Abstract
In this paper we study asymptotic behaviors of n-superharmonic functions at singularity using the Wolff potential and capacity estimates in nonlinear potential theory. Our results are inspired by and extend [6] of Arsove–Huber and [63] of Taliaferro in 2 dimensions. To study n-superharmonic functions we use a new notion of thinness in terms of n-capacity motivated by a type of Wiener criterion in [6]. To extend [63], we employ the Adams–Moser–Trudinger’s type inequality for the Wolff potential, which is inspired by the inequality used in [15] of Brezis–Merle. For geometric applications, we study the asymptotic end behaviors of complete conformally flat manifolds as well as complete properly embedded hypersurfaces in hyperbolic space. These geometric applications seem to elevate the importance of n-Laplace equations and make a closer tie to the classic analysis developed in conformal geometry in general dimensions.
Highlights
In this paper we will develop some understanding of isolated singularities of n-superharmonic functions in general dimensions and apply it to study some geometric problems
After adopting the definitions of n-harmonic functions and n-superharmonic functions from [48, Section 2], we present a review of what has been done in 2 dimensions to motivate what we want to do in general dimensions
We introduce the geometric problems that we expect to use n-superharmonic functions to study in this paper
Summary
In this paper we will develop some understanding of isolated singularities of n-superharmonic functions in general dimensions and apply it to study some geometric problems. Our second application is to study the asymptotic behavior at the end of properly embedded complete hypersurfaces with nonnegative Ricci curvature in hyperbolic space. It was shown in [12, Main Theorem] that such hypersurfaces have at most two ends, and are equidistant hypersurfaces if with two ends. Theorem 1.4 Suppose that n is a properly embedded, complete hypersurface with nonnegative Ricci curvature and one single end in hyperbolic space Hn+1.
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