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Abstract Two sharp comparison results are derived for 3D complete noncompact manifolds with scalar curvature bounded from below. The 1st one concerns the Green’s function. When the scalar curvature is nonnegative, it states that the rate of decay of an energy quantity over the level set is strictly less than that of the Euclidean space unless the manifold itself is isometric to the Euclidean space. The result is in turn converted into a sharp area comparison for the level set of the Green’s function when in addition the Ricci curvature of the manifold is assumed to be asymptotically nonnegative at infinity. The 2nd result provides a sharp upper bound of the bottom spectrum in terms of the scalar curvature lower bound, in contrast to the classical result of Cheng, which involves a Ricci curvature lower bound. 

We develop Green’s function estimates for manifolds satisfying a weighted Poincaré inequality together with a compatible lower bound on the Ricci curvature. This estimate is then applied to establish existence and sharp estimates of solutions to the Poisson equation on such manifolds. As an application, a Liouville property for finite energy holomorphic functions is proven on a class of complete Kähler manifolds. Consequently, such Kähler manifolds must be connected at infinity. 

Abstract A variant of Li–Tam theory, which associates to each end of a completeRiemannian manifold a positive solution of a given Schrödinger equation onthe manifold, is developed. It is demonstrated that such positive solutionsmust be of polynomial growth of fixed order under a suitable scalinginvariant Sobolev inequality. Consequently, a finiteness result for the number of endsfollows. In the case when the Sobolev inequality is of particular type, the finiteness resultis proven directly. As an application, an estimate on the number of ends for shrinkinggradient Ricci solitons and submanifolds of Euclidean space is obtained. 

Abstract. Oxygenated organic molecules (OOMs) are the crucial intermediates linkingvolatile organic compounds (VOCs) to secondary organic aerosols (SOAs) in theatmosphere, but comprehensive understanding of the characteristics of OOMsand their formation from VOCs is still missing. Ambient observations ofOOMs using recently developed mass spectrometry techniques are stilllimited, especially in polluted urban atmospheres where VOCs and oxidants areextremely variable and complex. Here, we investigate OOMs, measured by anitrate-ion-based chemical ionization mass spectrometer at Nanjing ineastern China, through performing positive matrix factorization on binnedmass spectra (binPMF). The binPMF analysis reveals three factors aboutanthropogenic VOC (AVOC) daytime chemistry, three isoprene-relatedfactors, three factors about biogenic VOC (BVOC) nighttime chemistry, andthree factors about nitrated phenols. All factors are influenced by NOxin different ways and to different extents. Over 1000 non-nitro moleculeshave been identified and then reconstructed from the selected solution ofbinPMF, and about 72 % of the total signals are contributed bynitrogen-containing OOMs, mostly regarded as organic nitrates formed throughperoxy radicals terminated by nitric oxide or nitrate-radical-initiatedoxidations. Moreover, multi-nitrates account for about 24 % of the totalsignals, indicating the significant presence of multiple generations,especially for isoprene (e.g., C5H10O8N2 andC5H9O10N3). Additionally, the distribution of OOMconcentration on the carbon number confirms their precursors are driven by AVOCsmixed with enhanced BVOCs during summer. Our results highlight the decisiverole of NOx in OOM formation in densely populated areas, and we encouragemore studies on the dramatic interactions between anthropogenic and biogenicemissions.