Otis Chodosh
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Registered User
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Stanford PhD student, interested in differential geometry and analysis.
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May 12 |
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The Isoperimetric problem for domains constrained to lie between two parallel planes fyi your title is very misleading: in common mathematical terminology a "minimal surface" minimizes the surface area with no constraints on volume (actually "minimal" only means a critical point of area). You are discussing "isoperimetric surfaces" . Also, could you clarify exactly what your question is, it seems like you've found an answer, up to integrating an ODE.... |
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May 3 |
answered | A simple and good reference about solitons |
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Apr 18 |
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Applications of pseudodifferential operators to PDE I'm not 100% sure, but I think you can do things like Schauder estimates and a lot of the other parts of "elliptic theory" with psiDO's. I've never read it, but I once glanced at Taylor's book "Pseudodifferential Operators and Nonlinear PDE": unc.edu/math/Faculty/met/nonlin.html which seems to have a bit of this. Anyways, it might also have some other topics you'd be interested in. |
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Apr 18 |
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Isoperimetric profile @Thomas Richard, thats exactly what you do. I think the only issue that might be a tiny bit tricky is getting a good expansion of $r$ in terms of $|B_r(p)|$, but I think that this is not too hard.. |
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Feb 25 |
awarded | ● Popular Question |
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Feb 19 |
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the left hand side of the Ricci flow equation at the initial value minor changes |
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Feb 19 |
answered | When a Riemannian manifold is of Hessian Typ |
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Feb 1 |
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fourier analytic proofs Central limit theorem? |
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Jan 29 |
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Geometric picture of scalar curvature Hi Renato, I have just filled out the registration, so I'll see you there! Also, I agree that the link is broken now. I think that it was working when I put it up, so perhaps MSRI is having some bug. If anyone is interested in the meantime I'd be happy to share the pdf of the worksheet with them. Just send me an email. |
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Jan 29 |
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Geometric picture of scalar curvature added 78 characters in body |
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Jan 28 |
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When is a Pseudo-differential operator trace class or in Dixmier ideal? edited tags |
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Jan 28 |
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Geometric picture of scalar curvature added 5 characters in body |
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Jan 28 |
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Geometric picture of scalar curvature @Deane Yang, Added some more! Feel free to add/modify what is there! |
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Jan 28 |
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Geometric picture of scalar curvature added more |
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Jan 28 |
answered | Geometric picture of scalar curvature |
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Jan 5 |
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What is known about analogous results of Kazdan and Warner in higher dimensions? *by my first sentence, I mean an identity between scalar curvature and the Gauss-Bonnet integrand (as @unknown comments) |
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Jan 5 |
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What is known about analogous results of Kazdan and Warner in higher dimensions? ...giving a contradiction. You may be interested in the following MO post: mathoverflow.net/questions/30035/…. |
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Jan 5 |
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What is known about analogous results of Kazdan and Warner in higher dimensions? @Ritwik, there is no possibility of such an identity holding. If $*e(TM)$ was some multiple of the scalar curvature $R$, this would imply that $\int R = \int *e(TM) dV = <e(TM),[M]>$, and this is a topological invariant of the (smooth structure) topology of $M$. (see en.wikipedia.org/wiki/…). However, one can show that in dimensions $\geq 3$, all manifolds admit a metric of negative scalar curvature, e.g. $S^3$. So, if the above identity held, we could evaluate $<e(TM),[M]>$ on the standard metric and this other one... |
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Jan 4 |
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Symbols of elliptic operators edited tags |
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Jan 4 |
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Which metric spaces have this superposition property? It seems that if you start with a Riemannian manifold with this property it must be highly symmetric. For example, it must be a symmetric space: take a unit speed geodesic through a point $p$ and consider the space $A = \gamma([-\epsilon,\epsilon))$ and $B = \gamma((-\epsilon,\epsilon])$. Then the associated isometry should be a inversion around $p$. I have no idea if this is sufficient, although it would be pretty cool if it was. Also, I'm not quite sure what to do if you modified your definition to demand that $A,B$ are closed. |
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Dec 29 |
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Intuition for mean curvature. added 15 characters in body |
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Dec 29 |
answered | Intuition for mean curvature. |
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Dec 6 |
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Parallel orthogonal complex structures on complexified tangent bundle. @Robert - I see, thanks! |
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Dec 6 |
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Parallel orthogonal complex structures on complexified tangent bundle. Have you thought about the "multiplication by $i$ endomorphism" $J \in End(T\mathbb{R}^3\otimes \mathbb{C})$? This clearly cannot be a "non-complexified" complex structure because $\mathbb{R}^3$ is odd dimensional. |
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Nov 30 |
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The mean curvature of a hypersurface The formula for graphical mean curvature is naturally in a divergence form. A hypersurfsce is locally graphical, so that gives what you want. Any book on minimal surfaces will give you the formula, the one I have in front of me is colding minicozzi eqn 1.6 on p 2. The 0 could replaced by mean curvature you're interested in nonminimal surfaces. |
