We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapion…
The paper studies properties of RCD(K,N) spaces and their boundaries.
problem Understanding the boundary structure and unit normal on RCD(K,N) spaces. method Proves concentration of boundary measure, discusses localization of unit normal, and develops tools for perimeter minimizers.
result Proves that the boundary measure of sets with finite perimeter is concentrated on the n-regular set Rn. We show that among sets of finite perimeter balls are the only volume-constrained critical points of the perimeter functional.
This note is devoted to the study of sets of finite perimeter over RCD(K,N) metric measure spaces. Its aim is to complete the picture about the generalization of De Giorgi's theorem within this framework. Starting from the results of [2] we obtain uniqueness of tangents and rectifiability for the reduced boundary of …
Given an elliptic integrand of class C3, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.
Paper proves isoperimetric inequality for Minkowski spacetime.
problem Maximizing volume of domain of dependence for finite lightcone.
method Lorentz polarisation to study variational problem.
result Maximal volume achieved by spacelike hyperplane truncated finite lightcone.
Study minimizers of quasi-perimeters in RCD spaces with volume constraints.
problem Regularity and topological properties of volume constrained minimizers in RCD spaces.
method New Deformation Lemma and study of interior and exterior points.
result Volume constrained minimizers are open bounded sets with Ahlfors regular boundary.
Study proves rigidity of critical points in hydrophobic capillary systems.
problem Rigidity of critical points in hydrophobic capillary systems.
method Proves rigidity among sets of finite perimeter in the half space, extending to full hydrophobic regime.
result Rigidity of critical points proven in hydrophobic capillary systems.
Generic level sets in mean curvature flow are BV solutions.
problem Understanding the behavior of level sets in mean curvature flow.
method Using the framework of sets of finite perimeter and distributional solutions, the paper extends Evans and Spruck's work.
result Generic level sets are distributional solutions with optimal energy dissipation rate.
In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is, possibly, weaker than the one introduced by Franchi, Serapioni, and Serra Cassano. Namely, we consider subsets Γ that, similarly…
We prove that the results regarding the Isoperimetric inequality and Cheeger constant formulated in terms of the Minkowski content, obtained by the authors in previous papers in the framework of essentially non-branching metric measure spaces verifying the local curvature dimension condition, also hold in the stronger …
Minimal partitions with minimal perimeter found in metric spaces.
problem Finding minimal partitions with minimal perimeter in metric spaces.
method Existence proof and regularity analysis of minimal domains.
result Existence and regularity of minimal partitions in various metric spaces.
The paper classifies energy-minimizing sets in specific domains.
problem Classifying volume-constraint local energy-minimizing sets.
method Proved a Poincaré-type inequality for stable sets.
result Relative boundary of energy-minimizing sets is smooth.
Infinite hyperbolic manifolds share same perimeter-to-volume ratio.
problem Finding hyperbolic manifolds with a fixed perimeter-to-volume ratio.
method Constructing infinitely many hyperbolic manifolds with nonempty boundaries.
result Existence of incommensurable hyperbolic manifolds with a fixed perimeter-to-volume ratio.
We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms Gp:TpM→[0,∞] are given. When we consider sub-Riemannian manifolds, our definition coincide with the one given in the more general context of metric measure spaces which are doub…
Given a positive lower semi-continuous density f on R2 the weighted volume Vf:=fL2 is defined on the L2-measurable sets in R2. The f-weighted perimeter of a set of finite perimeter E in R2 is written Pf(E). We study minimisers for the weighted isop…
Study shows limits of volume-constrained sets are finite unions of Wulff shapes.
problem Analyzing the behavior of sets with degenerating ellipticity.
method Proving rigidity of L1-accumulation points of volume-constrained almost-critical sets. result Limits of volume-constrained sets are finite unions of φ-Wulff shapes. Study fractional perimeter asymptotics on Riemannian manifolds as s approaches 0.
problem Asymptotics of fractional perimeter on Riemannian manifolds.
method Analysis of fractional Laplacian and existence of bounded harmonic functions.
result Asymptotics of fractional s-perimeter on all complete manifolds. This paper characterizes semigenerated Carnot groups and applies it to rectifiability of perimeter sets.
problem Characterizing semigenerated Carnot groups and their applications to rectifiability.
method Algebraic approach focusing on semigroup generation and Engel-type quotients.
result Complete characterization of semigeneration in Carnot groups of step 3 and sufficient criteria for semigeneration in Carnot groups of arbitrary step.
