Complete Minimal Submanifolds of Classical Non-Compact Riemannian Symmetric Spaces - Larsen,Lucas (Master’s Theses in Mathematical Sciences, 2025)
In this thesis we employ the method of eigenfamilies in order to construct explicit examples of complete minimal submanifolds in the classical non-compact Riemannian symmetric spaces $$\SLR n/\SO(n),\ \Sp(n,\R)/\UU(n),\ \SO^*(2n)/\UU(n),\ \SU^*(2n)/\Sp(n).$$ In each case, we produce a multidimensional parametrized family of complete minimal submanifolds of codimension two. The results form a continuation of the work in \cite{Geg-Gud-1}, which concerns the classical compact symmetric spaces. The approach is largely elementary, accessible to readers with a basic background in differential geometry and Lie groups. Chapters \ref{ch:eigen} and \ref{ch:symmetric} serve as an introduction to the theory of eigenfunctions and Riemannian symmetric spaces, respectively. In Chapter \ref{ch:matrix} we establish some useful notation and definitions for working with matrix Lie groups. Finally, each of the Chapters \ref{ch:slr}, \ref{ch:spr}, \ref{ch:sox} and \ref{ch:sux} are dedicated to one of the symmetric spaces in question.
Keywords: Riemannian symmetric space; eigenfunction; minimal submanifold; Lie group