In dimensions n≥62, there is a smooth metric on the spherical cap with positive Yamabe constant for which lower-energy positive solutions to the constant scalar curvature and constant boundary mean curvature equation exist but fail to be compact.
The Han-Li conjecture in constant scalar curvature and constant boundary mean curvature problem on compact manifolds
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abstract
The Han-Li conjecture states that: Let $(M,g_0)$ be an $n$-dimensional $(n\geq 3)$ smooth compact Riemannian manifold with boundary having positive (generalized) Yamabe constant and $c$ be any real number, then there exists a conformal metric of $g_0$ with scalar curvature $1$ and boundary mean curvature $c$. Combining with Z. C. Han and Y. Y. Li's results, we answer this conjecture affirmatively except for the case that $n\geq 8$, the boundary is umbilic, the Weyl tensor of $M$ vanishes on the boundary and has a non-zero interior point.
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math.DG 1years
2019 1verdicts
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Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation
In dimensions n≥62, there is a smooth metric on the spherical cap with positive Yamabe constant for which lower-energy positive solutions to the constant scalar curvature and constant boundary mean curvature equation exist but fail to be compact.