac{q}{2}})|_\varphi^2\,\omega_{\varphi}^n = (q-1)\int_{0}^{T}dt\int_M\; u^{q-2}|\nabla u|_\varphi^2\,\omega_{\varphi}^n{\nonumber}\\ &&=-\int_{0}^{T}dt\int_M\;u^{q-1}\Delta_{\varphi}u\,\omega_{\varphi}^n \leq-\int_{0}^{T}dt\int_M\;\Big(\hat{R} u^{q}+hu^{q-1}\Big)\,\omega_{\varphi}^n{\nonumber}\\ &&\leq\int_{0}^{T}dt\int_M\;|\hat{R}| u^{q}\,\omega_{\varphi}^n.{\nonumber} \end{align}\] Letting \(z=u^{\frac{q}{2}}\)and using the Sobolev inequality Lemma 4, we have \[\begin{align} \int_{0}^{T}\;dt\int_M\;|z|^{\beta}\;\omega_\varphi^n&\leq& C(n,\omega_g,\gamma,\|F\|_\infty)\sup_{t\in[0,T)}\;\|z\|_{\kappa,t}^{(1-\frac{2}{\gamma}) \kappa} \int_{0}^{T}dt\int_M\;\Big(|\nabla z|_\varphi^2+z^2\Big)\;\omega_\varphi^n{\nonumber}\\ &\leq& C\sup_{t\in[0,T)}\;\|z\|_{\kappa,t}^{(1-\frac{2}{\gamma})\kappa}\int_{0}^{T}\;dt\int_M\;(|\hat{R}|+1) u^{q}\,\omega_{\varphi}^n.{\nonumber} \end{align}\] by Lemma 18, we have \[\begin{align} \|z\|_{\kappa,t}^\kappa\leq \|z\|_{\kappa,0}^\kappa+Cq\kappa\|z\|_{b\kappa}^\kappa+Cq\kappa\|z\|_{2\kappa}^\kappa.{\nonumber} \end{align}\] Therefore, we have \[\begin{align} \|z\|_{\beta}&\leq& Cq^{\frac{\theta}{\kappa}}\kappa^{\frac{\theta}{\kappa}}\Big(\|z\|_{\kappa,0}^{\kappa}+\|z\|_{b\kappa}^{\kappa}+\|z\|_{2\kappa}^{\kappa}\Big)^{\frac{\theta}{\kappa}}\|z\|_{2r}^{\frac{2}{\beta}}{\nonumber}\\ &\leq& Cq^{\frac{\theta}{\kappa}}\kappa^{\frac{\theta}{\kappa}}\Big(\sup_{x\in M}\Big(e^A(|\nabla\varphi|_g^2(x,0)+10)\Big){\rm vol}_{\omega_g}(M)+2q\kappa\|z\|_{2\kappa}^{\kappa}\Big)^{\frac{\theta}{\kappa}}\|z\|_{2r}^{\frac{2}{\beta}}{\nonumber}\\ &\leq& Cq^{\frac{\theta}{\kappa}}\|z\|_{2\kappa}^{\theta}\|z\|_{2r}^{\frac{2}{\beta}},{\nonumber} \end{align}\] where \(C\) only depends on \(n,\kappa,\gamma,\omega_g,\|F\|_\infty,\|\varphi\|_\infty,A_{R,2p},B_{R,p+1},\varphi(0)\) and \(T\). By (22 ) we have \(\beta>\max\{2\kappa,2r\}.\) We conclude that if \(q\) is large enough, then \[\begin{align} \|z\|_{\beta}\leq Cq^{\frac{\theta}{\kappa}}\|z\|_{2\max\{\kappa,r\}},{\nonumber} \end{align}\] or equivalently, \[\begin{align} \|u\|_{\frac{q\beta}{2}}\leq C^{\frac{2}{ q}}q^{\frac{2\theta}{q\kappa}}\|u\|_{q\max\{\kappa,r\}}.