This paper contains some vanishing theorems for L 2 harmonic forms on complete Riemannian manifolds with a weighted Poincaré inequality and a certain lower bound of the curvature. The results are in the spirit of Li-Wang and Lam, but without assumptions of sign and growth rate of the weight function, so they can be applied to complete stable hypersurfaces.

In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if the ambient manifold is of bounded geometry, we prove that such a submanifold must have at least linear volume growth. In particular, we show that a properly immersed complete noncompact hypersurface in the Euclidean space with bounded Gaussian weighted mean curvature must have polynomial volume growth and at least linear volume growth.

In this paper we prove that on a complete smooth metric measure space with non-negative Bakry-Émery-Ricci curvature if the space of weighted L 2 harmonic one-forms is non-trivial then the weighted volume of the manifold is finite and universal cover of the manifold splits isometrically as the product of the real line with an hypersurface.

In this paper we prove some spectral properties of the drifted Laplacian of self-shrinkers properly immersed in gradient shrinking Ricci solitons. Then we use these results to prove some geometric properties of self-shrinkers. For example, we describe a collection of domains in the ambient space that cannot contain self-shrinkers.

Sarcoidosis is a multisystem disease with unknown etiology, marked by T lymphocytes and macrophages agglomeration, which leads to the formation of noncaseating granulomas in the affected tissues. We describe a case of a 40-year-old black patient referred to our service for evaluation of nephrolithiasis and persistent elevation of plasma creatinine. He reported important weight loss, fever episodes, and abdominal and low back intermittent pain in the past 6 months. The investigation revealed elevated serum calcium level, hepatosplenomegaly, retroperitoneal lymphadenopathy, anemia, thrombocytopenia, and nephrolithiasis. The initial diagnostic hypothesis was lymphoproliferative disease, but the laparoscopic propaedeutic showed multiple white lesions on the liver surface, which biopsy identified as noncaseating granulomas with asteroid corpuscles, suggestive of sarcoidosis. He was treated with corticosteroids with significant improvement in symptoms and in calcium and creatinine levels. Besides, the patient presented a long-term large joints arthropathy, especially on the knees (with bilateral prosthesis), wrists, and ankles, of unknown etiology. We discuss the systemic manifestations of sarcoidosis related to the reported case, as well as the possible overlapping of idiopathic juvenile arthritis with sarcoidosis.