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Ricci Flow on a Class of Noncompact Warped Product Manifolds

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Abstract

We consider the Ricci flow on noncompact \(n+1\)-dimensional manifolds M with symmetries, corresponding to warped product manifolds \(\mathbb {R}\times T^n\) with flat fibres. We show longtime existence and that the Ricci flow solution is of type III, i.e. the curvature estimate \(|{{\mathrm{Rm}}}|(p,t) \le C/t\) for some \(C > 0\) and all \(p \in M, t \in (1,\infty )\) holds. We also show that if M has finite volume, the solution collapses, i.e. the injectivity radius converges uniformly to 0 (as \(t \rightarrow \infty \)) while the curvatures stay uniformly bounded, and furthermore, the solution converges to a lower dimensional manifold. Moreover, if the (n-dimensional) volumes of hypersurfaces coming from the symmetries of M are uniformly bounded, the solution converges locally uniformly to a flat cylinder after appropriate rescaling and pullback by a family of diffeomorphisms. Corresponding results are also shown for the normalized (i.e. volume preserving) Ricci flow.

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Notes

  1. \(T^n\) denotes the n-dimensional torus.

  2. See the Appendix.

  3. That is, if O is an orbit, \(p \in O\) and if we consider the action of G restricted to O, then the induced representation of the isotropy group \(G_p \subset G\) on \(T_p O\) is irreducible.

  4. That is, \(k+g^2 g_{T^n}(x,q)(a+v,b+w) = k(x)(a,b) + g^2 (x)g_{T^n}(q) (v,w)\) for all \(x \in \mathbb {R}, q \in T^n\) and all \(a,b \in T_x \mathbb {R}, v,w \in T_q T^n\) (where we identified \(T_{(x,q)} (\mathbb {R}\times T^n) \cong T_x \mathbb {R}\oplus T_q T^n\)).

  5. More precisely, \(h(t), t \in [0,T_{\max })\) is the unique maximal solution of the Ricci flow in the class of families of metrics \(\widehat{h}(t), t \in [0,T)\), where \(0 < T \le \infty \), such that \(\widehat{h}(t)\) is complete for all \(t \in [0,T)\), \(\sup _{x \in M, t \in [0,S]}|{{\mathrm{Rm}}}_{\widehat{h}(t)}|_{\widehat{h}(t)}(x) < \infty \) for all \(0< S < T\) (i.e. the curvatures are bounded, when the family is restricted to a compact time interval) and \(\widehat{h}(0) = h_0\).

  6. In this case the orbits have to be replaced by sets \(\{x\} \times T^n\), where \(x \in \mathbb {R}\).

  7. Compare this and the next paragraph also with the introduction in [32] by Lott and Sesum.

  8. Here “orthogonal” means that the plane contains the direction orthogonal to O.

  9. Here and in the following we identify \(M/G \cong \mathbb {R}\), and the second component of the expression \(|K_V|(x,0)\) refers to time \(t = 0\).

  10. This time parametrized over O instead of \(M/G \times O\).

  11. Not every flat metric \(g_{T^n}\) can occur in (3.2): a necessary condition is that the isometry group of \((T^n, g_{T^n})\) acts isotropy irreducibly, which is for example not the case for \(\mathbb {R}^2/(\mathbb {Z}\times 2\mathbb {Z})\).

  12. The words “orbit” and “orbit space” are w.r.t. the induced G-action on \(\mathbb {R}\times T^n\), i.e. by (3.1) and the definition of \(\phi \) the orbits are the sets \(\{x\} \times T^n\), where \(x \in \mathbb {R}\), and the orbit space can be identified with \(\mathbb {R}\); in case \(k + g^2 g_{T^n}\) is not of the form (3.2), i.e. is not the pullback of a G-invariant metric under \(\phi \), by “orbit” we also mean a set \(\{x\} \times T^n\) and by “orbit space” we mean \(\mathbb {R}\).

  13. Here \(g_{\mathbb {R}^n}\) denotes the standard metric on \(\mathbb {R}^n\).

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Acknowledgements

This research was mostly carried out as part of my doctoral thesis, and at this point I would like to cordially thank my advisor Klaus Ecker! Also many thanks to Ahmad Afuni, Richard Bamler, Theodora Bourni, Bernhard Brehm, Bernold Fiedler, Lutz Habermann, Adrian Hammerschmidt, Gerhard Huisken, Tom Ilmanen, Felix Jachan, Dan Knopf, Ananda Lahiri, Markus Röser, Andreas Savas-Halilaj, Lars Schäfer, Oliver Schnürer, Felix Schulze, Brian Smith and Elmar Vogt! I also thank the SFB 647 “Space–Time–Matter. Analytic and Geometric Structures” of the DFG (German Research Foundation) for financial support.

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  1. Freie Universität Berlin, Berlin, Germany

    Tobias Marxen

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Appendix

Appendix

We show that the G-action on M descends to a G-action on \(M/T^n\): G is the semidirect product \(\mathrm {O}(n,\mathbb {Z}) \ltimes T^n\), where \(\mathrm {O}(n,\mathbb {Z})\) denotes the symmetry group of the n-cube, i.e. the group of all orthogonal \(n \times n\)-matrices with integral entries. The group structure of \(\mathrm {O}(n,\mathbb {Z}) \ltimes T^n\) is given by

$$\begin{aligned} (B,b) \circ (A,a) := (BA,Ba+b), \end{aligned}$$

where \(A,B \in \mathrm {O}(n,\mathbb {Z})\) and \(a,b \in T^n\).

Now let O be an orbit w.r.t. the \(T^n\)-action and let \(p,q \in O\), i.e. \(q = (I_n,c) \cdot p\) for some \(c \in T^n\), where \(I_n\) denotes the identity \(n \times n\)-matrix. Then, for any \((A,a) \in \mathrm {O}(n,\mathbb {Z}) \ltimes T^n\),

$$\begin{aligned} (A,a) \cdot q&= (A,a) \cdot ((I_n,c) \cdot p) = ((A,a) \circ (I_n,c)) \cdot p = (A,Ac+a) \cdot p \\&= ((I_n,Ac) \circ (A,a)) \cdot p = (I_n,Ac) \cdot ((A,a) \cdot p). \end{aligned}$$

Hence \((A,a) \cdot p,(A,a) \cdot q\) belong to the same \(T^n\)-orbit.

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Marxen, T. Ricci Flow on a Class of Noncompact Warped Product Manifolds. J Geom Anal 28, 3424–3457 (2018). https://doi.org/10.1007/s12220-017-9964-3

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