bio | website | |
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visits | member for | 4 years, 5 months |
seen | 11 hours ago | |
stats | profile views | 1,241 |
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 |
Mar 18 |
answered | In what condition is a conformal flat manifold flat? |
Mar 12 |
awarded | Yearling |
Feb 28 |
comment |
quantitative version of the rigidity of the 2-sphere
I am not an expert on the Ricci flow, but can't you use the existence of the flow for positive curvature metrics and corresponding estimates to deduce "how long" you need to flow, i.e., how close you are to the round sphere? |
Feb 27 |
answered | A question on differential forms and integral invariants |
Feb 7 |
awarded | Scholar |
Feb 7 |
accepted | Riemann's theorem on theta |
Jan 27 |
comment |
Normalizing the value of a principal connection at a point
You both are right, I was not clear enough with my phrasing "connection 1-form with respect to this section vanishes at p," which should mean pullback of the connection form by the section. And Robert's remark shows us how nicely differential calculus can be approximated linearly, I just oversaw it. |
Jan 27 |
answered | Normalizing the value of a principal connection at a point |
Dec 13 |
answered | How to compute the normals to Costa's minimal surface? |
Oct 29 |
comment |
Obstruction to this gauge choice of the connection of a vector bundle
Sorry, my mistake. Of course, you are right, as I should have been aware of if I had read the whole question. |
Oct 28 |
comment |
Obstruction to this gauge choice of the connection of a vector bundle
Somehow, a short computation shows me that your equation $\mathcal L_RA=d_A(I_R A)$ is not gauge invariant, and therefore not even well-defined on an arbitrary vector bundle |
Sep 22 |
awarded | dg.differential-geometry |
Aug 1 |
awarded | Nice Answer |
Jul 25 |
comment |
Formula for the curvature of an induced connection
If $a=0$ then $F_B=\lambda\cdot F_A$ where $\lambda\cdot\colon\mathfrak g\to \mathfrak h$ is the induced map on Lie algebras. For $a\neq0$ you would in general need also to know $f_*A.$ |