Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.
problem Characterizing totally geodesic submanifolds in geometrically finite manifolds.
method Analysis of totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds.
result Every maximal totally geodesic submanifold of dimension at least two in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur.
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.
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 solve the modified Kazdan-Warner problem of finding metrics with prescribed scalar curvature and unit total volume.
Study proves a new formula for capillary hypersurfaces and shows a flow converging to a special shape.
problem Understanding the behavior of capillary hypersurfaces in hyperbolic space.
method Developed a volume-preserving flow starting from a star-shaped initial hypersurface and proved its long-time existence and convergence.
result The flow converges to a θ-totally umbilical cap, which is an energy minimizer for a given enclosed volume. Positive simplicial volume found for certain non-positively curved manifolds with specific submanifolds.
problem Determining conditions for positive simplicial volume in non-positively curved manifolds.
method Analyzing isolated, closed totally geodesic submanifolds of codimension one and their impact on simplicial volume.
result Positive simplicial volume for certain non-positively curved manifolds with specific submanifolds.
Proof shows volume equals integral points for certain manifolds.
problem Counting integral points on affine manifolds.
method Rational Ehrhart theory and Fourier analysis.
result Volume equals number of integral points for integral-integral affine manifolds.
The paper extends volume comparison results to total σ_l-curvature.
problem Comparing total σ_l-curvature with σ_k-curvature.
method Volume comparison theorem extension to σ-curvature comparison.
result Comparison holds for metrics close to strictly stable positive Einstein metrics.
Study shows simplicial volume of certain fiber bundles is zero.
problem Understanding simplicial volume in fiber bundles with nonpositive curvature.
method Proved simplicial volume is zero for specific fiber bundles.
result Simplicial volume of certain fiber bundles is zero.
In this note, we describe the Hermitian metrics that leave the total Monge-Ampere volume invariant. In particular, we give several characterizations of the Hermitian metrics which satisfy the comparison principle for the complex Monge-Ampere operator
The authors showed in a preceding paper that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed on…
Two Morse-Bott volume forms are diffeomorphic if their cohomology classes are equal.
problem Establishing equivalence of Morse-Bott volume forms.
method Adapting Moser's trick to Morse-Bott volume forms.
result Two Morse-Bott volume forms with the same zero set are diffeomorphic if and only if they have equal total volumes.
It is shown that if the Holmes-Thompson volume definition is used, totally geodesic submanifolds of a Finsler space are minimal. The analogous result for the Hausdorff measure is known to be false. ----- Nous montrons que les sous-varietes totalement geodesiques d'une variete de Finsler sont minimales pour le volume de…
In this article we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally geodesic surface, this spectrum determines the commensurability class. In addition, we show that any finite volume hyperbolic 3-manifold …
We determine the Riemannian manifolds for which the group of exact volume preserving diffeomorphisms is a totally geodesic subgroup of the group of volume preserving diffeomorphisms, considering right invariant L2-metrics. The same is done for the subgroup of Hamiltonian diffeomorphisms as a subgroup of the group of…
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 on minimal submanifolds with finite curvature in Euclidean space.
problem Finite diffeomorphism types of complete immersed minimal submanifolds with finite total curvature.
method Adapted method from Chodosh, Ketover, and Maximo for hypersurfaces to submanifolds of arbitrary codimension.
result Proved finite diffeomorphism types for complete immersed minimal submanifolds with finite total curvature.
Estimates open sets for fibrations, leading to volume vanishing results.
problem Estimating open sets for fibrations.
method Straightforward estimate for open sets with fundamental group constraints.
result Vanishing results for simplicial volume and minimal volume entropy for certain mapping tori.
The study finds minimal volume hyperbolic 4-manifolds with embedded 3-manifolds.
problem Existence of small volume hyperbolic 4-manifolds with embedded 3-manifolds.
method Analysis of hyperbolic manifolds and their submanifolds.
result Minimal volume hyperbolic 4-manifolds with embedded 3-manifolds exist.
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.
In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic RP2m in CPn. Then it minimizes volume among the isotropic submanifolds in the same Z/2 homology class in CPn (but not among all submanifolds in this…
Infinite hyperbolic manifolds share same perimeter-to-volume ratio.
problem Finding hyperbolic manifolds with a fixed perimeter-to-volume ratio.
method Constructing infinitely many hyperbolic manifolds with nonempty boundaries.
result Existence of incommensurable hyperbolic manifolds with a fixed perimeter-to-volume ratio.
