Strong uniqueness of the Ricci flow.

*(English)*Zbl 1177.53036The author is interested in particular aspects of the Ricci flow equation, given in the form

\[ (\partial_t.g_{ij})(x,t)=-2R_{ij}(x,t),\quad (x,t)\in M\times{\mathbb R},\tag \(\clubsuit\) \]

where \(M\) is an \(n\)-dimensional smooth, noncompact and complete Riemannian manifold. In particular the assumption that \(M\) is not compact and complete, is a substantial difference with respect to the usual way in which such equation has been considered. In fact, the Ricci flow equation has been introduced to prove the Poincaré conjecture on compact closed \(3\)-dimensional manifolds homotopic equivalent to \(S^3\).

(See works quoted in references, and also the papers quoted below, where the Ricci flow equation has been studied in a new algebraic topologic framework for PDE’s. [1] A. Prástaro, “Geometry of PDE’s. I: Integral bordism groups in PDE’s”, J. Math. Anal. Appl. 319, No. 2, 547–566 (2006; Zbl 1100.35007). [2] R. Agarwal and A. Prástaro, “Geometry of PDE’s. III(II): Webs on PDE’s and integral bordism groups. Applications to Riemannian geometry PDE’s”, Adv. Math. Sci. Appl. 17, No. 1, 267–285 (2007; Zbl 1140.53005). [3] A. Prástaro, “Extended crystal PDE’s”, arXiv: 0811.3693]. In particular in the last two the Poincaré conjecture has been proved by using a method different to that of Hamilton and Perelmann.)

The main results in this paper are theorem 1.1 and corollary 1.2.

Theorem 1.1. Let \((M,g(x))\) be a complete noncompact three dimensional manifold with bounded and non-negative sectional curvature \(0\leq R_m\leq k_0\), for some fixed constant \(k_0\). Let \(g_1(x,t)\), \(g_2(x,t)\), \(t\in[0,T]\), be two smooth complete solutions to the Ricci flow with initial data \(g(x)\). Then, we have \(g_1(x,t)=g_2(x,t)\), for \(0\leq t<t_{\max}\equiv\min\{T,{{1}\over{4k_0}}\}\).

Corollary 1.2: In particular, when \((M,g(x))\) is the usual Euclidean space \(({\mathbb R}^3,\gamma_E)\), it follows that any smooth complete solution \(g(x,t)\) of the Ricci flow, starting from \(g_E\), is just \(g(x,t)=\gamma_E(x)\).

The paper, after a detailed introduction, where some previous works on the Ricci flow equation are recalled, and theorem 1.1 and corollary 1.2 introduced, splits into three more sections. 2. Local pinching estimate. 3. A priori estimates. 4. Concluding remarks.

Reviewer’s remark. This is a very interesting paper as it intersects a subject under focus of the international mathematical community. In particular, in section 4 there is put the following question. “Does the strong uniqueness of the Ricci flow hold on the Euclidean space \({\mathbb R}^n\), for \(n\geq 4\)?” We can see that the answer is: Yes it does! In the following we will go in some details about.

Let us also underline that the restriction to consider noncompact complete metrics does not permit flows with contractions, and neither simple homeomorphic ones. Isometric diffeomorphisms are instead permitted. Therefore, let us add to the equation (\(\clubsuit\)) the condition that \(g(x,t)\) is realized by means of a one parameter set of diffeomorphisms \(\phi_t:M\to M\), as pull-back, i.e, put \(g(x,t)=\phi_t^*\gamma\). One has for the Ricci tensor \(R_{ij}(x)\) of \(\gamma(x)\), the following induced deformation: \(R(x,t)=R(g(x,t))=\phi_t^*R(\gamma(x))\). In fact, for the natural covariance of the Riemannian metric and its Ricci tensor, we get that equations (\(\clubsuit\)) can be written as

\[ \begin{split} (\dot\phi_t)^a_i(x)(\phi_t)^b_j(x)\gamma_{ab}(\phi_t(x))+(\phi_t)^a_i(x)(\dot\phi_t)^b_j(x)\gamma_{ab}(\phi_t(x))\\ +(\phi_t)^a_i(x)(\phi_t)^b_j(x)(\dot\phi_t)^\alpha(x)(\partial_{x_\alpha}.\gamma_{ab})(\phi_t(x))\\ =-2(\phi_t)^a_i(x)(\phi_t)^b_j(x)R_{ab}(\phi_t(x))\end{split}\tag \(\clubsuit-1\) \]

Let us look for solutions of equations (\(\clubsuit-1\)) of the type \(\phi_t^a=e^{\omega t}h^a(x)\). Then, \(\omega\) and \(h^a(x)\) must satisfy the following equations:

\[ h^a_i(x)h^b_j(x)\left\{\omega\left[2\gamma_{ab}(\phi_t(x))+e^{\omega t}h^\alpha(x)(\partial_{x_\alpha}.\gamma_{ab})(\phi_t(x))\right]+2R_{ab}(\phi_t(x))\right\}=0.\tag{\(\clubsuit-2\)} \]

