Related papers: A mean curvature flow method for numerical cosmolo…
Currently, a method has been developed for the cosmological inflation model with a single scalar field to calculate cosmological parameters such as the power spectrum of scalar and tensor perturbations, their spectral indices, and the…
Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A corollary is that any area-decreasing map…
We consider a class of anisotropic curvature flows called a crystalline curvature flow. We present a survey on this class of flows with special emphasis on the well-posedness of its initial value problem.
In this paper we will discuss how one may be able to use mean curvature flow to tackle some of the central problems in topology in 4-dimensions. We will be concerned with smooth closed 4-manifolds that can be smoothly embedded as a…
We review the study of inhomogeneous perturbations about a homogeneous and isotropic background cosmology. We adopt a coordinate based approach, but give geometrical interpretations of metric perturbations in terms of the expansion, shear…
We investigate how various inflationary and bouncing cosmologies can be realized by imperfect fluids with a generalized equation of state, in the context of both classical and loop quantum cosmology. With regards to the inflationary…
We revisit how super-Hubble cosmological fluctuations induce, at any time in the cosmic history, a non-vanishing spatial curvature of the local background metric. The random nature of these fluctuations promotes the curvature density…
In clusters of galaxies, the specific entropy of intracluster plasma increases outwards. Nevertheless, a number of recent studies have shown that the intracluster medium is subject to buoyancy instabilities due to the effects of cosmic rays…
In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions…
We consider a diffused interface version of the volume-preserving mean curvature flow in the Euclidean space, and prove, in every dimension and under natural assumptions on the initial datum, exponential convergence towards single "diffused…
We consider the evolution of a $n$-dimensional convex hypersurface in the euclidean space under mean curvature flow with densities $e^{\varepsilon \frac12 n\mu^2 |x|^2}$, $\varepsilon =\pm 1$, and completely determine it depending on the…
We find initial data for numerical relativity simulations of inhomogeneous cosmologies. This involves treating an exceptional case of the general relativity constraint equations. We devise analytic and numerical methods to treat this…
We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a $n$-dimensional smooth compact manifold with boundary, where $n \geq 3$, assume the conformal…
The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation…
In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a…
Measurements of CMB anisotropy and, more recently, polarization have played a very important role allowing precise determination of various parameters of the `standard' cosmological model. The expectation of the paradigm of inflation and…
We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in $\mathbb R^n$. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the…
In these lectures I review the standard hot big-bang cosmology, emphasizing its successes, its shortcomings, and its major challenges-developing a detailed understanding of the formation of structure in the Universe and identifying the…
Average properties of general inhomogeneous cosmological models are discussed in the Newtonian framework. It is shown under which circumstances the average flow reduces to a member of the standard Friedmann--Lema\^\i tre cosmologies.…
We study the mean curvature flow of smooth $m$-dimensional compact submanifolds with quadratic pinching in the Riemannian manifold $\mathbb{C}P^n$. Our main focus is on the case of high codimension, $k\geq 2$. We establish a codimension…