Study compares volumes of hyperbolic 3-manifolds using Ricci-DeTurck flow.
problem Volume comparison on finite-volume hyperbolic 3-manifolds.
method Exponential convergence of Ricci-DeTurck flow to the hyperbolic metric.
result The hyperbolic metric minimizes volume among metrics with bounded scalar curvature.
New metrics found in hyperbolic manifolds as volume-minimizers.
problem Finding critical points of volume-renormalized mass.
method Critical points of the volume-renormalized mass over asymptotically hyperbolic manifolds.
result V-static metrics are critical points of volume-renormalized mass.
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.
The paper proves volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.
problem Investigating volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.
method Applying similar techniques to derive local rigidity theorems for strictly stable Ricci flat manifolds.
result Derives local rigidity theorems for strictly stable Ricci flat manifolds with respect to σ2-curvature.
The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.
problem Volume entropy rigidity in Cayley hyperbolic spaces.
method Repairing a gap in the proof of volume entropy rigidity theorem.
result Cayley hyperbolic space minimizes volume entropy.
The volume of the quantum mechanical state space over n-dimensional real, complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean measure is computed, and explicit formulas are presented for the expected value of the determinant in the general setting too. The case when the state space is endo…
The study shows conditions for larger volumes in the universal cover of a manifold.
problem Conditions for larger volumes in the universal cover of a manifold.
method Analyzes the relationship between the volume of a manifold and the volume of its universal cover.
result Guarantees the existence of balls with greater-than-hyperbolic volumes in the universal cover.
Proves rigidity for maps between manifolds using degree theory and current developments.
problem Lipschitz-volume rigidity for maps between metric manifolds and Riemannian manifolds.
method Degree theory and recent developments of Lipschitz-volume rigidity for integral currents.
result Proves a Lipschitz-volume rigidity result for 1-Lipschitz maps.
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…
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.
Natural volume forms defined for pseudo-Finslerian manifolds with specific metrics.
problem Defining natural volume forms on pseudo-Finslerian manifolds with m-th root metrics. method Definitions depend on the parity of m, expressed in terms of Cayley hyperdeterminants. result Volume forms computation simplified by avoiding integration over the indicatrix.
New definition of metric current yields Finsler geometry volume densities.
problem Defining volume functionals from Finsler geometry.
method Proposed a new definition of metric current and showed its utility.
result Obtained a family of extendibly convex volume densities.
Uniform volume estimate for Kähler metrics in big cohomology classes.
problem Estimating volume for singular Kähler metrics in big cohomology classes.
method Generalized mixed energy estimate for functions in complex Sobolev space to big cohomology classes.
result Uniform non-collapsing volume estimate for local Kähler metrics.
Uniform estimates for Kaehler metrics' diameters and volumes.
problem Estimating diameters and volumes of Kaehler metrics.
method Proving uniform diameter and volume estimates for a family of Kaehler metrics.
result Uniform estimates for diameters and volumes of Kaehler metrics.
Paper shows regions close to negatively curved metrics are minimal fillings and rigid.
problem Boundary rigidity and minimality of metrics near negatively curved ones.
method Generalizes previous work on filling volume minimality and boundary rigidity for almost hyperbolic metrics.
result Regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and boundary rigid.
We define and study the renormalized volume for geometrically finite hyperbolic 3-manifolds, including with rank-1 cusps. We prove a variation formula, and show that for certain families of convex co-compact hyperbolic metrics $g_\eps$ degenerating to a geometrically finite hyperbolic metric g0 with rank-1 cus…
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…
The study proves that certain manifolds can have metrics with specific volume growth.
problem Determining if manifolds with positive scalar curvature can have metrics with a given volume growth.
method Using Gromov-Lawson and Grimaldi-Pansu constructions, the study proves the existence of metrics with the desired volume growth on specific manifolds.
result The study positively answers the question for manifolds that are infinite connected sums of closed manifolds with positive scalar curvature.
Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.
problem Volume noncollapsing for Kähler metrics induced by complex Monge-Ampère equations.
method Proves local volume noncollapsing estimate with Ricci curvature lower bound.
result Establishes diameter and gradient estimates for Kähler metrics.
Constructs metrics with constant scalar curvature and unbounded volumes on spheres.
problem Creating metrics with constant scalar curvature on spheres with unbounded volumes.
method Constructs a sequence of metrics conformal to a given metric with scalar curvature 1 and unbounded volumes.
result Constructs metrics with constant scalar curvature and unbounded volumes on spheres.
