MathSciDoc: An Archive for Mathematician ∫

In this paper we construct a new time-periodic solution of the vacuum Einsteins field equations, this solution possesses physical singularities, i.e., the norm of the solutions Riemann curvature tensor takes the infinity at some points. We show that this solution is intrinsically time-periodic and describes a time-periodic universe with the time-periodic physical singularity. By calculating the Weyl scalars of this solution, we investigate new physical phenomena and analyze new singularities for this universal model.

We prove several rigidity theorems for elliptic genera associated to Toeplitz operators on odd dimensional spin manifolds. We also prove an equivariant odd index theorem for Dirac operators with involution parity and the Atiyah-Hirzebruch vanishing theorems for odd dimensional spin manifolds.

In this paper we investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential equations are strictly and uniformly parabolic. Based on this, we show that the generalized Ricci flow defined on an n-dimensional compact Riemannian manifold admits a unique short-time smooth solution. Moreover, we also derive the evolution equations for the curvatures, which play an important role in our future study.

In [LL-Y1, III: Sec. 5.4] on mirror principle, a method was developed to compute the integral \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d for a flag manifold $ X=\Fl_ {r_1,..., r_I}({\Bbb C}^ n) $ via an extended mirror principle diagram. This method turns the required localization computation on the augmented moduli stack $\bar {\cal M} _ {0, 0}(\CP^ 1\times X) $ of stable maps to a localization computation on a hyper-Quot-scheme $\HQuot ({\cal E}^ n) $. In this article, the detail of this localization computation on $\HQuot ({\cal E}^ n) $ is carried out. The necessary ingredients in the computation, notably, the \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d -fixed-point components and the distinguished ones \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d in $\HQuot ({\cal E}^ n) $, the \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d -equivariant Euler class of \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d in $\HQuot ({\cal E}^ n) $, and a push-forward formula of cohomology classes involved in the problem from the total space of a restrictive flag manifold bundle to its base manifold are given. With these, an exact expression of \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d is obtained. Comments on the Hori-Vafa conjecture are given in the end.

We present a new method to solve certain ar {\partial} -equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a ar {\partial} -lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at ar {\partial} -level, as well as certain injectivity theorem on compact Kahler manifolds.

In this paper we consider a purely algebraic result. Then given a circle or cyclic group of prime order action on a manifold, we will use it to estimate the lower bound of the number of fixed points. We also give an obstruction to the existence of \mathbb {Z} _p action on manifolds with isolated fixed points when \mathbb {Z} _p is a prime.

We derive an effective recursion for Witten's r-spin intersection numbers, using Witten's conjecture relating r-spin numbers to the Gel'fand-Dikii hierarchy (Theorem 4.1). Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions: Propositions 5.2 and 5.4, Corollary 5.5). We use these closed-form descriptions to prove Harer-Zagier's formula for the Euler characteristic of M_ {g, 1}. Finally in Section 6, we extend Witten's series expansion formula for the Landau-Ginzburg potential to study r-spin numbers in the small phase space in genus zero. Our key tool is the calculus of formal pseudodifferential operators, and is partially motivated by work of Brezin and Hikami.

By using Yau's Schwarz lemma and the Quillen determinant line bundles, several results about fibered algebraic surfaces and the moduli spaces of curves are improved and reproved.

By studying modular invariance properties of some characteristic forms, we prove some new anomaly cancellation formulas which generalize the Han-Zhang and Han-Liu-Zhang anomaly cancellation formulas

We briefly survey our recent results about the Mumford goodness of several canonical metrics on the moduli spaces of Riemann surfaces, including the Weil-Petersson metric, the Ricci metric, the Perturbed Ricci metric and the Kahler-Einstein metric. We prove the dual Nakano negativity of the Weil-Petersson metric. As applications of these results we deduce certain important results about the <i>L</i> ²-cohomology groups of the logarithmic tangent bundle over the compactified moduli spaces.