\label{eq:34699} \end{align}\tag{55}\] Letting \(\theta_1=\frac{\beta{\max\{2\kappa,2r\}}}{>}1\) and \(q_n=\frac{2}{\max\{r,\kappa\}}\theta_1^n\), the inequality (55 ) implies that \[\begin{align} \|u\|_{q_{n+1}\max\{r,\kappa\}}\leq C^{\frac{2}{q_n}}q_n^{\frac{2\theta}{ q_n \kappa}}\|u\|_{q_n\max\{r,\kappa\}}.{\nonumber} \end{align}\] Since \(\kappa<\frac{2}{1+\frac{2}{\gamma}}<\frac{2n}{2n-1}<2\) and \(q_0=\frac{2}{\max\{r,\kappa\}}>1\), the standard Moser iteration argument shows that \[\|u\|_\infty\leq C\|u\|_2\label{eq:346100}\tag{56}\] for some constant \(C\) depending on \(n,\kappa,\gamma,\omega_g,\|F\|_\infty,\|\varphi\|_\infty,A_{R,2p},B_{R,p+1},\varphi(0)\) and \(T\). By the interpolation inequality Lemma 2, we have \[\begin{align} \|u\|_2\leq \|u\|_1^{\frac{1}{2}}\|u\|_\infty^\frac{1}{2}.\label{eq:346101} \end{align}\tag{57}\] Combining (56 ) and (57 ), we get \[\begin{align} \|u\|_\infty \leq C\|u\|_1.\label{eq:1465} \end{align}\tag{58}\] Next we show that \(\|u\|_1\) is bounded. \[\begin{align} \|u\|_1&=&\int_{0}^{T}dt\int_Me^A(|\nabla\varphi|_g^2+10)\,\omega_\varphi^n{\nonumber}\\ &\leq& C(\|\varphi\|_\infty,\|F\|_\infty)\int_{0}^{T}dt\int_M(|\nabla\varphi|_g^2+10)\,\omega_g^n{\nonumber}\\ &\leq& 10CT\cdot {\rm vol}_{\omega_g}(M)+C\int_{0}^{T}dt\int_M|\varphi\Delta_g\varphi|\,\omega_g^n.\label{eq:346103} \end{align}\tag{59}\] Since \(|\Delta_g\varphi|\leq |\nabla^2\varphi|_\varphi (n+\Delta_g\varphi)\), we have \[\begin{align} \int_{0}^{T}dt\int_M|\varphi\Delta_g\varphi|\,\omega_g^n&\leq& C(\|\varphi\|_\infty)\Big(\int_{0}^{T}dt\int_M|\nabla^2\varphi|_\varphi^2\,\omega_\varphi^n\Big)^\frac{1}{2}\Big(\int_{0}^{T}dt\int_M(n+\Delta_g\varphi)^2\,\omega_\varphi^n\Big)^\frac{1}{2}{\nonumber}\\ &=&C\Big(\int_{0}^{T}dt\int_M|\Delta_\varphi \varphi|^2\,\omega_\varphi^n\Big)^\frac{1}{2}\Big(\int_{0}^{T}dt\int_M(n+\Delta_g\varphi)^2\,\omega_\varphi^n\Big)^\frac{1}{2}.{\nonumber}\\ \quad\label{eq:34694} \end{align}\tag{60}\] Since \(\Delta_\varphi \varphi=n-{\rm tr}_\varphi\omega_g\leq n+\tilde{v}^{n-1}e^{-F}\) by (29 ), we conclude that the right-hand side of (60 ) is bounded. Therefore, (59 ) implies that \(\|u\|_1\) is bounded and by (58 ) we have (?? ). The lemma is proved. ◻
In this section, we show the estimate of \(\Delta_{g}\varphi\). First, we recall the following result from Chen-Cheng , see also Lemma 2.8 in Li-Zhang-Zheng .
