Timeline for Laplace-Beltrami of the mean curvature
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 11, 2021 at 14:53 | comment | added | Lightmann | Thank @RobertBryant. Are there some references or books on this topic? | |
| Aug 10, 2021 at 15:23 | answer | added | Robert Bryant | timeline score: 4 | |
| Aug 10, 2021 at 12:16 | comment | added | Robert Bryant | The Laplacian of $H$ with respect to the induced metric is 4-th order invariant of the surface, while $H$ and $K$ are only 2-nd order invariants. It's easy to show that there cannot be any generally valid formula of the form $\Delta_SH = F(H,K)$, though, of course, there are plenty of examples of local surfaces that satisfy this equation for a given function $F$. | |
| Aug 10, 2021 at 0:21 | comment | added | Lightmann | Thanks Ben McKay and Deane Yang. My aim is to approximate $\Delta_S H$ by some polynomials of $H, K$. I don't know how to derive it. Is there some reference on this direction? Why the result is not a function of $H,K$? @BenMcKay | |
| Aug 9, 2021 at 19:00 | comment | added | Deane Yang | A reasonable analogy of your question would be the following: Suppose $f$ and $g$ are scalar functions (on $\mathbb{R}^2$ for simplicity) such that $\Delta f = g$. What's the formula for $\Delta g$? There's not much you can say beyond the fact that $\Delta g = \Delta^2f$. The answer here would be similar. | |
| Aug 9, 2021 at 16:49 | comment | added | Ben McKay | What sort of quantities do you allow in the dots? Obviously not the Laplace Beltrami of $H$, so maybe not derivatives. Perhaps polynomials in $H,K$? Clearly it is not a function of $H,K$. | |
| Aug 9, 2021 at 16:36 | review | First posts | |||
| Aug 9, 2021 at 16:53 | |||||
| Aug 9, 2021 at 16:31 | history | asked | Lightmann | CC BY-SA 4.0 |