In this paper we investigate the connection between the index and the geometry and topology of capillary surfaces. We prove an index estimate for compact capillary surfaces immersed in general 3-manifolds with boundary. We also study noncompact capillary surfaces with finite index and show that, under suitable curvature assumptions, such surface is conformally equivalent to a compact Riemann surface with boundary, punctured at finitely many points. We then prove that a strongly stable capillary surface immersed in a half-space of R 3 which is minimal or has a contact angle less than or equal to π/2 must be a half-plane. Using this uniqueness result we obtain curvature estimates for strongly stable capillary surfaces immersed in a 3-manifold with bounded geometry.
We show that every effective action of a compact Lie group K on a unit sphere S n admits an explicit orbit whose principal curvatures are bounded from above by 4 √ 14.
Morphology is an essential phenotype in the characterization of cells and their states, as it is closely related to cell function. Recent advances like Patch-seq enable simultaneously profiling the morphology, gene expression, and physiological properties of individual cells. However, computational methods that can summarize the great diversity of complex cell morphologies found in tissues and infer associations with other single-cell data modalities remain scarce. Here we report a computational framework, named CAJAL, for the morphometric and multi-modal analysis of single-cell data. CAJAL uses concepts from metric geometry to accurately build and visualize cell morphology summary spaces, integrate cellular morphologies across technologies, and establish associations between morphological, molecular, and physiological cellular processes. We demonstrate the utility of CAJAL by applying it to published Patch-seq, patch-clamp, serial electron, and two-photon microscopy data, and show that it represents a substantial improvement in functionality, scope, and accuracy with respect to current methods for cell morphometry.
Using the local picture of the degeneration of sequences of minimal surfaces developed by Chodosh, Ketover and Maximo [3] we show that in any closed Riemannian 3-manifold (M, g), the genus of an embedded CMC surface can be bounded only in terms of its index and area, independently of the value of its mean curvature. We also show that if M has finite fundamental group, the genus and area of any non-minimal embedded CMC surface can be bounded in term of its index and a lower bound for its mean curvature.
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