A note on a Sung-Wang's paper

TL;DR

This paper investigates the connectedness at infinity of manifolds using $p$-harmonic functions, showing that maximal eigenvalues imply connectedness unless the manifold is a specific warped product cylinder.

Contribution

It establishes a link between the maximal eigenvalue of the $p$-Laplacian and the topological structure at infinity of Kähler and quaternionic Kähler manifolds.

Findings

Maximal $oldsymbol{ ext{first eigenvalue}}$ implies connectedness at infinity.

Manifolds are either connected at infinity or are a specific warped product cylinder.

Results apply to Kähler and quaternionic Kähler manifolds.

Abstract

The purpose of this note is to study the connectedness at infinity of manifold by using the theory of $p$-harmonic functions. We show that if the first eigenvalue $\lambda_{1,p}$ for the $p$-Laplacian achievies its maximal value on a K\"{a}hler manifold or a quaternionic K\"{a}hler manifold then such a manifold must be connected at infinity unless it is a topological cylinder with an explicit warped product metric.