Constant $\sigma_{k}$-curvature metrics with Delaunay type ends

TL;DR

This paper constructs new complete non-compact Riemannian metrics with positive constant 3- curvature by connecting Delaunay type solutions, addressing a complex nonlinear elliptic PDE problem.

Contribution

It introduces a method to produce families of 3- curvature metrics with Delaunay type ends via connected sums, expanding the known solutions in geometric analysis.

Findings

Successfully constructed non-compact metrics with constant 3- curvature

Extended the class of known solutions with Delaunay type ends

Solved a challenging nonlinear elliptic PDE in geometric context

Abstract

In this paper we produce families of complete non compact Riemannian metrics with positive constant $\sigma_k$-curvature by performing the connected sum of a finite number of given $n$-dimensional Delaunay type solutions, provided $2 \leq 2k < n$. The problem is equivalent to solve a second order fully nonlinear elliptic equation.