arXiv research

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

New formula for spherical polygon area via prequantization.

Researchers prove formulas for flag area measures, extending previous work.

Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.

We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.

New formulas limit minimal submanifolds' area in curved spaces.

Formula for Heisenberg group surface areas derived.

We present the area and coarea formulas for Lipschitz maps, valid for general volume densities. As applications, we give a short, "euclidean" proof of the anisotropic Sobolev inequality and describe an anisotropic tube formula for hypersurfaces in $mathbb{R}^n$. A discussion about the first variation of the anisotropic…

Study area and coarea formulas for graphs and submanifolds in Carnot groups.

New geometric measure simplifies complex analysis.

A formula for triangle area in Deep Sets form.

The coarea formula is proven for Heisenberg group maps, addressing open questions.

Paper derives formulas for static Einstein spaces, linking Neumann data to stability.

Formula for renormalized area of hypersurfaces in hyperbolic spaces.

The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the…

We study polygonal analogues of several moving boundary problems and their time discretization which preserves the constant area speed property. We establish various polygonal analogues of geometric formulas for moving boundaries and make use of the geometric formulas for our numerical scheme and its analysis of genera…

We prove that higher moment maps on area measures of a euclidean vector space are injective, while the kernel of the centroid map equals the image of the first variation map. Based on this, we introduce the space of smooth dual area measures on a finite-dimensional euclidean vector space and prove that it admits a natu…

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

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. R…

Formula derived for renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.

We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmi…

The paper proves a statement about surfaces diffeomorphic to annuli.

The paper explores connections between perimeter, area, and visual angle of convex sets.

We establish the equivalence of the Tuynman midpoint area formula for a spherical triangle to the classical area formulas of Euler and of Cagnoli. The derivation also yields a variant of the Cagnoli formula in terms of the medial triangle. We introduce the three barycentric coordinates of a point within the spherical t…

We revisit the contact measures introduced by Firey, and further developed by Schneider and Teufel, from the perspective of the theory of valuations on manifolds. This reveals a link between the kinematic formulas for area measures studied by Wannerer and the integral geometry of curved isotropic spaces. As an applicat…

Study uniformly differentiable graphs in Carnot groups, proving area formulas.

We prove an analogue of the classical Steiner formula for the $L_p$ affine surface area of a Minkowski outer parallel body for any real parameters $p$. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Stein…

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

Ancient formula connects volume forms and infinitesimal square volumes in manifolds.

New weighted surface area measures for convex bodies with applications.

The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.

This paper studies rectifiability in Carnot groups and proves geometric area formulas.

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…

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…

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

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…

In this paper we study areas (called p-areas) and volumes for parametric surfaces in the 3D-Heisenberg group $\mathbb{H}_1$, which is considered as a flat model of pseudo-hermitian manifolds. We derive the formulas of p-areas and volumes for parametric surfaces in $\mathbb{H}_1$ and show that the classical result of Pa…

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, $P$ can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or…

Study on spin random fields using chaos decomposition for cosmic microwave background modeling.

We establish an area-type formula for the intrinsic spherical Hausdorff measure of every regular curve embedded in an arbitrary graded group.

This note generalizes the visual angle to convex sets in 3D space.

We describe a conjectural formula via intersection numbers for the Masur-Veech volumes of strata of quadratic differentials with prescribed zero orders, and we prove the formula for the case when the zero orders are odd. For the principal strata of quadratic differentials with simple zeros, the formula reduces to compu…

We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric. We also construct explicitly some conical metrics whose curvature is not integrable.

Generalizes Gauss-Bonnet to metrics with logarithmic singularities.

A new curve flow preserves area and converges to a circle.

New theory for area of Legendrian surfaces, proving smoothness and variational results.

The paper shows how heat flow approximates area functional on specific geometric spaces.

The paper extends geometric inequalities from Euclidean space to Riemannian manifolds.