New proofs given for space curves with totally positive torsion.
problem Description of convex hulls of space curves with totally positive torsion.
method New proofs of parametric representation, surface area, and volume formulas.
result Recovery of formulas for convex hull's surface area and volume.
Revisits conformal metrics with finite Q-curvature, providing necessary and sufficient conditions.
problem Understanding conformal metrics with finite total Q-curvature.
method Introduces conformal mass and provides necessary and sufficient conditions for normality.
result Derives volume comparison theorems and proves a positive mass type theorem related to Q-curvature.
Study simplicial volume in fiber bundles with connected groups.
problem Understanding simplicial volume in fiber bundles.
method Investigated fiber bundles with connected structure groups.
result Simplicial volume of total space matches trivial bundle under certain conditions.
On a compact n-dimensional manifold M, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume, is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvat…
Sharp inequality proved in 3D hyperbolic spaces using flow methods.
problem Proving a sharp Minkowski-type inequality in Cartan-Hadamard 3-spaces.
method Using harmonic mean curvature flow to prove the inequality.
result Sharpened estimates for total mean curvature in hyperbolic 3-space.
In this paper we analyze and classify the totally geodesic subspaces of finite volume quaternionic hyperbolic orbifolds and their generalizations, locally symmetric orbifolds arising from irreducible lattices in Lie groups of the form $(\mathbf{Sp}_{2n}(\mathbb{R}))^q \times \prod_{i=1}^r \mathbf{Sp}(p_i,n-p_i) \times …
S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its pert…
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…
We prove that, among metrics on a compact quotient of (H2)n (product of hyperbolic planes) of prescribed total volume, the product of hyperbolic metrics has minimal volume entropy.
We prove that the product of equators S1×S1 in S2×S2 is globally volume minimizing under Hamiltonian deformations.
For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in Cn with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian n-varifold locally uniformly in $C^{k…
We prove that the 8^4_2 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice of the volume V_8 of the ideal regular octahedron, i.e. 7.32... = 2V_8. The proof relies on Agol's argument used to determine the minimal volume hyperbolic 3-manifolds with 2 cusps. We also nee…
On a compact n-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature o…
Smooth isotopy on cube saves energy with extra dimensions.
problem Constructing an isotopy with infinite kinetic energy.
method Explicit construction of isotopy on [0,1]n. result Existence of isotopy with infinite kinetic energy.
Paper shows how to evenly distribute intersections in hyperbolic spaces.
problem Equidistribution of intersections in hyperbolic manifolds.
method Properly immersed totally geodesic submanifolds, hyperbolic volume measure.
result Effective equidistribution of intersection points as submanifolds grow.
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.
Study new conjectures linking knot volume and knot cohomology.
problem Understanding the relationship between knot volume and knot cohomology.
method Analyzing 12-17-crossing knots, establishing three conjectures.
result Conjectures linking knot volume and knot cohomology.
The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equ…
On a compact n-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature o…
Totally geodesic hypersurfaces in hyperbolic manifolds are rigid under certain conditions.
problem Conditions under which totally geodesic hypersurfaces in hyperbolic manifolds are rigid.
method Study of homotopy equivalence and sectional curvature properties.
result Conditions for rigidity of totally geodesic hypersurfaces in hyperbolic manifolds.
Gradient flow converges to a minimal convex structure.
problem Finding the minimal convex structure in hyperbolic manifolds.
method Weil-Petersson gradient vector field of renormalized volume.
result The flow converges to the structure with minimum convex core volume.
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.
A dynamic herding model with interactions of trading volumes is introduced. At time t, an agent trades with a probability, which depends on the ratio of the total trading volume at time t−1 to its own trading volume at its last trade. The price return is determined by the volume imbalance and number of trades. The …
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.
We study totally geodesic planes in hyperbolic 3-manifolds M having incompressible core and degenerate ends. We prove a Ratner-type phenomenon: a closed minimal PSL(2,R)−invariant subset of M is either an immersed totally geodesic surface or all of M. We also show that for an arbitrary infinite volume hyperboli…
The Hessian of the renormalized volume of geometrically finite hyperbolic 3-manifolds without rank-1 cusps, computed at the hyperbolic metric g with totally geodesic boundary of the convex core, is shown to be a strictly positive bilinear form on the tangent space to Teichmüller space. The metric g is known fro…