On generalized Gauss maps of minimal surfaces sharing hypersurfaces in a projective variety

TL;DR

This paper investigates the uniqueness of generalized Gauss maps of minimal surfaces in real space sharing hypersurfaces within a projective variety, marking the first such study in this context and extending prior results.

Contribution

It introduces the first analysis of the unicity of generalized Gauss maps of minimal surfaces sharing hypersurfaces in a projective variety, generalizing previous work.

Findings

Established conditions for the uniqueness of generalized Gauss maps.

Extended previous results to more general settings in projective varieties.

Provided new insights into the geometric properties of minimal surfaces and their Gauss maps.

Abstract

In this article, we study the uniqueness problem for the generalized gauss maps of minimal surfaces (with the same base) immersed in $\mathbb R^{n+1}$ which have the same inverse image of some hypersurfaces in a projective subvariety $V\subset\mathbb P^n(\mathbb C)$. As we know, this is the first time the unicity of generalized gauss maps on minimal surfaces sharing hypersurfaces in a projective varieties is studied. Our results generalize and improve the previous results in this field.