In the case \((M,\gamma)=({\mathbb R}^n,\gamma_E)\), one has \(R_{ab}=0\) and the equations (\(\clubsuit-2\)) reduce to the following ones:

\[ h^a_i(x)h^b_j(x)\omega\left[2\gamma_{ab}(\phi_t(x))+e^{\omega t}h^\alpha(x)(\partial_{x_\alpha}.\gamma_{ab})(\phi_t(x))\right]=0.\tag{\(\clubsuit-3\)} \]

Let us add the condition \(\phi_0=id_M\). We get that necessarily \(h^a(x)=x^a\), for \(a=1,\dots, n\). Therefore, equations (\(\clubsuit-3\)) reduce to the following ones:

\[ \omega\left[2\gamma_{ij}(\phi_t(x))+e^{\omega t}x^\alpha(\partial_{x_\alpha}.\gamma_{ij})(\phi_t(x))\right]=0.\tag{\(\clubsuit-4\)} \]

In a cartesian coordinate system, we have \(\gamma_{ij}=\delta_{ij}\). Therefore, equations (\(\clubsuit-4\)) reduce to \(2\omega\delta_{ij}=0\). Thus we get the unique trivial solution for \(\omega\), i.e., \(\omega=0\) and the unique solution for \(\phi_t(x)\), i.e., \(\phi_t^a=x^a\), that gives the unique solution \(g(x,t)=\gamma_E(x)\).

This result does not depend on the particular assumption made on the type of solutions. In fact, considering equation (\(\clubsuit-1\)) in the case \(M={\mathbb R}^n\), with \(\gamma=\delta_{ij}\), we get the following equations

\[ \begin{cases} A^{rs}_{abij}(\partial t\partial x_r.\phi_t^a)(\partial x_s.\phi_t^b)=0\cr A^{rs}_{abij}\equiv\delta_{ab}(\delta^{rs}_{ij}+\delta^{rs}_{ji}).\end{cases}\tag{\(\clubsuit-5\)} \]

Then, one can easily see that the unique smooth solution (up to rigid flows) of (\(\clubsuit-5\)) that satisfies the initial condition \(\phi^a_0=x^a\) is just \(\phi_t^a=x^a\), for all \(t\in{\mathbb R}\).

\[ (\partial_t.g_{ij})(x,t)=-2R_{ij}(x,t),\quad (x,t)\in M\times{\mathbb R},\tag \(\clubsuit\) \]

where \(M\) is an \(n\)-dimensional smooth, noncompact and complete Riemannian manifold. In particular the assumption that \(M\) is not compact and complete, is a substantial difference with respect to the usual way in which such equation has been considered. In fact, the Ricci flow equation has been introduced to prove the Poincaré conjecture on compact closed \(3\)-dimensional manifolds homotopic equivalent to \(S^3\).

(See works quoted in references, and also the papers quoted below, where the Ricci flow equation has been studied in a new algebraic topologic framework for PDE’s. [1] A. Prástaro, “Geometry of PDE’s. I: Integral bordism groups in PDE’s”, J. Math. Anal. Appl. 319, No. 2, 547–566 (2006; Zbl 1100.35007). [2] R. Agarwal and A. Prástaro, “Geometry of PDE’s. III(II): Webs on PDE’s and integral bordism groups. Applications to Riemannian geometry PDE’s”, Adv. Math. Sci. Appl. 17, No. 1, 267–285 (2007; Zbl 1140.53005). [3] A. Prástaro, “Extended crystal PDE’s”, arXiv: 0811.3693]. In particular in the last two the Poincaré conjecture has been proved by using a method different to that of Hamilton and Perelmann.)

The main results in this paper are theorem 1.1 and corollary 1.2.

Theorem 1.1. Let \((M,g(x))\) be a complete noncompact three dimensional manifold with bounded and non-negative sectional curvature \(0\leq R_m\leq k_0\), for some fixed constant \(k_0\). Let \(g_1(x,t)\), \(g_2(x,t)\), \(t\in[0,T]\), be two smooth complete solutions to the Ricci flow with initial data \(g(x)\). Then, we have \(g_1(x,t)=g_2(x,t)\), for \(0\leq t<t_{\max}\equiv\min\{T,{{1}\over{4k_0}}\}\).

Corollary 1.2: In particular, when \((M,g(x))\) is the usual Euclidean space \(({\mathbb R}^3,\gamma_E)\), it follows that any smooth complete solution \(g(x,t)\) of the Ricci flow, starting from \(g_E\), is just \(g(x,t)=\gamma_E(x)\).

