Upper bound on geodesic ball volume in Riemannian manifolds.
problem Bounding geodesic ball volume in Riemannian manifolds.
method Techniques to provide an upper bound on geodesic ball volume.
result Upper bound on geodesic ball volume in Euclidean space.
Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.
problem Characterizing the behavior of volume functions associated with quadratic differentials.
method Analyzing the Thurston volume of unit balls in measured lamination spaces.
result The volume function is not proper and is p-integrable for any 0<p<1. Round balls minimize liquid drop model volumes ≤ 1.
problem Minimizing volumes in liquid drop models.
method Proved uniqueness of minimizers for small volumes.
result Round balls uniquely minimize volumes ≤ 1.
If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most c(n)Volume(M,hyp), then we prove that the universal cover of (M,g…
Lower bounds for geodesic ball volume in 3D with Ricci curvature constraints.
problem Finding volume bounds in 3D manifolds with Ricci curvature limits.
method Providing lower bounds for geodesic ball volume with upper bounds on Ricci curvature.
result Established lower bounds for geodesic ball volume under Ricci curvature constraints.
Study calculates volume of small sub-Riemannian balls in 3D contact manifolds.
problem Computing the volume of small sub-Riemannian balls in 3D contact manifolds.
method Asymptotic expansion and geometric invariants of the sub-Riemannian structure.
result Expressed first geometric coefficients in terms of sub-Riemannian structure invariants.
This paper confirms volumes of geodesic balls can identify 4D space forms.
problem Determining if a 4D manifold is a space form using geodesic ball volumes.
method Tensor calculus and classical theorems, not topological characterizations.
result Similar results for 4D manifold space forms confirmed.
Balls are the only volume-constrained critical points of perimeter.
problem Finding volume-constrained critical points of perimeter.
method Analyzing sets of finite perimeter and using Alexandrov's theorem.
result Balls are the only volume-constrained critical points of perimeter.
The study calculates the volumes of 3-ball quotients and their orbifold Euler characteristics.
problem Calculating the volumes and orbifold Euler characteristics of 3-ball quotients.
method Explicit description and presentation of 3-ball quotients, deduction of orbifold Euler characteristics.
result Explicit values for the orbifold Euler characteristics of 3-ball quotients.
Paper finds critical metrics with pinched curvature are geodesic balls.
problem Identifying critical metrics with specific curvature constraints.
method Proved isometry to geodesic balls in S^n and provided conditions for the gradient of the potential function.
result Critical metrics with pinched curvature are isometric to geodesic balls in S^n.
In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group G⊂PU(n,1) acting on complex hyperbolic space. Consequently the volume of all complex hyperbolic n-manifolds is bounded below by the volume of this bal…
New method uses relative capacities of geodesic balls to determine scalar curvature.
problem Determining scalar curvature from geodesic ball volumes.
method Using relative capacities of concentric small geodesic balls.
result Scalar curvature is determined by relative capacities of geodesic balls.
We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In…
Uniform comparison of hyperbolic ball volumes on universal cover.
problem Comparing hyperbolic ball volumes on universal cover.
method Proving a constant δ_n exists such that for any metric g on M, if the volume ratio is less than δ_n, the volume of hyperbolic balls is at least as large as in hyperbolic space.
result Every Riemannian metric g on M with a specific volume ratio satisfies the volume of hyperbolic balls on the universal cover is at least as large as in hyperbolic space.
Smooth surface encloses less volume than a ball.
problem Can a smooth surface enclose less volume than a ball?
method Example of a smooth closed surface in R3 with specific curvature constraints. result No, a supersqueezed sphere encloses less volume than a unit ball.
The paper compares volumes near specific metrics and proves volume comparison for geodesic balls and closed manifolds.
problem Investigating volume comparison with scalar curvature for specific metrics.
method Analyzing small geodesic balls and closed manifolds near V-static and strictly stable Einstein metrics.
result Volume comparison holds for small geodesic balls and for metrics near strictly stable Einstein metrics.
The paper proves bounded domains are biholomorphic to the unit ball.
problem Covering manifolds with finite volume.
method Analyzes C2 and C1,ε boundary conditions for bounded domains. result Bounded domains are biholomorphic to the unit ball under certain conditions.
Compactness theorem for manifolds with scalar curvature and entropy bounds.
problem Understanding the structure of manifolds with specific curvature and entropy bounds.
method Using volume upper bounds to prove Gromov-Hausdorff closeness to Euclidean balls.
result Unit balls in such manifolds are bi-Hölder and bi-W1,p homeomorphic to Euclidean balls. Proves stability of cone-volume measure with nearly constant density.
problem Stability of cone-volume measure with near constant density.
method Proves stability of cone-volume measure with near constant density.
result Homothetic copy of the body is close to the unit ball in the L2-distance. New findings on isoperimetric shapes in n-dimensional space.
problem Determining isoperimetric shapes with specific volume and perimeter densities.
method Analyzing the stability and convexity of generating curves.
result Balls are uniquely isoperimetric under certain conditions.
If (Mn,g) is a closed Riemannian manifold where every unit ball has volume at most εn (a sufficiently small constant), then the (n−1)-dimensional Uryson width of (Mn,g) is at most 1.
Study shows diffused interface flows to single diffused balls over time.
problem Volume-preserving mean curvature flow in Euclidean space.
method Diffused interface version, exponential convergence proof.
result Exponential convergence to single diffused balls.
In a compact orbifold, for small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.
