Timeline for On a closed manifold whose curvature is close to "hyperbolic"
Current License: CC BY-SA 4.0
14 events
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| Feb 16, 2022 at 20:33 | comment | added | Igor Belegradek | @Bo_Y: you can read more about the weak curvature tensor in Nikolaev's paper "Stability problems in a theorem of F. Schur". | |
| Feb 16, 2022 at 8:22 | comment | added | Bo_Y | @DeaneYang Following your suggestion. Using the $C^{1, \alpha}$ convergence, I think it holds that any sufficiently small geodesic triangle on the limit should have the same interior angle with that in the space form $K=-1$. Is the metric smooth hyperbolic? Not sure how to go back to curvature. It sounds a special case of the rigdity result of Alexandrov mentiond by Igor. | |
| Feb 16, 2022 at 8:13 | comment | added | Bo_Y | The regularity of limit seems to appear in many similar pinching problems. Another example is Berger's famous $\frac{1}{4}-\epsilon$ positive pinching (Ann. Inst. Fourier (Grenoble) (1983)). There he proves the limit is smooth by some comparison geometry and use diamater, cut locus and other stuff. I don't know French, not sure if his method is helpful to negative pinching. | |
| Feb 16, 2022 at 7:56 | comment | added | Bo_Y | @IgorBelegradek Yes, I saw it but it seems it deals with $C^{1, \alpha}$ convergence away from the bubble. I did not see exactly the weak curvature defined. Back to sectional curvature pinching, I found Fukaya has another approach in Gromov's situation in his survey "Hausdorff convergence of Riemannian manifolds and its applications" (1990). On p.198 Lemma 15.3 there he proved the limit is smooth by showing the limit metric is symmetric and its isometric group is a Lie group, then it is smooth as we write local coordinates $g_{ij}$ in terms of smooth data from this Lie group. | |
| Feb 15, 2022 at 3:48 | comment | added | Igor Belegradek | @Bo_Y: p.360 says "from the proof of Theorem 0.1, the weak curvature tensor of g is well defined and we have...". | |
| Feb 15, 2022 at 3:31 | vote | accept | Bo_Y | ||
| Feb 15, 2022 at 3:30 | comment | added | Bo_Y | Sorry for my delay! Thanks to both of you for bringing new references. I sounds to me that Gao did not discuss the weak curvature explicitly in his work? In any case, I agree that some weak notion of curvature (bypassing the Alexandrov curvature) works on the limit since we have C^{1,1} convergence. I will try to figure out an argument following your suggestions. | |
| Feb 11, 2022 at 18:13 | comment | added | Deane Yang | Another approach is not to use the $C^{1,\alpha}$ convergence theory but the $W^{p,2}$ convergence theory, including the existence of harmonic coordinates (Anderson-Cheeger do this very nicely). I think that might suffice. | |
| Feb 11, 2022 at 18:08 | comment | added | Deane Yang | Is $C^1$ convergence not enough to show that geodesic segments converge to geodesic segments strongly enough to use geodesic triangles to show the limit is hyperbolic? | |
| Feb 11, 2022 at 17:38 | history | edited | Igor Belegradek | CC BY-SA 4.0 |
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| Feb 11, 2022 at 14:21 | comment | added | Igor Belegradek | @Bo_Y: maybe I am misremembering the flow part. Let me think this through. | |
| Feb 11, 2022 at 3:02 | comment | added | Bo_Y | then I don't know why $h(0)$ can be defined and is hyperbolic. Did I miss sth? | |
| Feb 11, 2022 at 3:02 | comment | added | Bo_Y | Thanks for your answer! Regarding the 2nd approach, I did tried before asking the question. Assume there is a sequence $(M_i, g_i)$ with the above assumption. there we may construct a sequence of flows $(M_i, g_i(t))$ on $[0, T]$. Then the bounds of covariant derivatives is like $\frac{C}{t}$ and $K(g_i(t)) \in [-1-ct, -1+\epsilon_i+ct]$. It seems that we can not take a sequence like $(M_i, g_i(t_i))$ with $t_i$ converging to zero. Instead we have to do it for a fix $t_o$. This will produce a limit flow $(N, h(t))$ where $t \in (0, T]$ with $K(h(t)) \in [-1-ct, -1+ct]$. | |
| Feb 10, 2022 at 16:50 | history | answered | Igor Belegradek | CC BY-SA 4.0 |