arXiv research

Simple proof for Cauchy's surface area formula.

Generalized Cauchy's surface area formula to arbitrary submanifolds in R^n.

Formula for Heisenberg group surface areas derived.

Formula calculates index for CR operators on surfaces with boundary punctures.

In 3-d the average projected area of a convex solid is 1/4 the surface area, as Cauchy showed in the 19th century. In general, the ratio in n dimensions may be obtained from Cauchy's surface area formula, which is in turn a special case of Kubota's theorem. However, while these latter results are well-known to those wo…

We address the geometric Cauchy problem for surfaces associated to the membrane shape equation describing equilibrium configurations of vesicles formed by lipid bilayers. This is the Euler-Lagrange equation of the Canham-Helfrich-Evans elastic curvature energy subject to constraints on the enclosed volume and the surfa…

The article develops a mathematical theory for uniformly distributing points on surfaces.

This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof makes use of an existence result for marginal surfaces, in the presence of barriers…

Paper provides a formula for translating solitons and singular minimal surfaces.

Paper discusses existence of CMC surfaces in cosmological spacetimes.

Minimal surfaces connect to horizons and electrostatic systems.

New Steiner formula for $L_p$ affine surface area in Minkowski theory.

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy-Riemann equation. A lot of classical results in Rie…

The paper proves formulas for capillary surfaces and applies them to inequalities and area estimates.

Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.

Solves Minkowski problem for affine invariant convex domains.

Study Schiffer operators on Riemann surfaces, linking conformal and topological invariants.

New $L_p$-Steiner quermassintegrals defined from Steiner formula.

We develop two types of integral formulas for the perimeter of a convex body K in planar geometries. We derive Cauchy-type formulas for perimeter in planar Hilbert geometries. Specializing to H^2 we get a formula that appears to be new. We show that it implies the standard Cauchy-Santalo formula involving a central ang…

The paper estimates area and volume for spacetimes with integral mean curvature bounds.

Study proves Chern-Osserman type equality for surfaces in Euclidean space.

New weighted surface area measures for convex bodies with applications.

New cosmological spacetimes without CMC Cauchy surfaces found.

We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator fo…

The paper proves a statement about surfaces diffeomorphic to annuli.

Formula for renormalized area of hypersurfaces in hyperbolic spaces.

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-spac…

In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $p\neq -n$. Our definitions are motivated from an equivalent formula for the mixed $p$-affine surface area. Some properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are prove…

To every Darboux integrable system there is an associated Lie group $G$ which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial differential equation can be reduced to solving an equation of Lie type for the …

Generalized Gauss-Bonnet formula for conical metrics on compact Riemann surfaces.

Consider a random smooth Gaussian field $G(x):F\to\mathbb{R}$, where $F$ is a compact in $\mathbb{R}^d$. We derive a formula for average area of a surface generated by the equation $G(x)=0$ and give some applications. As an auxiliary result we obtain an integral expression for area of a surface induced by zeros of a \e…

The paper extends Pappus-Guldin theorems to 3D-Heisenberg group surfaces.

My main results are simple formulas for the surface area of d-dimensional lattice polytopes using Ehrhart theory.

Researchers study horospherical transforms on hyperboloids.

Solves a Cauchy problem for minimal spacelike surfaces in 4D spacetime.

Researchers prove Fredholm property for Dirac operator on specific spacetimes.

The paper finds transformation formulas for quaternionic complex structures.

We give a local integral formula, valid on general curved space-times, for the characteristic Cauchy problem for the Dirac equation with arbitrary spin using the method developed by Friedlander in his book "the wave equation on a curved spacetime" (1975). The results obtained by Penrose in the flat case in "Null hypers…

We establish a Cauchy type inequality for the geometric intersection number between two 1-dimensional submanifolds in a surface. Some of the basic results in Thurston's theory of measured laminations on surfaces are derived from the Cauchy inequality.

Generalizes Gauss-Bonnet to metrics with logarithmic singularities.

Proves symmetry in vacuum spacetimes with compact Cauchy horizons.

For arbitrary quantizable compact Kaehler manifolds, relations between the geometry given by the coherent states based on the manifold and the algebraic (projective) geometry realised via the coherent state mapping into projective space, are studied. Polar divisors, formulas relating the scalar products of coherent vec…

Revisits contact measures in isotropic spaces, linking kinematic formulas to curved spaces.

Anti-de Sitter spacetimes embed cone-metrics as bent Cauchy surfaces.

We prove several global existence theorems for spacetimes with toroidal or hyperbolic symmetry with respect to a geometrically defined time. More specifically, we prove that generically, the maximal Cauchy development of $T^2$-symmetric initial data with positive cosmological constant $Λ>0$, in the vacuum or with Vlaso…

Continuous family of elliptic operators' projections maintain Cauchy data spaces.

New vacuum spacetimes without CMC Cauchy surfaces found.

Researchers derive expressions for metric perturbations of extremal surfaces.