Generalized Cauchy's surface area formula to arbitrary submanifolds in R^n.
problem Finding surface area of arbitrary submanifolds in R^n.
method Defining natural projected areas and volumes, deriving a recursive formula.
result Derived a new surface area formula that coincides with Crofton's and De Jong's formulas.
New formula for spherical polygon area via prequantization.
problem Traditional area formula for spherical polygons requires measuring angles.
method Uses prequantization to create a new formula that doesn't require angle measurement.
result New formula applicable to a wider range of degenerate curves and polygons.
Researchers prove formulas for flag area measures, extending previous work.
problem Proving additive kinematic formulas for flag area measures.
method Introducing an algebraic framework to compute these formulas explicitly.
result Existence and explicit computation of additive kinematic formulas for flag area measures.
Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
problem Calculating the renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
method Derives a Gauss-Bonnet formula for renormalized area in terms of scalar invariants and characterizes minimal hypersurfaces in terms of conformal geometry.
result Derives a formula expressing renormalized area in terms of integrals of scalar Riemannian invariants.
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.
problem Bounding minimal submanifolds' area in curved spaces.
method Developed new monotonicity formulae involving energy-like integrals over non-geodesic sets.
result Imply sharp area bounds for minimal submanifolds through a prescribed point.
Formula for Heisenberg group surface areas derived.
problem Deriving a formula for surface areas in Heisenberg groups.
method Analogy of Cauchy's surface area formula in Heisenberg groups.
result Formula for p-area of compact hypersurfaces in Heisenberg groups.
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 mathbbRn. A discussion about the first variation of the anisotropic…
Study area and coarea formulas for graphs and submanifolds in Carnot groups.
problem Understanding geometric properties of submanifolds in Carnot groups.
method Developed area and coarea formulas for CH1 intrinsic graphs and submanifolds. result Deduced density properties for Hausdorff measures and coarea formula for Carnot groups.
New geometric measure simplifies complex analysis.
problem Complex geometric analysis challenges.
method Geometric integration and convergence methods.
result Smallest measure satisfying Area Formula.
A formula for triangle area in Deep Sets form.
problem Finding a polynomial formula for triangle area in Deep Sets form.
method Expressing area as a permutation-invariant function and finding a suitable Deep Sets form.
result Explicit polynomial formula for triangle area in Deep Sets form.
The coarea formula is proven for Heisenberg group maps, addressing open questions.
problem Proving the coarea formula for Lipschitz maps from the Heisenberg group to Euclidean space.
method Introducing a new integral to define symplectic area of curves and proving convergence conditions.
result The coarea formula is established for CH1 maps from the Heisenberg group to R2n. Paper derives formulas for static Einstein spaces, linking Neumann data to stability.
problem Stability of conformally compact static spaces.
method First and second variation formulas for renormalized area.
result Negativity of Neumann data implies instability.
Formula for renormalized area of hypersurfaces in hyperbolic spaces.
problem Calculating the renormalized area of asymptotically minimal hypersurfaces in hyperbolic spaces.
method Combining Chen's conformal invariant quantity and Chern-Gauss-Bonnet formulas.
result Extension of renormalized area formulas to higher dimensions and non-minimal cases.
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 Lp-Steiner quermassintegrals defined from Steiner formula.
problem Defining new Lp-Steiner quermassintegrals. method Analogy to classical Steiner formula, investigating properties in convex bodies.
result Rotation and reflection invariant valuations in convex bodies.
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.
problem Calculating the renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
method Decomposition of extrinsic Q-curvature and application to renormalized area. result Renormalized area formula expressed as a linear combination of Euler characteristic and scalar conformal submanifold invariant.
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.
problem Proving a statement about surfaces diffeomorphic to annuli in Perelman's paper.
method Uses extrinsic techniques, co-area formula, and is potentially generalizable.
result Potential generalizability to higher dimensions.
The paper explores connections between perimeter, area, and visual angle of convex sets.
problem Understanding geometric properties of convex sets through visual angle and related measurements.
method Establishing universal formulas and characterizing convex sets of constant width.
result Crofton's formula is the unique universal formula relating visual angle, length, and area.
