A Negative Mass Theorem for the 2-Torus

Let M be a closed surface. For a metric g on M, denote the area element by dA and the Laplace-Beltrami operator by Δ = Δ _g . We define the Robin mass m(p) at the point to be the value of the Green function G(p, q) at q = p after the logarithmic singularity has been subtracted off, and we define trace . This regularized trace can also be obtained by regularization of the spectral zeta function and is hence a spectral invariant which heuristically measures the total wavelength of the surface.We define the Δ-mass of (M, g) to equal , where is the Laplacian on the round sphere of area A. This scale invariant quantity is a non-trivial analog for closed surfaces of the ADM mass for higher dimensional asymptotically flat manifolds.In this paper we show that in each conformal class for the 2-torus, there exists a metric with negative Δ-mass. From this it follows that the minimum of the Δ-mass on is negative and attained by some metric . For this minimizing metric g, one gets a sharp logarithmic Hardy-Littlewood-Sobolev inequality and an Onofri-type inequality.We remark that if the flat metric in is sufficiently long and thin then the minimizing metric g is non-flat. The proof of our result depends on analyzing the ordinary differential equation which is equivalent to h′′ = 1 − 1/h. The solutions are periodic and we need to establish quite delicate, asymptotically sharp inequalities relating the period to the maximum value. The author was supported by the National Science Foundation #DMS-0302647.     

Remarks on Positive Mass Theorem

We establish a positive mass theorem for time-symmetric initial data sets by relaxing the nonnegative scalar curvature to the case where the first Neumann eigenvalue of conformal Laplacian operators is eventually nonnegative. Received: 4 May 1999 / Accepted: 28 June 1999     

A Positive Mass Theorem for Spaces with Asymptotic SUSY Compactification

We prove a positive mass theorem for spaces which asymptotically approach a flat Euclidean space times a Calabi-Yau manifold (or any special honolomy manifold except the quaternionic Kähler). This is motivated by the very recent work of Hertog-Horowitz-Maeda [HHM].     

Angular Momentum and Positive Mass Theorem

Total angular momentum for asymptotically flat manifolds is defined. Positive mass theorem for initial (spin) data set (M, g _ij , p _ij ) with nonsymmetric p _ij is proved. As an application, we establish positive mass theorems involving total linear momentum and total angular momentum. This gives an answer to a problem of S. T. Yau in his Problem Section [Ya2] and a partial answer to his recent conjecture on the relationship among total energy, total linear momentum, total angular momentum and entropy of black hole. Received: 26 October 1998 / Accepted: 10 March 1999     

On the Positive Mass Theorem for Manifolds with Corners

We study the positive mass theorem for certain non-smooth metrics following P. Miao’s work. Our approach is to smooth the metric using the Ricci flow. As well as improving some previous results on the behaviour of the ADM mass under the Ricci flow, we extend the analysis of the zero mass case to higher dimensions.     

A Strong Central Limit Theorem for a Class of Random Surfaces

This paper is concerned with d = 2 dimensional lattice field models with action ${V(\nabla\phi(\cdot))}$ , where ${V : \mathbf{R}^d \rightarrow \mathbf{R}}$ is a uniformly convex function. The fluctuations of the variable ${\phi(0) - \phi(x)}$ are studied for large |x| via the generating function given by ${g(x, \mu) = \ln \langle e^{\mu(\phi(0) - \phi(x))}\rangle_{A}}$ . In two dimensions ${g'' (x, \mu) = \partial^2g(x, \mu)/\partial\mu^2}$ is proportional to ${\ln\vert x\vert}$ . The main result of this paper is a bound on ${g''' (x, \mu) = \partial^3 g(x, \mu)/\partial \mu^3}$ which is uniform in ${\vert x \vert}$ for a class of convex V. The proof uses integration by parts following Helffer–Sjöstrand and Witten, and relies on estimates of singular integral operators on weighted Hilbert spaces.     

A Definition of Total Energy-Momenta and the Positive Mass Theorem on Asymptotically Hyperbolic 3-Manifolds. I

Two sets of asymptotically hyperbolic initial data are defined, which correspond to the spatial infinity in asymptotically AdS spacetimes and to the null infinity in asymptotically Minkowski spacetimes respectively. The positive mass theorem involving the total energy, the total linear momentum and the total angular momentum is established for these initial data sets.Research partially supported by National Natural Science Foundation of China under grant 10231050 and the innovation project of the Chinese Academy of Sciences.     

Structured Surfaces for Generation of Negative Index of Refraction

Materials with negative index of refraction have properties that are not naturally available. Such properties can be used to develop novel devices like the superlens which can surpass the diffraction limit. Optical cloaking can be achieved through this negative refractive index method. This article reviews the progress made in the area of negative refractive index materials from the first generation of negative electrical permittivity to the demonstration of negative index of refraction at optical frequency, with the relevant discussion on the physics of these materials. The prime focus of this article is on experimental demonstrations and fabrication related issues of negative refractive index materials which makes use of structured surfaces.     

