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Throughout thispaper,we refertothisequation as "the equation(/,M)". Now, we areinterestedin thestructureofthe moduli spaceof(complete) conformal metricson M with scalarcurvature/.In thiswork, we study the equation(f,M) in the case when (M,g) isa subdomain ofa compact Rie- mannian manifold (M,g). More precisely,we considermainly the case when h(Lg) > 0,(M,g) isthecomplement Af\Sofa compact submanifold E, and/ isnonpositive. Under this assumption, Mazzeo (12) proved that, when d = dimS < (n- 2)/2 and / = 0 on M, "the fullsolutionspace of scalarflatcomplete conformal metrics on M is parametrized by the space of strictlypositive measures on S." This fact means that is the Martin boundary of the Laplacian with respectto a scalarflatcomplete conformal metricon M. When / has a compact support,any conformal metricuq~1gon M with scalarcurvature/is bounded above by some scalarflatconformal metric

Structure theorems of the scalar curvature equation on subdomains of a compact Riemannian manifold