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Geometric Analysis Seminar

On the stability of the Yamabe invariant of S3

Xuan Yao

Location:  Hill-705
Date & time: Tuesday, 09 April 2024 at 2:50PM - 3:50PM

Let g be a complete, asymptotically flat metric on ℝ3 with vanishing scalar curvature. Moreover, assume that (ℝ3,g) supports a nearly Euclidean L2 Sobolev inequality. We prove that (ℝ3,g) must be close to Euclidean space with respect to the dp-distance defined by Lee-Naber-Neumayer. We then discuss some consequences for the stability of the Yamabe invariant of S3. More precisely, we show that if such a manifold (ℝ3,g) carries a suitably normalized, positive solution to Δgw+λw5=0 then w must be close, in a certain sense, to a conformal factor that transforms Euclidean space into a round sphere.

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