A compactness theorem for scalar-flat metrics on manifolds with boundary

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this set is compact for dimensions greater than or equal to 7 under the generic condition that the trace-free 2nd fundamental form of the boundary is nonzero everywhere.Comment: 49 pages. Final version, to appear in Calc. Var. Partial Differential Equation

Constraints on singlet right-handed neutrinos coming from the Z0Z^0-width

We study the constraints on masses and mixing angles imposed by the measured $Z^0$ invisible width, in a model in which a singlet right-handed neutrino mixes with all the Standard Model neutrinos.Comment: Revtex, 7 pages, two figures available from the authors, preprint IFT-P.040/92 IFUSP/P-1023/9

Sharp constants in weighted trace inequalities on Riemannian manifolds

We establish some sharp weighted trace inequalities W^{1,2}(\rho^{1-2\sigma}, M)\hookrightarrow L^{\frac{2n}{n-2\sigma}}(\pa M) on $n+1$ dimensional compact smooth manifolds with smooth boundaries, where $\rho$ is a defining function of $M$ and $\sigma\in (0,1)$. This is stimulated by some recent work on fractional (conformal) Laplacians and related problems in conformal geometry, and also motivated by a conjecture of Aubin.Comment: 34 page

Quantification of Genome Editing and Transcriptional Control Capabilities Reveals Hierarchies among Diverse CRISPR/Cas Systems in Human Cells

CRISPR/Cas technologies have revolutionized the ability to redesign genomic information and tailor endogenous gene expression. Nevertheless, the discovery and development of new CRISPR/Cas systems has resulted in a lack of clarity surrounding the relative efficacies among these technologies in human cells. This deficit makes the optimal selection of CRISPR/Cas technologies in human cells unnecessarily challenging, which in turn hampers their adoption, and thus ultimately limits their utility. Here, we designed a series of endogenous testbed systems to methodically quantify and compare the genome editing, CRISPRi, and CRISPRa capabilities among 10 different natural and engineered Cas protein variants spanning Type II and Type V CRISPR/Cas families. We show that although all Cas protein variants are capable of genome editing and transcriptional control in human cells, hierarchies exist, particularly for genome editing and CRISPRa applications, wherein Cas9 ≥ Cas12a > Cas12e/Cas12j. Our findings also highlight the utility of our modular testbed platforms to rapidly and systematically quantify the functionality of practically any natural or engineered genomic-targeting Cas protein in human cells

On Vanishing Theorems For Vector Bundle Valued p-Forms And Their Applications

Let $F: [0, \infty) \to [0, \infty)$ be a strictly increasing $C^2$ function with $F(0)=0$. We unify the concepts of $F$-harmonic maps, minimal hypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, and introduce $F$-Yang-Mills fields, $F$-degree, $F$-lower degree, and generalized Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on manifolds. When $F(t)=t, \frac 1p(2t)^{\frac p2}, \sqrt{1+2t} -1,$ and $1-\sqrt{1-2t},$ the $F$-Yang-Mills field becomes an ordinary Yang-Mills field, $p$-Yang-Mills field, a generalized Yang-Mills-Born-Infeld field with the plus sign, and a generalized Yang-Mills-Born-Infeld field with the minus sign on a manifold respectively. We also introduce the $E_{F,g}-$energy functional (resp. $F$-Yang-Mills functional) and derive the first variational formula of the $E_{F,g}-$energy functional (resp. $F$-Yang-Mills functional) with applications. In a more general frame, we use a unified method to study the stress-energy tensors that arise from calculating the rate of change of various functionals when the metric of the domain or base manifold is changed. These stress-energy tensors, linked to $F$-conservation laws yield monotonicity formulae. A "macroscopic" version of these monotonicity inequalities enables us to derive some Liouville type results and vanishing theorems for $p-$forms with values in vector bundles, and to investigate constant Dirichlet boundary value problems for 1-forms. In particular, we obtain Liouville theorems for $F-$harmonic maps (e.g. $p$-harmonic maps), and $F-$Yang-Mills fields (e.g. generalized Yang-Mills-Born-Infeld fields on manifolds). We also obtain generalized Chern type results for constant mean curvature type equations for $p-$forms on $\Bbb{R}^m$ and on manifolds $M$ with the global doubling property by a different approach. The case $p=0$ and $M=\mathbb{R}^m$ is due to Chern.Comment: 1. This is a revised version with several new sections and an appendix that will appear in Communications in Mathematical Physics. 2. A "microscopic" approach to some of these monotonicity formulae leads to celebrated blow-up techniques and regularity theory in geometric measure theory. 3. Our unique solution of the Dirichlet problems generalizes the work of Karcher and Wood on harmonic map

ff-minimal surface and manifold with positive mm-Bakry-\'{E}mery Ricci curvature

In this paper, we first prove a compactness theorem for the space of closed embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-\'{E}mery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the $f$-Laplacian on compact manifold with positive $m$-Bakry-\'{E}mery Ricci curvature, and prove that the lower bound is achieved only if the manifold is isometric to the $n$-shpere, or the $n$-dimensional hemisphere. Finally, for compact manifold with positive $m$-Bakry-\'{E}mery Ricci curvature and $f$-mean convex boundary, we prove an upper bound for the distance function to the boundary, and the upper bound is achieved if only if the manifold is isometric to an Euclidean ball.Comment: 15 page

Estimates for the Sobolev trace constant with critical exponent and applications

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality S\|u\|^p_{L^{p_*}(\partial\Omega) \hookrightarrow \|u\|^p_{W^{1,p}(\Omega)} that are independent of $\Omega$. This estimates generalized those of [3] for general $p$. Here $p_* := p(N-1)/(N-p)$ is the critical exponent for the immersion and $N$ is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of [16]. Finally, we study an optimal design problem with critical exponent.Comment: 22 pages, submitte

Prevalence of phase variable epigenetic invertons among host-associated bacteria.

Type I restriction-modification (R-M) systems consist of a DNA endonuclease (HsdR, HsdM and HsdS subunits) and methyltransferase (HsdM and HsdS subunits). The hsdS sequences flanked by inverted repeats (referred to as epigenetic invertons) in certain Type I R-M systems undergo invertase-catalyzed inversions. Previous studies in Streptococcus pneumoniae have shown that hsdS inversions within clonal populations produce subpopulations with profound differences in the methylome, cellular physiology and virulence. In this study, we bioinformatically identified six major clades of the tyrosine and serine family invertases homologs from 16 bacterial phyla, which potentially catalyze hsdS inversions in the epigenetic invertons. In particular, the epigenetic invertons are highly enriched in host-associated bacteria. We further verified hsdS inversions in the Type I R-M systems of four representative host-associated bacteria and found that each of the resultant hsdS allelic variants specifies methylation of a unique DNA sequence. In addition, transcriptome analysis revealed that hsdS allelic variations in Enterococcus faecalis exert significant impact on gene expression. These findings indicate that epigenetic switches driven by invertases in the epigenetic invertons broadly operate in the host-associated bacteria, which may broadly contribute to bacterial host adaptation and virulence beyond the role of the Type I R-M systems against phage infection