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 …
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48 results for “volume constrained minimizers”
Study minimizers of quasi-perimeters in RCD spaces with volume constraints.
problem Regularity and topological properties of volume constrained minimizers in RCD spaces.
method New Deformation Lemma and study of interior and exterior points.
result Volume constrained minimizers are open bounded sets with Ahlfors regular boundary.
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.
Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy 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…
Study on capillarity minimizers with nonlocal repulsion and gravity, proving existence and nonexistence.
problem Minimizing an energy functional with capillarity, nonlocal repulsion, and gravity.
method Quantitative isoperimetric inequalities applied to capillarity problem in a half-space.
result Existence and nonexistence of minimizers for various nonlocal kernels and masses.
We prove that the conformal immersions of complex two tori into which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fi…
Since the pioneering work of Canham and Helfrich, variational formulations involving curvature-dependent functionals, like the classical Willmore functional, have proven useful for shape analysis of biomembranes. We address minimizers of the Canham-Helfrich functional defined over closed surfaces enclosing a fixed volu…
We address the minimization of the Canham-Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers…
A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Planemath.DG
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 -totally umbilical cap, which is an energy minimizer for a given enclosed volume.
We introduce a notion of generalized Willmore functionals motivated by the Hawking energy of General Relativity and bending energies of membranes. An example of a bending energy is discussed in detail. Using results of Y. Chen and J. Li, we present a compactness result for branched, immersed, haunted, stratified surfac…
We study the stability of partitions in convex domains involving simultaneous coexistence of three phases, viz. triple junctions. We present a careful derivation of the formula for the second variation of area, written in a suitable form with particular attention to boundary and spine terms, and prove, in contrast to t…
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.
Capillarity functionals are parameter invariant functionals defined on classes of two-dimensionals parametric surfaces in R3 as the sum of the area integral with an anisotropic term of suitable form. In the class of parametric surfaces with the topological type of S2 and with fixed volume, extremals of capillarity func…
Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in R3 as the sum of the area integral and a non homogeneous term of suitable form. Here we consider the case of a class of non homogenous terms vanishing at infinity for which the corresponding capillar…
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.
Paper proves unique energy-minimizing curves in constrained spaces.
problem Uniqueness of energy-minimizing curves in constrained spaces.
method Investigated energy-minimizing curves with fixed endpoints in a constrained space.
result Proved that the set of points for which the energy-minimizing curve is not unique has no interior points.
Study shows limits of volume-constrained sets are finite unions of Wulff shapes.
problem Analyzing the behavior of sets with degenerating ellipticity.
method Proving rigidity of -accumulation points of volume-constrained almost-critical sets.
result Limits of volume-constrained sets are finite unions of -Wulff shapes.
Essential minimal volume bounds for Einstein 4-manifolds.
problem Bounding the essential minimal volume of Einstein 4-manifolds.
method Introduced a new volume concept, essential minimal volume, and showed bounds for closed Einstein 4-manifolds.
result Closed Einstein 4-manifolds satisfy linear bounds on essential minimal volume.
We show that among sets of finite perimeter balls are the only volume-constrained critical points of the perimeter functional.
Beating level-set methods for 3D seismic data interpolation: a primal-dual alternating approachmath.OC
Acquisition cost is a crucial bottleneck for seismic workflows, and low-rank formulations for data interpolation allow practitioners to `fill in' data volumes from critically subsampled data acquired in the field. Tremendous size of seismic data volumes required for seismic processing remains a major challenge for thes…
Round balls minimize liquid drop model volumes ≤ 1.
problem Minimizing volumes in liquid drop models.
method Proved uniqueness of minimizers for small volumes.
result Round balls uniquely minimize volumes ≤ 1.
Extended characterization of RAAGs with zero minimal volume entropy.
problem Characterizing RAAGs with vanishing minimal volume entropy.
method Extended characterization from geometric dimension 2 to higher dimensions.
result Extended characterization of RAAGs with zero minimal volume entropy.
Minimal volume entropy vanishes or is positive under certain fiber growth conditions.
problem Conditions for minimal volume entropy to be zero or positive.
method Analyzes topological conditions related to fiber growth of maps.
result Examples of finite simplicial complexes with zero simplicial volume and large minimal volume entropy.
