1,735 reputation
627
bio website www-ian.math.uni-magdeburg.de/…
location Magdeburg, Germany
age 31
visits member for 3 years, 7 months
seen Jul 16 at 0:38

I am an Australian mathematician.


Jul
11
comment Soft question: How does basic differential geometry “fit together”?
You're not looking for a book recommendation, you just want a high-level map? The problem is that there is no way to fit even just the names and connections between all topics in diff geom on one page. You're going to have to be more specific.
Sep
10
comment How to solve this quasilinear parabolic evolution equation (result of curve shortening flow)?
It is just a one dimensional quasilinear parabolic PDE... this is covered by standard theory. Can you explain more which texts you looked at? Did you check Lieberman for example?
Sep
6
comment Geometric Interpretation: Parallel forms are harmonic
See the answer for the first question. For the second: It isn't an arbitrary solution of the Laplacian, it's the Laplacian acting on the volume form.
Sep
2
comment Geometric Interpretation: Parallel forms are harmonic
The fact that the volume form is parallel implies that infinitesimal deformations of the volume form act exactly linearly. Since the Levi-Civita connection controls also the cotangent deformations, this implies that the volume form expands linearly in every possible direction. This is stronger than $\Delta \mu = 0$!
Apr
12
comment Periodic polynomial?
This is wrong -- consider $\prod (1/n)$.
Nov
15
comment characterizing semi-Riemannian spaces of constant curvature
Hi Jason. Your example of $S^2\times S^2$ is not with the metric I normally consider (I normally take the flat metric on $S^2\times S^2$). With the round metric, is this still a smooth manifold? Does the condition $\nabla R = 0$ make sense at the transition lines from positive to zero curvature? It seems to me that there are regions with constant positive curvature, and constant zero curvature, and nothing in-between.... I'm probably being dense, but could you shed some light on this?
Oct
14
comment An inequality in Evans' PDE
It is just $ab \le \frac12a^2 + \frac12b^2$.
Oct
14
comment inverting a cone to a torus
You should just edit the one answer...
Oct
13
comment An inequality in Evans' PDE
This is just Peter-Paul: the C is not really the same C as on the left. It is $ C^2/2$.
Oct
10
comment A positive “Fourier transform” is integrable
What does the complex plane in the subscript mean?
Oct
8
comment signed curvature
If this is homework, please add the homework tag.
Oct
6
comment Total mean curvature in $L^2$ and minimal surfaces in spaces with non-positive sectional curvature
Oh! I had a notational mix-up. That's one problem I don't see ever being solved in differential geometry: some semblance of unification of notation. Nice answer.
Oct
5
comment Total mean curvature in $L^2$ and minimal surfaces in spaces with non-positive sectional curvature
I don't understand the application of G-B here. I'm probably being dense, but I thought we had $\int K = 4\pi$ and not $\int S = 8\pi$. (The $S$ is the scalar curvature of the background space, and should be negative, right?) Perhaps you could add a bit more detail there for the slowpokes like me.
Oct
5
comment Total mean curvature in $L^2$ and minimal surfaces in spaces with non-positive sectional curvature
Thanks for the answer. I think you'll find that the mean curvature scales like $r^{-1}$; this is easy to remember from the scale invariance of the Willmore energy in Euclidean space: the measure scales like $r^2$ and so the integrand $H^2$ must scale like $r^{-2}$. Or just from thinking about the prinicipal curvatures, which must scale like $r^{-1}$.
Sep
28
comment How do I interpret $F_1, \neg F_1 \vdash F_2$
@Jiew If you know both $F_1$ and $not F_1$ then you have a contradiction. A falsehood. An example is that both the propositions $1+1 = 2$ and $1+1 = 3$ are valid for whatever reason. The rule in your question then gives ANYTHING as a valid consequence. For example, you could conclude $1+1 = 73$, $pigs fly$ or whatever you like.
Sep
18
comment Constant tensors and covariant derivatives
@Yuri Yes, you're right, it doesn't vanish and will involve second fundamental forms. Also, I think the answer to my other question is no. I guess this question can be closed or if it is preferable I can answer it myself.
Sep
18
comment Constant tensors and covariant derivatives
@Yuri I mean for $\text{Ric}$ to act on $T\Sigma$, although I'm not sure if it turns out to be better for it to act on the pullback bundle. Can you explain why you don't think the expression is well-defined either way? Let's just take $X\in\{e_1,e_2\}$. What I mean by "connection in the normal bundle" is just the projection of $\nabla$ onto $(T\Sigma)^\perp$. Sorry that I am being confusing.
Sep
18
comment Constant tensors and covariant derivatives
@Yuri (1) By restricting $\text{Ric}$ to $\Sigma$. (This involves composition with the immersion map.) (2) $D$ induces a connection in the normal bundle, and this acts "in the normal direction".
Sep
3
comment existence of a map between $\mathbb R^2$ and $\mathbb R$
I guess I was too subtle in my suggestion that the addition of a note to the effect of "The existence of a bijection is clear, it is the additional continuity hypothesis that is at work here." could improve the answer.
Sep
1
comment existence of a map between $\mathbb R^2$ and $\mathbb R$
And if we drop the continuity hypothesis?