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Register a new userExistence of maximal surface containing given curve and special singularity
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We give a different formulation for describing maximal surfaces in Lorentz-Minkowski space, $mathbb{L}^3$, using the identification of $mathbb L^3$ with $mathbb Ctimes mathbb R$. Further we give a different proof for the singular Bjorling problem for the case of closed real analytic null curve. As an application, we show the existence of maximal surface which contains a given curve and has a special singularity.
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Let $H^4$ denote the hyperbolic four-space. Given a bordered Riemann surface, $M$, we prove that every smooth conformal superminimal immersion $overline Mto H^4$ can be approximated uniformly on compacts in $M$ by proper conformal superminimal immersions $Mto H^4$. In particular, $H^4$ contains properly immersed conformal superminimal surfaces normalised by any given open Riemann surface of finite topological type without punctures. The proof uses the analysis of holomorphic Legendrian curves in the twistor space of $H^4$.
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.
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 prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold $M$ with only one end if $M$ has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and $p$-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures.
Let $G$ be a simple graph with $ngeq4$ vertices and $d(x)+d(y)geq n+k$ for each edge $xyin E(G)$. In this work we prove that $G$ either contains a spanning closed trail containing any given edge set $X$ if $|X|leq k$, or $G$ is a well characterized graph. As a corollary, we show that line graphs of such graphs are $k$-hamiltonian.
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