MaxVol NMF maximizes the volume of in NMF for better sparse and interpretable solutions.
problem Finding interpretable and unique NMF solutions.
method Dual approach to MinVol NMF, maximizing the volume of .
result MaxVol NMF solutions correspond to clustering columns in disjoint clusters.
In a remarkable article published in 1982, M. Gromov introduced the concept of minimal volume, namely, the minimal volume of a manifold is defined to be the greatest lower bound of the total volumes of with respect to complete Riemannian metrics whose sectional curvature is bounded above in absolute value b…
Minimal volume entropy vanishes for mapping tori over 3-manifolds.
problem Volume entropy of mapping tori over 3-manifolds.
method A variation of amenable category and minimal volume entropy of a homology class.
result Minimal volume entropy vanishes.
Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.
problem Understanding the group of volume preserving diffeomorphisms through symplectic geometry.
method Using cotangent bundles of spaces of smooth embeddings, symplectic reduction, and nonlinear Grassmannians of augmented submanifolds.
result Descriptions of coadjoint orbits of the group of volume preserving diffeomorphisms in terms of submanifolds of augmented spaces.
We show that the conjectural cusped complex hyperbolic 2-orbifolds of minimal volume are the two smallest arithmetic complex hyperbolic 2-orbifolds. We then show that every arithmetic cusped complex hyperbolic 2-manifold of minimal volume covers one of these two orbifolds. We also give all minimal volume manifolds that…
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.
Renormalized volume of minimally bounded regions in asymptotically hyperbolic Einstein spacesmath.DG
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.
We show that complete uniform visibility manifolds of finite volume with sectional curvature have positive simplicial volumes. This implies that their minimal volumes are non-zero.
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.
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.
Hexagonal tilings minimize perimeter with unequal volumes.
problem Finding optimal tessellations with unequal cell volumes.
method Minimizing perimeter functionals for different classes of problems.
result Hexagonal tilings are optimal among partitions with almost equal areas.
In this paper we consider a ``flow'' of nonparametric solutions of the volume constrained Plateau problem with respect to a convex planar curve. Existence and regularity is obtained from standard elliptic theory, and convexity results for small volumes are obtained as an immediate consequence. Finally, the regularity i…
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 study examines conditions for minimal volume entropy of simplicial complexes.
problem Conditions for minimal volume entropy of simplicial complexes.
method Topological conditions and growth of fundamental groups.
result Examples of simplicial complexes with zero simplicial volume and large minimal volume entropy.
Study simplicial volume for fixed fundamental groups, finding gaps.
problem Understanding simplicial volume for manifolds with fixed fundamental group.
method Relate gap problem to rationality questions in bounded (co)homology.
result Show existence of gaps in simplicial volume spectrum at zero.
In this article, we show that, for any compact 3-manifold, there is a volume-minimizing one-dimensional foliation. More generally, we show the existence of mass-minimizing rectifiable sections of sphere bundles without isolated "pole points" in the base manifold. This same analysis is used to show that the exam…
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.
Researchers found the minimum volume of a 3-cusped hyperbolic 3-manifold.
problem Finding the minimum volume of a 3-cusped orientable hyperbolic 3-manifold.
method Using guts in sutured and pared manifolds.
result The volume of a 3-cusped orientable hyperbolic 3-manifold is at least 5.49... = 6 × Catalan's constant.
Upper bounds for essential spectrum of minimal submanifolds linked to volume growth.
problem Estimating the essential spectrum of minimal submanifolds.
method Using volume growth to bound the bottom of the essential spectrum.
result Improved essential spectrum estimate for minimal submanifolds.
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…
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.
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.
Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.
problem Existence and classification of conical Calabi-Yau structures on horospherical cones.
method Variational approach to establish equivalence between volume minimization and existence of conical Calabi-Yau structures.
result Existence of many irregular horospherical cones with mild singularities.
Study minimal volume entropy for free-by-cyclic groups and 2D right-angled Artin groups.
problem Characterize minimal volume entropy for aspherical simplicial complexes with these groups as fundamental groups.
method Algebraic and geometric characterization, using fiber -growth collapse and non-collapsing assumptions.
result Provide bounds and criteria for minimal volume entropy in aspherical simplicial complexes.
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.