Lemma 20. (cf. , ) Let \[v=e^{-\alpha(F+\lambda\varphi)}(n+\Delta_g\varphi).{\nonumber}\] Let \(q>1\) and \(\alpha>1\). There exists a constant \(C(\omega_g)\) such that for \(\lambda>C(\omega_g)\) we have \[\begin{align} && \frac{3(q-1)}{q^2}\int_M\; |\nabla v^{\frac{q}{2}}|_{\varphi}^2\;\omega_{\varphi}^n\leq \int_M\; \Big( \tilde{f}+\frac{\alpha\lambda}{\alpha-1}+\frac{1}{n} e^{-\frac{F}{n}}R_g \Big)v^{q}\omega_{\varphi}^n{\nonumber}\\ &&+2q\int_M\; v^{q}|\nabla F|_\varphi^2\,\omega_\varphi^n+\frac{2\alpha^2\lambda^2q}{(\alpha-1)^2}\int_M\; e^B v^{q-1}|\nabla\varphi|_g^2\,\omega_g^n,\label{eq:B01} \end{align}\qquad{(15)}\] where \(B=(1-\alpha)F-\alpha\lambda\varphi\) and \(\tilde{f}=\alpha (\lambda n-R).\)
Combining Lemma 20, Lemma 12 with Lemma 4, we have the result.
Lemma 21. If \(A_{R,2p}\), \(B_{R,p+1}\) are bounded for some \(p>n\), and \(Q_F\) is bounded, then there exists a constant C depending on \(n,\omega_g,Q_F,A_{R,2p},B_{R,p+1},\|\varphi\|_\infty,\|F\|_\infty,\varphi(0)\) and \(T\) such that\[\begin{align} n+\Delta_g\varphi\leq C. \end{align}\]
Proof. Since \(n+\Delta_g \varphi\geq n e^{\frac{F}{n}}\), we have \[v^{q-1}=e^{\alpha(F+\lambda\varphi)}\frac{v^q}{n+\Delta_g\varphi}\leq \frac{1}{n} e^{\alpha(F+\lambda\varphi)-\frac{F}{n}}\,v^q.\label{eq:B08}\tag{61}\] Taking \(z=v^{\frac{q}{2}}\) and \(\alpha=2\) in the inequality (?? ), we have \[\begin{align} && \frac{3(q-1)}{q^2}\int_{0}^{T}\;dt\int_M\; |\nabla z|_{\varphi}^2\;\omega_{\varphi}^n\leq \int_{0}^{T}\;dt\int_M\; \Big(\tilde{f}+2\lambda+\frac{1}{n} e^{-\frac{F}{n}}R_g\Big)z^2\omega_{\varphi}^n{\nonumber}\\ &&+2q\int_{0}^{T}\;dt\int_M\; z^{2}|\nabla F|_\varphi^2\,\omega_\varphi^n+8\lambda^2q\int_{0}^{T}\;dt\int_M\; e^B v^{q-1}|\nabla\varphi|_g^2\,\omega_g^n{\nonumber}\\ &\leq&\int_{0}^{T}\;dt\int_M\; \Big(\tilde{f}+2\lambda+\frac{1}{n} e^{-\frac{F}{n}}R_g\Big)z^2\omega_{\varphi}^n{\nonumber}\\ &&+2q\int_{0}^{T}\;dt\int_M\; z^{2}|\nabla F|_\varphi^2\,\omega_\varphi^n+C(n, \omega_g, \|F\|_\infty, \|\varphi\|_\infty)q\int_{0}^{T}\;dt\int_M\; v^q\,\omega_\varphi^n,{\nonumber} \end{align}\]where we used (61 ) and Lemma 19 in the last inequality. Thus, we have \[\frac{3(q-1)}{q^2}\int_{0}^{T}\;dt\int_M\; |\nabla z|_{\varphi}^2\;\omega_{\varphi}^n\leq q\int_{0}^{T}\;dt\int_M\; Gz^2\,\omega_{\varphi}^n+ 2q\int_{0}^{T}\;dt\int_M\; z^{2}|\nabla F|_\varphi^2\,\omega_\varphi^n,{\nonumber}\]where \[G=\tilde{f}+2\lambda+\frac{1}{n} e^{-\frac{F}{n}}R_g+C(g, \|F\|_\infty, \|\varphi\|_\infty).