67 research outputs found
Harmonic maps for Hitchin representations
Let be a hyperbolic surface, be a Hitchin representation for
, and be the unique -equivariant harmonic map from
to the corresponding symmetric space. We show
its energy density satisfies and equality holds at one point only
if and is the base -Fuchsian representation of
. In particular, we show given a Hitchin representation for
, every -equivariant minimal immersion from a
hyperbolic plane into the corresponding symmetric space is
distance-increasing, i.e. . Equality holds at
one point only if it holds everywhere and is an -Fuchsian
representation.Comment: 14 pages, comments are welcom
On the Uniqueness of Vortex Equations and Its Geometric Applications
We study the uniqueness of a vortex equation involving an entire function on the complex plane. As geometric applications, we show that there is a unique harmonic map u : C β H^2 satisfying βu β 0 with prescribed polynomial Hopf differential; there is a unique affine spherical immersion u : C β R^3 with prescribed polynomial Pick differential. We also show that the uniqueness fails for non-polynomial entire functions with finitely many zeros
Minimal surfaces for Hitchin representations
Given a reductive representation Ο:Ο_1(S) β G, there exists a Ο-equivariant harmonic map f from the universal cover of a fixed Riemann surface Ξ£ to the symmetric space G/K associated to G. If the Hopf differential of f vanishes, the harmonic map is then minimal. In this paper, we investigate the properties of immersed minimal surfaces inside symmetric space associated to a subloci of Hitchin component: the q_n and q_(nβ1) cases. First, we show that the pullback metric of the minimal surface dominates a constant multiple of the hyperbolic metric in the same conformal class and has a strong rigidity property. Secondly, we show that the immersed minimal surface is never tangential to any flat inside the symmetric space. As a direct corollary, the pullback metric of the minimal surface is always strictly negatively curved. In the end, we find a fully decoupled system to approximate the coupled Hitchin system
- β¦