Kiselman Minimum Principle and Rooftop Envelopes in Complex Hessian Equations

TL;DR

This paper explores the properties of m-subharmonic functions related to complex Hessian equations, introducing new principles and decompositions that advance understanding of geodesic connectivity in complex analysis.

Contribution

It introduces a rooftop equality and an analogue of the Kiselman minimum principle for m-subharmonic functions, advancing the theory of complex Hessian equations.

Findings

Decomposition of m-subharmonic solutions

A general comparison principle for singular Hessian measures

A criterion for geodesic connectivity based on asymptotic envelopes

Abstract

We initiate the study of $m$-subharmonic functions with respect to a semipositive $(1,1)$-form in Euclidean domains, providing a significant element in understanding geodesics within the context of complex Hessian equations. Based on the foundational Perron envelope construction, we prove a decomposition of $m$-subharmonic solutions, and a general comparison principle that effectively manages singular Hessian measures. Additionally, we establish a rooftop equality and an analogue of the Kiselman minimum principle, which are crucial ingredients in establishing a criterion for geodesic connectivity among $m$-subharmonic functions, expressed in terms of their asymptotic envelopes.