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Ann. 289 (1991), 631–662. [J] K. Johnson, On a ring of invariant polynomials on a Hermitian symmetric space, J. Algebra 67 (1980), 72–81. [K] V. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), 190–213. [Ko] T. Kobayashi, Multiplicity-free representations and visible actions on complex manifolds, Publ. RIMS, Kyoto Univ. 41 (2005), 497–549. [KVS] F. Knop, B. Van Steirteghem, Classiﬁcation of smooth aﬃne spherical varieties, Transform. Groups 11 (2006), 495–516. [L] A. S. Leahy, A classiﬁcation of multiplicity free representations, J.

METZGER trapped surfaces – here the above mentioned relation of Jang’s equation to the MOTS equation comes into play. ˆ of a solution of Jang’s equation Secondly, the induced geometry of the graph M can be confomally changed to a metric with zero scalar curvature without increasing the mass. , the linearization of Jang’s equation, has, in a certain sense, non-negative for M spectrum. Equation (4) is translation invariant in the vertical direction. One of the consequences of this fact is that the non-negative lapse for the foliation of M × R arising by this translation can be viewed as a principal eigenfunction of the linearization of equation (4), with eigenvalue zero.

The above argument can then be used to show that a MOTS Σ exists in the modiﬁed data set. The maximum principle shows that this Σ cannot intersect the region where the data has been modiﬁed so that Σ is also a MOTS with respect to the original data set. The change of the geometry and the data are local but ˜ ¯ that many of the geometric large. 4 depend on is hard to control explicitly. 6. The curvature estimates of [2], which do not require a priori area bounds and which depend only on the original data, are then available for Σ.