Communications in Analysis and Geometry

Volume 25 (2017)

Number 2

Uniqueness of photon spheres via positive mass rigidity

Pages: 303 – 320



Carla Cederbaum (Fachbereich Mathematik,Universität Tübingen, Germany)

Gregory J. Galloway (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)


In a recent paper the first author established the uniqueness of photon spheres, suitably defined, in static vacuum asymptotically flat spacetimes by adapting Israel’s proof of static black hole uniqueness. In this note we establish uniqueness of photon spheres by adapting the argument of Bunting and Masood-ul-Alam, which then allows certain assumptions to be relaxed. In particular, multiple photon spheres are allowed a priori.

As a consequence of our result, we can rule out the existence of static configurations involving multiple “very compact” bodies and black holes.

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The first author is indebted to the Baden-Württemberg Stiftung for the financial support of this research project by the Eliteprogramme for Postdocs.

The second author was partially supported by NSF grant DMS-1313724.

Paper received on 24 April 2015.