arXiv subjects

Gobinda Sau

Publications and source records attributed to Gobinda Sau.

4 recordsLinked to original sources

arXiv2026

Commutators of finite multiplicative order

This article studies the equation $[A,B]^k = \Id_n$ for matrices over $\CC$,characterizing the pairs $(k,n)$ for which solutions exist via a classical result of Lam and Leung on sums of roots of unity. The problem is next generalized to matrix rings $M_n(S)$ over arbitrary unital rings $S$, where a sufficient condition on the unity of $S$ is established and explicit constructions of solutions are provided. Beyond matrix rings, the structural implications of the equation $[a,b]^n = 1$ in a general unital ring $R$ are investigated, yielding a collection of idempotents whose properties govern the ring's structure. We prove that under a suitable condition on these idempotents, $[a,b]^n = 1$ implies $R$ is isomorphic to $M_n(S)$ for some unital ring $S$. We also provide an alternative proof using a result on characterisation of matrix rings by Goyal and Khurana. These results together establish a framework connecting commutator equations and classical criteria for recognizing full matrix rings.

math.RA↗

arXiv2025

Harmonic maps and framed $\mathrm{PSL}_2(\mathbb{C})$-representations

We show that given an element $X$ of the enhanced Teichmüller space $\mathcal{T}^\pm(\mathbb{S}, \mathbb{M})$ and a type-preserving framed $\mathrm{PSL}_2(\mathbb{C})$-representation $\hatρ = (ρ,β)$, there is a $ρ$-equivariant harmonic map $f:\mathbb{H}^2 \to \mathbb{H}^3$ that is asymptotic to the framing $β$. Here, the domain is the universal cover of the punctured Riemann surface obtained from a conformal completion of $X$. Moreover, such a harmonic map is unique if one prescribes, in addition, the principal part of the Hopf differential at each puncture. The proof uses the harmonic map heat flow.

math.DG↗

arXiv2024

The p-triviality of stunted projective spaces

In this article, we introduce the notion of $\mathcal P$-triviality of topological manifolds and give a complete description of the $\mathcal P$-triviality of stunted real and complex projective spaces.

math.AT↗

arXiv2024

On harmonic maps from the complex plane to hyperbolic 3-space

For any twisted ideal polygon in $\mathbb{H}^3$, we construct a harmonic map from $\mathbb{C}$ to $\mathbb{H}^3$ with a polynomial Hopf differential, that is asymptotic to the given polygon, and is a bounded distance from a pleated plane. Our proof uses the harmonic map heat flow. We also show that such a harmonic map is unique once we prescribe the principal part of its Hopf differential.

math.DG↗