We show that, the solutions of the isoperimetric problem for small volumes are C2,α-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a compact riemannian manifol…
The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{po…
Study σ2-curvature and volume of compact manifolds, proving conditions for Einstein metrics and geodesic balls.
problem Understanding σ2-curvature and volume in compact manifolds. method Critical point analysis, volume comparison, variational properties, geodesic balls.
result Sufficient and necessary condition for a critical metric to be Einstein, volume comparison results.
Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.
problem Understanding isoperimetric constants in metric measure spaces with measure contraction property.
method Proves local isoperimetric inequalities on essentially non-branching MCP(K,N) spaces with volume constraints and geometric conditions.
result Establishes bounds on isoperimetric constants in smaller geodesic balls.
Constructs simplified or complexified simplicial complexes.
problem Efficiently simplifying or complexifying complex spaces.
method Embeddings of simplicial complexes into a simplicial ball with bounded degrees and low volume.
result Realizes complicated spaces as parts of a ball/sphere or gives spheres specific metrics.
Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.
problem Developing an ABP approach to Sobolev and Michael-Simon inequalities under volume noncollapsing assumptions.
method Using a refinement of Brendle's contact-set argument to derive lower bounds for the volumes of geodesic balls.
result A Michael-Simon type inequality for immersed submanifolds with nonnegative sectional curvature and volume noncollapsing.
The smallest r so that a metric r-ball covers a metric space M is called the radius of M. The volume of a metric r-ball in the space form of constant curvature k is an upper bound for the volume of any Riemannian manifold with sectional curvature ≥k and radius ≤r. We show that when such a manifo…
We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of 38π2, i.e., they attain all possible volumes of c…
New metric properties show volume constraints in collapsing spaces.
problem Volume constraints in collapsing spaces.
method Generalization of recent progress in metric geometry involving the volume of balls of radius in a certain range with collapsing at different scales.
result For every Riemannian metric on a manifold of sufficiently small volume, there is a point with volume constraints in the universal cover.
We prove a so called κ non-inflating property for Ricci flow, which provides an upper bound for volume ratio of geodesic balls over Euclidean ones, under an upper bound for scalar curvature. This result can be regarded as the opposite statement of Perelman's κ non-collapsing property for Ricci flow. These two resul…
Study cohomology of ball quotients and their compactifications.
problem Cohomology of symmetric power of cotangent bundles of ball quotients and their compactifications.
method Hodge theory for complete hermitian manifolds, Green's operator, extension of results.
result Established existence of Hodge decomposition and Green's operator for ball quotients and their compactifications.
The study provides volume growth estimates for specific types of manifolds.
problem Estimating volume growth for Ricci solitons and quasi-Einstein manifolds.
method Similar to classical results, the study proves volume growth estimates for gradient Ricci solitons and quasi-Einstein manifolds.
result Sharp volume growth estimates for gradient shrinking Ricci solitons and upper bound volume growth estimates for quasi-Einstein manifolds.
Constructs ε-splitting maps for geodesic balls with non-negative Ricci curvature.
problem Constructing ε-splitting maps for geodesic balls with non-negative Ricci curvature.
method Induction and stratified almost Gou-Gu Theorem for finding directional points; error estimates for projections.
result Constructs ε-splitting maps on concentric geodesic balls with uniformly small radius. We consider the rate of volume growth of large Carnot-Carathéodory metric balls on a class of unbounded model hypersurfaces in C2. When the hypersurface has a uniform global structure, we show that a metric ball of radius δ≫1 either has volume on the order of δ3 or δ4. We also give necessary and …
The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.
problem Investigating the geometric properties of Riemannian manifolds influenced by average scalar curvature.
method Analyzing the conjugate radius, average area of geodesic spheres, average volume of metric balls, and total volume of closed manifolds.
result Improves the Bishop-Gromov estimate on the average volume of metric balls and proves monotone decreasing properties of certain geometric integrals.
Study a flow in a ball that preserves volume and converges to spherical caps.
problem Preserving volume in a flow with a capillary boundary.
method Mean curvature flow with capillary boundary.
result The flow has longtime existence and converges to spherical caps.
New theorem shows shapes close to balls, flow converges to balls in 2D and 3D.
problem Understanding the asymptotic behavior of volume-preserving mean curvature flow.
method Proved a new quantitative Alexandrov theorem and used it to show flow convergence.
result Weak solutions of volume-preserving mean curvature flow converge to disjoint balls in R^2 and R^3.
We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.
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 give upper and lower bounds for the ratio of the volume of metric ball to the area of the metric sphere in Finsler-Hadamard manifolds with pinched S-curvature. We apply these estimates to find the limit at the infinity for this ratio. Derived estimates are the generalization of the well-known result in Riemannian ge…
The ball does not maximize the first Steklov eigenvalue in higher dimensions, unlike in two dimensions.
problem Optimizing the first nonzero Steklov eigenvalue in higher dimensions.
method Analyzing contractible domains of fixed boundary volume and boundary components.
result Increasing boundary components does not increase the normalized first Steklov eigenvalue in higher dimensions.
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.
We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary minimal submanifolds in the ball we show that the boundary volume is reduced up…
The paper proves a theorem linking ball volume and Uryson width, with applications to 3-sphere metrics.
problem Relating volume of balls to Uryson width and Hausdorff content.
method Short proof and generalization of Guth's theorem, using Riemannian metrics and map diameter constraints.
result For any C>0, there exists a 3-sphere metric with volume 1 and a map violating diameter constraints. We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff sense to a countably Hm rectifiable metric space of the…