Paper finds formulas for minimizing perimeter in special spaces, proving key dimensions and existence.
problem Understanding the structure of perimeter minimizing sets in specific metric spaces.
method Established a monotonicity formula and proved rigidity for perimeter minimizers in RCD(0,N) spaces.
result Sharp Hausdorff dimension estimates for singular strata and existence of blow-down cones.
A lens cluster minimizes perimeter in the plane with given area constraints.
problem Minimizing perimeter in the plane with given area constraints.
method Analyzing lens clusters consisting of circular arcs with specific geometric properties.
result Lens clusters are local minimizers of the total perimeter functional.
The paper examines perimeter minimizing sets in curved spaces and finds conditions for their boundary to match a specific structure.
problem Conditions for perimeter minimizing sets in curved spaces to have a boundary matching a product structure.
method Analyzes Riemannian manifolds with non-negative sectional curvature and quadratic volume growth.
result The boundary of a perimeter minimizing set in such manifolds is identified with a slice in the product structure.
Proof shows smooth minimal hypersurfaces for perimeter-minimizing sets in low-dimensional Riemannian manifolds.
problem Finding sets of least perimeter in Riemannian manifolds.
method Short proof using de Giorgi and Miranda's tradition in flat space.
result Reduced boundary of least perimeter sets is a smooth minimal hypersurface in low dimensions.
Perimeter minimizers in curved spaces have a singular set no more than 5 dimensions.
problem Understanding the structure of minimizers in spaces with bounded Ricci curvature.
method Analysis of non-collapsed Ricci limit spaces with two-sided curvature bounds.
result The Hausdorff dimension of the singular set is at most \(N-5\).
Let D be a closed disk centered at the origin in the horizontal hyperplane {t=0} of the sub-Riemannian Heisenberg group $\hh^n$, and C the vertical cylinder over D. We prove that any finite perimeter set E such that D⊂E⊂C has perimeter larger than or equal to the one of the rotationally symm…
Flow preserves volume on flat torus, converging to stable set.
problem Volume preservation in discrete mean curvature flow on flat torus.
method Discrete mean curvature flow, quantitative Alexandrov estimate, characterization in 2D.
result Flow converges exponentially fast to stable set.
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.
Geometric proof shows regularity of anisotropic minimal surfaces in 2D.
problem Regularity of anisotropic minimal surfaces in 2D.
method Geometric proof using surface energy and strict convexity.
result All anisotropic surface minimizers in 2D are locally disjoint unions of line segments.
Study on droplet flow on uneven surfaces, proving existence and properties.
problem Understanding droplet movement on irregular surfaces.
method Existence of smooth flow and 1/2-Hölder continuous minimizing movement solutions.
result Properties of minimizing movements including comparison principles and uniform boundedness.
This study proves the Half Space Property for RCD(0,N) and RCD(K,N) spaces.
problem Proving the Half Space Property for RCD(K,N) spaces.
method Analyzing locally perimeter minimizing sets and extending Green's functions results.
result The Half Space Property holds for RCD(K,N) spaces under specific conditions.
Study finds critical points in perimeter functional for fixed volume sets.
problem Finding critical points in perimeter functional for sets of fixed volume.
method Utilizes Mazurwoski--Zhou techniques and new Cacciopoli set connectedness results.
result Constructs smooth almost embedded hypersurfaces with non-zero constant mean curvature.
