We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constructed using superconnections. In the case of a general limit space X , we expr...

In this paper, we derive a partial result related to a question of Professor Yau: “Does a simply-connected complete Kähler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?” Main Theorem. Let M be a simply-connected complete Kähler manifold M with negative sectional curvature ≤ −1 and S∞(M) be the sphere at infinity of M . Then there is an explicit ...

We establish some inequalities for invariant submanifolds involving totally real sectional curvature and the scalar curvature in a quaternion space form. The equality cases are also discussed. AMS Mathematics Subject Classification (2010): 53C40

We give lower bounds estimates for the first Dirichilet eigenvalues for domains Ω in submanifolds M ⊂ N with locally bounded mean curvature. These lower bounds depend on the local injectivity radius, local upper bound for sectional curvature of N and local bound for the mean cuvature of M . For sumanifolds with bounded mean curvature of Hadamard manifolds these lower bounds depends only on the ...

We obtain an estimate for the norm of the second fundamental form of stable H-surfaces in Riemannian 3-manifolds with bounded sectional curvature. Our estimate depends on the distance to the boundary of the surface and on the bound on the sectional curvature but not on the manifold itself. We give some applications, in particular we obtain an interior gradient estimate for H-sections in Killing...

An indecomposable Riemannian symmetric space which admits nontrivial twistor spinors has constant sectional curvature. Furthermore, each homogeneous Riemannian manifold with parallel spinors is at. In the present paper we solve the twistor equation on all indecomposable Lorentzian symmetric spaces explicitly. In particular, we show that there are-in contrast to the Riemannian case-indecom-posab...

It is shown that the integral of the scalar curvature of a closed Riemannian manifold can be bounded from above in terms of the manifold’s dimension, diameter, and a lower bound for the sectional curvature. §

The purpose of this paper is fourfold: (1) to introduce and study a second order PDE, determined accidentally by a Riemann wave, reflecting the connection between oriented parallelograms area and sectional curvature on Riemannian manifolds; (2) to introduce and study the asymptotic behavior of oriented parallelograms area controlled by the sectional curvature; (3) to study some partial differen...

Hadamard (or complete $CAT(0)$) spaces are complete, non-positive curvature, metric spaces. Here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. Our results extend the standard non-linear ergodic theorems for non-expansive maps on real Hilbert spaces, to non-expansive maps on Ha...

In this paper we give a partial affirmative answer to a conjecture of Greene-Wu and Yau. We prove that a complete noncompact Kähler surface with positive and bounded sectional curvature and with finite analytic Chern number c 1 (M) 2 is biholomorphic to C 2. The celebrated theorem of Cheeger–Gromoll–Meyer [3], [10] states that a complete noncompact Riemannian manifold with positive sectional cu...