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

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

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

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

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

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

An SU(N) Mott insulator of an atomic Fermi gas realized by large-spin Pomeranchuk cooling

The Hubbard model, containing only the minimum ingredients of nearest neighbor hopping and on-site interaction for correlated electrons, has succeeded in accounting for diverse phenomena observed in solid-state materials. One of the interesting extensions is to enlarge its spin symmetry to SU(N>2), which is closely related to systems with orbital degeneracy. Here we report a successful formation of the SU(6) symmetric Mott insulator state with an atomic Fermi gas of ytterbium (173Yb) in a three-dimensional optical lattice. Besides the suppression of compressibility and the existence of charge excitation gap which characterize a Mott insulating phase, we reveal an important difference between the cases of SU(6) and SU(2) in the achievable temperature as the consequence of different entropy carried by an isolated spin. This is analogous to Pomeranchuk cooling in solid 3He and will be helpful for investigating exotic quantum phases of SU(N) Hubbard system at extremely low temperatures.Comment: 20 pages, 6 figures, to appear in Nature Physic

Disease progression in Plasmodium knowlesi malaria is linked to variation in invasion gene family members.

Emerging pathogens undermine initiatives to control the global health impact of infectious diseases. Zoonotic malaria is no exception. Plasmodium knowlesi, a malaria parasite of Southeast Asian macaques, has entered the human population. P. knowlesi, like Plasmodium falciparum, can reach high parasitaemia in human infections, and the World Health Organization guidelines for severe malaria list hyperparasitaemia among the measures of severe malaria in both infections. Not all patients with P. knowlesi infections develop hyperparasitaemia, and it is important to determine why. Between isolate variability in erythrocyte invasion, efficiency seems key. Here we investigate the idea that particular alleles of two P. knowlesi erythrocyte invasion genes, P. knowlesi normocyte binding protein Pknbpxa and Pknbpxb, influence parasitaemia and human disease progression. Pknbpxa and Pknbpxb reference DNA sequences were generated from five geographically and temporally distinct P. knowlesi patient isolates. Polymorphic regions of each gene (approximately 800 bp) were identified by haplotyping 147 patient isolates at each locus. Parasitaemia in the study cohort was associated with markers of disease severity including liver and renal dysfunction, haemoglobin, platelets and lactate, (r = ≥ 0.34, p =  <0.0001 for all). Seventy-five and 51 Pknbpxa and Pknbpxb haplotypes were resolved in 138 (94%) and 134 (92%) patient isolates respectively. The haplotypes formed twelve Pknbpxa and two Pknbpxb allelic groups. Patients infected with parasites with particular Pknbpxa and Pknbpxb alleles within the groups had significantly higher parasitaemia and other markers of disease severity. Our study strongly suggests that P. knowlesi invasion gene variants contribute to parasite virulence. We focused on two invasion genes, and we anticipate that additional virulent loci will be identified in pathogen genome-wide studies. The multiple sustained entries of this diverse pathogen into the human population must give cause for concern to malaria elimination strategists in the Southeast Asian region

Deletion of parasite immune modulatory sequences combined with immune activating signals enhances vaccine mediated protection against filarial nematodes

&lt;p&gt;Background: Filarial nematodes are tissue-dwelling parasites that can be killed by Th2-driven immune effectors, but that have evolved to withstand immune attack and establish chronic infections by suppressing host immunity. As a consequence, the efficacy of a vaccine against filariasis may depend on its capacity to counter parasite-driven immunomodulation.&lt;/p&gt; &lt;p&gt;Methodology and Principal Findings: We immunised mice with DNA plasmids expressing functionally-inactivated forms of two immunomodulatory molecules expressed by the filarial parasite Litomosoides sigmodontis: the abundant larval transcript-1 (LsALT) and cysteine protease inhibitor-2 (LsCPI). The mutant proteins enhanced antibody and cytokine responses to live parasite challenge, and led to more leukocyte recruitment to the site of infection than their native forms. The immune response was further enhanced when the antigens were targeted to dendritic cells using a single chain Fv-αDEC205 antibody and co-administered with plasmids that enhance T helper 2 immunity (IL-4) and antigen-presenting cell recruitment (Flt3L, MIP-1α). Mice immunised simultaneously against the mutated forms of LsALT and LsCPI eliminated adult parasites faster and consistently reduced peripheral microfilaraemia. A multifactorial analysis of the immune response revealed that protection was strongly correlated with the production of parasite-specific IgG1 and with the numbers of leukocytes present at the site of infection.&lt;/p&gt; &lt;p&gt;Conclusions: We have developed a successful strategy for DNA vaccination against a nematode infection that specifically targets parasite-driven immunosuppression while simultaneously enhancing Th2 immune responses and parasite antigen presentation by dendritic cells.&lt;/p&gt