Derive Dirichlet scalar curvature energy functional variation formula
problem Dirichlet scalar curvature energy functional
method First variation formula
result Introduce Dirichlet-Einstein metrics
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48 results for “curvature variation”
The paper studies variations of -curvature for submanifolds in Riemannian manifolds.
problem Understanding the behavior of -curvature under variations of submanifolds.
method Analyzes the functional of -curvature for submanifolds of arbitrary codimension in Riemannian manifolds.
result Provides insights into the variational properties of -curvature.
Boosting Variational Inference improves posterior approximations with adaptive step-sizes.
problem Limited resources hinder the widespread adoption of Boosting Variational Inference.
method Characterized global curvature impact, introduced local curvature, and developed an approximate backtracking algorithm.
result New theoretical convergence rates and experimental validation demonstrate improved performance.
A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as the geometry varies in a particular way, which we call a conformal variation. This…
Some variational properties of the weighted σ_{r}-curvature for submanifolds in Riemannian manifoldsmath.DG
The paper studies variations of weighted curvature on submanifolds.
problem Variational properties of weighted curvature on submanifolds.
method Analysis of a functional with integrant r-th weighted curvature.
result Applications to hypersurfaces in Euclidean space and the unit sphere.
Study variations of Riemannian submersions to maintain geodesic fibers and positive curvatures.
problem Maintain geodesic fibers and positive sectional curvatures in Riemannian submersions.
method Vary Riemannian metrics while keeping fibers totally geodesic and horizontal distribution fixed.
result Conditions for making sectional curvatures positive and existence of fat submersions.
New curves defined by curvature powers studied for variational properties.
problem Characterizing translating solitons in curve flows.
method Variational characterization of generalized elastic curves.
result New variational characterization of grim reaper curve.
Variational properties of the total inverse mean curvature in the plane under boundary constraintsmath.DG
Study variational properties of curves in half-plane with area constraints.
problem Characterize critical points of inverse mean curvature.
method Variational analysis of curves with boundary constraints.
result Existence and stability of critical points with prescribed area.
Study finds loops with specific curvature exist using Hardy's inequality.
problem Existence of closed planar loops with prescribed curvature.
method Variational approach, Hardy's inequality and associated functional space.
result Existence of loops with specific curvature proven.
New formulas for coassociative submanifolds' volume variation.
problem Understanding volume changes in coassociative submanifolds.
method Proved new variation formulae using data.
result Highlight the role of ambient torsion and Ricci curvature in volume changes.
We derive variational formulas for the total Q-prime curvature under the deformation of strictly pseudoconvex domains in a complex manifold. We also show that the total Q-prime curvature agrees with the renormalized volume of such domains with respect to the complete Einstein-Kähler metric. In the appendix, by Rod Gove…
A new framework for Einstein-Hilbert action with topological variations.
problem Understanding critical points and dimensionality in Einstein-Hilbert action.
method Localized Einstein-Hilbert variational principle, topology on Sobolev configurations, topological variations.
result No critical points in dimension 4, higher dimensions free of this problem.
Liouville Theorem for curvature equation in half space with fully nonlinear boundary conditionmath.DG
Study proves Liouville theorem for specific curvature equations with boundary conditions.
problem Proving Liouville theorem for -curvature equations in half spaces with nonlinear boundary conditions.
method Established using positive constant curvature equations and variational functional approach.
result Proved Liouville theorem for positive constant -curvature equations in and boundary conditions.
First variation of fractional -dimensional measure for submanifolds
problem Computing the first variation of a fractional -dimensional measure for submanifolds
method First variation computation
result Definition of a nonlocal mean-curvature vector for embedded submanifolds
Proves existence of planar curves with specific curvature.
problem Existence of planar closed curves with prescribed curvature.
method Variational methods, adding a parameter, monotonicity trick.
result Existence of planar closed curves with prescribed curvature for some curvature functions.
Study variations of metrics on Riemannian submersions to preserve fiber geometry.
problem Preserving specific geometries of fibers under metric variations on Riemannian submersions.
method Formulated conditions for preserving fiber geometry (totally geodesic, umbilical, minimal) and examined variations of sectional curvatures.
result Conditions for metric to be a critical point of integrated squared norms of fiber curvatures, with non-negative second variation.
Nonnegative sectional curvature linked to matrix displacement convexity.
problem Nonnegative sectional curvature in Riemannian manifolds.
method Matrix displacement convexity as a criterion for nonnegative sectional curvature.
result Entropy functional matrix displacement convexity implies nonnegative sectional curvature.
Paper solves curvature assignment on surfaces with sharp points and edges.
problem Prescribing curvatures on surfaces with conical singularities and corners.
method New variational formulation for surfaces with singularities.
result First results for prescribed curvatures on surfaces with singularities.
New optimal surfaces found in Heisenberg group defy Euclidean sphere optimality.
problem Optimizing mean curvature in Heisenberg group sub-Riemannian setting.
method Developed variational theory, established first and second variation formulas, introduced new critical surfaces.
result Identified and characterized a new family of rotationally invariant critical surfaces, the Pansu-Minkowski spheres.
A PDE approach to the existence and regularity of surfaces of minimum mean curvature variationmath.DG
Analyzes surfaces minimizing mean curvature variation using PDEs.
problem Finding surfaces of minimum mean curvature variation.
method Develops an analytic theory using partial differential equations.
result Establishes existence and regularity of minimizers.
