Ann. of Math. (2), Vol. 155 (2002), no. 3, 709--787 We formulate natural conformally invariant conditions on a 4-manifold for the
existence of a metric whose Schouten tensor satisfies a quadratic ...inequality.
This inequality implies that the eigenvalues of the Ricci tensor are positively
pinched.
The study of the $k$-th elementary symmetric function of the Weyl-Schouten
curvature tensor of a Riemannian metric, the so called $\sigma_k$ curvature,
has produced many fruitful results in conformal ...geometry in recent years. In
these studies, the deforming conformal factor is considered to be a solution of
a fully nonlinear elliptic PDE. Important advances have been made in recent
years in the understanding of the analytic behavior of solutions of the PDE.
However, the singular behavior of these solutions, which is important in
describing many important questions in conformal geometry, is little
understood. This note classifies all possible radial solutions, in particular,
the \emph{singular} solutions of the $\sigma_k$ Yamabe equation, which
describes conformal metrics whose $\sigma_k$ curvature equals a constant.
Although the analysis involved is of elementary nature, these results should
provide useful guidance in studying the behavior of singular solutions in the
general situation.
Most attempts to relate changes in patterns of sunlight exposure to the rise in incidence of malignant melanoma have concentrated on the positive association between intermittent exposure to sunlight ...and risk of melanoma. The Western Canada Melanoma Study, however, also detected a significant inverse association between melanoma and chronic or long-term occupational sun exposure in men, with the lowest risk (OR = 0.5) in those with maximum occupational exposure, suggesting that chronic exposure may be protective. Data obtained from Canadian census figures indicated that since 1951 there has been a substantial reduction in the number of males engaged in outdoor occupations in Canadian society. These observations suggest that part of the increase in the incidence of melanoma in low-sunlight areas may be due to a reduction over the past 40 years of the size of this group of individuals "protected" by their exposure to UV light.
Hepatocellular carcinoma (HCC, hepatoma) is one of the most lethal cancers in humans. The incidence of HCC is as high as 30 new cases per 100,000 men per year in high risk regions such as Qi Dong of ...China and Mozambique. The incidence of HCC is rising in the United States. Epidemiological studies suggest that chronic hepatitis related liver injury and aflatoxin contamination in food and drinking water are the major risk factors in the development of liver cancer. HCC from areas with high hepatitis virus carrier rate and high aflatoxin exposure often has high p53 mutation rate at codon 249. Previous experiments demonstrate that p53 loss, p53 mutation and aflatoxin exposure each contribute to the development of liver cancer in a transgenic mouse model. Further studies using proliferating cell nuclear antigen (PCNA) immunostaining indicate that loss of p53 increases hepatocyte proliferation in adult mouse livers as compared to wild-type controls, and mutation of p53 at codon 246 (equivalent to p53 mutation at codon 249 in humans) leads to accumulation of hepatocytes in the G1 phase of the cell cycle as compared to the p53 null mice. Here, experiments were undertaken to: (1) examine the possible mechanisms underlying the increased hepatocyte proliferation in p53 deficient mice. (2) investigate G1 phase hepatocyte accumulation in p53 mutant mice. (3) study the effect of neonatal aflatoxin exposure on hepatocyte proliferation in p53 wild-type mice. Mice of different p53 genotypes at three months of age were produced and used for the first two aims. For the third aim, mice were injected peritoneally with AFB1 at 7 days of age and sacrificed at 6 weeks of age. Our results suggest that p27 downregulation may be the major factor contributing to the hepatocyte proliferation in p53 deficient mice at 3 months of age. Although p21 is a direct downstream mediator of p53, its expression is low in p53 wild type mouse livers and is not significantly decreased in p53 deficient mouse livers. Introduction of a p53 mutant transgene increases expression of p21 and p27 as compared to their null counterparts. However, only a small percentage of hepatocytes express mutant p53 protein. Cells with increased expression of p21 and p27 could be blocked in G1 and account for the hepatocyte accumulation in G1 phase in p53 mutant mice. Our results also demonstrate that neonatal aflatoxin exposure can lead to prolonged hepatocyte and ductual cell proliferation in adult mice. The potential molecular interplay of p53 loss, p53 mutation, chronic hepatitis related liver injury and aflatoxin exposure in eventually causing dysregulated cell cycle control and the development of liver cancer is discussed in the context.
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze
(5), 4 (2005) 129-177. We consider surfaces immersed in three-dimensional pseudohermitian manifolds.
We define the notion of (p-)mean ...curvature and of the associated (p-)minimal
surfaces, extending some concepts previously given for the (flat) Heisenberg
group. We interpret the p-mean curvature not only as the tangential
sublaplacian of a defining function, but also as the curvature of a
characteristic curve, and as a quantity in terms of calibration geometry.
As a differential equation, the p-minimal surface equation is degenerate
(hyperbolic and elliptic). To analyze the singular set, we formulate some {\em
extension} theorems, which describe how the characteristic curves meet the
singular set. This allows us to classify the entire solutions to this equation
and to solve a Bernstein-type problem (for graphs over the $xy$-plane) in the
Heisenberg group $H_1$. In $H_{1}$, identified with the Euclidean space
$R^{3}$, the p-minimal surfaces are classical ruled surfaces with the rulings
generated by Legendrian lines. We also prove a uniqueness theorem for the
Dirichlet problem under a condition on the size of the singular set in two
dimensions, and generalize to higher dimensions without any size control
condition.
We also show that there are no closed, connected, $C^{2}$ smoothly immersed
constant p-mean curvature or p-minimal surfaces of genus greater than one in
the standard $S^{3}.$ This fact continues to hold when $S^{3}$ is replaced by a
general spherical pseudohermitian 3-manifold.
In this paper we provide a sharp characterization of the smooth
four-dimensional sphere. The assumptions of the theorem are conformally
invariant, and can be reduced to an L^2 inequality of the Weyl ...tensor and
positivity of the Yamabe invariant.
