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.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.
problem Characterizing properties of ball-convex shapes.
method Introducing illumination bodies and weighted illumination bodies, proving convexity, and establishing surface area measures.
result Illumination bodies are convex and provide surface area measures for ball-convex shapes.
Entropy study of geodesic flow on convex projective surfaces.
problem Entropy of Sinai-Ruelle-Bowen measure on convex projective surfaces.
method Analysis of Hilbert area and Blaschke metric.
result Entropy tends to zero if and only if the Hilbert area tends to infinity.
The paper defines surface area for graphs and derives spectral estimates.
problem Understanding connectivity measures and spectral properties of graphs.
method Introducing surface area concepts related to inverse degree and deriving spectral bounds.
result An upper bound on the second eigenvalue for planar graphs.
Minimal surfaces and average area ratio found to be maximized by hyperbolic metrics.
problem Finding sharp relations between minimal surface entropy and average area ratio.
method Ricci flow with surgery and invariant measures.
result Minimal surface entropy maximized by hyperbolic metrics among metrics with scalar curvature ≥ -6.
Consider a closed marked flat surface S of genus g≥2 and area 1 and its universal covering S~. We show that the measure class of the Hausdorff measure of the Gromov boundary of S~ uniquely determines S.
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…
Given a measured lamination on a finite area hyperbolic surface we consider a natural measure Mon the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure M gives summation id…
Study shows limits of Fuchsian surfaces in hyperbolic 3-manifolds.
problem Understanding the limits of Fuchsian surfaces in hyperbolic 3-manifolds.
method Analyzing asymptotically Fuchsian maps and their induced probability area measures.
result Weak-* limits of induced area measures are convex combinations of Haar and totally geodesic surface measures.
The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Fefferman's invariant hypersurface measure.
We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes account of area, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces. The BV and measure properties of functions with finite relaxed energy are studied…
Study on Lp affine surface areas and their inequalities for convex bodies.
problem Understanding weighted Lp affine surface areas in convex bodies. method Investigating valuations, isoperimetric inequalities, and connections to f divergences. result Established isoperimetric inequalities for weighted Lp affine surface areas. Explains the history and challenges of minimal surfaces.
problem Understanding the regularity of minimal surfaces.
method Historical overview and technical analysis.
result Outlines the evolution and current state of minimal surfaces.
For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…
We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space. Our main re…
Loewner inequality proven for curved surfaces.
problem Proving Loewner's inequality for nonpositively curved surfaces.
method Combining Gauss-Bonnet formula with averaging argument using geodesic flow invariance.
result Found a disk with large total curvature around its center, leading to large area.
We show that for any weakly convergent sequence of ergodic SL2(R)-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech constants converge to the Siegel-Veech constant of the limit measure. Together with a measure equidistribution result due to Eskin-M…
Paper solves capillary Orlicz-Minkowski problem with new inequalities.
problem Finding capillary convex bodies with prescribed Orlicz surface area measures.
method Continuity method and inequalities to solve the capillary even Orlicz-Minkowski problem.
result Volume-normalized smooth solutions and inequalities established.
Stationary measures on hyperbolic surfaces with cusps are singular and stable under quasi-symmetries.
problem Understanding stationary measures on hyperbolic surfaces with cusps.
method Analyzing exponential decay of cusp excursions and proving quasi-symmetry stability.
result Stationary measures on hyperbolic surfaces with cusps are quasi-symmetrically stable and singular.
Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.
problem Investigating curvature measures in spherical, hyperbolic, and de Sitter spaces.
method Establishing a unifying framework for curvature measures in real-analytic spaces of constant curvature.
result Floating bodies and duality in non-Euclidean spaces are connected to curvature measures in Euclidean space.
Study spherical convex bodies using Lp-floating areas and curvature entropy.
problem Analogous isoperimetric inequalities for spherical convex bodies.
method Introduced Lp-floating areas and curvature entropy for spherical convex bodies. result Established isoperimetric inequalities and dual isoperimetric inequalities.
The paper proves that symmetric sets with minimal Gaussian surface area are nearly convex cylinders.
problem Finding the shape of symmetric sets with minimal Gaussian surface area.
method Analyzing the boundary of symmetric sets and applying isoperimetric inequalities.
result Symmetric sets with minimal Gaussian surface area are nearly convex cylinders.
We prove the existence of a continuous BV minimizer with C0 boundary value for the p-area (pseudohermitian or horizontal area) in a parabolically convex bounded domain. We extend the domain of the area functional from BV functions to vector-valued measures. Our main purpose is to study the first and second v…
We introduce a natural stratification of the space of projective classes of measured laminations on a complete hyperbolic surface of finite area. We prove a rigidity result, namely, the group of self-homeomorphisms of the space of projective measured laminations that preserve such a stratification is in general identif…
Solves a long-standing convex geometry problem about mixed volumes.
problem Characterizing the support of mixed area measures.
method Geometric approach to convex bodies in R^n and R^3.
result Resolved one direction of Schneider's conjecture for arbitrary convex bodies.
