In this paper, we introduce the geominimal surface area for all , which extends the classical geominimal surface area () by Petty and the geominimal surface area by Lutwak (). Our extension of the geominimal surface area is motivated by recent work on the extension of the a…
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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.
Affine surface areamath.DG
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.
General affine surface areasmath.MG
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 affine surface areas are established.
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.
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.
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
Finite area surfaces have cyclic hyperbolic Veech groups.
problem Understanding Veech groups of flat surfaces with finite area.
method Analyzing specific flat surfaces to find cyclic hyperbolic Veech groups.
result Finite area surfaces can have Veech groups that are infinite cyclic and hyperbolic.
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.
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 relative surface areas, proving invariance and inequalities, and using geometric interpretations.
result Established inequalities and a new notion of entropy for ball-convex bodies.
A bound on surface area in Riemannian manifolds without totally geodesic surfaces.
problem Bounding the area of surfaces in Riemannian manifolds without totally geodesic surfaces.
method Using extrinsic curvature energy to bound surface area.
result The area of any complete surface immersed into is bounded by a multiple of its extrinsic curvature energy.
A Sharp Isoperimetric Property of the Renormalized Area of a Minimal Surface in Hyperbolic Spacemath.DG
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.
The study finds large area minimal surfaces in certain manifolds.
problem Existence of minimal surfaces with large area in manifolds.
method Analyzes bumpy closed Riemannian n-manifolds for n between 3 and 7.
result Proves the existence of a sequence of minimal surfaces with unbounded area.
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.
Study estimates area of non-compact surfaces in 3-manifolds, proving rigidity under certain conditions.
problem Estimating area of non-compact -surfaces in 3-manifolds with negative curvature.
method Analyzes area estimates and rigidity conditions for -surfaces embedded in 3-manifolds of negative curvature.
result Proves rigidity for equality in area estimate under specific conditions, but provides a counterexample for minimal surfaces.
We study the classification of area-stationary and stable regular surfaces in the space of the rigid motions of the Minkowski plane E(1,1), equipped with its sub-Riemannian structure. We construct examples of area-stationary surfaces that are not foliated by sub-Riemannian geodesics. We also prove that there exis…
New proof of high genus Lawson surfaces with area estimates.
problem Existence and area estimates of high genus Lawson surfaces.
method Deforming DPW potential to prove existence and calculating area.
result Estimates on area of Lawson surfaces in terms of genus.
Study proves conditions for area-minimizing surfaces in a specific space.
problem Existence and non-existence of area-minimizing surfaces in .
method Analyzes sufficient conditions for curves to be the asymptotic boundary of area-minimizing surfaces.
result Presented sufficient conditions for a curve to admit a solution to the asymptotic Plateau problem.
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.
We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carathéodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero (or constant if a volume-preservin…
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 minimal surfaces grow area very quickly.
problem Understanding minimal surfaces with rapid area growth.
method Examples of minimal immersions in Euclidean space.
result Proper minimal surfaces with rapid area growth found.
In this paper, we introduce several mixed geominimal surface areas for multiple convex bodies for all . Our definitions are motivated from an equivalent formula for the mixed -affine surface area. Some properties, such as the affine invariance, for these mixed geominimal surface areas are prove…
The paper develops inequalities for log-concave functions and related surface areas.
problem Understanding log-concave functions and their inequalities.
method Establishing new inequalities through f-divergences and functional affine surface areas.
result New inequalities on functional affine surface area and bounds for Kullback-Leibler divergence.
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.
Bounds on the index of free boundary minimal surfaces.
problem Understanding the index of free boundary minimal surfaces.
method Comparison of energy and area indices, combining with previous work.
result Area index is bounded by a linear function of genus and boundary components.
We study surfaces in TN that are area-stationary with respect to a neutral Kaehler metric constructed on TN from a riemannian metric g on N. We show that holomorphic curves in TN are area-stationary, while lagrangian surfaces that are area-stationary are also holomorphic and hence totally null. However, in general, are…
Geodesics on hyperbolic surfaces become evenly spread over time.
problem Distribution of geodesics on hyperbolic surfaces.
method Equidistribution analysis of geodesics with length less than T as T approaches infinity.
result Closed geodesics on hyperbolic surfaces of finite area become evenly spread over time.
We obtain a bound for the area of a capillary surface in a three-manifold with umbilic boundary and controlled sectional curvature. We then analyze the geometry when this area bound is realized, and obtain rigidity theorems. As a side product, we obtain existence of totally geodesic embedded surfaces in hyperbolic …
New bounds on genus and area for CMC surfaces in 3-manifolds.
problem Bounding genus and area of CMC surfaces in 3-manifolds.
method Local degeneration of minimal surfaces and index-area bounds.
result Genus and area of CMC surfaces are bounded by index and area.
Solves area-minimizing surface problem for finite curves in H^2xR.
problem Asymptotic Plateau problem for area-minimizing surfaces.
method Complete solution for finite curves in $\BHH$.
result Fairly complete solution for finite curves in $\BHH$.
Estimates area and spectrum of stable minimal surfaces in Euclidean and hyperbolic spaces.
problem Estimating the growth of area and spectrum of stable minimal surfaces.
method Elementary argument and stability inequality for Euclidean space; explicit area growth estimate for hyperbolic space; scalar curvature lower bound for spectrum.
result Minimal surfaces in Euclidean space grow like the Euclidean plane, and in hyperbolic space, explicit area growth estimates are derived.
The study finds area estimates for specific surface types in a flat 3-torus.
problem Finding surfaces with constant mean curvature in a flat 3-torus.
method Analyzes closed surfaces with specific genus and constant mean curvature in a closed flat 3-torus.
result Establishes area estimates for surfaces with constant mean curvature, contrasting with minimal surfaces.
We show that area minimizing polyhedral surfaces are saddle.
Study of large area-constrained Willmore surfaces in Schwarzschild-like manifolds.
problem Understanding Willmore surfaces in asymptotically Schwarzschild 3-manifolds.
method Application of Lyapunov-Schmidt reduction method.
result End of the manifold is foliated by area-constrained Willmore spheres.
Characterizes area-minimizing maps for surfaces of genus ≥ 2.
problem Equivariant area-minimizing maps on surface covers.
method Classifies minimal surfaces in Hilbert spheres with constant negative Gaussian curvature.
result Characterizes all equivariantly area-minimizing maps from the universal cover of a surface to a Hilbert sphere.
Computes circular area and spherical volume invariants via integrals.
problem Computing circular area and spherical volume invariants for curves and surfaces.
method Using the Divergence Theorem, express area and volume integrals as line and surface integrals against kernels, then compute analytically on triangulated meshes.
result Simple algorithm for computing spherical volume invariant for triangulated surfaces without discretizing ambient space.
Study on geodesic surfaces in hyperbolic 3-orbifolds, proving overlaps in area sets.
problem Understanding overlaps in geometric genus spectra of non-commensurable hyperbolic 3-orbifolds.
method Defined geometric genus spectrum and totally geodesic area set, proving results on overlaps.
result Arithmetic hyperbolic 3-orbifolds can have large overlaps in their totally geodesic area sets.
New metric creates a surface with infinite topology.
problem Constructing a surface with infinite topology.
method Constructing a Riemannian metric and a curve, then finding an area-minimizing surface.
result The area-minimizing surface has infinite topology.
Proves unique continuation for area minimizing currents.
problem Ensuring area minimizing currents match minimal surfaces.
method Analyzes infinite order contact between currents and minimal surfaces.
result Currents and minimal surfaces coincide in a neighborhood.
Improved bound on the product of first Laplacian eigenvalue and area for genus three surfaces.
problem Bounding the product of the first eigenvalue of the Laplacian and the area for compact surfaces of genus three.
method Improved the bound established by Yang and Yau, using numerical computations for the hyperbolic Klein quartic surface.
result Showed that the product of the first eigenvalue of the Laplacian and the area is bounded above by approximately 21.668π.
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.
The paper finds metrics for surfaces with boundaries that match specific eigenvalues and areas.
problem Finding metrics for surfaces with boundaries that match specific eigenvalues and areas.
method The approach involves constructing a metric on a compact surface with boundary that satisfies given eigenvalues and area constraints.
result A metric can be constructed on a compact surface with boundary that matches a given sequence of eigenvalues and area.
The paper proves a diastolic inequality linking surface area and loop length.
problem Finding short geodesics on Riemannian surfaces.
method Proving a universal inequality between diastole and area of closed surfaces.
result Every Riemannian surface can be decomposed into two domains with bounded boundary length.
Study counts minimal surfaces in curved 3D spaces, finding hyperbolic space minimizes area.
problem Counting minimal surfaces in negatively curved 3-manifolds.
method Introduced an asymptotic quantity to count area-minimizing surfaces and showed minimization by hyperbolic metric.
result Hyperbolic metric minimizes the quantity of area-minimizing surfaces in negatively curved 3-manifolds.