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visits | member for | 5 years, 1 month |
seen | 18 hours ago | |
stats | profile views | 1,339 |
Mar 12 |
awarded | Yearling |
Dec 12 |
answered | Moduli spaces of connections as representation spaces |
Nov 8 |
comment |
Is the heat kernel more spread out with a smaller metric?
The only thing I know about the $\theta$'s is that they can be expressed in terms of connection data along the diagonal, see for example the book of John Roe: Elliptic operators,... . But if I remember correctly, the series $(\theta_0(p,q)+t\theta_1(p,q)+...)$ converges for small t and nearby $p,q$ to a smooth map which is the identity for $t=0$ and $p=q.$ Therefore, you can get the inequality in a neighbourhood of the diagonal times the $t=0$ slice. |
Nov 7 |
comment |
Is the heat kernel more spread out with a smaller metric?
Are you aware of the asymptotic expansion for the heat kernel in terms of $t:$ $H_t(p,q)= h_t(p,q)(\theta_0(p,q)+t\theta_1(p,q)+...),$ where $h_t(p,q)=\frac{1}{(4 \pi t)^{dim M/2}}exp(-d(p,q)^2/4t)$ and $\theta_0(p,p)=1.$ At least for small $t$ and nearby points $p,q$ this should give your inequality. |
Nov 3 |
answered | Is there a non-abelian version of the Torelli map? |
Oct 31 |
comment |
Marten's proof of torelli theorem
I do not think that the proof is really mysterious: The idea is to identify the image $W^1$ in the intersection of $W^{g-1}$ with some of its shifts. I think it would be a good idea to do the proof for the genus 3 case by hand. |
Oct 22 |
comment |
Calculating Exterior Distance from Measurements of Inner Geometry
An upper bound is easy. For a lower bound it might be useful to take a look into Brendle's proof of the Lawson conjecture and related literature (e.g. arxiv.org/pdf/1402.1748.pdf,...). |
Oct 21 |
comment |
Calculating Exterior Distance from Measurements of Inner Geometry
I guess that is why Thomas wrote in his answer "a piece of a cylinder", and why I explicitly specified my subsets of the plane and of the cylinder in my answer below. |
Oct 21 |
answered | Calculating Exterior Distance from Measurements of Inner Geometry |
Oct 20 |
comment |
Tangent space describes the manifold's first order characteristic. Is there something like tangent space describes higher order characteristic?
@Michael: You are of course right, that one can recover the multi-linear algebra. Nevertheless, this is not totally transparent, and quite often people prefer to use connections in order to work with higher order derivatives. |
Oct 17 |
answered | Tangent space describes the manifold's first order characteristic. Is there something like tangent space describes higher order characteristic? |
Oct 14 |
comment |
An example of mean curvature flow that does not preserve embeddedness
Have you tried a trefoil knot? I am also pretty sure that you can connected your 2 circles "far out" such that you can get an example of the mean curvature flow which does not stay embedded (before singularities occur). |
Oct 7 |
comment |
Symplectic form on moduli space of connections
I do not know an answer for the general linear case, but for compact groups like $SU(2)$ there is a nice answer, see for example "Some comments on CHern-Simons Gauge Theory" by Ramada, Singer and Weitsman. |
Sep 26 |
answered | Factors of automorphy from Chern connection |
Sep 23 |
revised |
Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature
added 429 characters in body |
Jul 2 |
awarded | Curious |
Jun 16 |
answered | Examples of “Unusual” Classifications |
May 8 |
comment |
Deformation of Hitchin-Simpson correspondence
3. A holomorphic quadratic differential $\alpha$ determines in a natural way a holomorphic Higgs field $\Psi_\alpha\colon L\oplus L^*\to KL\oplus KL^*$ whenever $L^2=K.$ Then you add this Higgs field onto the Higgs field $\theta.$ |
May 8 |
comment |
Deformation of Hitchin-Simpson correspondence
2. You should try to compute what happens to first order in $t$ when you look at the solution to the self-duality equations corresponding to $t\alpha.$ This gives you a section $\bar\alpha\in\Gamma(\Sigma,\bar K K^{-1})$ (w.r.t. the hyperbolic metric) which can be considered as the tangent vector given by the variation of Riemann surface structures. |
May 8 |
comment |
Deformation of Hitchin-Simpson correspondence
1.in every conformal class of metrics on a cpt. or. surface of genus $g\geq2$, there is a unique metric of constant curvature -4. Moreover, every metric which is not compatible with a given complex structure on your smooth surface gives a different Riemann surface structure on the surface. |