{\nonumber}\] By Lemma 4, we have \[\begin{align} \int_{0}^{T}\;dt\int_M\;|z|^\beta\;\omega_{\varphi}^n&\leq& C(n,\omega_g,\|F\|_\infty,\gamma)q^2\sup_{[0,T)}\|z\|_{\kappa,t}^{(1-\frac{2}{\gamma})\kappa}\int_{0}^{T}\;dt\int_{M}\;\Big(G+|\nabla F|_\varphi^2\Big)z^2\;\omega_{\varphi}^n.{\nonumber}\\ \quad \label{eq:2461} \end{align}\tag{62}\] By Lemma 12, we have \[\begin{align} \|z\|_{\kappa,t}^{\kappa}-\|z\|_{\kappa,0}^{\kappa}\leq Cq\kappa\Big(\|z\|_{r\kappa}^{\kappa}+\|z\|_{\kappa b}^{\kappa}+\|z\|_{2\kappa}^{\kappa}\Big)\leq Cq\kappa\|z\|_{2\kappa}^{\kappa}.\label{eq:2462} \end{align}\tag{63}\] According to Lemma 14, \(|\nabla F|_\varphi^2\in L^{s}(M\times[0,T),\omega_\varphi^n\wedge dt)\) for \(2n<s<2p\). Combining (62 ) with (63 ), we have \[\begin{align} \|z\|_{\beta}&\leq& C^{\frac{1}{\beta}}q^{\frac{2}{\beta}}\Big(\|z\|_{\kappa,0}^\kappa+Cq\|z\|_{2\kappa}^{\kappa}\Big)^{\frac{\theta}{{\kappa}}}\Big(\|z\|_{2r}^2+\|z\|_{2h}^{2}\Big)^{\frac{1}{\beta}}{\nonumber}\\ &\leq& Cq^{\frac{2}{\beta}}q^{\frac{\theta}{\kappa}}\|z\|_{2\kappa}^{\theta}\Big(\|z\|_{2r}^2+\|z\|_{2h}^{2}\Big)^{\frac{1}{\beta}},{\nonumber} \end{align}\] where \(C\) depends on \(n,\omega_g,\kappa,\gamma,A_{R,2p},B_{R,p+1},\|\varphi\|_\infty,\|F\|_\infty\) and \(\varphi(0)\). Here, \(h\) and \(s\) satisfy the equality \(\frac{1}{h}+\frac{1}{s}=1\). We need that \[\begin{align} \beta>\max\Big\{2\kappa,2r,2h\Big\},{\nonumber} \end{align}\] or equivalently, \[\max\Big\{\frac{2r-2}{1-\frac{2}{\gamma}},\frac{2h-2}{1-\frac{2}{\gamma}}\Big\}<\kappa<\frac{2}{1+\frac{2}{\gamma}}.{\nonumber}\] Note that \(s< 2p\), we need the inequality \[\begin{align} \frac{2h-2}{1-\frac{2}{\gamma}}<\frac{2}{1+\frac{2}{\gamma}}.\label{eq:B09} \end{align}\tag{64}\] We can choose \(\gamma\) close to \(\frac{2n}{n-1}\) such that (64 ) holds. Then we have \[\begin{align} \|v\|_{\frac{q\beta}{2}}\leq C^{\frac{2}{q}}q^{\frac{2}{q}(\frac{2}{\beta}+\frac{\theta}{\kappa})}\|v\|_{q\max\{h,\kappa\}}.\label{eq:346118} \end{align}\tag{65}\] Letting \(\theta_2=\frac{\beta}{2\max\{h,\kappa\}}>1\) and taking \(q_n=\frac{2}{\max\{h,\kappa\}}\theta_2^n\), the inequality (65 ) implies that \[\begin{align} \|v\|_{q_{n+1}\max\{h,\kappa\}}\leq C^{\frac{2}{q_n}}q_n^{\frac{2}{q_n}(\frac{2}{\beta}+\frac{\theta}{\kappa})}\|v\|_{q_n\max\{h,\kappa\}}.{\nonumber} \end{align}\] Since \(h<\frac{2n}{2n-1}<2\) and \(q_0=\frac{2}{\max\{h,\kappa\}}>1\), the standard Moser iteration shows \[\begin{align} \|v\|_\infty\leq C\|v\|_{q_0\max\{h,\kappa\}}.\label{eq:346120}=C\|v\|_2.{\nonumber} \end{align}\tag{66}\] Since \(\|v\|_2\) is bounded by Lemma 13 we know that \(v\) is bounded and the lemma is proved. ◻
