Sharp integral inequalities for harmonic functions

Motivared by Carleman's proof of the isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper halfspace. We also derive the regularity for nonnegative solutions of the associated integral system and some Liouville type theorems.Comment: 35 page

Curvature Pinching Estimate And Singularities Of The Ricci Flow

In this paper, we first derive a pinching estimate on the traceless Ricci curvature in term of scalar curvature and Weyl tensor under the Ricci flow. Then we apply this estimate to study finite-time singularity behavior. We show that if the scalar curvature is uniformly bounded, then the Weyl tensor has to blow up, as a consequence, the corresponding singularity model must be Ricci flat with non-vanishing Weyl tensor.Comment: 12 page

On the dominated splitting of Lyapunov stable aperiodic classes

Recent works related to Palis conjecture of J. Yang, S. Crovisier, M. Sambarino and D. Yang showed that any aperiodic class of a $C^1$-generic diffeomorphism far away from homoclinic bifurcations (or homoclinic tangencies) is partially hyperbolic. We show in this paper that, generically, a non-trivial dominated splitting implies partial hyperbolicity for an aperiodic class if it is Lyapunov stable. More precisely, for $C^1$-generic diffeomorphisms, if a Lyapunov stable aperiodic class has a non-trivial dominated splitting $E\oplus F$, then one of the two bundles is hyperbolic (either $E$ is contracted or $F$ is expanded)

Transversally Elliptic Operators

We construct certain spectral triples in the sense of A. ~Connes and H. \~Moscovici (``The local index formula in noncommutative geometry'' {\it Geom. Funct. Anal.}, 5(2):174--243, 1995) that is transversally elliptic but not necessarily elliptic. We prove that these spectral triples satisfie the conditions which ensure the Connes-Moscovici local index formula applies. We show that such a spectral triple has discrete dimensional spectrum. A notable feature of the spectral triple is that its corresponding zeta functions have multiple poles, while in the classical elliptic cases only simple poles appear for the zeta functions. We show that the multiplicities of the poles of the zeta functions have an upper bound, which is the sum of dimensions of the base manifold and the acting compact Lie group. Moreover for our spectral triple the Connes-Moscovici local index formula involves only local transverse symbol of the operator.Comment: Updated 11/25/2003 with corrected format, and in 12pt fonts Updated 5/20/2004, major reorganizatio

Equations of Motion with Multiple Proper Time: A New Interpretation of Basic Quantum Physics

Equations of motion for single particle under two proper time model and three proper time model have been proposed and analyzed. The motions of particle are derived from pure classical method but they exhibit the same properties of quantum physics: the quantum wave equation, de Broglie equations, uncertainty relation, statistical result of quantum wave-function. This shows us a possible new way to interpret quantum physics. We will also prove that physics with multiple proper time does not cause causality problem.Comment: 6 pages, 4 figure

Compactifications of Complete Riemannian manifolds and Their Applications

To study a noncompact Riemannian manifold, it is often useful to find a compactification. We discuss several common compactifications and survey some recent results

First Eigenvalues of Geometric Operators under the Ricci Flow

In this paper, we prove that the first eigenvalues of $-\Delta + cR$ ($c\geq \frac14$) is nondecreasing under the Ricci flow. We also prove the monotonicity under the normalized flow for the case $c=1/4$, and $r\le 0$.Comment: 5 pages, add one more referenc

On the pluricanonical maps of varieties of intermediate Kodaira dimension

In this paper we will prove a uniformity result for the Iitaka fibration $f:X \rightarrow Y$, provided that the generic fiber has a good minimal model and the variation of $f$ is zero or that $\kappa(X)=\rm{dim}(X)-1$.Comment: 21 page

Hyperbolicity versus weak periodic orbits inside homoclinic classes

We prove that, for $C^1$-generic diffeomorphisms, if the periodic orbits contained in a homoclinic class $H(p)$ have all their Lyapunov exponents bounded away from 0, then $H(p)$ must be (uniformly) hyperbolic. This is in sprit of the works of the stability conjecture, but with a significant difference that the homoclinic class $H(p)$ is not known isolated in advance, hence the "weak" periodic orbits created by perturbations near the homoclinic class have to be guaranteed strictly inside the homoclinic class. In this sense the problem is of an "intrinsic" nature, and the classical proof of the stability conjecture does not pass through. In particular, we construct in the proof several perturbations which are not simple applications of the connecting lemmas

Spin-Flavor Decomposition in Polarized Semi-Inclusive Deep Inelastic Scattering Experiments at Jefferson Lab

A Jefferson Lab experiment proposal was discussed in this talk. The experiment is designed to measure the beam-target double-spin asymmetries $A_{1n}^h$ in semi-inclusive deep-inelastic $\vec n({\vec e}, e^\prime \pi^+)X$ and $\vec n({\vec e}, e^\prime \pi^-)X$ reactions on a longitudinally polarized $^3$He target. In addition to $A_{1n}^h$, the flavor non-singlet combination $A_{1n}^{\pi^+ - \pi^-}$, in which the gluons do not contribute, will be determined with high precision to extract $\Delta d_v(x)$ independent of the knowledge of the fragmentation functions. The data will also impose strong constraints on quark and gluon polarizations through a global NLO QCD fit.Comment: 5 pages, 4 figures, to appear in the proceedings of the First Workshop on Quark-Hadron Duality and the Transition to pQCD, Frascati, Italy. June 6-8, 200