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.
Sharp upper bound for minimal graph area in unit ball established.
problem Determining the exact upper limit for the area of minimal graphs intersecting a unit ball.
method Constructing a sequence of minimal graphs via solutions to a Dirichlet problem.
result The areas of constructed minimal graphs tend to the upper bound of 2π. Minimal surfaces' area bounds proven equivalent, extending known results.
problem Equivalence of area bounds for minimal surfaces.
method Combining recent breakthroughs, extending known results.
result Equivalence of intrinsic and extrinsic area density bounds for minimal immersions.
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.
Study area-minimizing subgraphs in integer lattices.
problem Finding the most efficient subgraphs in integer lattices.
method Formulated functions of bounded variations, classified subgraphs in 2D, proved properties in higher dimensions.
result Classified area-minimizing subgraphs in 2D integer lattice up to isomorphisms.
Sharp lower bound found for area of vector fields on spherical annuli.
problem Finding the minimum area of unit vector fields on spherical annuli.
method Established a sharp lower bound through mathematical analysis.
result Sharp lower bound for the area of unit vector fields on spherical annuli.
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.
Study on CMC hypersurfaces with bounded index and area, proving multiplicity one convergence and bounds on genus.
problem Understanding CMC hypersurfaces with bounded index and area.
method Bubble-compactness theory for embedded CMC hypersurfaces in low dimensions.
result Minimal blow-ups are all catenoids, and bounds on genus provided.
This note shows surfaces in stable 3D data are bounded by area and diameter.
problem Bounding stable surfaces in 3D initial data sets.
method Demonstrating area and diameter bounds for stable, marginally outer trapped surfaces.
result Stable surfaces in 3D data sets are bounded by both area and diameter.
Sharp bounds found on shortest geodesic on punctured spheres.
problem Finding the shortest closed geodesic on punctured spheres.
method Sharp curvature-free upper bounds expressed in terms of area, extremal metrics described.
result Optimal bounds for spheres with up to four ends, extended to larger numbers of punctures.
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.
Lower bounds for surface area and volume of convex hypersurfaces.
problem Establishing bounds for surface area and volume of convex hypersurfaces.
method Using displacement under continuous maps to establish lower bounds.
result Proves a lower bound for the volume of a Riemannian n-sphere in all dimensions.
Study shows area-minimizing submanifolds are mostly rough, not smooth.
problem Understanding the smoothness of area-minimizing submanifolds.
method Proved non-smoothness by contradiction and established Hausdorff dimension bounds.
result Area-minimizing submanifolds are not generically smooth, resolving a conjecture.
Study examines Hilbert area of inscribed polygons in projective geometry.
problem Understanding Hilbert area of inscribed polygons in projective geometry.
method Examined correspondence between Fock-Goncharov and Cartesian coordinates, analyzed degeneration and Hilbert area of inscribed quadrilaterals, developed microlocal condition.
result Sequence of strictly convex domains with bounded Hilbert area and divergent Goldman parameters.
We obtain a bound for the area of a capillary H−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 …
We show that a smooth unknotted curve in R^3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giv…
Proves upper bound on systolic ratio for circle fillings.
problem Bounding systolic ratio for circle fillings.
method Proved upper bound on systolic ratio depending on genus.
result Filling Area Conjecture holds for large genus.
Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.
problem Understanding the relationship between intrinsic and extrinsic metrics and surface area.
method Examined surfaces within a unit ball in R3, provided lower bounds on the ratio in terms of area, and showed non-existence of global lower bounds.
result Found that the ratio of intrinsic to extrinsic metrics has a lower bound in terms of surface area, but no global lower bound exists.
Upper bound on singular set dimension for area-minimizing currents.
problem Bounding the dimension of singular points in area-minimizing currents.
method Using upper Minkowski dimension and properties of blow-up scales.
result Upper Minkowski bound of m−2 for the interior singular set. 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…
Suppose C is a compact, n-edged two-cell of the centered dual decomposition of a locally finite set in the hyperbolic plane, a coarsening of the Delaunay tessellation which was introduced in the author's prior work. We describe an effectively computable lower bound on the area of C, given an n-tuple of positive…
Consider a sequence of minimal varieties M_i in a Riemannian manifold N such that the boundary measures are uniformly bounded on compact sets. Let Z be the set of points at which the areas of the M_i blow up. We prove that Z behaves in some ways like a minimal variety without boundary: in particular, it satisfies the s…
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 confirms Gromov's speculation and provides bounds for taming symplectic structures.
problem Understanding the relationship between taming symplectic structures and the area of pseudoholomorphic curves.
method Analyzes the numerical cone of taming symplectic structures and characterizes coarsely holomorphic curves.
result An almost complex manifold with an area bound admits a taming symplectic structure, confirming Gromov's speculation.
