Related papers: Conformal Scalar Curvature Equations in Open Space…
We present several problems and results relating the scalar curvatures of manifolds with mean curvatures of their boundaries
We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.
We review the relation between scale and conformal symmetries in various models and dimensions. We present a dimensional reduction from relativistic to non-relativistic conformal dynamics.
It is known that some cosmological perturbations are conformal invariant. This facilitates the studies of perturbations within some gravitational theories alternative to general relativity, for example the scalar-tensor theory, because it…
This is a short essay about some fundamental results on scalar curvature and the two key methods that are used to establish them.
A space curve is determined by conformal arc-length, conformal curvature, and conformal torsion, up to M\"obius transformations. We use the spaces of osculating circles and spheres to give a conformally defined moving frame of a curve in…
We review recent compactness and non-compactness results for the Yamabe equation. We also discuss the asymptotic behavior of the parabolic Yamabe flow.
This article is a survey of results involving conformal deformation of Riemannian metrics and fully nonlinear equations.
We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete…
The conformal geometry of surfaces in the conformal space $\mathbf Q^n_1$ is studied. We classify the space-like surfaces in $\mathbf Q^n_1$ with vanishing conformal form up to conformal equivalence.
This article is a survey of recent results about scalar curvature and contractible open $3$-manifolds. It is dedicated to the memory of Professor S. S. Chern.
We explain how to derive largeness constraints in scalar curvature geometry using some basic splitting results and the potential theory on singular area minimizing hypersurfaces. This includes a variety of results like the non-existence of…
This paper investigates conformal deformations of the scalar curvature and mean curvature on complete Riemannian manifolds with boundary. We establish sufficient conditions for the existence of conformal deformations to complete metrics…
We collect a few guesses on possible implications of a lower bound on the scalar curvature of a Riemannian manifold on the size and shape of this manifold.
This is not in any way meant to be a complete survey on positive curvature. Rather it is a short essay on the fascinating changes in the landscape surrounding positive curvature. In particular, details and many results and references are…
Conformal invariance plays a significant role in many areas of Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an important role in geometry: theory of Riemannian…
We study solutions to conformally invariant equations with isolated singularties.
This is a brief introduction to the subject of Conformal Field Theory on surfaces with boundaries and crosscaps, which describes the perturbative expansion of open string theory.
The main intention of the paper is to investigate an osculating curve under the conformal map. We obtain a sufficient condition for the conformal invariance of an osculating curve. We also find an equivalent system of a geodesic curve under…
In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature…