On Manifolds with Lower Quadratic Curvature Decay and Volume Pinching

We give some conditions for a complete noncompact Riemannian manifold with lower quadratic curvature decay to have finite topological type. Most of our curvature conditions are much weaker than the assumptions in Lott [5] and in Sha–Shen [10], hence our results can be viewed as natural generalizations of those works.

The topology of an open manifold with radial curvature bounded from below by a model surface with finite total curvature and examples of model surfaces

We construct distinctive surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we prove that a complete noncompact Riemannian manifold M is homeomorphic to the interior of a compact manifold with boundary if the manifold M is not less curved than a noncompact model surface of revolution and if the total curvature of the model surface is finite and less than 2π. By the first result mentioned above, the second result covers a much wider class of manifolds than that of complete noncompact Riemannian manifolds whose sectional curvatures are bounded from below by a constant.

Pseudolocality for the Ricci Flow and Applications

Perelman established a differential Li-Yau-Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds. As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete noncompact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. The conditions are satisfied by asymptotically flatmanifolds. We also prove a long time existence result for the Kähler-Ricci flow on complete nonnegatively curved Kähler manifolds.

Existence of Infinitely Many Distinct Solutions to the Quasirelativistic Hartree-Fock Equations

We establish existence of infinitely many distinct solutions to the semilinear elliptic Hartree-Fock equations forN-electron Coulomb systems with quasirelativistic kinetic energy−α−2Δxn+α−4−α−2for thenthelectron. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total chargeZtotofKnuclei is greater thanN−1and thatZtotis smaller than a critical chargeZc. The proofs are based on a new application of the Fang-Ghoussoub critical point approach to multiple solutions on a noncompact Riemannian manifold, in combination with density operator techniques.