The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight

We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau from dimension $n=3$ to dimensions $3 \leq n <8$. This requires us to address several technical difficulties that are not present when $n=3$. The regularity and decay assumptions for the initial data sets to which our argument applies are weaker than those of R. Schoen and S.-T. Yau. In recent joint work with L.-H. Huang, D. Lee, and R. Schoen we have given a different proof of the full positive mass theorem in dimensions $3 \leq n < 8$. We pointed out that this theorem can alternatively be derived from our density argument and the positive energy theorem of the present paper.Comment: All comments welcome! Final version to appear in Comm. Math. Phy

A generalization of Hawking's black hole topology theorem to higher dimensions

Hawking's theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking's results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology $S^2 \times S^1$. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).Comment: 8 pages, latex2e, references updated, minor corrections, to appear in Communications in Mathematical Physic

Existence, Regularity, and Properties of Generalized Apparent Horizons

We prove a conjecture of Tom Ilmanen's and Hubert Bray's regarding the existence of the outermost generalized apparent horizon in an initial data set and that it is outer area minimizing.Comment: 16 pages, thoroughly revised, no major changes, to appear in Comm. Math. Phy

Effects of Environmental Zoning on Household Sorting: Empirical Evidence and Ecological Implications

In this paper we present a preliminary analysis of whether and how spatial variation in environmental attributes affects the residential sorting of households with heterogeneous preferences. An important implication of such sorting arises if variation in preferences over environmental attributes is correlated with household activities affecting the local ecosystem, such as the replacement of native vegetation with lawns, and the removal of course woody habitat from a lake. In this case the sorting process may engender differential evolution of local ecosystems (lakes) with the same initial ecological state. The model examined in the paper has the potential to statistically examine this issue, and therefore holds promise for understanding the behavioral implications of land use policies designed to protect local ecosystems. By facilitating the grouping of different types of households onto different lakes, for instance, lakeshore zoning policies may engender differentiation in the ecological evolution of lakes beyond what would be expected from the zoning policies themselves.Environmental Economics and Policy, Land Economics/Use,

Positive mass theorem for the Paneitz-Branson operator

We prove that under suitable assumptions, the constant term in the Green function of the Paneitz-Branson operator on a compact Riemannian manifold $(M,g)$ is positive unless $(M,g)$ is conformally diffeomophic to the standard sphere. The proof is inspired by the positive mass theorem on spin manifolds by Ammann-Humbert.Comment: 7 page

The 2007-2009 Financial Crisis: An Erosion of Ethics - A Case Study

This case study examines five dimensions of the 2007-2009 financial crisis in the United States: (1) the devastating effects of the financial crisis on the U.S. economy, including unparalleled unemployment, massive declines in gross domestic product (GDP), and the prolonged mortgage foreclosure crisis; (2) the multiple causes of the financial crisis and panic, such as the housing and bond bubbles, excessive leverage, lax financial regulation, disgraceful banking practices, and abysmal rating agency performance; (3) the extraordinary efforts of the Federal Reserve, the Federal Reserve Bank of New York, and the Department of the Treasury to stem the financial freefall triggered by the crisis and resuscitate financial institutions, (4) the ethical implications of the unprecedented actions by government institutions to rescue financial institutions and drag the country back from the brink of global financial collapse, and the conduct of the various parties contributing to the financial crisis, such as the shoddy behavior of mortgage brokers, the massive securitization of mortgages into overly complex bonds, the excessive leverage of financial institutions, the disgraceful work of bond rating firms, the abysmal risk management systems employed by financial institutions, and the massive operations of the shadow banking and over-the-counter derivatives markets; and (5) the major provisions of the Dodd-Frank Wall Street Reform and Consumer Protection Act signed into law to in response to the financial crisis and for the purpose of correcting the egregious conduct of major financial institutions