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Manifolds with positive curvature operators are space forms. (English) Zbl 1185.53073

For each point \(p\) of a Riemannian manifold \(M\) the curvature tensor defines a self adjoint curvature operator \(R_{p} : \Lambda^{2}(T_{p}M) \rightarrow \Lambda^{2}(T_{p}M)\) such that \(\langle R_{p}(v \wedge w), x \wedge y \rangle = \langle R(v,w) x, y \rangle\) for \(x,y,z,w \in T_{p}M\). Here \(\Lambda^{2}(T_{p}M)\) is equipped with the inner product induced from the inner product on \(T_{p}M\). The curvature operator is said to be 2-positive if the sum of the two smallest eigenvalues is positive. In this article the authors use Hamilton’s maximum principle and an injectivity radius estimate of Klingenberg to obtain the following
Theorem: On a compact Riemannian manifold of dimension \(n \geq 3\) the normalized Ricci flow evolves a Riemannian metric with 2-positive curvature operator at every point to a limit metric with constant sectional curvature. This result, previously known in dimensions \(\leq 5\), carries over to orbifolds of dimension \(\geq 3\).
Let \(O(n,\mathbb{R})\) denote the group of real orthogonal \(n\times n\) matrices, and let \(\mathfrak{s} \mathfrak{o} (n, \mathbb{R})\) denote its Lie algebra of real \(n\times n\) skew symmetric matrices. To generalize the arguments of R. S. Hamilton [J. Differ. Geom. 24, 153–179 (1986; Zbl 0628.53042)] the authors begin by considering the vector space \(S(\mathfrak{s} \mathfrak{o} (n, \mathbb{R})\) of linear operators on \(\mathfrak{s} \mathfrak{o} (n, \mathbb{R})\) that are self adjoint with respect to the inner product given by \(\langle A , B \rangle = - \frac{1}{2}~\text{trace}(AB)\) for A,B \(\in \mathfrak{s} \mathfrak{o} (n, \mathbb{R}\). For a point p \(\in\) M an orthonormal basis \(\{e_{1},\dots , e_{n} \}\) of \(T_{p}M\) defines a natural linear isometry \(\varphi : \Lambda^{2}(T_{p}M) \rightarrow \mathfrak{s} \mathfrak{o} (n, \mathbb{R})\), and the self adjoint curvature operator \(R_{p} : \Lambda^{2}(T_{p}M) \rightarrow \Lambda^{2}(T_{p}M)\) gives rise to an element \(\overline{R_{p}} = \varphi \circ R_{p} \circ \varphi^{-1}\) of \(S(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\). For g \(\in O(n, \mathbb{R})\) the orthonormal basis \(\{g(e_{1}), ... , g(e_{n}) \}\) of \(T_{p}M\) gives rise to the operator \(\overline{R_{p}}(g) = \varphi_{g} \circ R_{p} \circ \varphi_{g}^{-1} = g^{-1} \circ \overline{R_{p}} \circ g = g^{-1}(\overline{R_{p}})\). The operators \(g(\overline{R_{p}}), g \in O(n, \mathbb{R})\), lie in the \(O(n, \mathbb{R})\)-invariant subspace \(S_{B}^{2}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\) of curvature operators in \(S(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\) that is defined by the Bianchi identities for \(R_{p}\).
Let \(ad : \Lambda^{2}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R})) \rightarrow \mathfrak{s} \mathfrak{o} (n, \mathbb{R})\) be the linear map defined by \(ad(R \wedge S) = [R,S]\) for \(R,S \in \mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\), and for \(p \in M\) let \(R_{p}^{\#} = ad \circ (R_{p} \wedge R_{p}) \circ ad^{*} : \mathfrak{s} \mathfrak{o} (n, \mathbb{R})) \rightarrow \mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\), where \(ad^{*}\) denotes the metric adjoint of ad and \(R_{p}\) is identified with \(\overline{R_{p}}\). Call a subset \(C\) of \(S_{B}^{2}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))~ \mathit{invariant}\) if it is invariant under \(O(n, \mathbb{R})\) and also invariant under the ordinary differential equation \(\frac{dR}{dt} = R^{2} + R^{\#}\). Hamilton’s maximum principle says that if \(C\) is a closed, convex invariant subset of \(S_{B}^{2}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\) and if \(R_{p} \in C\) for all p \(\in\) M, then \(R_{p}(t) \in C\) for all p \(\in\) M and all t in the domain of the Ricci flow given by \(\frac{\partial g}{\partial t} = - 2 Ric(g)\). Here \(R_{p}(t)\) is the curvature operator, or more precisely, its orbit under \(O(n, \mathbb{R})\) defined by the metric g(t).
Given an appropriate closed, convex subset \(C\) of \(S_{B}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\) defined by explicit conditions on the curvature operators \(R_{p}\) the authors construct a family of closed, convex subsets \(C_{a,b}\) in \(S_{B}^{2}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\), where \(a \geq 0\), \(b \geq 0\) are explicitly defined and \(C_{0,0} = C\). This allows the authors to construct a continuous family of invariant cones in \(S_{B}^{2}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\) that join the invariant cone of 2-positive curvature operators to the invariant cone of positive multiples of the identity. The authors then generalize the arguments of Hamilton for 4-manifolds to obtain the theorem stated above.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)

Citations:

Zbl 0628.53042
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