Timeline for Constructing a Sobolev space containing the differential k-forms of a Riemannian manifold
Current License: CC BY-SA 3.0
10 events
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| May 17, 2013 at 7:37 | comment | added | alvarezpaiva | Oh, "unexpectedly" because of the title. | |
| May 17, 2013 at 7:36 | comment | added | alvarezpaiva | There is a very nice book where this is (somewhat unexpectedly) REALLY nicely done: Schrödinger Operators: With Applications to Quantum Mechanics and Global Geometry by Barry Simon et al. As a bonus, they have a good exposition of the relation between Hodge theory and Morse theory that was first uncovered by Witten. | |
| May 17, 2013 at 4:34 | comment | added | user34072 | Great, thank you for clarifying Deane! I'll keep your advice about the professors in mind. These guys can be pretty intimidating though :) | |
| May 17, 2013 at 2:14 | comment | added | Deane Yang | And you shouldn't be shy about interrupting these busy professors. No matter how much research they're doing, they have a responsibility to teach their students and answer questions. You can tell them I said so. | |
| May 17, 2013 at 2:12 | comment | added | Deane Yang | Actually, the only detail that you didn't get right in the summary is that the Laplacian is not an operator from $H$ to $H$, where $H$ is a fixed Sobolev space. The Sobolev space you want to work with is $W^{2,k}$, where the norm of $\omega$ is the sum of the $L^2$ norms of the derivatives of $\omega$ up to order $k$. Then the Laplacian is a bounded linear map from $W^{2,k}$ to $W^{2,k-2}$. You construct the Sobolev space as the completion of smooth forms on $M$. | |
| May 16, 2013 at 23:59 | comment | added | Liviu Nicolaescu | Maybe Section 2.1.4 of the notes below might help www3.nd.edu/~lnicolae/ind-thm.pdf | |
| May 16, 2013 at 23:45 | comment | added | user34072 | Um it would not be easier. They're all very busy doing research, and I've already asked Melrose and my topology professor for some clarification. I am definitely speaking with others though. Please keep in mind that my grasp of this material is incomplete, the course I'm taking is at the undergraduate level and somewhat unrelated to this stuff. Would you briefly point out precisely where I went wrong above so that I can understand the material properly? Thanks! | |
| May 16, 2013 at 23:29 | comment | added | Deane Yang | And you're at MIT! Wouldn't it be easier to ask a professor or graduate student? Guillemin, Melrose, Colding, Minicozzi, Mrowka, Staffilani, and probably others could help you with this. | |
| May 16, 2013 at 23:25 | comment | added | Deane Yang | What you have written is not quite right. I'm sure someone will provide an answer with the correct details better than I can. But I'm curious about where you got the information you've posted. Surely, whatever your source is has the correct details, too. Some places where this can be found are Warner, Foundations of Differentiable Manifolds and Lie Groups; Griffiths-Harris, Principles of Algebraic Geometry. | |
| May 16, 2013 at 23:05 | history | asked | user34072 | CC BY-SA 3.0 |