Study of p-adic simplicial volumes and their properties.
problem Understanding simplicial volumes over p-adic seminormed rings. method Definition and study of p-adic simplicial volumes, homology bounds, and computation of volumes for surfaces. result Established homology bounds and computed p-adic simplicial volumes for surfaces. Minimal volume vector fields on surfaces via calibrations.
problem Finding minimal volume vector fields on Riemann surfaces.
method Theory of calibrations to write the equation of minimal volume vector fields.
result Equation of minimal volume vector fields on Riemann surfaces.
New examples show no upper bounds on link volumes on incompressible surfaces.
problem Finding upper bounds on volumes of links on incompressible surfaces.
method Examined weakly generalised alternating and fully augmented links on incompressible surfaces.
result Found infinite families of links on incompressible surfaces with no upper bounds on volume.
We calculate finite volumes of moduli spaces of flat surfaces with conical singularities.
problem Calculating volumes of moduli spaces of flat surfaces with prescribed conical singularities.
method Induction on the Euler characteristics of the punctured surface for almost all orders of the singularities.
result Explicit computation of volumes is possible.
We define the ``volume'' contained by pointed k-surfaces, first studied by the author in [9], and we show that this volume is always finite. Likewise, we show that the surface area of a pointed k-surface is always finite.
Infinite volume moduli spaces of hyperbolic surfaces are redefined with exponential forms.
problem Infinite volume of moduli spaces of hyperbolic surfaces with cusps.
method Introduce exponential volume form exp(-W)Vol(K,L) where W is a function of hyperbolic areas.
result Exponential volume forms make moduli spaces finite and relevant to open string theory.
Develops new methods for Epstein surfaces and W-volume.
problem Constructing and understanding Epstein surfaces and W-volume.
method Alternate construction using Osgood-Stowe differential; variational formulas.
result Generalizations of Epstein's univalence criterion and variational formulas for W-volume.
New pseudo-Anosovs on surfaces with punctures have infinite volume.
problem Volume of pseudo-Anosovs on surfaces with punctures is not bounded.
method Constructing pseudo-Anosovs with minimal entropy and showing volume tends to infinity.
result Volume of pseudo-Anosovs on surfaces with punctures can be arbitrarily large.
The Willmore flow preserves surface volume, leading to convergence to a sphere.
problem Long-term behavior of volume-preserving Willmore flow on surfaces.
method Volume-preserving Willmore flow, blow-up analysis, constrained Lojasiewicz-Simon inequality.
result Smooth solutions exist for spherical surfaces with Willmore energy below 8π and converge to a sphere.
A convex projective surface is the quotient of a properly convex open Ω of P(R) by a discret subgroup Γ of SL3(R). We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann's measure. We deduce of this that if Ω is not a triangle then …
The paper calculates the volume growth of hyperbolic surfaces with short geodesics.
problem Understanding the volume growth of hyperbolic surfaces with short geodesics.
method Introduced a function L(g) to measure the length of geodesics and computed the volume growth rate.
result The volume of surfaces with short geodesics is equal to V_g almost surely as g approaches infinity.
Formula calculates volume of CMC surfaces with translational periods, disproving isoperimetric conjecture.
problem Isoperimetric problem for CMC surfaces with translational periods.
method General formula relating volume, surface area, and curvature term.
result Disproved isoperimetric conjecture in T2imesR, providing counterexample. For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the other one is the power of the Atiyah-Bott-Goldman symplectic form. We introduce an h…
Maximal representations into SO0(2,3) have bounded volume.
problem Bounding the volume of maximal representations into SO0(2,3). method Uniform upper and lower bounds on the volume for different surface groups.
result Volume is bounded from above and below for maximal representations into SO0(2,3). The volume conjecture is extended for surface diffeomorphisms with quantum invariants.
problem Extending the volume conjecture for quantum invariants of surface diffeomorphisms.
method Relating asymptotics of quantum invariants to hyperbolic cone structures on mapping tori.
result The conjecture is proven for a specific case of the once-punctured torus bundle.
Paper connects volumes of moduli spaces of super Riemann surfaces to integrals over stable Riemann surfaces.
problem Relating volumes of moduli spaces of super Riemann surfaces to integrals over stable Riemann surfaces.
method Relates volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces Mg,n. result Proves recursion between volumes of moduli spaces of super hyperbolic surfaces using algebraic geometry.
Totally umbilic surfaces in hyperbolic 3-manifolds are constructed and characterized.
problem Characterizing totally umbilic surfaces in hyperbolic 3-manifolds of finite volume.
method Construction and characterization of surfaces based on their properties and embedding conditions.
result Characterization of totally umbilic surfaces in hyperbolic 3-manifolds of finite volume.
We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our r…
The volume conjecture for surface diffeomorphisms connects volumes to polynomial evaluations.
problem Connecting surface volumes to polynomial evaluations of quantum invariants.
method Developing combinatorial and algebraic techniques to compute isomorphisms between representations.
result Numerical evidence supports a conjecture linking surface volumes to polynomial evaluations.
Mirzakhani volumes of moduli spaces are polylogarithmic.
problem Understanding the volume of moduli spaces of hyperbolic surfaces.
method Expressed as a sum of polylogarithms evaluated at specific points.
result Mirzakhani volumes are polylogarithmic.
The study shows how certain surfaces can be filled by hyperbolic manifolds.
problem Conditions for a Riemann surface to be filled by hyperbolic manifolds.
method Analyzes conditions involving short hyperbolic curves on the surface.
result Riemann surfaces can be filled by hyperbolic manifolds of negative volume.
