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.
Proves generic nondegeneracy for solutions under volume constraint in closed manifolds.
problem Proving nondegeneracy for solutions of the Van der Waals-Allen-Cahn-Hilliard equation.
method Adapting techniques from previous research to prove nondegeneracy.
result Generic nondegeneracy for solutions of the Van der Waals-Allen-Cahn-Hilliard equation under a volume constraint in closed manifolds.
In this paper we consider the problem of minimizing area subject to a volume constraint in a given convex set.
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.
Paper derives inequalities for eigenvalues of Witten-Laplacian under fixed volume constraint.
problem Eigenvalue inequalities of Witten-Laplacian on bounded domains.
method Rearrangement technique and trial functions under fixed weighted volume constraint.
result Several isoperimetric inequalities for eigenvalues of Witten-Laplacian.
The paper proves uniqueness of a solution in general relativity.
problem Uniqueness of solutions in the conformal method for Einstein's constraint equations.
method Analyzes solutions with arbitrary mean curvature and volume constraint.
result The Holst-Nagy-Tsogtgerel--Maxwell solution is unique for volumes below a certain threshold.
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.
We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals-Allen--Cahn--Hilliard type, satisfying a linear volume constraint, on a bounded Lipschitz domain $Ω\subset\mathds R^N$. The number of solutions is estimated in terms of topological …
Motivated by Brendle-Marques-Neves' counterexample to the Min-Oo's conjecture, we prove a volume constrained scalar curvature rigidity theorem which applies to the hemisphere.
We establish existence of compact minimizers of the prescribed mean curvature problem with volume constraint in periodic media. As a consequence, we construct compact approximate solutions to the prescribed mean curvature equation. We also show convergence after rescaling of the volume-constrained minimizers towards a …
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.
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 study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some …
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.
We prove that the Whitehead link complement and the (-2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... = 4 x Catalan's constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on vo…
The paper explores generalizations of Mirzakhani's recursion and computes volumes for physical gravity models.
problem Computing volumes for physical gravity models.
method Topological recursion and physical two-dimensional gravity models.
result Derivation of Virasoro constraints and cut-and-join equations for generalized Mirzakhani's recursions.
Non-Negative Matrix Factorization, NMF, attempts to find a number of archetypal response profiles, or parts, such that any sample profile in the dataset can be approximated by a close profile among these archetypes or a linear combination of these profiles. The non-negativity constraint is imposed while estimating arch…
The paper proves stability of manifolds with boundary under volume and distance constraints.
problem Stability of manifolds with boundary under volume and distance constraints.
method Volume preserving intrinsic flat convergence of metrics with boundary constraints.
result The stability of manifolds with boundary under volume and distance constraints is proven.
Study shows convergence of volumes on manifolds with boundary under area constraints.
problem Volume convergence on manifolds with boundary under area constraints.
method Doubling with necks procedure and area constraints.
result Only a bound on boundary area is needed for volume preserving intrinsic flat convergence.
Study on predicting sequences with Gaussian constraints, linking to intrinsic volumes and metric complexity.
problem Predicting sequences almost as well as the best Gaussian distribution with mean in a given subset.
method Expressed minimax regret in terms of intrinsic volumes, established comparison inequality for Wills functional, characterized global covering numbers and local Gaussian widths.
result Sharp estimates on the log-Laplace transform of intrinsic volume sequence for a general nonconvex set.
Study finds multiple solutions for Van der Waals-Cahn-Hilliard equation on manifolds.
problem Finding multiple solutions for a specific equation on manifolds.
method Combines Lusternik-Schnirelman and Morse theory with a photography method.
result Establishes multiplicity of solutions using topological invariants.
New constraints rule out some optimal domains for helicity maximisation.
problem Finding a smooth domain of fixed volume that maximizes helicity.
method Established additional geometric constraints on optimal domains.
result Ruled out the optimality of a broad class of solid tori.
The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.
problem Geometric obstructions and volume bounds in singular spaces.
method Developed new degree theorem for Alexandrov spaces using integral currents.
result Entropy-volume minimization prevents metric singularities in Gromov-Hausdorff limits.
Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy W=∫H2 under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the crit…
In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a conically bounded convex set, i.e., an unbounded convex body admitting an \emph{exterior} asymptotic cone. Results concerning existence of isoperimetric regions, the behavior of the isoperimetric pr…
The paper studies 4D Ricci flow manifolds with curvature constraints.
problem Investigating 4D Ricci flow manifolds with specific curvature conditions.
method Analyzing 4D manifolds with curvature constraints via Ricci flow.
result Proves topological and geometric gap theorems for maximal volume growth.
We study the variation of a smooth volume form along extremals of a variational problem with nonholonomic constraints and an action-like Lagrangian. We introduce a new invariant describing the interaction of the volume with the dynamics and we study its basic properties. We then show how this invariant, together with c…
The paper studies Ricci flow with specific curvature and volume constraints, proving convergence and curvature bounds.
problem Analyzing Ricci flow with Ricci curvature and volume constraints.
method Proving convergence and curvature bounds for Ricci flow with specific constraints.
result Ricci flow with specified constraints converges to a flat cone or static flow.
