Abstract
We consider the question whether a static potential on an asymptotically flat 3-manifold can have nonempty zero set which extends to the infinity. We prove that this does not occur if the metric is asymptotically Schwarzschild with nonzero mass. If the asymptotic assumption is relaxed to the usual assumption under which the total mass is defined, we prove that the static potential is unique up to scaling unless the manifold is flat. We also provide some discussion concerning the rigidity of complete asymptotically flat 3-manifolds without boundary that admit a static potential.
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Communicated by James A. Isenberg.
P. Miao’s research was partially supported by Simons Foundation Collaboration Grant for Mathematicians #281105.
L.-F. Tam’s research was partially supported by Hong Kong RGC General Research Fund #CUHK 403108.
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Miao, P., Tam, LF. Static Potentials on Asymptotically Flat Manifolds. Ann. Henri Poincaré 16, 2239–2264 (2015). https://doi.org/10.1007/s00023-014-0373-x
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DOI: https://doi.org/10.1007/s00023-014-0373-x