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Register a new userMaximal Sp(4,R) surface group representations, minimal immersions and cyclic surfaces
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Authors
Brian Collier
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Let $S$ be a closed surface of genus at least $2$. For each maximal representation $rho: pi_1(S)rightarrowmathsf{Sp}(4,mathbb{R})$ in one of the $2g-3$ exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric space $mathsf{Sp}(4,mathbb{R})/mathsf{U}(2)$ is a minimal immersion. Using a Higgs bundle parameterization of these components, we give a mapping class group invariant parameterization of such components as fiber bundles over Teichmuller space. Unlike Labouries recent results on Hitchin components, these bundles are not vector bundles.
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In this article, we interpolate a given real analytic spacelike curve $a$ in Lorentz-Minkowski space $mathbb{L}^3$ to another real analytic spacelike curve $c$, which is close enough to $a$ in a certain sense, by a maximal surface using inverse function theorem for Banach spaces. Using the same method we also interpolate a given real analytic curve $a$ in Euclidean space $mathbb{E}^3$ to another real analytic curve $c$, which is close enough to $a$ in a certain sense, by a minimal surface. The Bjorling problem and Schwartzs solution to it play an important role.
In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank 2.
We study the character variety of representations of the fundamental group of a closed surface of genus $ggeq2$ into the Lie group SO(n,n+1) using Higgs bundles. For each integer $0<dleq n(2g-2),$ we show there is a smooth connected component of the character variety which is diffeomorphic to the product of a certain vector bundle over a symmetric product of a Riemann surface with the vector space of holomorphic differentials of degree 2,4,...,2n-2. In particular, when d=n(2g-2), this recovers Hitchins parameterization of the Hitchin component. We also exhibit $2^{2g+1}-1$ additional connected components of the SO(n,n+1)-character variety and compute their topology. Moreover, representations in all of these new components cannot be continuously deformed to representations with compact Zariski closure. Using recent work of Guichard and Wienhard on positivity, it is shown that each of the representations which define singularities (i.e. those which are not irreducible) in these $2^{2g+1}-1$ connected components are positive Anosov representations.
Kirigami is the art of cutting paper to make it articulated and deployable, allowing for it to be shaped into complex two and three-dimensional geometries. The mechanical response of a kirigami sheet when it is pulled at its ends is enabled and limited by the presence of cuts that serve to guide the possible non-planar deformations. Inspired by the geometry of this art form, we ask two questions: (i) What is the shortest path between points at which forces are applied? (ii) What is the nature of the ultimate shape of the sheet when it is strongly stretched? Mathematically, these questions are related to the nature and form of geodesics in the Euclidean plane with linear obstructions (cuts), and the nature and form of isometric immersions of the sheet with cuts when it can be folded on itself. We provide a constructive proof that the geodesic connecting any two points in the plane is piecewise polygonal. We then prove that the family of polygonal geodesics can be simultaneously rectified into a straight line by flat-folding the sheet so that its configuration is a (non-unique) piecewise affine planar isometric immersion.
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