Timeline for History of Laplacian comparison theorem
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5, 2021 at 6:28 | comment | added | Quarto Bendir | @DeaneYang Bishop announced his work in the 1963 Notices of the AMS | |
| Dec 1, 2021 at 5:46 | review | Suggested edits | |||
| Dec 1, 2021 at 7:35 | |||||
| Dec 1, 2021 at 3:57 | comment | added | Deane Yang | @IgorBelegradek, I forgot about Rauch. To my surprise, few people cite Rauch's original paper. It appears to be this one: jstor.org/stable/1969309 | |
| Nov 30, 2021 at 15:53 | comment | added | Igor Belegradek | @DeaneYang: the usual proof of the Jacobi field comparison theorem (which I think is due to Rauch) reduces to the Sturm comparison theorem. This is explained in many places, e.g. Lee's "Riemannian geometry". This is how one shows that e.g. an upper sectional curvature bound gives a lower bound on the conjugate radius. Toponogov's theorem is more involved. | |
| Nov 30, 2021 at 14:45 | comment | added | Deane Yang | Using Sturm for Jacobi fields is perhaps due to Toponogov? As for volume comparison, when did Bishop prove his inequality? | |
| Nov 30, 2021 at 2:25 | comment | added | Igor Belegradek | Modern history can be traced from the references on p.1 of Eschenburg's "Comparison Theorems in Riemannian Geometry", see math.toronto.edu/~vtk/eschenburg-comparison.pdf. I think the subject originates from ODE comparison theorems, see en.wikipedia.org/wiki/Comparison_theorem. For example, Sturm's comparison theorem is in J. Math. Pures Appl. 1 (1836), 106–186. | |
| Nov 30, 2021 at 1:36 | comment | added | Quarto Bendir | Thanks, that looks like a great paper. I see there's also an extension of the usual Hopf-Rinow theorem. Although that does answer the version of the theorem I put in my question, I am still curious about the more general version. | |
| Nov 30, 2021 at 1:24 | comment | added | Igor Belegradek | It seems this goes back to Calabi's "An extension of E. Hopf’s maximum principle with an application to Riemannian geometry". Duke Math. J., 25:45–56, 1958. See projecteuclid.org/journals/duke-mathematical-journal/volume-25/…. | |
| Nov 30, 2021 at 1:12 | history | edited | LSpice | CC BY-SA 4.0 |
Capitalise Laplace and Hesse to match Riemann and Ricci
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| Nov 30, 2021 at 0:58 | history | asked | Quarto Bendir | CC BY-SA 4.0 |