On the ergodicity of the frame flow on even-dimensional manifolds

Abstract It is known that the frame flow on a closed $n$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> -dimensional Riemannian manifold with negative sectional curvature is ergodic if $n$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> is odd and $n \neq 7$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>7</mml:mn> </mml:math> . In this paper we study its ergodicity in the remaining cases. For $n$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> even and $n \neq 8, 134$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>8</mml:mn> <mml:mo>,</mml:mo> <mml:mn>134</mml:mn> </mml:math> , we show that: if $n \equiv 2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>2</mml:mn> </mml:math> mod 4 or $n=4$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:math> , the frame flow is ergodic if the manifold is $\sim 0.3$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∼</mml:mo> <mml:mn>0.3</mml:mn> </mml:math> -pinched, if $n \equiv 0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>0</mml:mn> </mml:math> mod 4, it is ergodic if the manifold is $\sim 0.6$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∼</mml:mo> <mml:mn>0.6</mml:mn> </mml:math> -pinched. In the three dimensions $n=7,8,134$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>7</mml:mn> <mml:mo>,</mml:mo> <mml:mn>8</mml:mn> <mml:mo>,</mml:mo> <mml:mn>134</mml:mn> </mml:math> , the respective pinching bounds that we need in order to prove ergodicity are 0.4962..., 0.6212..., and 0.5788.... This is a significant improvement over the previously known results and a step forward towards solving a long-standing conjecture of Brin asserting that 0.25-pinched even-dimensional manifolds have an ergodic frame flow.