Compactness for immersions of prescribed Gaussian curvature. II: Geometric aspects
type: Item
instance of: scholarly article
title: Compactness for immersions of prescribed Gaussian curvature. II: Geometric aspects
zbMATH Open document ID: 1304.53061
published in: Geometriae Dedicata
publication date: 2014-10-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1002.2982
review text: 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, s…
reviewed by: Vamsi Pritham Pingali
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 6358012
zbMATH Keywords: Gaussian curvature
zbMATH Keywords: Plateau problem
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: non-linear elliptic PDEs
MaRDI profile type: Publication
OpenAlex ID: W2108082706
arXiv ID: 1002.2982
cites work: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order
cites work: The dirichlet problem for nonlinear second-order elliptic equations I. Monge-ampégre equation
cites work: The dirichlet problem for nonlinear second-order elliptic equations. II. Complex monge-ampère, and uniformaly elliptic, equations
cites work: The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian
cites work: Nonlinear second-order elliptic equations V. The dirichlet problem for weingarten hypersurfaces
cites work: Q5643306
cites work: Elliptic Partial Differential Equations of Second Order
cites work: Boundary-value problems on \(S^ n\) for surfaces of constant Gauss curvature
cites work: The existence of hypersurfaces of constant Gauss curvature with prescribed boundary.
cites work: Q4134602
cites work: Dirichlet duality and the nonlinear Dirichlet problem
cites work: Q4110164
cites work: A Morse lemma for convex surfaces
cites work: Degree Theory of Immersed Hypersurfaces
cites work: On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolic space
cites work: An Infinite Dimensional Version of Sard's Theorem
cites work: An Arzela-Ascoli theorem for immersed submanifolds
cites work: Compactness results for immersions of prescribed Gaussian curvature. I: Analytic aspects
cites work: On locally convex hypersurfaces with boundary
cites work: Q3974681
DOI: 10.1007/S10711-013-9921-7
author: Graham Smith
description: scientific article; zbMATH DE number 6358012