Bing-Long Chen

Sharp dimension estimates of holomorphic functions and rigidity

Let M n be a complete noncompact Kahler manifold of complex dimension n with nonnegative holomorphic bisectional curvature. Denote by O d (M n ) the space of holomorphic functions of polynomial growth of degree at most d on M n . In this paper we prove that dimcod(M n ) ≤ dim c O [d] (C n ), for all d > 0, with equality for some positive integer d if and only if M n is holomorphically isometric to C n . We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.

Local foliations and optimal regularity of Einstein spacetimes

Abstract We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the existence of a local coordinate chart defined in a ball with definite size in which the metric coefficients have optimal regularity. The proof is based on quantitative estimates for, on one hand, a constant mean curvature (CMC) foliation by spacelike hypersurfaces defined locally near the observer and, on the other hand, the metric in local coordinates that are spatially harmonic in each CMC slice. The results and techniques in this paper should be useful in the context of general relativity for investigating the long-time behavior of solutions to the Einstein equations.

Compact Kähler manifolds homotopic to negatively curved Riemannian manifolds

In this paper, we show that any compact Kähler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a Kähler–Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold X homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure J compatible with the symplectic form, there is no non-constant J-holomorphic entire curve

Strong uniqueness of the Ricci flow

f:{\mathbb C \,}\rightarrow X

Ricci flow with surgery on four-manifolds with positive isotropic curvature

f:C→X.

Complete Riemannian manifolds with pointwise pinched curvature

Self-pairings are a special subclass of pairings and have interesting applications in cryptographic schemes and protocols. In this paper, we speed up the computation of the self-pairing by using a simple final exponentiation on supersingular elliptic curves with embedding degree k = 3 . We also compare the efficiency of self-pairing computations on different curves over large characteristic. We indicate that supersingular elliptic curves with k = 3 may be more attractive for implementing the self-pairings.