Study of limits of Einstein-Bogomol'nyi metrics on P^1 in two regimes.
problem Understanding limits of Einstein-Bogomol'nyi metrics on P^1.
method Analysis of two regimes: dissolving limit and large volume limit.
result Recovery of Einstein-Bogomol'nyi metrics on C with total string number N' for each N'.
We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least -6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter-Schwarzschild metrics.
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 article, we investigate the volume comparison with respect to scalar curvature. In particular, we show volume comparison holds for small geodesic balls of metrics near a V-static metric. For closed manifold, we prove the volume comparison for metrics near a strictly stable Einstein metric. As applications, we g…
Study on volume continuity of Lagrangian submanifolds.
problem Lower semi-continuity of Lagrangian volume.
method Analysis of volume properties with respect to Hofer- and γ-distances.
result Volume is γ-lower semi-continuous in two specific cases.
The study examines metrics with unit volume or area on manifolds with boundaries, finding critical points and solving curvature problems.
problem Finding metrics with prescribed curvature on manifolds with boundaries.
method Variational properties of volume and boundary area functionals, using critical metrics and curvature conditions.
result Sufficient and necessary conditions for metrics to be critical points and for scalar/mean curvature functions.
Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.
problem Understanding shared properties of geodesically equivalent Finsler metrics.
method Computing first integrals as coefficients of a characteristic polynomial.
result Geodesically invariant functions are first integrals of geodesically equivalent Finsler metrics.
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.
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.
Study finds critical points of volume functionals on Sasaki manifolds.
problem Finding Kähler-Einstein metrics on Sasaki manifolds.
method Revisited moment polytopes, applied to volume minimization.
result Transverse coupled Kähler-Einstein metrics found as critical points.
This paper carries out a renormalization of the volume of the Loewner-Nirenberg singular Yamabe metric in a given conformal class on a compact manifold-with-boundary. This generalizes the usual volume renormalization for Poincare-Einstein metrics. The coefficient of the log term in the volume expansion defines a confor…
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.
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…
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.
New stability theorem for hyperbolic metrics without volume bounds.
problem Stability of finite volume hyperbolic metrics without upper volume bounds.
method Abstract axiomatic framework and bootstrap argument to extend stability result.
result Weaker exponential control of the metric allows for a broader application of the stability theorem.
Second part of Q-curvature research focusing on volume comparison.
problem Volume control and rigidity of Q-curvature.
method Volume comparison and local rigidity analysis.
result Volume comparison theorem for metrics close to strictly stable positive Einstein metrics.
The study establishes equivalence of conditions on metric manifolds with finite volume.
problem Characterizing metric spaces with a metric fundamental class.
method Analyzing three conditions on metric manifolds with finite volume.
result Conditions (1), (2), and (3) are equivalent for metric manifolds with finite Nagata dimension.
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.
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 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…
Grimaldi-Pansu metrics are constructed for manifolds with multiple ends.
problem Volume growth on manifolds with more than one end.
method Constructing Riemannian metrics with bounded geometry and uniform bounds for volume growth.
result Uniform bounds for volume growth of Grimaldi-Pansu metrics in certain manifolds.
We solve the modified Kazdan-Warner problem of finding metrics with prescribed scalar curvature and unit total volume.
Study calculates volumes of Fano K-moduli spaces in various dimensions.
problem Computing volumes of Fano K-moduli spaces in different dimensions.
method Computed volumes of Fano K-moduli spaces of Quartic del Pezzo and log Fano hyperplane arrangements in dimensions one and two.
result Relates computed volumes to Weil-Petersson volumes, extending the Weil-Petersson metric to the log case.
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…
Study on collapsing Calabi-Yau manifolds and their metrics.
problem Understanding degenerations of Calabi-Yau manifolds with Ricci-flat Kahler metrics.
method Survey of recent developments, focusing on volume collapsing metrics.
result New insights into the behavior of Calabi-Yau manifolds under volume collapse.
Alternative proof of simplicial volume bound using area-minimizing sets.
problem Bounding simplicial volume of manifolds with restricted ball volumes.
method Area-minimizing separating sets method.
result Explicit constants for simplicial volume bound derived.
We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.
Study on Berwald-Weyl curvature with projective invariance and vanishing results.
problem Characterizing Berwald-Weyl curvature for spray/Finsler metrics.
method Analyzing expressions and proving vanishing conditions for curvature.
result Berwald-Weyl curvature vanishes for certain spray/Finsler metrics.