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A Monge-Ampère equation in conformal geometry. (English) Zbl 1192.53045
Fully nonlinear equations arise naturally in differential geometry when studying equations of prescribed curvature, for example, of hypersurfaces in Euclidean space or Riemannian manifolds under conformal changes of metric. Typically one imposes a constraint on the solution space, often referred to as admissibility, in order to guarantee that the equations are elliptic. For example, when studying the Monge-Ampère equation one considers convex solutions. While technically advantageous these conditions can sometimes be geometrically unnatural, or at least overly restrictive. To give an example which is relevant to the main results of the present paper, consider the so-called \(k\)-Yamabe problem.
One goal in this paper is to propose a generalization of the \(k\)-Yamabe problem which retains some of the geometric flavor of the original while bypassing many of the non-essential technical deficiencies. The author considers the Monge-Ampère equation \(\det(A+ \lambda g)= const.\), where \(A\) is the Schouten tensor of a conformally related metric and \(\lambda>0\) is a constant suitably chosen. When the scalar curvature is non-positive, he gives necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero, he also establishes existence. Moreover, by adapting a construction of Schoen, he shows that solutions are not unique in general.
MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53A30 Conformal differential geometry (MSC2010)
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