We consider the volume entropy of closed flat surfaces of genus g≥2 and area 1. We show that a sequence of flat surfaces diverges in the moduli space if and only if the volume entropy converges to infinity. Equivalently the Hausdorff dimension of the Gromov boundary of the isometric universal cover tends to infin…
Study Liouville equation on Riemannian surfaces, linking volume growth to classification results.
problem Classifying solutions and manifolds of the Liouville equation on Riemannian surfaces.
method Analyzing the Liouville equation −Δu=eu on Riemannian surfaces with non-negative Ricci curvature, considering asymptotic volume growth. result Established classification results for solutions and manifolds, revealing a connection between volume growth and classification.
We show that surface bundles over surfaces with base and fiber of genus at least 2 have non-vanishing simplicial volume.
Study volumes of Klein surfaces, extending Mirzakhani's recursion.
problem Volumes of moduli spaces of bordered Klein surfaces.
method Generalization of Mirzakhani's recursion, integration over regularised moduli space.
result Explicit formula for Klein bottle moduli space volumes, recursion for arbitrary topologies.
The paper estimates the volume of singular points in evolving surfaces.
problem Estimating the volume of singular points in evolving surfaces.
method Uniform and sharp volume estimates for singular sets of mean curvature flows.
result Uniform and sharp volume estimates for singular sets of mean curvature flows.
Defines renormalized volume for bounded regions in asymptotically hyperbolic Einstein spaces.
problem Calculating the volume of bounded regions in complex geometries.
method Defines renormalized volume, proves Gauss-Bonnet theorem, computes derivative under variations.
result Derives a Gauss-Bonnet theorem for the renormalized volume.
In this paper, we are interested in flat metric structures with conical singularities on surfaces which are obtained by deforming translation surface structures. The moduli space of such flat metric structures can be viewed as some deformation of the moduli space of translation surfaces. Using geodesic triangulations, …
Integral filling volume of mapping tori grows sublinearly with complexity.
problem Characterizing mapping classes with vanishing integral filling volume.
method Analyzing Dehn twists and mapping tori, using simplicial volume and complexity.
result Integral simplicial volume of mapping tori grows sublinearly with respect to the monodromy power.
New theorem bounds link volume using surface coefficients.
problem Bounding hyperbolic volume of links on surfaces.
method Analogue of Dasbach-Lin theorem for surface links.
result Bounds on link volume from surface polynomial coefficients.
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. We provide sharp lower bounds for the simplicial volume of compact 3-manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of 3-manifolds, including handlebodies and products of surfaces with the interval. Our results provide the firs…
We construct an algorithm that lists all closed essential surfaces in the complement of a knot that lies on the fiber of a trefoil or figure eight knot. Such knots are Berge knots and hence admit lens space surgeries. Furthermore they may have arbitrarily large hyperbolic volume. Using this algorithm we concoct large v…
Formula derived for enclosed volume of CMC surfaces in 3-sphere.
problem Calculating the enclosed volume of constant mean curvature surfaces in the 3-sphere.
method Using Chern-Simons gauge theory and holonomy on the Chern-Simons bundle.
result Formula for enclosed volume only depends on gauge classes of flat connections.
The study examines surfaces in hyperbolic 3-manifolds that become nearly flat.
problem Characterizing surfaces in hyperbolic 3-manifolds that become nearly flat.
method Analyzes asymptotically geodesic surfaces in hyperbolic 3-manifolds with finite and infinite volume.
result For finite volume, asymptotically geodesic surfaces are dense; for infinite volume, they do not exist.
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 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.
The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact lo…
No CMC surfaces exist in certain hyperbolic 3-manifolds.
problem Existence of constant mean curvature (CMC) surfaces in hyperbolic 3-manifolds.
method Proof by contradiction using hyperbolic geometry and topology.
result Nonexistence of CMC surfaces with mean curvature ≥ 1.
We prove a Gauss-Bonnet type formula for Riemann-Finsler surfaces of non-constant indicatrix volume and with regular piecewise smooth boundary. We give a Hadamard type theorem for N-parallels of a Landsberg surface.
The paper contains a new proof that a complete, non-compact hyperbolic 3-manifold M with finite volume contains an immersed, closed, quasi-Fuchsian surface.
We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through a ramified covering. These are the ma…
Study shows volumes of knot complements are bounded by linear functions of geodesic periods.
problem Volume calculation of knot complements associated with geodesics on modular surfaces.
method Analyzes geodesics on modular surfaces, their associated knots, and their complements' volumes.
result Volumes of knot complements are bounded linearly by the period of geodesic continued fractions.
Energy quantization for surfaces with area, volume, and mean curvature constraints.
problem Energy quantization for constrained Willmore surfaces.
method Established through strong compactness under energy thresholds.
result Strong compactness of constrained Willmore surfaces, including minimizers.
Study shows bounds on volumes of weakly generalised alternating knots.
problem Volume bounds for weakly generalised alternating knots.
method Analysis of weakly generalised alternating knots in 3-manifolds.
result Upper volume bound does not hold for weakly generalised alternating knots.
The paper derives formulas for symplectic volume forms on surface representation varieties.
problem Calculating symplectic volume forms on surface representation varieties.
method Multiplicative gluing formulas and Heusener-Porti results.
result Symplectic volume form on Σg,0 is a product of forms on Σ2,1 and Σ2,2. We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic 3-manifold N. We also obtain a least area, incompressible, properly embedded, finite topology, 2-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This dete…
Measures neural network decision boundary volume to predict model performance.
problem Understanding the geometry of deep learning models for better performance.
method Local surface volumes to measure decision boundary, applying Weyl's tube formula.
result Smaller surface volume correlates with higher classification accuracy.