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. Paper introduces a new cosmological volume function and its properties.
problem Introducing a new cosmological volume function.
method Introduces and analyzes the cosmological volume function τ_V, showing it's continuously differentiable.
result τ_V leads to a canonical splitting of the metric tensor and a canonical Wick-rotated Riemannian metric.
In this paper, we study volume growth, Liouville theorem and the local gradient estimate for f-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such manifolds.
New properties are derived of renormalized volume functionals, which arise as coefficients in the asymptotic expansion of the volume of an asymptotically hyperbolic Einstein (AHE) manifold. A formula is given for the renormalized volume of an even-dimensional AHE manifold in terms of an arbitrary totally geodesic compa…
New length functions on mapping class groups linked to simplicial volumes of mapping tori.
problem Understanding the relationship between mapping class groups and simplicial volumes of mapping tori.
method Introducing filling volumes as length functions and proving their properties.
result Real filling volumes equal the simplicial volume of mapping tori, while integral filling volumes are not smaller than the stable integral simplicial volume.
Winterbottom shape minimizes capillary functional under volume constraint.
problem Finding the shape that minimizes capillary functional under volume constraint.
method Proved minimality of Winterbottom shape using anisotropic capillary functional.
result Winterbottom shape is a volume-constraint minimizer of the anisotropic capillary functional.
Study critical metrics on manifolds with boundary using integral and boundary estimates.
problem Investigate geometry of critical metrics on compact manifolds with boundary.
method Use generalized Reilly's formula to derive integral and boundary estimates.
result Establish new boundary estimates for critical metrics of the volume functional.
Sharp heat kernel and Green's function estimates on Euclidean volume growth manifolds.
problem Estimating heat kernels and Green's functions on specific Riemannian manifolds.
method Analyzing noncompact Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Sharp Moser-Trudinger inequalities on manifolds with Euclidean volume growth.
Study intrinsic volume forms on complex hypersurfaces.
problem Computing volume functionals on pseudoconvex hypersurfaces.
method Compute first and second variation formulae, explore infinite dimensional aspects.
result Discuss possible analogues of the affine isoperimetric inequality.
The minimizer of a volume function is unique for klt singularities.
problem Uniqueness of the minimizer of the normalized volume function for klt singularities.
method Defining stability thresholds for valuations and showing K-semistability.
result The minimizer of the normalized volume function for a klt singularity is unique up to rescaling.
Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.
problem Proving uniqueness of asymptotic limits for noncollapsed Ricci flat manifolds with linear volume growth.
method Relating uniqueness to the existence of a harmonic function asymptotic to a Busemann function, proving uniqueness via a monotone quantity.
result Proves uniqueness of the asymptotic limit and establishes a polynomial convergence rate.
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
The study links Ricci curvature and convexity in complex tori.
problem Characterizing Ricci curvature signs in toric manifolds.
method Characterization through convexity of volume functional.
result Relationships between Ricci curvature, volume, submanifolds, and pluri-subharmonic functions.
Study critical metrics on manifolds, proving specific isometries.
problem Investigating critical metrics on complete manifolds.
method Analyzing volume functional and proving isometries.
result Critical metrics on specific manifolds are isometric to standard models.
Study the relative volume function on AH manifolds and its applications.
problem Characterize the height of geodesic defining functions and capacity of balls.
method Define and analyze the relative volume function, proving its boundedness and regularity.
result Uniformly bounded relative volume function at infinity, bound dependent only on dimension.
The filling volume functions of the n-th quaternionic Heisenberg group grow, up to dimension n, as fast as the ones of the Euclidean space. We identify the growth rate of the filling volume function in dimension n+1, which is strictly faster than the growth rate of the (n+1)-dimensional filling volume function of the E…
In the context of Synthetic Differential Geometry, we describe the square volume of a ``second-infinitesimal simplex'', in terms of square-distance between its vertices. The square-volume function thus described is symmetric in the vertices. The square-volume gives rise to a characterization of the volume form in the t…
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.
Estimates volume of convex Alexandrov spaces with boundary.
problem Estimating volume of Alexandrov spaces with convex boundaries.
method Gradient flow of semi-concave functions.
result Volume upper bound achieved implies Boundary Conjecture.
The signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron. If we consider these up to translation and scaling, the resulting quotient space is diffeomorphic to a sphere. The mean volume function restricted …
Ricci curvature links volume convexity and minimal submanifolds.
problem Volume functional convexity and minimal submanifolds in Kaehler geometry.
method Toric Kaehler geometry and quasi-homogeneous manifolds.
result Sign of Ricci curvature correlates with volume functional convexity.
The main purpose of this paper is to present a number of analytic and geometric properties of the l-function and the reduced volume of Perelman, including in particular the monotonicity, the upper bound and the rigidities of the reduced volume.
Entropy derived from Colding's volume on Ricci-flat manifolds.
problem Deriving Perelman's entropy from Colding's monotonic volume.
method Applying Colding's monotonic volume to Perelman's N-space for harmonic functions on Ricci-flat manifolds.
result Entropy is the limit of Colding's monotonic volume.
Volume gaps for minimal submanifolds in spheres are proven.
problem Volume gaps for minimal submanifolds in spheres.
method Analyzing height functions and multiplicity in direction p.
result Volume inequalities for minimal submanifolds in spheres.