Let C be a real-analytic Jordan curve in R3. Then C cannot bound infinitely many disk-type minimal surfaces which provide relative minima of area.
We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.
Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above by the area and the topology of the surface. Combining these results with work of…
The paper estimates area and volume for spacetimes with integral mean curvature bounds.
problem Estimating area and volume for spacetimes with specific curvature conditions.
method Using strong energy condition and norms of second fundamental form/mean curvature.
result Established area and volume estimates for spacetimes.
We give an a priori bound on the (n-7)-dimensional measure of the singular set for an area-minimizing n-dimensional hypersurface, in terms of the geometry of its boundary.
We prove existence and a.e. regularity of an area minimizing soap film with a bound on energy spanning a given Jordan curve in R^3. The energy of a film is defined to be the sum of its surface area and the length of its singular branched set. The class of surfaces over which area is minimized includes images of disks, …
New bounds found for minimal surfaces in hyperbolic 3-manifolds.
problem Bounding the maximum principal curvatures of minimal surfaces in hyperbolic 3-manifolds.
method Short argument using work of Uhlenbeck and families of fibered hyperbolic 3-manifolds.
result Uniform lower bound for maximum principal curvatures greater than one.
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.
As discussed in the paper, in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, its area must be bounded above by 4π/c, where c>0 is a lower bound on a natural energy momentum term. In this note we cons…
New method to bound Laplacian eigenvalues of geodesic balls.
problem Computing upper bounds for the first eigenvalue of Laplacian on geodesic balls.
method Transforming metric tensor into rotationally symmetric form preserving geodesic sphere areas.
result Upper bound for Laplacian eigenvalues is sharp and computable using geodesic sphere areas.
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.
In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in the boundary of M. Our main result is that for any g>=0…
In this paper, We define a F-functional and study F-stability of λ-hypersurfaces, which extend a result of Colding-Minicozzi. Lower bound growth and upper bound growth of area for complete and non-compact λ-hypersurfaces are studied.
Injective construction proves bounded cohomology dimensions.
problem Injectivity of Gambaudo--Ghys construction on bounded cohomology.
method Generalized Gambaudo--Ghys construction on bounded cohomology.
result Injectivity of the construction and infinite-dimensional bounded cohomology.
In this paper, we prove that for any closed 4-dimensional Riemannian manifold M with trivial first homology group, if the Ricci curvature ∣Ric∣≤3, the diameter diam(M)≤D and the volume vol(M)>v>0, then the area of a smallest 2-dimensional stationary integral varifold in M is bounded by F(v,D), for some…
We extend to higher dimensions earlier sharp bounds for the area of two dimensional free boundary minimal surfaces contained in a geodesic ball of the round sphere. This follows work of Brendle and Fraser-Schoen in the euclidean case.
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 area-minimizing hypersurfaces in manifolds with controlled curvature.
problem Characterize area-minimizing hypersurfaces in manifolds with Ricci curvature bounds.
method Apply Cheeger-Colding theory and blow-up techniques to analyze hypersurfaces.
result Proves continuity of volume functions and existence of area-minimizing limits.
In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a sharp lower bound on the hilbert area of ideal triangles, independant of the con…
In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by 4π/c, where c>0 is a lower bound on a natural energy-momentum term. We then consider th…
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.
We introduce a Z--coefficient version of Guth's macroscopic stability inequality for almost-minimizing hypersurfaces. In manifolds with a lower bound on macroscopic scalar curvature, we use the inequality to prove a lower bound on areas of hypersurfaces in terms of the Gromov simplicial norm of their homolog…
Let Σbe a k-dimensional minimal surface in the unit ball B^n which meets the unit sphere orthogonally. We show that the area of Σis bounded from below by the volume of the unit ball in R^k. This answers a question posed by R. Schoen.
Extends diameter bounds for submanifolds with boundary and minor curvature restrictions.
problem Bounding the diameter of submanifolds with boundary and minor curvature restrictions.
method Applies bounds dependent on mean curvature and area to minimal, constant mean curvature, and prescribed mean curvature surfaces.
result Diameter bounds for submanifolds with boundary and minor curvature restrictions.