The study proves properties of optimizers for sets maximizing perimeter under fixed volume constraints.
problem Existence and properties of bounded convex sets in Riemannian manifolds maximizing perimeter under fixed volume constraints.
method Analyzes the properties of optimizers for sets maximizing perimeter under fixed volume constraints in Euclidean, spherical, and hyperbolic spaces.
result Proves that there are no C2-maximisers of perimeter with prescribed volume and that the smallest principal curvature is constant in regions where the set is of class C2. Study non-local isoperimetric energies on spheres using a Riemannian autocorrelation function.
problem Analyse non-local isoperimetric energies on spheres.
method Introduce Riemannian autocorrelation function and use it to reformulate and compute the energies.
result Show that the non-local isoperimetric energies can be reformulated and computed using the Riemannian autocorrelation function.
Geodesic spheres in certain symmetric spaces are quantitatively stable under small perturbations.
problem Stability of geodesic spheres in symmetric spaces under perturbations.
method Quantitative stability analysis using spectral gap of the Laplacian on geodesic spheres.
result Geodesic spheres are uniformly stable with respect to small C1-volume preserving perturbations. We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the BV fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regul…
A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to whic…
Study shows how flat flow solutions in 2D converge to disks.
problem Understanding the asymptotics of area-preserving mean curvature flow in 2D.
method Analyzes flat flow solutions starting from bounded sets of finite perimeter.
result Flat flow solutions converge to a union of equally sized disks with exponential rate.
We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite sub-Riemannian perimeter. The purpose of this paper is threefold. First, we prove some mild regularity and structural results in arbitrary Carnot groups. Namely, we show that for every co…
Hexagonal tilings minimize perimeter with unequal volumes.
problem Finding optimal tessellations with unequal cell volumes.
method Minimizing perimeter functionals for different classes of problems.
result Hexagonal tilings are optimal among partitions with almost equal areas.
Study optimizes perimeter in convex domains with anisotropic constraints.
problem Optimizing perimeter in convex domains with anisotropic constraints.
method Analytical properties, topological features, and geometric measure theory results.
result Sharp isoperimetric inequalities and existence of minimizers.
This work is an investigation of perimeter measures in the metric measure space given by the Heisenberg group with Haar measure and a Carnot-Carathéodory metric, which is in general a sub-Finsler metric. Included is a reduction of Minkowski content in any CC-metric to an integral formula in terms of Lebesgue surface ar…
We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface ∂Ω0, we show that there exists a weak solution to the null mean curvatu…
The paper studies isoperimetric inequalities on warped product manifolds.
problem Investigating isoperimetric inequalities on warped product manifolds.
method Exploiting the interplay between isoperimetric inequalities and warped product structures, deriving necessary and sufficient conditions.
result Established a quantitative lower bound for the first nonzero Dirichlet eigenvalue of geodesic balls centered at the pole.
We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and relevant homomorphisms are also calculable in quadratic time. The algorithm also dec…
Study sharp inequalities for perimeter functionals in capillarity and convex cones.
problem Quantitative isoperimetric inequalities for perimeter functionals in capillarity and convex cones.
method Derivation of Fuglede-type estimates and application of selection principle.
result Sharp quantitative isoperimetric inequalities in strong and barycentric forms.
Perimeter on manifolds leads to new symmetrization methods.
problem Applying symmetrization methods to quasilinear elliptic problems on RN. method Generalization of perimeter to manifolds, using hear kernel regularization.
result New symmetrization method on spheres for quasilinear elliptic problems.
We study the localization of sets with constant nonlocal mean curvature and prescribed small volume in a bounded open set with smooth boundary, proving that they are {\em sufficiently close} to critical points of a suitable non-local potential. We then consider the fractional perimeter in half-spaces. We prove the exis…
Locally isoperimetric partitions minimize perimeter in space.
problem Finding minimal perimeter partitions in space.
method Proving closure theorem to limit sequences of isoperimetric clusters.
result Examples of isoperimetric partitions in various dimensions.