New adapted renormalized volume for hyperbolic 3-manifolds with compressible boundary.
problem Analyzing convex co-compact hyperbolic 3-manifolds with compressible boundaries.
method Defining and analyzing a new version of the renormalized volume.
result The adapted renormalized volume is bounded and has properties analogous to the classical renormalized volume.
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.
Ancient caloric functions on manifolds with polynomial growth are studied under volume doubling barrier.
problem Analyzing ancient caloric functions on manifolds beyond volume doubling.
method Time polynomial structure result on ancient caloric functions with polynomial growth.
result Finiteness result for ancient caloric functions is essentially sharp, except for multi-end cases.
We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami …
The paper studies Q-curvature and volume entropy on manifolds, proving polynomial growth polyharmonic function finiteness and rigidity.
problem Investigating Q-curvature and volume entropy on conformally flat manifolds.
method Introducing a new volume entropy, establishing identities, and proving rigidity results.
result Each polynomial growth polyharmonic function on such manifolds is of finite dimension, and the Cohn-Vossen inequality achieves equality under specific conditions.
Explicit bounds found for shortest orthogeodesics and volumes of hyperbolic manifolds.
problem Finding explicit bounds for shortest orthogeodesics and volumes of hyperbolic manifolds.
method Derived explicit estimates for functions related to volumes and orthospectra, using a new approach.
result Explicit lower bound for the length of the shortest orthogeodesic in terms of volume.
Study on hyperbolic manifolds and their boundary data, focusing on volume functions.
problem Determining the hyperbolic metric from boundary data of convex co-compact hyperbolic manifolds.
method Analysis of volume functions and their relation to boundary data, using first variations.
result New connections with physics and probability theory, with open questions remaining.
In this article, we investigate the geometry of critical metrics of the volume functional on an n-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifol…
We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenböck type formula for critical metrics of the volume functional on four-dimensional manifolds. As an application, we obtain a clas…
Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this resul…
This paper studies the geometry of minimum-volume confidence sets for multinomial parameters.
problem Determining if minimum-volume confidence sets for multinomial outcomes are disjoint.
method Enumerating and covering the continuous regions of the exact p-value function to study the geometry of minimum-volume confidence sets.
result The geometry of minimum-volume confidence sets for multinomial parameters is studied, providing insights into their structure and properties.
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.
Characterizes metrics with finite total Q-curvature and introduces new volume entropy.
problem Understanding metrics with finite total Q-curvature and their geometric properties.
method Characterization of metrics through total Q-curvature and introduction of new volume entropy.
result Controlled volume growth for complete metrics with finite total Q-curvature and bounded scalar curvature.
Two rigidity theorems for manifolds with nonnegative Ricci curvature and specific volume growth.
problem Characterizing manifolds with nonnegative Ricci curvature and specific volume growth properties.
method Rigidity theorems based on volume growth and existence of harmonic functions.
result Conditions for the Riemannian universal cover to have Euclidean volume growth and existence of nonconstant linear growth harmonic functions.
New formula calculates volumes of ideal hyperbolic drums.
problem Computing volumes of ideal hyperbolic drums.
method Proved a volume formula for arbitrary ideal hyperbolic antiprisms (drums).
result Volume formula for ideal hyperbolic drums.
Study shows volume density in central harmonic spaces can vary arbitrarily.
problem Volume density in central harmonic spaces can vary arbitrarily.
method Analyzes asymptotics of volume density function in central harmonic manifolds.
result Volume density in central harmonic spaces can be specified arbitrarily and does not determine geometry.
Study of n-cylinder surfaces to calculate Masur-Veech volumes.
problem Calculating Masur-Veech volumes for hyperbolic surfaces.
method Combinatorial approach using metric ribbon graphs and plane trees.
result Found generating function for n-cylinder contributions. The present paper regards the volume function of a doubly truncated hyperbolic tetrahedron. Starting from the previous results of J. Murakami, U. Yano and A. Ushijima, we have developed a unified approach to express the volume in different geometric cases via dilogarithm functions and to treat properly the many analyti…
We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finit…
New methods for computing volumes and constructing Fano fibrations.
problem Understanding Fano fibrations and their weighted volumes.
method New methods for computing weighted volumes, including Laplace transforms and incomplete Gamma-functions. Conjectural construction of Fano fibrations.
result Conjectural construction of Fano fibrations with asymptotically conical bases from degenerating Fano fibrations.
Proves volume conjecture for twist knots using complex analysis.
problem Volume conjecture for twist knots.
method Equivalence relation, complex analysis, analytic continuation, function of several complex variables.
result Proves volume conjecture for twist knots.
Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.
problem Analyzing properties of Kähler manifolds with nonnegative bisectional curvature.
method Established precise relations among minimal degree, volume growth, and scalar curvature decay.
result Unified understanding of Kähler-Ricci flow through polynomial growth holomorphic functions.
In this paper we study volume growth of gradient steady Ricci solitons. We show that if the potential function satisfies a uniform condition, then the soliton has at most Euclidean volume growth.
Given a space Y in X, a cycle in Y may be filled with a chain in two ways: either by restricting the chain to Y or by allowing it to be anywhere in X. When the pair (G,H) acts on (X,Y), we define the k-volume distortion function of H in G to measure the large-scale difference between the volumes of…