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A mathematician specializing in nonlinear hyperbolic PDEs, geometric analysis of pseudo-Riemannian manifolds, and general relativity.


18h
awarded  mp.mathematical-physics
1d
answered Minkowski spacetime in Newman Penrose formalism
1d
answered The Laplacian of an expression involving the Ricci tensor
1d
comment The Laplacian of an expression involving the Ricci tensor
I am slightly confused by your notation: do you mean to first fix some $x$ and consider the function $y\mapsto \triangle_y (Ric_y(\exp_y^{-1} x, \exp_y^{-1} x))$ (well-defined in some neighbourhood of $x$), and then evaluating it at $y = x$?
1d
answered Reference request for the focussing example
Aug
21
comment Vector Fields in a Riemannian Manifold
@DeaneYang: yes. But then you need the fact that the Laplacian uniquely determines the metric. Which you get by looking at its principal part, and then we come back to Tobias' comments. (In any case, if you are talking about my tendency to provide lowbrow answers on MO, I am definitely guilty as charged. :-p)
Aug
21
comment Inequality for a gradient of a function in Holder space
Just take $X$ any fixed invertible linear transformation. Then we have that $\nabla X$ is a constant matrix, and so is the inverse transformation. You then have that $\|\nabla X\|_\gamma = |\nabla X|_0$. Then do scaling as Anthony Quas suggested.
Aug
21
answered Vector Fields in a Riemannian Manifold
Aug
13
answered Conditions on a Lorentzian manifold to ensure existence of global proper-time foliation?
Aug
13
comment Conditions on a Lorentzian manifold to ensure existence of global proper-time foliation?
What do you mean by "tangent vector field" $\vec{t}$ of a "time function"? Do you just mean the metric dual vector to the differential?
Aug
13
comment Conditions on a Lorentzian manifold to ensure existence of global proper-time foliation?
... in the diamond.
Aug
13
comment Conditions on a Lorentzian manifold to ensure existence of global proper-time foliation?
Flowing is not enough. After normalizing the flow may not be complete. Furthermore the incompleteness may occur at different "times". For an extremely simple example, look at the diamond $|x| + |t| < 1$ in 1+1 Minkowski space which is globally hyperbolic. Pick any GH foliation such that along $\{x = 0\}$ the normal vector lines up with the $t$-axis. A propertime foliation built from this GH foliation will have total proper time length of $2$. But for every $\epsilon$ there exists a point $p$ in the diamond such that any timelike curve of length $>\epsilon$ through $p$ cannot be contained...
Jul
21
awarded  Popular Question
Jun
30
revised Does strong convergence in $W_p^1$ imply strong convergence of derivatives of absolute values in $L_p$?
edited title
Jun
30
answered Equivalent Norms on Sobolev Spaces
Jun
30
answered An inequality with critical Sobolev exponent
Jun
30
comment An inequality with critical Sobolev exponent
For the example Pietro gave, doesn't $\|v_\epsilon\|_p$ for $p < 2^*$ (where $v_\epsilon = \epsilon^{-n/2^*} u(x / \epsilon)$) go to zero for $\epsilon \to 0$? How is adding $\|u\|_p^{p/2}$ supposed to help? (With the $\epsilon$ in front it can only help if it diverges...)
Jun
22
answered Positive solutions to Yamabe problem?
Jun
22
comment Positive solutions to Yamabe problem?
I don't understand how your displayed equation has anything to do with the paragraphs above. The conformal change equation for scalar curvature is well-known (see Wikipedia or this article), and is definitely non-linear. It differs from your linear equation. Did you mean to have a term $u^q$ for some $q$ on the right hand side?
Jun
18
answered Fourier series and transform related to Epicycles