Lagrangian translating solitons and special lagrangians in $$\mathbb {C}^{m}$$ with symmetries

Abstract We construct novel families of exact immersed and embedded Lagrangian translating solitons and special Lagrangian submanifolds in $$\mathbb {C}^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> </mml:math> that are invariant under the action of various admissible compact subgroups $$G \le {{\,\textrm{SU}\,}}(m-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>≤</mml:mo> <mml:mrow> <mml:mspace/> <mml:mtext>SU</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with cohomogeneity-two. These examples are obtained via an Ansatz generalising a construction of Castro–Lerma in $$\mathbb {C}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . We give explicit examples of admissible group actions, including a full classification for G simple. We also describe novel Lagrangian translators symmetric with respect to non-compact subgroups of the affine special unitary group $${{\,\textrm{SU}\,}}(m)\ltimes \mathbb {C}^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mtext>SU</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⋉</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> </mml:math> , including cohomogeneity-one examples.