On The Capacity of Surfaces in Manifolds with Nonnegative Scalar Curvature

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at infinity. Even in the special case of Euclidean space, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.Comment: 18 page

Dual Actions for Born-Infeld and Dp-Brane Theories

Dual actions with respect to U(1) gauge fields for Born-Infeld and $Dp$-brane theories are reexamined. Taking into account an additional condition, i.e. a corollary to the field equation of the auxiliary metric, one obtains an alternative dual action that does not involve the infinite power series in the auxiliary metric given by ref. \cite{s14}, but just picks out the first term from the series formally. New effective interactions of the theories are revealed. That is, the new dual action gives rise to an effective interaction in terms of one interaction term rather than infinite terms of different (higher) orders of interactions physically. However, the price paid for eliminating the infinite power series is that the new action is not quadratic but highly nonlinear in the Hodge dual of a $(p-1)$-form field strength. This non-linearity is inevitable to the requirement the two dual actions are equivalent.Comment: v1: 11 pages, no figures; v2: explanation of effective interactions added; v3: concision made; v4: minor modification mad

New Born-Infeld and DpDp-Brane Actions under 2-Metric and 3-Metric Prescriptions

The parent action method is utilized to the Born-Infeld and $Dp$-brane theories. Various new forms of Born-Infeld and $Dp$-brane actions are derived by using this systematic approach, in which both the already known 2-metric and newly proposed 3-metric prescriptions are considered. An auxiliary worldvolume tensor field, denoted by ${\omega}_{{\mu}{\nu}}$, is introduced and treated probably as an additional worldvolume metric because it plays a similar role to that of the auxiliary worldvolume (also called {\em intrinsic}) metric ${\gamma}_{{\mu}{\nu}}$. Some properties, such as duality, permutation and Weyl invariance as a local worldvolume symmetry of the new forms are analyzed. In particular, a new symmetry, i.e. the double Weyl invariance is discovered in 3-metric forms.Comment: v1: 30 pages, 4 figures; v2: 31 pages, 4 figures, final version with some modifications to appear in Phys. Rev.

Well-posedness for the viscous shallow water equations in critical spaces

In this paper, we prove the existence and uniqueness of the solutions for the 2D viscous shallow water equations with low regularity assumptions on the initial data as well as the initial height bounded away from zero.Comment: 32 page

Variable Temptations and Black Market Reputations

In a world of imperfect information, reputations often guide the sequential decisions to trust and to reward trust. We consider two-player situations, where one player--the truster--decides whether to trust, and the other player--the temptee--has a temptation to betray when trusted. The strength of the temptation to betray varies from encounter to encounter. We refer to a recorded betrayal as a black mark and focus on mechanisms that only reveal the number of black marks of a temptee. We show that the greater the number of black marks, the less likely the temptee is to betray. We then study the different equilibria that emerge, depending on which side of the market has the ability to specify the equilibrium. In closing, we generalize to cases where the number of encounters is also recorded.