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.
problem Characterize uniformly differentiable intrinsic graphs in Carnot groups.
method Characterize uniform intrinsic differentiability via Hölder properties of projections of vector fields.
result Explicit area formula for uniformly intrinsically differentiable maps in Carnot groups.
We prove an analogue of the classical Steiner formula for the Lp 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.
problem Understanding capillary surfaces and their properties.
method Established monotonicity formulas for capillary surfaces in half-space and unit ball.
result Extended Li-Yau-type inequalities and optimal area estimates for capillary surfaces.
Ancient formula connects volume forms and infinitesimal square volumes in manifolds.
problem Relating volume forms and infinitesimal square volumes in Riemannian manifolds.
method Uses Heron's formula to link these concepts.
result Established a connection between volume forms and infinitesimal square volumes.
New weighted surface area measures for convex bodies with applications.
problem Generalizing surface area measures to weighted Borel measures.
method Formulating and analyzing weighted surface area measures, proving integral formula and Bézout-type inequality.
result New integral formula for mixed measure of three bodies, generalizing Bézout-type inequality.
The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.
problem Calculating area Siegel--Veech constants for affine invariant submanifolds of REL zero.
method Using volumes of the principal boundary strata and intersection theory.
result Proves a conjectural formula for the area Siegel--Veech constant in the case of REL zero.
This paper studies rectifiability in Carnot groups and proves geometric area formulas.
problem The study of rectifiability in Carnot groups and related geometric properties.
method Analysis of rectifiable measures in Carnot groups, geometric area formulas, and rectifiability of geodesic spheres.
result Geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in Carnot groups.
Consider a random smooth Gaussian field G(x):F→R, where F is a compact in Rd. 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 Lp geominimal surface areas for multiple convex bodies for all p=−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 Lp geominimal surface areas are prove…
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 of polygon spaces, characterizing critical points of area function.
problem Characterizing critical points of area function in polygon spaces.
method Geometric characterization of critical points and calculation of Morse indices.
result Generalization of isoperimetric theorems for polygons in the plane.
Study on spin random fields using chaos decomposition for cosmic microwave background modeling.
problem Modeling polarization of Cosmic Microwave Background using spin random fields.
method Explicit Wiener-Itô chaos decomposition of area measures of level sets.
result Reveals a clear difference between high frequency regime and zero spin case.
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.
problem Analyzing geometric properties of convex sets in 3D space.
method Generalizing the visual angle to convex sets in Euclidean space and expressing geometric quantities in terms of integrals of functions related to the solid angle.
result Invariant quantities of the original convex set can be expressed by integrals of functions related to the solid angle.
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.
problem Calculating curvature for metrics with singularities on compact surfaces.
method Proves a generalized Gauss-Bonnet formula under Lebesgue integrability condition.
result Establishes formula for special Kähler metrics with meromorphic cubic differentials.
A new curve flow preserves area and converges to a circle.
problem Preserving area in centro-equiaffine geometry.
method Fourth-order centro-equiaffine invariant curve flow via affine Minkowski formula.
result The flow preserves area and converges to a round circle.
Formula for Masur-Veech volumes in quadratic differentials with odd zeros.
problem Calculating volumes of specific quadratic differential strata.
method Intersection theory, topological recursion, Hodge integrals.
result Conjectural formula for volumes proved for odd zero orders.
The paper extends Pappus-Guldin theorems to 3D-Heisenberg group surfaces.
problem Extending classical theorems to a new geometric setting.
method Deriving formulas for p-areas and volumes in the Heisenberg group.
result Pappus-Guldin theorems hold for surfaces in the Heisenberg group.
New theory for area of Legendrian surfaces, proving smoothness and variational results.
problem Understanding the area of Legendrian surfaces under constraints.
method Introducing PHSLVs, proving sequential compactness, regularity, and variational results.
result Generalized regularity theory for Legendrian surfaces, achieving variational minima.
The paper shows how heat flow approximates area functional on specific geometric spaces.
problem Approximating the area functional on $\RCD(K,\infty)$ spaces.
method Using heat flow and properties of $\RCD(K,\infty)$ spaces.
result The area functional coincides with its relaxation in $\RCD(K,\infty)$ spaces.