Cohomogeneity one 4-Dimensional Gradient Ricci Solitons

Abstract Simply-connected four-dimensional gradient Ricci solitons that are invariant under a compact cohomogeneity one group action have been studied extensively. However, the special case where the group is $$\textrm{SU}(2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SU</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> (the smallest possible example) has received comparatively little attention. The purpose of this article is to give a comprehensive study of simply-connected $$\textrm{SU}(2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SU</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -invariant expanding cohomogeneity one gradient Ricci solitons. The main result is the construction of new 3-parameter families of complete $$\textrm{SU}(2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SU</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -invariant asymptotically conical expanding gradient Ricci solitons and new 2-parameter families of complete $$\textrm{SU}(2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SU</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -invariant Einstein metrics.