Seungsu Hwang

Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature

Abstract: It is well known that critical points of the total scalar curvature functional ? on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of ? is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also Einstein or isometric to a standard sphere. In this paper we prove that n-dimensional critical points have vanishing n− 1 homology under a lower Ricci curvature bound for dimension less than 8.

Relative equilibria of point vortices on the hyperbolic sphere

Relative equilibria of point vortices on a hyperbolic sphere with constant negative curvature is formulated and considered. Known symmetries for the vortex motion on the hyperbolic sphere are expounded by setting up a geometrical formulation of dynamics. In particular, the configuration of rings of vortices is investigated in detail.

CRITICAL POINT METRICS OF THE TOTAL SCALAR CURVATURE

In this paper, we deal with a critical point metric of the total scalar curvature on a compact manifold M. We prove that if the criti- cal point metric has parallel Ricci tensor, then the manifold is isometric to a standard sphere. Moreover, we show that if an n-dimensional Rie- mannian manifold is a warped product, or has harmonic curvature with non-parallel Ricci tensor, then it cannot be a critical point metric.

Fuzzy Lipschitz maps and fixed point theorems in fuzzy metric spaces

In this paper, we introduce the notion of dilation and fuzzy Lipschitz of a map from a fuzzy metric space into a fuzzy metric space and we prove continuity properties for such maps. We also define the notion of the fuzzy Lipschitz distance between two fuzzy metric spaces and show that two compact fuzzy metric spaces whose Lipschitz distance is zero is fuzzy isometric to each other. On the other hand, we introduce the concept of minimal slope of a map between fuzzy metric spaces, which is defined by the ratio of two fuzzy metrics and derive some properties on it and relations with the dilation. In particular, we show that if the dilation of a map from a fuzzy metric space which is complete in George and Veeramani sense into itself is less than the minimal slope, then the map must have a fixed point. In case that a fuzzy metric space is considered in the sense of Kramosil and Michalek and that the completeness in the sense of Grabiec, the same result holds.

CRITICAL POINTS AND WARPED PRODUCT METRICS

It has been conjectured that, on a compact orient able manifold M, a critical point of the total scalar curvature functional restricted the space of unit volume metrics of constant scalar curvature is Einstein. In this paper we show that if a manifold is a 3-dimensional warped product, then (M, g) cannot be a critical point unless it is isometric to the standard sphere.

Bach-flat h-almost gradient Ricci solitons

On an

THREE DIMENSIONAL CRITICAL POINT OF THE TOTAL SCALAR CURVATURE

n

A note on the circle actions on Einstein manifolds

-dimensional complete manifold

THE CRITICAL POINT EQUATION ON A FOUR DIMENSIONAL WARPED PRODUCT MANIFOLD

M

ON THE SOLUTIONS OF THE (λ, n + m)-EINSTEIN EQUATION

, consider an