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.
Computes volumes of metric maps on surfaces, linking to Weil-Petersson volumes.
problem Computing volumes of specific metric maps on surfaces.
method Using recent results on discrete maps with irreducibility constraints, computes volumes as homogeneous polynomials.
result Identifies volumes as homogeneous polynomials and satisfies string and dilaton equations.
In this paper we investigate the flow of surfaces by a class of symmetric functions of the principal curvatures with a mixed volume constraint. We consider compact surfaces without boundary that can be written as a graph over a sphere. The linearisation of the resulting fully nonlinear PDE is used to prove a short time…
The goal of this paper is to study weakly Einstein critical metrics of the volume functional on a compact manifold M with smooth boundary ∂M. Here, we will give the complete classification for an n-dimensional, n=3 or 4, weakly Einstein critical metric of the volume functional with nonnegative scalar …
In curved spaces, isoperimetric sets don't exist for small volumes.
problem Nonexistence of isoperimetric sets in spaces of positive curvature.
method Constructing specific noncompact smooth Riemannian manifolds with positive curvature.
result Nonexistence of isoperimetric sets for small volumes in spaces of positive curvature.
Study approximates Plateau's laws using the Allen-Cahn equation.
problem Approximating Plateau's laws with the Allen-Cahn equation.
method Minimizing the Allen-Cahn energy under volume and spanning constraints.
result Energy minimizing solutions approximate Plateau-type singularities.
The Allen-Cahn system on manifolds yields multiple phase distributions.
problem Finding the number of solutions to the Allen-Cahn system on manifolds.
method Volume-fixing variations approach to classify isoperimetric clusters.
result The number of solutions is bounded by topological invariants for parallelizable manifolds.
An investor with constant relative risk aversion and an infinite planning horizon trades a risky and a safe asset with constant investment opportunities, in the presence of small transaction costs and a binding exogenous portfolio constraint. We explicitly derive the optimal trading policy, its welfare, and implied tra…
Proves existence of multiple solutions to a multiphasic equation on manifolds.
problem Existence of multiple solutions to a multiphasic equation with a small volume constraint.
method Lusternik-Schnirelmann and infinite-dimensional Morse theories, combined with isoperimetric theory and transversality theorem.
result Lower bound for the number of solutions depending on topological invariants.
Study finds knots with ideal length need not have smallest volume.
problem Tackles the conjecture that ideal knot length equals smallest volume.
method Measures convex hull volume of knots during length annealing.
result Identifies knots with non-ideal global minimum volume.
We consider the flow of closed convex hypersurfaces in Euclidean space Rn+1 with speed given by a power of the k-th mean curvature Ek plus a global term chosen to impose a constraint involving the enclosed volume Vn+1 and the mixed volume Vn+1−k of the evolving hypersurface. We prove that i…
The paper proves rigidity results for Einstein manifolds with specific geometric constraints.
problem Understanding the rigidity of Einstein manifolds under bounded covering geometry.
method Analyzing Einstein manifolds with bounded covering geometry to prove rigidity results.
result Compact Einstein manifolds with specific geometric properties are isometric to space forms.
New metrics on 3D manifolds with large Steklov eigenvalues.
problem Finding metrics with large Steklov eigenvalues on compact manifolds.
method Expressed Steklov spectrum of warped products and applied to metrics with fixed volume.
result Examples of metrics on 3D manifolds with arbitrarily large first non-zero Steklov eigenvalue.
The paper solves reverse isoperimetric problems for convex bodies with curvature constraints.
problem Finding the smallest volume among λ-convex bodies of a given surface area. method Using λ-convex bodies and analyzing their properties in model spaces of constant curvature. result The λ-convex lens is the unique minimizer of volume among all λ-convex bodies of given surface area in R3. 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.
We show that a steady-state stock-flow consistent macro-economic model can be represented as a Constraint Satisfaction Problem (CSP).The set of solutions is a polytope, which volume depends on the constraintsapplied and reveals the potential fragility of the economic circuit,with no need to study the dynamics. Several …
We prove existence of regions minimizing perimeter under a volume constraint in contact sub-Riemannian manifolds such that their quotient by the group of contact transformations preserving the sub-Riemannian metric is compact.
We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows - surfaces of revolution - in R3 with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additio…
The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.
problem Proving finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.
method The approach removes constraints of sectional curvature or conjugate radius and extends to previous related studies.
result Theorems are proven for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume, without the need for triangle comparison of Toponogov type.
In this paper, we consider a free boundary problem with volume constraint. We show that positive minimizer is locally Lipschitz and the free boundary is analytic away from a singular set with Hausdorff dimension at most n−8.