Conformal Positive Mass Theorems

We show the following two extensions of the standard positive mass theorem (one for either sign): Let ,g) and ,g' be asymptotically flat Riemannian 3-manifolds with compact interior and finite mass, such that g and g' are C ^2, and related via the conformal rescaling g' = ⁴ g with a C ^2, -function > 0. Assume further that the corresponding Ricci scalars satisfy R ± ⁴ R'; 0. Then the corresponding masses satisfy m±m' 0. Moreover, in the case of the minus sign, equality holds iff g and g' are isometric, whereas equality holds for the plus sign iff both ,g) and ,g') are flat Euclidean spaces. While the proof of the case with the minus signs is rather obvious, the case with the plus signs requires a subtle extension of Witten's proof of the standard positive mass theorem. The idea for this extension is due to Masood-ul-Alam who, in the course of an application, proved the rigidity part m+m' = 0 of this theorem, for a special conformal factor. We observe that Masood-ul-Alam's method extends to the general situation.     

On Genus Two Riemann Surfaces Formed from Sewn Tori

We describe the period matrix and other data on a higher genus Riemann surface in terms of data coming from lower genus surfaces via an explicit sewing procedure. We consider in detail the construction of a genus two Riemann surface by either sewing two punctured tori together or by sewing a twice-punctured torus to itself. In each case the genus two period matrix is explicitly described as a holomorphic map from a suitable domain (parameterized by genus one moduli and sewing parameters) to the Siegel upper half plane . Equivariance of these maps under certain subgroups of is shown. The invertibility of both maps in a particular domain of is also shown. Support provided by the National Science Foundation DMS-0245225, and the Committee on Research at the University of California, Santa Cruz. Supported by The Millenium Fund, National University of Ireland, Galway.     

Mass Spectrometry of Some Benzylidene Imidazolidinediones and Thiazolidinediones. III- Positive and Negative Electron Impact Mass Spectra of Chlorobenzyl Imidazolidinedione or Thiazolidinedione Compounds

In the present investigation, a study of the electron-impact mass spectrometry in positive- and negative-ion modes is reported for a series of 3-chlorobenzyl-5-benzylidene-imidazolidine-2,4-dione and -thiazolidine-2,4-dione derivatives previously synthetized.     

Yamabe Solitons, Determinant of the Laplacian and the Uniformization Theorem for Riemann Surfaces

In this letter, we prove that non-trivial compact Yamabe solitons or breathers do not exist. In particular our proof in the two dimensional case depends only on properties of the determimant of the Laplacian and turns out to be independent of the classical uniformization theorem (UT). Using this remarkable fact we are able to explain how an independent proof of the UT for Riemann surfaces can be obtained using the Yamabe–Ricci flow. Luca Fabrizio Di Cerbo was partially supported by a Renaissance Technology Fellowship.     

Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I

We define the partition and n-point functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice vertex operator algebras and a continuous orbifolding of the rank two fermion vertex operator super algebra. We compute the genus two Heisenberg vector n-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity for these theories.     

巧用质心运动定理

中学阶段学过两种运动形式：平动、转动．日常生活中，遇到的不可能都是单一的运动形式．当我们面对一个既有平动又有转动的复杂运动时往往束手无策．例如，投出的手榴弹一面翻转，一面前进，其中各点的运动情况相当复杂．引入质量中心后讨论就简便了．     

Real-time Optical Wavelet-Transform with Positive and Negative Signals

An opto-electronic hybrid system that is based on the joint transform correlator (JTC) is suggested for the implementation of real-time wavelet transform. Holographic encoding of both the object signal and wavelet function enables the JTC-based real-time correlator to execute the optical wavelet transformation involving positive and negative values. We suggest an interferometric method to retain the information about the polarity of the final wavelet transform. The principle of this method is verified by experiments.Presented at 1996 International Topical Meeting on Optical Computing (OC ‘96), April 21-25, Sendai, Japan.     

正、负电性纳米银吸附阴、阳离子型分子的SERS比较

使用表面带负电的胶态纳米银,分别进行阴,阳离子型分子的SERS效应研究,并与表面带正电的胶态纳米银比较,当阳离子型分子盐酸副玫瑰苯胺吸附在正,负电性纳米银表面上时,后者SERS较强,当阴离子型分子吲哚丁酸分别吸附在这两种胶态纳米银上时,后者无SERS效应。     

Generating Functional in CFT and Effective Action for Two-Dimensional Quantum Gravity on Higher Genus Riemann Surfaces

We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus g > 1, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action functional uses various double complexes naturally associated with a Riemann surface, with computations that are quite similar to descent calculations in BRST cohomology theory. We also provide an interpretation of the action functional in terms of the geometry of different fiber spaces over the Teichmüller space of compact Riemann surfaces of genus g > 1. Received: 12 September 1996 / Accepted: 6 January 1997