Related papers: Geometric Analysis and General Relativity
In a recent paper [D. Philipp, V. Perlick, D. Puetzfeld, E. Hackmann, and C. L\"ammerzahl, Phys. Rev. D 95, 104037 (2017)], a definition of a general-relativistic geoid, restricted to stationary spacetimes, was presented in terms of…
This thesis concerns the split of Einstein's field equations (EFE's) with respect to nowhere null hypersurfaces. Areas covered include A) the foundations of relativity, deriving geometrodynamics from relational first principles and showing…
In this paper I present a kind of proof for classical Euclidean geometric problems which relies on both synthetic and analytic geometry. Using the elementary tools of polynomial algebra and multivariate calculus we manage to reduce the…
The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the stan- dard parametric form, but it can also be in the implicit form which is commonly used in…
Gravitational lensing has developed into one of the most powerful tools for the analysis of the dark universe. This review summarises the theory of gravitational lensing, its main current applications and representative results achieved so…
We obtain a series of results in the global theory of free boundary minimal surfaces, which in particular provide a rather complete picture for the way different complexity criteria, such as area, topology and Morse index compare, beyond…
This is a brief review where some basic elements of non-commutative geometry are given. The rules and ingredients that enter in the construction of the standard model and grand unification models in non-commutative geometry are summarized.…
We explore the concentration properties of the ratio between the geometric mean and the arithmetic mean, showing that for certain sequences of weights one does obtain concentration, around a value that depends on the sequence.
In this papaer, we put forward some new definitions and integral inequalities by using fairly elementary analysis.
The null-surface formulation of general relativity -- recently introduced -- provides novel tools for describing the gravitational field, as well as a fresh physical way of viewing it. The new formulation provides ``local'' observables…
Geometric complexity theory (GCT) is an approach to the $P$ vs. $NP$ and related problems through algebraic geometry and representation theory. This article gives a high-level exposition of the basic plan of GCT based on the principle,…
The geometric foundations of General Relativity are revisited, with particular attention to its gauge invariance, as a key to understanding the true nature of spacetime. Beyond the common image of spacetime as a deformable 'fabric' filling…
We define a new type of valuation of a ring that combines the notion of Krull valuation with that of multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations. This…
This article introduces the subject of quasi-local horizons at a level suitable for physics graduate students who have taken a first course on general relativity. It reviews properties of trapped surfaces and trapped regions in some simple…
Global hyperbolicity is a central concept in Mathematical Relativity. Here, we review the different approaches to this concept explaining both, classical approaches and recent results. The former includes Cauchy hypersurfaces, naked…
We present a description of two dimensional Yang-Mills gauge theory on the plane and on compact surfaces, examining the topological, geometric and probabilistic aspects.
We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify…
The analytic hyperbolic geometric viewpoint of Einstein's special theory of relativity is presented.
Trapped surface formation in general relativity can be studied through a coupled set of nonlinear equations, where various terms can be neglected, as was proved by a rigorous mathematical analysis of Christodoulou. This paper is devoted to…
Some basic geometric properties related to connectedness and topological dimension 0 are discussed, especially in connection with the ultrametric version of the triangle inequality.