Compactness for immersions of prescribed Gaussian curvature. II: Geometric aspects
The main result of this paper is an affirmative answer to the following question. Given a compact manifold with boundary and an immersion of the boundary into a simply-connected manifold with nonpositive curvature (i.e., a Hadamard manifold), is it possible to extend this immersion to an immersion of the entire compact manifold with boundary in such a way that the Gaussian curvature is a prescribed function. The assumptions made in the main result (theorem 1.1.1) are quite natural (essentially the existence of a supersolution and ellipticity). This problem is actually classical and quite well studied in the context of Euclidean space. For graphs over convex domains in the Euclidean space the answer follows from the standard theory of Monge-Ampere equations. For manifolds in the Euclidean space other than graphs, the results of Guan-Spruck and Trudinger-Wang solve the problem. The idea is to deform the ``supersolution'' into a solution using a Perron-like method. Since the author is dealing with manifolds other than the Euclidean space (where the geometry is quite nice because it is a space form) the corresponding Perron method breaks down. The author (motivated partially by a result of Labourie) manages to make some of the techniques work in this fairly difficult situation. As nicely summarized in Section 1.2 of the paper, the proof of Theorem 1.1 relies on degree theory (which is standard but is written clearly in this paper with all details). In order to use degree theory, one needs a priori estimates (i.e., compactness). The author (motivated by Guan-Spruck) proves lower bounds for the size of balls over which certain immersions are embeddings, Lipschitz bounds and lower derivative bounds at the boundary (this is the main technical content of the paper) and then uses standard theory to jack up the regularity. Next, the author defines an appropriate notion of ``bounding'' (a concept that is clear in Euclidean space) and proceeds to use degree theory. For Part I, see [\textit{G. Smith}, Adv. Math. 229, No. 2, 731--769 (2012; Zbl 1236.53053)].
scientific article; zbMATH DE number 6358012
scientific article