Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in $S^4$, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them holds interesting duality theorem. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.Comment: 19 page

Complete stationary surfaces in R14\mathbb{R}^4_1 with total curvature −∫KdM=4π-\int K\mathrm{d}M=4\pi

Applying the general theory about complete spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space $\mathbb{R}^4_1$, we classify those regular algebraic ones with total Gaussian curvature $-\int K\mathrm{d}M=4\pi$. Such surfaces must be oriented and be congruent to either the generalized catenoids or the generalized enneper surfaces. For non-orientable stationary surfaces, we consider the Weierstrass representation on the oriented double covering $\widetilde{M}$ (of genus $g$) and generalize Meeks and Oliveira's M\"obius bands. The total Gaussian curvature are shown to be at least $2\pi(g+3)$ when $\widetilde{M}\to\mathbb{R}^4_1$ is algebraic-type. We conjecture that there do not exist non-algebraic examples with $-\int K\mathrm{d}M=4\pi$.Comment: 22 page

The Moebius geometry of Wintgen ideal submanifolds

Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. They are Moebius invariant objects. The mean curvature sphere defines a conformal Gauss map into a Grassmann manifold. We show that any Wintgen ideal submanifold has a Riemannian submersion structure over a Riemann surface with the fibers being round spheres. Then the conformal Gauss map is shown to be a super-conformal and harmonic map from the underlying Riemann surface. Some of our previous results are surveyed in the final part.Comment: This is a survey of our recent work on the Moebius geometry of Wintgen ideal submanifolds, which also include two new important results. Submitted to the the conference "ICM 2014 Satellite Conference on Real and Complex Submanifolds

Functional Implications of DNA Methylation in Adipose Biology.

The twin epidemics of obesity and type 2 diabetes (T2D) are a serious health, social, and economic issue. The dysregulation of adipose tissue biology is central to the development of these two metabolic disorders, as adipose tissue plays a pivotal role in regulating whole-body metabolism and energy homeostasis (1). Accumulating evidence indicates that multiple aspects of adipose biology are regulated, in part, by epigenetic mechanisms. The precise and comprehensive understanding of the epigenetic control of adipose tissue biology is crucial to identifying novel therapeutic interventions that target epigenetic issues. Here, we review the recent findings on DNA methylation events and machinery in regulating the developmental processes and metabolic function of adipocytes. We highlight the following points: 1) DNA methylation is a key epigenetic regulator of adipose development and gene regulation, 2) emerging evidence suggests that DNA methylation is involved in the transgenerational passage of obesity and other metabolic disorders, 3) DNA methylation is involved in regulating the altered transcriptional landscape of dysfunctional adipose tissue, 4) genome-wide studies reveal specific DNA methylation events that associate with obesity and T2D, and 5) the enzymatic effectors of DNA methylation have physiological functions in adipose development and metabolic function

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in $S^4$, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them holds interesting duality theorem. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.Comment: 19 page