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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.
May
7
answered Deformation of Hitchin-Simpson correspondence