Rigidity of four-dimensional gradient shrinking Ricci solitons

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mi>f</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {{(M,g,f)}} be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>Ric</m:mi> <m:mo>+</m:mo> <m:mrow> <m:msup> <m:mo>∇</m:mo> <m:mn>2</m:mn> </m:msup> <m:mo>⁡</m:mo> <m:mi>f</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo>⁢</m:mo> <m:mi>g</m:mi> </m:mrow> </m:mrow> </m:math> {{Ric+∇²f=λ g}} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> {{λ}} is a positive real number. We prove that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>M</m:mi> </m:math> {{M}} has constant scalar curvature <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>S</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>λ</m:mi> </m:mrow> </m:mrow> </m:math> {{S=2λ}} , it must be a quotient of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>𝕊</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> {{S²²}} . Together with the known results, this implies that a four-dimensional complete gradient shrinking Ricci soliton has constant scalar curvature if and only if it is rigid, that is, it is either Einstein, or a finite quotient of Gaussian shrinking soliton <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mn>4</m:mn> </m:msup> </m:math> {{R⁴}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>𝕊</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> {{S²²}} or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>𝕊</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo>×</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> {{S³}} .