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Mehmood Ur Rehman

Publications and source records attributed to Mehmood Ur Rehman.

3 recordsLinked to original sources
arXiv2026

Constantly curved minimal immersions of the two-sphere in unitary groups

In this article, we investigate rigidity results for constantly curved minimal immersions of the two-sphere $S^2$ into the unitary group $\mathrm{U}(n)$. Using loop group methods for harmonic maps, we establish a correspondence between such immersions and a distinguished class of constantly curved holomorphic immersions of $S^2$ into finite-dimensional Grassmannians. In the case $\mathrm{U}(3)$, we classify the constantly curved minimal immersions of $S^2$ with uniton number one and prove that, under a natural unramifiedness condition, those of uniton number two are $S^1$-invariant; as a consequence, every constantly curved totally unramified minimal immersion $S^2\to \mathrm{U}(3)$ of uniton number two is unitarily congruent to the composition of the first Gauss map of the Veronese curve in $\mathbb{C}P^2$ with the Cartan embedding $\mathbb{C}P^2\hookrightarrow \mathrm{U}(3)$.

math.DG↗
arXiv2025

Primitive immersions of constant curvature of surfaces into flag manifolds

We investigate certain immersions of constant curvature from Riemann surfaces into flag manifolds equipped with invariant metrics, namely primitive lifts associated to pseudoholomorphic maps of surfaces into complex Grassmannians. We prove that a primitive immersion from the two-sphere into the full flag manifold which has constant curvature with respect to \emph{at least one} invariant metric is unitarily equivalent to the primitive lift of a Veronese map, hence it has constant curvature with respect to \emph{all} invariant metrics. We prove a partial generalization of this result to the case where the domain is a general simply connected Riemann surface. On the way, we consider the problem of finding the invariant metric on the flag manifold, under a certain normalization condition, that maximizes the induced area of the two-sphere by a given primitive immersion.

math.DG↗