On partial uniqueness of complete non-compact Ricci flat metrics

TL;DR

This paper proves the uniqueness of bounded solutions to the Monge-Ampère equation on certain non-compact Kähler manifolds, using Caccioppoli inequalities without assuming decay conditions.

Contribution

It establishes partial uniqueness results for solutions to the Monge-Ampère equation on non-compact Kähler manifolds with sub-quadratic volume growth, without decay assumptions.

Findings

Uniqueness of bounded $C^{1,1}$ solutions established

Applicable to a broad class of non-compact Kähler manifolds

Uses Caccioppoli inequality techniques

Abstract

Using techniques for Caccioppoli inequality, on a fairly general class of complete non-compact K\"ahler manifolds with sub-quadratic volume growth, we show uniqueness of bounded $C^{1,1}$ solution to Monge-Ampere equation. This does not a priori require any decay of the solution.