Differential Harnack Estimates for Parabolic Equations

Let $(M,g(t))$ be a solution to the Ricci flow on a closed Riemannian manifold. In this paper, we prove differential Harnack inequalities for positive solutions of nonlinear parabolic equations of the type \ppt f=\Delta f-f \ln f +Rf. We also comment on an earlier result of the first author on positive solutions of the conjugate heat equation under the Ricci flow.Comment: 10 page

On quasi-isomorphic DGBV algebras

One of the methods to obtain Frobenius manifold structures is via DGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra construction. An important problem is how to identify Frobenius manifold structures constructed from two different DGBV algebras. For DGBV algebras with suitable conditions, we show the functorial property of a construction of deformations of the multiplicative structures of their cohomology. In particular, we show that quasi-isomorphic DGBV algebras yield identifiable Frobenius manifold structures.Comment: 16 pages, AMS-LaTe

Degenerate Chern-Weil Theory and Equivariant Cohomology

We develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. This provides a method of changing an equivariant closed form within its cohomological class to a form more suitable to yield localization results. This work is motivated by our work on reproving wall crossing formulas in Seiberg-Witten theory, where the Lie group is the circle. As applications, we derive two localization formulas of Kalkman type for G = SU(2) or SO(3)-actions on compact manifolds with boundary. One of the formulas is then used to yield a very simple proof of a localization formula due to Jeffrey-Kirwan in the case of G = SU(2) or SO(3).Comment: 23 pages, AMSLaTe

Frobenius Manifold Structure on Dolbeault Cohomology and Mirror Symmetry

We construct a differential Gerstenhaber-Batalin-Vilkovisky algebra from Dolbeault complex of any close Kaehler manifold, and a Frobenius manifold structure on Dolbeault cohomology.Comment: 10 pages, AMS LaTe

On Quantum de Rham Cohomology

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds, of which symplectic manifolds are an important class of examples. Quantum de Rham cohomology is defined as the cohomology of d_h. We also define quantum Dolbeault cohomology. Quantum hard Lefschetz theorem is proved. We also define a version of quantum integral, and prove the quantum Stokes theorem. By the trick of replacing d by d_h and wedge by wedge_h in the usual definitions, we define many quantum analogues of important objects in differential geometry, e.g. quantum curvature. The quantum characteristic classes are then studied along the lines of classical Chern-Weil theory, i.e., they can be represented by expressions of quantum curvature. Quantum equivariant de Rham cohomology is defined in a similar fashion. Calculations are done for some examples, which show that quantum de Rham cohomology is different from the quantum cohomology defined using pseudo-holomorphic curves.Comment: 36 pages, AMS LaTe

DGBV Algebras and Mirror Symmetry

We describe some recent development on the theory of formal Frobenius manifolds via a construction from differential Gerstenhaber-Batalin-Vilkovisk (DGBV) algebras and formulate a version of mirror symmetry conjecture: the extended deformation problems of the complex structure and the Poisson structure are described by two DGBV algebras; mirror symmetry is interpreted in term of the invariance of the formal Frobenius manifold structures under quasi-isomorphism.Comment: 11 pages, to appear in Proceedings of ICCM9

Formal Frobenius manifold structure on equivariant cohomology

For a closed K\"{a}hler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model.Comment: AMS-LaTex, 14 page

Mass shifts of 3PJ^3P_J heavy quarkonia due to the effect of two-gluon annihilation

In this work, we calculate the nonrelativistic asymptotic behavior of the amplitudes of $q\overline{q}\rightarrow 2g \rightarrow q\overline{q}$ in the leading order of $\alpha_s$ (LO-$\alpha_s$) with $q\overline{q}$ in the $^3P_J$ channels. In the practical calculation we take the momenta of quarks and antiquarks on-shell and expand the amplitudes on the three-momentum of the quarks and antiquarks to order 6 and get three nonzero terms. The imaginary parts of the first term and the second term are the old. The real parts of the results have IR divergence. When applying the results to the heavy quarkonia, the corresponding amplitude of $q\overline{q}\rightarrow1g\rightarrow q\overline{q}$ with $q\overline{q}$ in the color octet $^3S_1$ channel is considered to absorb the IR divergence in a unitary way in the leading order of $v$ (LO-$v$). The finial results can be used to estimate the mass shifts of the $^3P_J$ heavy quarkonia due to the effect of two-gluon annihilation. The numerical estimation shows that the contributions to the mass shifts of $\chi_{c0,c1,c2}$ are about $1.23\sim1.58$ MeV, $1.57\sim1.86$ MeV and $5.92\sim5.45$ MeV when taking $\alpha_s\approx 0.25\sim0.35$

On Locally Conformally Flat Gradient Shrinking Ricci Solitons

In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvature is bounded from below and the Riemannian curvature tensor has at most exponential growth. As a consequence of this identity, we classify complete locally conformally flat gradient shrinking Ricci solitons with Ricci curvature bounded from below.Comment: 13 pages, revised version, third author adde

Exact Relations for Two-Photon-Exchange Effect in Elastic epep Scattering by Dispersion Relation and Hadronic Model

The two-photon-exchange (TPE) effect plays a key role to extract the form factors (FFs) of the proton. In this work, we present some exact properties on the TPE effect in the elastic $ep$ scattering based on four types of typical and general interactions. The possible kinematical singularities, the asymptotic behaviors and the branch cuts of the TPE amplitudes are analyzed. The analytic expressions clearly indicate some exact relations between the dispersion relation (DR) method and the hadronic model (HM) method. It suggests that the two methods should be modified to general forms, respectively. After the modifications the new forms give the same results. Furthermore, they automatically and correctly include the contributions due to the seagull interaction, the meson-exchange effect, the contact interactions and the off-shell effect. To analyze the elastic $e^{\pm}p$ scattering data sets, the new forms should be used.Comment: 3 figur