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.
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.
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.
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 \(N-5\).
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.
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 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. 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.
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.
We prove that a family of entire intrinsic minimal graphs in the Heisenberg group are not perimeter minimizing.
Study fractional Allen-Cahn equation and nonlocal minimal surfaces, improving energy and perimeter estimates.
problem Properties of solutions to fractional Allen-Cahn equation and stationary nonlocal minimal surfaces.
method Quantitative stratification principle applied to fractional Allen-Cahn equation, leading to optimal estimates.
result Sharp potential energy and perimeter estimates for fractional Allen-Cahn equation and nonlocal minimal surfaces.
Almost perimeter-minimizing boundaries in plentiful groups can be approximated by Lipschitz graphs.
problem Regularity of boundaries in plentiful groups.
method Lipschitz approximation of boundaries.
result Boundary of almost minimizers can be approximated by intrinsic Lipschitz graphs.
An elementary proof found for the double bubble problem in a specific norm.
problem Finding the shapes that minimize perimeter for given volumes in a specific norm.
method Direct comparison to a small family of parameterized sets for analysis.
result Existence of minimizing sets for any volume ratio parameter.
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.
Study shows zero level sets of solutions to Allen-Cahn equation are minimal surfaces with zero mean curvature.
problem Understanding phase transitions through entire solutions of the Allen-Cahn equation.
method Proving minimality of the zero level set with respect to a perimeter functional with density and showing zero mean curvature.
result The zero level set of entire solutions of the Allen-Cahn equation has zero mean curvature and is minimal.
The generalized soap bubble problem seeks the least perimeter way to enclose and separate n given volumes in R^m. We study the possible configurations for perimeter minimizing bubble complexes enclosing more than two regions. We prove that perimeter minimizing planar bubble complexes with equal pressure regions and wit…
Hexagonal norm double bubble problem solved with minimal configurations.
problem Finding the optimal shapes for minimizing perimeter in hexagonal geometry.
method Elementary proof and geometric exclusions to simplify minimizer search.
result Existence of minimizing sets for volume ratio parameter α in (0,1].
We prove that, among all convex hyperbolic polygons with given angles, the perimeter is minimized by the unique polygon with an inscribed circle. The proof relies on work of J.-M.\ Schlenker.
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.
We address the double bubble problem for the anisotropic Grushin perimeter Pα, α≥0, and the Lebesgue measure in R2, in the case of two equal volumes. We assume that the contact interface between the bubbles lays on either the vertical or the horizontal axis. Since no regularity theory is available i…
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.
Study proves existence of regions minimizing perimeter in specific geometric structures.
problem Existence of isoperimetric regions in sub-Finsler nilpotent groups.
method Analyzes nilpotent Lie groups with a bracket-generating distribution and asymmetric norms.
result Proves existence of minimizers of perimeter under volume constraint.
Investigates convexity of minimizers under mass constraint using nonlocal perimeter and potential.
problem Convexity of minimizers under mass constraint.
method Nonlocal free energy with nonlocal perimeter and convex potential.
result Quantitative stability theorem for nonlocal free energy assuming symmetry on the potential.
New mathematical surfaces without boundaries found.
problem Existence of nonlocal free boundary minimal surfaces.
method Fractional perimeter critical points with invariant boundary.
result Existence of nonlocal free boundary minimal surfaces without boundaries.
We prove the existence of a perimeter-minimizing partition of R^n into regions of unit volume. We conclude with a short tribute to the late Manuel A. Fortes.
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.
This paper solves a variation of the isoperimetric problem in higher dimensions.
problem Minimizing a weighted perimeter functional with given half-space volumes.
method Introduced a weighted perimeter functional with three weights.
result Characterized two types of minimizers made of spherical domes.
New geometric insights reveal properties of adversarial training problems.
problem Adversarial training in binary classification.
method Equivalence with regularized risk minimization and convex relaxations.
result Existence of minimal and maximal solutions, and regular solutions.
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…
The paper proves isoperimetric regions on Riemannian manifolds with Ricci bounded below.
problem Proving the existence of isoperimetric regions in Riemannian manifolds.
method Gromov-Hausdorff asymptotic analysis to study perimeter-minimizing sequences.
result Existence of isoperimetric regions in noncollapsed Riemannian manifolds with Ricci curvature bound.
The study examines stable regions in weighted manifolds with boundary properties.
problem Studying stable regions in weighted manifolds with boundary properties.
method Using deformations constructed from parallel vector fields tangent to the boundary, the study deduces rigidity properties for stable sets.
result The classification of stable sets in some Riemannian cylinders and uniqueness results for minimizers.
We prove that, among the polygons in a punctured disc with fixed angles, the perimeter is minimized by the polygon with an inscribed horocycle centered at the puncture. We generalize this to a disc with a cone point and to an annulus with a geodesic boundary component and a complete end. Then we apply this result to de…
The study proves a unique perimeter minimizer for area in simply connected space forms and proves the isoperimetric inequality.
problem Finding the perimeter minimizer for area in simply connected space forms.
method Uniform geometric proof for all three space forms.
result There exists a unique perimeter minimizer for area in Mκ2 and it is a circle. We consider isotropic non lower semicontinuous weighted perimeter functionals defined on partitions of domains in Rn. Besides identifying a condition on the structure of the domain which ensures the existence of minimizing configurations, we describe the structure of such minima, as well as their regularity…
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.
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 consider the configuration space of planar n-gons with fixed perimeter, which is diffeomorphic to the complex projective space CPn−2. The oriented area function has the minimal number of critical points on the configuration space. We describe its critical points (these are regular stars) and compute …
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…
We show that among sets of finite perimeter balls are the only volume-constrained critical points of the perimeter functional.
Study nonlocal minimal surfaces for minimal surfaces in 3D manifolds.
problem Existence and regularity of minimal surfaces in 3D manifolds.
method Min-max methods, fractional perimeters, uniform estimates.
result Uniform estimates for min-max s-minimal surfaces in 3-manifolds, convergence to smooth minimal surfaces. Here a new notion of fractional length of a smooth curve, which depends on a parameter σ, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length u…
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 solves the isoperimetric problem for Heisenberg group norms.
problem Solving the isoperimetric problem for anisotropic norms in the Heisenberg group.
method Representation formula for perimeter, foliation property, differential equation characterization, approximation procedure.
result Characterization of isoperimetric sets as sub-Finsler analogues of Pansu's bubbles.
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. We prove some stability results for smooth H-minimal hypersurfaces immersed in a sub-Riemannian k-step Carnot group G. The main tools are the formulas for the 1st and 2nd variation of the H-perimeter measure.
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 …
Study on capillarity minimizers with nonlocal repulsion and gravity, proving existence and nonexistence.
problem Minimizing an energy functional with capillarity, nonlocal repulsion, and gravity.
method Quantitative isoperimetric inequalities applied to capillarity problem in a half-space.
result Existence and nonexistence of minimizers for various nonlocal kernels and masses.