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Jul
10
comment Prove that the holonomies along any two homotopic paths are the same if the curvature of the connection vanishes
Dear QIAOJIAXIN, your question is not research level, and you should be able to extract the answer from any differential geometry book dealing with the Frobenius theorem, e.g. Lee's "Manifolds and Differential Geometry".
Jul
10
comment Prove that the holonomies along any two homotopic paths are the same if the curvature of the connection vanishes
Dear Dimitri, I think this is not only too complicated, but also circular in the following sense: In order to prove Theorem 4.4. and Lemma 4.5 you have (at some point) to prove a (stronger) version of the assertion, see for example Proposition 3.8 in the mentioned paper.
Jul
1
comment Can every hyperelliptic genus 3 surface be minimally immersed in flat $T^3$
Dear Piojo, I do not understand your comment. Whenever you have a minimal surface with translational periods only, you get two holomorphic spinors, and vice versa. The problem is to find two spinors such that the corresponding real parts of the 2g many $\mathbb C^3$ valued periods span a lattice in euclidean 3-space.
Jul
1
revised Can every hyperelliptic genus 3 surface be minimally immersed in flat $T^3$
added 16 characters in body
Jul
1
answered Can every hyperelliptic genus 3 surface be minimally immersed in flat $T^3$
Apr
30
accepted Periods of translation surfaces
Apr
30
comment Periods of translation surfaces
Dear Matheus, somehow, I have not realized that my question was answered until now. Thanks.
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.