The paper, after a detailed introduction, where some previous works on the Ricci flow equation are recalled, and theorem 1.1 and corollary 1.2 introduced, splits into three more sections. 2. Local pinching estimate. 3. A priori estimates. 4. Concluding remarks.

Reviewer’s remark. This is a very interesting paper as it intersects a subject under focus of the international mathematical community. In particular, in section 4 there is put the following question. “Does the strong uniqueness of the Ricci flow hold on the Euclidean space \({\mathbb R}^n\), for \(n\geq 4\)?” We can see that the answer is: Yes it does! In the following we will go in some details about.

Let us also underline that the restriction to consider noncompact complete metrics does not permit flows with contractions, and neither simple homeomorphic ones. Isometric diffeomorphisms are instead permitted. Therefore, let us add to the equation (\(\clubsuit\)) the condition that \(g(x,t)\) is realized by means of a one parameter set of diffeomorphisms \(\phi_t:M\to M\), as pull-back, i.e, put \(g(x,t)=\phi_t^*\gamma\). One has for the Ricci tensor \(R_{ij}(x)\) of \(\gamma(x)\), the following induced deformation: \(R(x,t)=R(g(x,t))=\phi_t^*R(\gamma(x))\). In fact, for the natural covariance of the Riemannian metric and its Ricci tensor, we get that equations (\(\clubsuit\)) can be written as

\[ \begin{split} (\dot\phi_t)^a_i(x)(\phi_t)^b_j(x)\gamma_{ab}(\phi_t(x))+(\phi_t)^a_i(x)(\dot\phi_t)^b_j(x)\gamma_{ab}(\phi_t(x))\\ +(\phi_t)^a_i(x)(\phi_t)^b_j(x)(\dot\phi_t)^\alpha(x)(\partial_{x_\alpha}.\gamma_{ab})(\phi_t(x))\\ =-2(\phi_t)^a_i(x)(\phi_t)^b_j(x)R_{ab}(\phi_t(x))\end{split}\tag \(\clubsuit-1\) \]

Let us look for solutions of equations (\(\clubsuit-1\)) of the type \(\phi_t^a=e^{\omega t}h^a(x)\). Then, \(\omega\) and \(h^a(x)\) must satisfy the following equations:

\[ h^a_i(x)h^b_j(x)\left\{\omega\left[2\gamma_{ab}(\phi_t(x))+e^{\omega t}h^\alpha(x)(\partial_{x_\alpha}.\gamma_{ab})(\phi_t(x))\right]+2R_{ab}(\phi_t(x))\right\}=0.\tag{\(\clubsuit-2\)} \]

In the case \((M,\gamma)=({\mathbb R}^n,\gamma_E)\), one has \(R_{ab}=0\) and the equations (\(\clubsuit-2\)) reduce to the following ones:

\[ h^a_i(x)h^b_j(x)\omega\left[2\gamma_{ab}(\phi_t(x))+e^{\omega t}h^\alpha(x)(\partial_{x_\alpha}.\gamma_{ab})(\phi_t(x))\right]=0.\tag{\(\clubsuit-3\)} \]

Let us add the condition \(\phi_0=id_M\). We get that necessarily \(h^a(x)=x^a\), for \(a=1,\dots, n\). Therefore, equations (\(\clubsuit-3\)) reduce to the following ones:

\[ \omega\left[2\gamma_{ij}(\phi_t(x))+e^{\omega t}x^\alpha(\partial_{x_\alpha}.\gamma_{ij})(\phi_t(x))\right]=0.\tag{\(\clubsuit-4\)} \]

In a cartesian coordinate system, we have \(\gamma_{ij}=\delta_{ij}\). Therefore, equations (\(\clubsuit-4\)) reduce to \(2\omega\delta_{ij}=0\). Thus we get the unique trivial solution for \(\omega\), i.e., \(\omega=0\) and the unique solution for \(\phi_t(x)\), i.e., \(\phi_t^a=x^a\), that gives the unique solution \(g(x,t)=\gamma_E(x)\).

This result does not depend on the particular assumption made on the type of solutions. In fact, considering equation (\(\clubsuit-1\)) in the case \(M={\mathbb R}^n\), with \(\gamma=\delta_{ij}\), we get the following equations

\[ \begin{cases} A^{rs}_{abij}(\partial t\partial x_r.\phi_t^a)(\partial x_s.\phi_t^b)=0\cr A^{rs}_{abij}\equiv\delta_{ab}(\delta^{rs}_{ij}+\delta^{rs}_{ji}).\end{cases}\tag{\(\clubsuit-5\)} \]

Then, one can easily see that the unique smooth solution (up to rigid flows) of (\(\clubsuit-5\)) that satisfies the initial condition \(\phi^a_0=x^a\) is just \(\phi_t^a=x^a\), for all \(t\in{\mathbb R}\).

Reviewer: Agostino Prástaro (Roma)