Four-manifolds with \(1/4\)-pinched flag curvatures. (English) Zbl 1187.53066

The authors aim to characterize solutions of the Ricci-flow equation on a connected compact Riemannian manifold \((M,g_0)\) with respect to the so-called pinching constant \(\lambda=\text{min sec/max sec}\), i.e., by adding an upper as well as a lower bound on the sectional curvature (denoted {sec}). With respect to this scalar characterization of Riemannian manifolds, there is some interesting information on their global structure. Let us recall some of them. 6mm
(Brendle and Schoen) If \(M\) is a compact \(n\)-dimensional Riemannian manifold with \(\lambda\in({{1}\over{4}},1)\) (strict pinching), \(M\) is diffeomorphic to a spherical space form. If \(\lambda\in[{{1}\over{4}},1]\) \(M\) is diffeomorphic to a spherical space form or isometric to a locally symmetric space.
(Berger-Klingenberg) If \(M\) is a compact simply connected manifold with \(\lambda\geq{{1}\over{4}}\) then \(M\) is either homeomorphic to \(S^n\) or isometric to \({\mathbb CP}^n\), \({\mathbb HP}^n\) or \(Ca{\mathbb P}^2\), with their standard Fubini metric.
(Synge) A manifold with positive curvature does not necessarily be simply connected. In fact if \(\dim M=n=2k\), one has \(\pi_1(M)=0\) if orientable and \(\pi_1(M)={\mathbb Z}_2\) if non-orientable. If \(\dim M=n=2k+1\) and positively curved \(M\) is orientable.
(Cheeger) Given a constant \(\epsilon>0\), there are only finitely many diffeomorphism types of compact simply connected \(2n\)-dimensional manifolds \(M\) with \(\lambda\geq\epsilon\).
(Fang-Rong-Petrunin-Tuschmann) Given a constant \(\epsilon>0\), there are only finitely many diffeomorphism types of compact \((2n+1)\)-dimensional manifolds \(M\) with \(\pi_1(M)=\pi_2(M)=0\) and \(\lambda\geq\epsilon\).
In this paper, one considers four-dimensional manifolds with \(\lambda\)-pinched flag curvature, (\(0\leq \lambda<1\)), i.e., \(R_u(v,v)\geq\lambda(x)R_u(w,w)\), where \(R_u(.,.)=R(u,.,u,.)\) is the symmetric bilinear form, defined by the curvature tensor \(R\) and a nonzero vector \(u\in T_xM\). Here \(v,w\in T_x^\perp \langle{\mathbb R}u\rangle\), \(|v|=|w|\). The main result in this paper is that the class of positively curved compact connected \(4\)-dimensional manifolds, with \(\lambda\)-pinched flag curvature, \(\lambda\geq {{1}\over{4}}\), is invariant under Ricci-flow. Moreover, any such manifold is either diffeomorphic to a spherical space form or isometric to \({\mathbb CP}^2\) with Fubini-Study metric (up to scaling). The paper, after a detailed introduction, splits in six sections: 2. Notation and preliminary results. 3. Preserving flag pinching. 4. Lower bound. 5. Improving flag pinching. 6. Convergence. 7. Weak quarter pinched flag curvature.
Remark. In this very interesting paper, the authors refer to a uniqueness existence theorem for local solutions of the Cauchy problem for the Ricci-flow equation. A first proof was given by R. S. Hamilton [1982–1986]. Different proofs were obtained also by D. T. DeTurk [1983] and B. Chow and D. Knopp [2004]. However, it is important to underline that such uniqueness is strictly related to the class of regular solutions considered there. In fact, by recasting the Cauchy problem in the geometric theory of PDE’s, one can see that the Ricci-flow equation defines an analytic submanifold \((RF)\subset J{\mathcal D}^2(W)\) of the second jet-derivative space for sections of the fiber bundle \(\pi:W\equiv {\mathbb R}\times \widetilde{S^0_2(M)}\to {\mathbb R}\times M\), where \(\widetilde{S^0_2(M)}\subset S^0_2(M)\) is the open subbundle of non-degenerate symmetric tensors of type \((0,2)\) on \(M\). However, in order to describe singularities in the flow, it is useful to consider the embeddings \((RF)\subset J{\mathcal D}^2(W)\subset J^2_{n+1}(W)\), where \(J^2_{n+1}(W)\) is the \(2\)-jet-space for \((n+1)\)-dimensional submanifolds of \(W\), \(\dim M=n\). Since the Ricci-flow equation is formally integrable and completely integrable, there exist also singular solutions satisfying the Cauchy problem considered above. Then the characterization of global solutions is obtained by means of the singular integral bordism groups of \((RF)\). For details see works on the geometry of PDE’s by the reviewer of this paper.
For example, in the particular case of \(4\)-dimensional closed, compact, connected, Riemannian manifolds \(M\), one obtains \(\Omega_4^{(RF)}/K^{(RF)}_4\cong{\mathbb Z}_2\oplus{\mathbb Z}_2\cong \Omega_4\), where \(K^{(RF)}_4\) is the kernel of the canonical projection \(p:\Omega_4^{(RF)}\to\Omega_4\cong {\mathbb Z}_2\oplus{\mathbb Z}_2\). In view of this, taking into account the well-known theorems by A. Dold [Math. Z. 65, 200–206 (1956; Zbl 0071.17502)] and C. T. C. Wall [Bull. Am. Math. Soc. 65, 329-331 (1959; Zbl 0128.16801), Ann. Math. (2) 72, 292–311 (1960; Zbl 0097.38801)] on the (oriented) cobordism ring (\({}^{+}\Omega_\bullet\)) \(\Omega_\bullet\), we can get from the results of this paper the following interesting new issue: The \(4\)-dimensional Riemannian manifolds preserving \(\lambda\)-pinched flag curvature, \(\lambda\geq {{1}\over{4}}\), in a Ricci-flow, belong to the cobordism classes in the image \(p^{-1}(r({}^{+}\Omega_4\cong{\mathbb Z}))\subset \Omega_4^{(RF)}\), where \(r:{}^{+}\Omega_\bullet\to\Omega_\bullet\) is the forgetting orientation natural mapping.


53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K55 Nonlinear parabolic equations
58J35 Heat and other parabolic equation methods for PDEs on manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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