Rigidity and L2L^2 cohomology of hyperbolic manifolds

When X=\Gamma\backslash \H^n is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of $L^2$ harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results

Harmonic Functions On Manifolds Whose Large Sphere Are Small

We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the gradient of harmonic functions

Apes with a Moral Code? Primatology, Moral Sentimentalism, and the Evolution of Morality in The Planet of the Apes

This essay examines the recent Planet of the Apes films through the lens of recent research in primatology. The films lend imaginary support to primatologist Frans de Waal’s evolutionary moral sentimentalism; however, the movies also show that truly moral motions outstrip the cognitive capacities of the great apes. The abstract moral principles employed by the ape community in the movie require the ability to understand and apply a common underlying explanation to perceptually disparate situations; in contrast, recent research in comparative psychology demonstrates that the great apes lack this capacity. Since the capacity for abstraction is required on even the most basic version of moral sentimentalism—Shaun Nichols’ sentimental rules account—the lack of the capacity for abstraction reveals a qualitative distinction between primate social behavior and human morality

Riesz transform on manifolds with quadratic curvature decay

We investigate the $L^p$-boundness of the Riesz transform on Riemannian manifolds whose Ricci curvature has quadratic decay. Two criteria for the $L^p$-unboundness of the Riesz transform are given. We recover known results about manifolds that are Euclidean or conical at infinity

Riesz transforms on connected sums

We investigate the boundness of the Riesz transform on $L^p$ for connected sum of manifolds where the Riesz transform is bounded on $L^p$