This paper presents hyperbolic rank rigidity results for rank 1, nonpositively curved spaces. Let M be a compact, rank 1 manifold with nonpositive sectional curvature and suppose that along every geodesic in M there is a parallel vector field making curvature −a2 with the geodesic direction. We prove that M has constant curvature equal to −a2 if M is odd dimensional, or if M is even dimensional...

In this paper, we proved that a compact Sasakian manifold [Formula: see text] with negative transverse holomorphic sectional curvature must have structure Ricci curvature. Similarly, nonpositive curvature, then the first basic Chern class is nef and Miyaoka–Yau-type inequality. When quasi-negative, obtain number

Let (M, g, J) be a compact Hermitian manifold with a smooth boundary. Let ∆p,B and ⊓ ⊔p,B be the realizations of the real and complex Laplacians on p forms with either Dirichlet or Neumann boundary conditions. We generalize previous results in the closed setting to show that (M, g, J) is Kaehler if and only if Spec(∆p,B) = Spec(2 ⊓ ⊔p,B) for p = 0, 1. We also give a characterization of manifold...

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum of the L2-norm of Ricci curvature for all complex surfaces of general type. We are also able to show that the standard metric on any complex hyperbolic 4manif...

We show that the pseudohermitian sectional curvature Hθ(σ) of a contact form θ on a strictly pseudoconvex CR manifold M measures the difference between the lengthes of a circle in a plane tangent at a point of M and its projection on M by the exponential map associated to the Tanaka-Webster connection of (M, θ). Any Sasakian manifold (M, θ) whose pseudohermitian sectional curvature Kθ(σ) is a p...

We study subgroups of fundamental groups of real analytic closed 4-manifolds with nonpositive sectional curvature. In particular, we are interested in the following question: if a subgroup of the fundamental group is not virtually free abelian, does it contain a free group of rank two ? The technique involves the theory of general metric spaces of nonpositive curvature.

We establish the existence of Euclidean tangent cones on Wasserstein spaces over compact Alexandrov spaces of curvature bounded below. By using this Riemannian structure, we formulate and construct gradient flows of functions on such spaces. If the underlying space is a Riemannian manifold of nonnegative sectional curvature, then our gradient flow of the free energy produces a solution of the l...

Harmonic maps are natural generalizations of harmonic functions and are critical points of the energy functional defined on the space of maps between two Riemannian manifolds. The Liouville type properties for harmonic maps have been studied extensively in the past years (Cf. [Ch], [C], [EL1], [EL2], [ES], [H], [HJW], [J], [SY], [S], [Y1], etc.). In 1975, Yau [Y1] proved that any harmonic funct...

Let M 2n be a compact Riemannian manifold of non-positive sectional curvature. It is shown that if M 2n is homeomorphic to a KK ahler manifold, then its Euler number satisses the inequality (?1) n (M 2n) 0. Introduction The results of this paper are related to a well-known problem, attributed sometimes to Hopf and sometimes to Chern, to the eeect that the Euler number (M 2n) of a compact Rieman...

This paper constructs a class of complete Kähler metrics of positive holomorphic sectional curvature on C and finds that the constructed metrics satisfy the following properties: As the geodesic distance ρ → ∞, the volume of geodesic balls grows like O(ρ 2(β+1)n β+2 ) and the Riemannian scalar curvature decays like O(ρ − 2(β+1) β+2 ), where β ≥ 0.