Proof of Theorem 1. Firstly we show that \(Q_F\) is bounded along the Calabi flow. Without loss of generality, we may assume that \(\varphi(0)\in{\mathcal{H}}_0\). Then we have that \(\varphi(t)\in{\mathcal{H}}_0\) by (7 ). According to Lemma 4.4 of , we have \[\begin{align} |J_{-{Ric(\omega_g)}}(\varphi)|\leq C(n, g)d_1(0,\varphi). \end{align}\] Combining (4.1) with the proof of Lemma 8, we conclude that \(J_{-{Ric(\omega_g)}}(\varphi)\) is uniformly bounded along Calabi flow. Since \(\int_MF\,\omega_\varphi^n={\mathcal{K}}(\varphi)-J_{-{Ric(\omega_g)}}(\varphi)\), we know that \(\int_MF\,\omega_\varphi^n\) is uniformly bounded under Calabi flow. Therefore, \(Q_F\) is bounded.
By the assumption, we have that \(A_{R,2p}^n,B_{R,p+1}^n\) are bounded for \(p> n\). Combining this with the boundedness of \(Q_F\), we know that \(\|\varphi\|_\infty\) and \(\|F\|_\infty\) are bounded by Theorem 5. Moreover, combining Lemma 13, Lemma 14, Lemma 19 and Lemma 21 we conclude that \(\|n+\Delta\varphi\|_\infty\) is bounded. Therefore, there exists a constant \(C>0\) such that for any \(t\in [0, T)\) \[\begin{align} \frac{1}{C}\omega_g\leq \omega_\varphi\leq C\omega_g. \label{eq:1468} \end{align}\tag{67}\] Note that \(F\) satisfies the parabolic equation \[\frac{\partial F}{\partial t}-\Delta_{\varphi}F=K, \quad K:=\Delta_{\varphi}R+R-{\rm tr}_{\varphi}Ric(\omega_g).\] By the assumption of Theorem 1, the inequality (36 ) and Lemma 21, we have \[\begin{align} \int_0^T\,dt\int_M\;|K|^{p+1}\,\omega_g^n&\leq& C(p, \|F\|_{\infty})\int_0^T\,dt\int_M\;\Big(|\Delta_{\varphi}R|^{p+1}+|R|^{p+1}+\tilde{v}^{(n-1)(p+1)}\Big)\,\omega_{\varphi}^n\\ &\leq&C,\quad p>n. \end{align}\] Since \(\omega_{\varphi}\) satisfies (67 ), by the Hölder estimates of parabolic equations (cf. Theorem 23 in the appendix), we know that \(F\in C^{\alpha}(M\times [\frac{1}{2}T, T), \omega_g)(\alpha\in (0, 1))\). This together with (67 ) implies that \(\varphi\in C^{2, \alpha'}(M\times [\frac{1}{2}T, T), \omega_g)\) for any \(\alpha'\in (0, \alpha)\) (cf. Chen-Wang , Y. Wang ). Therefore, by He the Calabi flow can be extended past time \(T\). The theorem is proved. ◻
In the appendix, we recall the Hölder estimates of parabolic equations. The readers are referred to Lieberman , Guerand , or Vasseur for details.
We use the notations in Guerand . Let \(r>0\) and \(x_0\in {\mathbb{R}}^d\). We denote by \(B_r(x_0)\) the ball of radius \(r\) centered at \(x_0\). For \((x_0, t_0)\in {\mathbb{R}}^d\times {\mathbb{R}}\) we define the parabolic cylinder \(Q_r(x_0, t_0)=B_r(x_0)\times (t_0-r^2, t_0)\) and \(Q_r=B_r(0)\times (-r^2, 0)\).
