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.
Brendle, Blow-up phenomena for the Y amabe equation , J
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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.