A general existence proof for non-linear elliptic equations in semi-Riemannian spaces

2007/09/20 by Gerhardt, Claus
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.0709.3123

Abstract

We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular solutions. As an application we prove the existence of closed hypersurfaces with curvature prescribed in the tangent bundle of an ambient Riemannian manifold N without supposing any sign condition on the sectional curvatures K_N. A curvature flow wouldn't work in this situation, neither the method of successive approximation.

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