Theorem 22. Let \(u: Q_2\rightarrow{\mathbb{R}}\) be a solution of \[\frac{\partial u}{\partial t}=\nabla_x\cdot (A\nabla_x u) +B\cdot\nabla_xu+g,\]where \(A(x, t), B(x, t)\) and \(g(x, t)\) satisfy the following conditions:
\(A(x, t)\) is a bounded measurable matrix and satisfies an ellipticity condition for two positive constants \(\lambda, \Lambda\), \[0<\lambda I\leq A\leq \Lambda I,\]
\(B(x, t)\) is bounded, measurable and \(|B|\leq \Lambda,\)
\(g(x, t)\) is bounded, measurable and satisfies \[\|g\|_{ L^q(Q_2)}\leq 1, \quad q>\max\Big\{2, \frac{d+2}{2}\Big\}.\label{eq:a1}\qquad{(16)}\]
Then we have \[\|u\|_{C^{\alpha}(Q_1)}\leq C(d, \lambda, \Lambda)(\|u\|_{L^2(Q_2)}+1), \label{eq:a2}\qquad{(17)}\] where \(\alpha\) depends only on \(d, \lambda\) and \(\Lambda.\)
We can easily remove the bound (?? ). In fact, letting \(\tilde{g}=K^{-1}g\) with \(K:=\|g\|_{ L^q(Q_2)}\) and \(\tilde{u}=K^{-1}u\), by (?? ) we have \[\|\tilde{u}\|_{C^{\alpha}(Q_1)}\leq C(d, \lambda, \Lambda)(\|\tilde{u}\|_{L^2(Q_2)}+1).\]Therefore, we have \[\|u\|_{C^{\alpha}(Q_1)}\leq C(d, \lambda, \Lambda)(\| u\|_{L^2(Q_2)}+\|g\|_{ L^q(Q_2)}).\]
Theorem 23. Let \((M, g)\) be a Riemannian manifold of dimension \(d\) and \(Q_r=B_r(x_0)\times (t_0-r^2, t_0)\), where \(B_r(x_0)\subset M\) denotes the ball centered at \(x_0\in M\) of radius \(r>0\) with respect to the metric \(g\). If \(u: Q_2\rightarrow{\mathbb{R}}\) be a solution of \[\frac{\partial u}{\partial t}=\Delta_{h}u +f,\]where \(h(x, t)\) and \(f(x, t)\) satisfy the following conditions:
\(h(x, t)\) is a metric equivalent to \(g\), i.e. there exist two constants \(\lambda, \Lambda>0\) such that \[0<\lambda g\leq h\leq \Lambda g,\]
\(f(x, t)\) is a bounded, measurable function and satisfies \(f\in L^q(Q_2)\) with \(q>\max\{2, \frac{d+2}{2}\}\).
Then we have \[\|u\|_{C^{\alpha}(Q_1)}\leq C(d, \lambda, \Lambda, g)(\|u\|_{L^2(Q_2)}+\|f\|_{ L^q(Q_2)}), \label{brpfazim}\qquad{(18)}\] where \(\alpha\) depends only on \(d, \lambda\) and \(\Lambda.\)
Proof. We can choose a good coordinate chart with respect to the metric \(g\), and the theorem follows from Theorem 22 by the standard argument. See, for example, Hebey or Metsch for more details. ◻
Haozhao Li, Institute of Geometry and Physics, and Key Laboratory of Wu Wen-Tsun Mathematics, School of Mathematical Sciences, University of Science and Technology of China, No. 96 Jinzhai Road, Hefei, Anhui Province, 230026, China;
hzli@ustc.edu.cn.
Linwei Zhang, School of Mathematical Sciences, University of Science and Technology of China, No. 96 Jinzhai Road, Hefei, Anhui Province, 230026, China; zhanglinwei@mail.ustc.edu.cn.
Supported by NSFC grant No. 12471058, the CAS Project for Young Scientists in Basic Research (YSBR-001), and the Fundamental Research Funds for the Central Universities.↩︎