arXiv subjects
Publications and source records attributed to Rivu Bardhan.
This article investigates the global structure of maximal surfaces (space-like immersions with zero mean curvature) in Lorentz-Minkowski $3$-space $\mathbb{E}^3_1$, especially focusing on the interplay between genus, the number of singular components--loci, and simple ends. We construct complete maximal maps with arbitrarily many singular components for any genus $p\geq 0$ with simple ends.
Let $S_g$ be a closed oriented surface of genus $g\ge2$. For a reductive representation $ρ:π_1(S_g)\to\PU(2,1)$, let $E_ρ$ be the energy function on Teichmüller space associated to equivariant harmonic maps into $\CH^2$. For every positive integer $d$ with $3\nmid d$, all sufficiently large $h$, and every $g>h$, we construct an irreducible reductive representation \[ ρ_{g,h,d}:π_1(S_g)\to\PU(2,1) \] with \[ τ(ρ_{g,h,d})=2h-2-\frac{2d}{3}\notin\mathbb Z, \qquad \operatorname{Crit}(E_{ρ_{g,h,d}})=\varnothing. \] Consequently, the associated branched-minimal-surface forgetful map is not surjective in these nonintegral Toledo components.
We prove the existence of complete minimal surfaces in $\mathbb{R}^3$ of arbitrary genus $p\, \ge\, 1$ and least total absolute curvature with precisely two ends -- one catenoidal and one Enneper-type -- thereby solving, affirmatively, a problem posed by Fujimori and Shoda. These surfaces, which are called \emph{Angel surfaces}, generalize some examples numerically constructed earlier by Weber. The construction of these minimal surfaces involves extending the orthodisk method developed by Weber and Wolf \cite{weber2002teichmuller}. A central idea in our construction is the notion of \emph{partial symmetry}, which enables us to introduce controlled symmetry into the surface.
We have proven the existence of new higher-genus maxfaces with Enneper end. These maxfaces are not the companions of any existing minimal surfaces, and furthermore, the singularity set is located away from the ends. The nature of the singularities is systematically investigated.
For a given zero mean curvature surface $X$ (in the Lorentz Minkowski space) having folded singularity, we construct a family of maxface and minface, having increasing cuspidal crosscaps, converging to $X$. We include a general discussion of this.
In this paper, we discuss about the independent types of infinite extensions to a general version of Kannan [5] and Fisher [3] of which the well-known Kannan and Fisher theorems come as a corollaries. We also provide a strong connection between continuous Kannan operator and Fisher operator in a restricted type of metric space. We also provide an application of the main theorem of this paper, in the field of integral equations.