In this paper, we introduce the Lp geominimal surface area for all −n=p<1, which extends the classical geominimal surface area (p=1) by Petty and the Lp geominimal surface area by Lutwak (p>1). Our extension of the Lp geominimal surface area is motivated by recent work on the extension of the Lp a…
Overview of affine surface area and its history.
problem None explicitly stated; focuses on overview.
method None explicitly stated; focuses on overview.
result None explicitly stated; focuses on overview.
New findings show that not all area-minimizing surfaces are calibrated, even on complex manifolds.
problem Understanding when area-minimizing surfaces cannot be calibrated.
method Analyzing homology classes and metrics on manifolds to determine if area-minimizers are calibrated.
result Calibrated area-minimizers are non-generic, challenging the common assumption that they are typical.
The paper connects least area surfaces to quasi-normal surfaces in 3-manifolds.
problem Understanding the properties of least area surfaces in 3-manifolds.
method Introducing quasi-normal surfaces and showing their relationship to least area surfaces in fine triangulations.
result Least area surfaces in 3-manifolds are quasi-normal with respect to fine triangulations, and this quasi-normality leads to piecewise flat approximations.
Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for Lφ affine surface areas are established.
New foliations found for critical surfaces of Hawking energy, resolving discrepancies.
problem Finding consistent critical surfaces for the Hawking energy in non-totally geodesic spacelike hypersurfaces.
method Constructing a unique local foliation of area constrained critical surfaces of the Hawking energy in the general case of non-totally geodesic spacelike hypersurfaces.
result Discrepancy found in the small sphere limit of the Hawking energy, explained and resolved.
Random translation surfaces converge to a Poisson plane as genus grows.
problem Understanding the geometric behavior of high genus translation surfaces.
method Proving convergence of random translation surfaces to a Poisson plane using statistical local geometric properties.
result The radius-r neighborhood of a random point in an MSV-distributed random translation surface converges in distribution to the radius r neighborhood of the root in a Poisson translation plane. Introduces new weighted floating functions and affine surface areas.
problem Developing new mathematical concepts for convex bodies.
method Introducing weighted floating functions and weighted functional affine surface areas.
result New relations to traditional and classical affine surface areas.
In this paper, we define a new metric structure on the shape space of a high genus surface. We introduce a rigorous definition of a shape of a surface and construct a metric based on two energies measuring the area distortion and the angle distortion of a quasiconformal homeomorphism. We show that the energy minimizer …
In this paper we introduce the stochastic Ricci flow (SRF) in two spatial dimensions. The flow is symmetric with respect to a measure induced by Liouville Conformal Field Theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on a flat torus, in the full…
Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.
problem Finding area constraints for surfaces with constant mean curvature in hyperbolic 3-manifolds.
method Established an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and genus bounds.
result Area bound implies compactness for such surfaces, with specific proportional bounds for Bryant surfaces.
We study an area minimization problem for spacelike zero mean curvature surfaces in four dimensional Lorentz-Minkowski space. The areas of these surfaces are compared of with the areas of certain marginally trapped surfaces having the same boundary values.
The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily …
Study on existence and structure of P-area surfaces in Heisenberg group.
problem Existence and structure of P-area minimizing surfaces in the Heisenberg group.
method Characterization of existence and structure using an underlying vector field N, proving existence even without satisfying boundary conditions, and applying Barrier condition.
result Existence of P-area minimizing surfaces under certain conditions, providing new understanding of the Heisenberg group.
Minimal surfaces in hyperbolic space have a renormalized area criterion.
problem Minimal surfaces in hyperbolic space
method Renormalized area criterion
result Y must be a totally geodesic disk
We consider a smooth closed surface M of fixed genus ⩾2 with a Riemannian metric g of negative curvature with fixed total area. The second author has shown that the topological entropy of geodesic flow for g is greater than or equal to the topological entropy for the metric of constant negative curvatu…
New geodesic rays found on hyperbolic surfaces with unique halo.
problem Existence of geodesic rays not asymptotic to lamination leaves.
method Analyzing intersections and lifts of measured geodesic laminations.
result Uncountable halo of geodesic endpoints disjoint from lamination leaves.
Classifies area-minimizing surfaces in R^4 as algebraic.
problem Classifying entire area-minimizing surfaces in R^4.
method Using quadratic area growth and holomorphic polynomials to cut out surfaces.
result Entire 2-dimensional area-minimizing or stable surfaces in R^4 are algebraic.
Minimal surfaces in hyperbolic space have a sharp area bound.
problem Bounding the renormalized area of minimal surfaces.
method Proving an inequality using conformal length of ideal boundary.
result Sharp isoperimetric property of renormalized area.
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.
Minimal surfaces in a ball have limited area.
problem Bounding the area of genus zero minimal surfaces in a unit ball.
method Proving an area inequality and showing convergence of saturating sequences.
result The area of each nonflat surface is less than its radial projection, with sharp asymptotic bounds.
SRNF framework extends surface distance to Lipschitz surfaces.
problem Defining a distance metric for unparametrized surfaces.
method Square Root Normal Fields (SRNF) and Wasserstein Fisher Rao (WFR) metric.
result SRNF distance on Lipschitz surfaces is equivalent to WFR metric.