Conformally variational Riemannian invariants (CVIs), such as the scalar curvature, are homogeneous scalar invariants which arise as the gradient of a Riemannian functional. We establish a wide range of stability and rigidity results involving CVIs, generalizing many such results for the scalar curvature.
Splitting theorem for non-positively curved Lorentzian spaces.
problem Understanding curvature in Lorentzian spaces.
method Proving a splitting theorem with global non-positive timelike curvature and extending first variation formula.
result Splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature.
In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the …
The paper studies stability of discrete planar curves using variational methods.
problem Stability of discrete planar curves under area constraints.
method Unified interpretation of discrete curvatures, determination of equilibrium curves, stability analysis.
result Equilibrium curves for the length functional under area-constraint conditions are determined and their stability is studied.
Study on surfaces in Heisenberg group with constant mean curvature.
problem Constant mean curvature surfaces in sub-Lorentzian Heisenberg group.
method First-variation formula derivation and isoperimetric candidates classification.
result Characterization and conjecture of isoperimetric maximizers.
The Variational Principle for the Uniform Acceleration and Quasi-Spin in Two Dimensional Space-Timemath-ph
The variational principle and the corresponding differential equation for geodesic circles in two dimensional (pseudo)-Riemannian space are being discovered. The relationship with the physical notion of uniformly accelerated relativistic particle is emphasized. The known form of spin-curvature interaction emerges due t…
We use variational methods and a modified curvature flow to give an alternative proof of the existence of a self-shrinking torus under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.
Localized deformation of scalar curvature and mean curvature on manifolds.
problem Deforming scalar curvature and mean curvature on compact manifolds with boundary.
method Proving localized surjection of scalar curvature and mean curvature map, handling non-variational linearized problem.
result Localized deformations of scalar curvature and mean curvature on compact manifolds are possible.
Paper proves Toponogov's theorem in Alexandrov geometry.
problem Proving Toponogov's theorem in Alexandrov geometry with lower curvature bound.
method Inspired by Riemannian geometry, uses second variation formula.
result Elementary proof of Toponogov's theorem in Alexandrov geometry.
Euclidean geometry has historically been the typical "workhorse" for machine learning applications due to its power and simplicity. However, it has recently been shown that geometric spaces with constant non-zero curvature improve representations and performance on a variety of data types and downstream tasks. Conseque…
The paper studies constant mean curvature hypersurfaces in Finsler manifolds.
problem Understanding geometric properties of hypersurfaces in Finsler manifolds.
method Using volume preserving variation and homothetic navigation.
result Deduced a Heintze-Karcher type inequality and proved an Alexandrov type theorem.
Study on surface configurations with curvature and elasticity.
problem Equilibrium configurations of surfaces with curvature and elasticity.
method Investigates the Euler-Helfrich functional, focusing on axially symmetric surfaces and their variational problems.
result Critical surfaces for the Euler-Helfrich functional, if axially symmetric, satisfy a simpler second order variational problem.
The paper classifies surfaces with constant skew curvature in 3-space forms.
problem Classifying surfaces with constant skew curvature in 3-space forms.
method Variational characterization and flow of binormal vector field.
result Classification of rotational surfaces with constant skew curvature.
Duality structures and discrete conformal variations of piecewise constant curvature surfacesmath.GT
A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Ad…
Variational Laplace improves Bayesian neural networks performance.
problem Improving Bayesian neural networks performance.
method Develops variational Laplace for BNNs, exploiting curvature of likelihood.
result Variational Laplace outperforms other inference methods.
Classifies soap film surfaces with vertical potentials.
problem Classifying soap film surfaces with vertical potentials.
method Variational characterization of -elastic curves.
result Obtains a full description of -elastic curves.
After establishing some new global facts (like a measure theoretic structure theorem and approximation results) about complex-valued functions with bounded variation on arbitrary noncompact Riemannian manifolds, we extend results of Miranda/the second author/Paronetto/Preunkert and of Carbonaro/Mauceri on the heat semi…
Graphs with non-negative Ollivier-Ricci curvature cannot be expanders.
problem Understanding the relationship between graph curvature and expansion properties.
method Proving an inequality linking isoperimetric profiles to total variation decay of random walks.
result Graphs with non-negative Ollivier-Ricci curvature cannot be expanders.
The paper proves rigidity of bordered polyhedral surfaces using variational principles.
problem Determining the rigidity of bordered polyhedral surfaces.
method Using the variational principle, the paper shows that bordered polyhedral surfaces are determined by boundary values and discrete curvatures on interior edges.
result The paper re-proves the classical result that two Euclidean or hyperbolic cyclic polygons are congruent if their side lengths are equal.
We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, , on Teichmüller space. We then use this formula to determine the asymptotic behavior as of the second variation. As a consequence, for , we obtain the complete expansio…
Study variational problems in Kähler geometry to construct metrics.
problem Maximizing/minimizing Monge--Ampère energy on Kähler potentials.
method Prove existence and uniqueness of extremals, use them to construct metrics.
result Existence and uniqueness of extremals with simple characterization.
In this note we continue the analysis of metric measure space with variable ricci curvature bounds. First, we study -convex functions on metric spaces where is a lower semi-continuous function, and gradient flow curves in the sense of a new evolution variational inequality that captures the information that …
Directly applies Kazdan--Warner results to prescribe scalar curvature on bundles.
problem Prescribing scalar curvature functions on bundles.
method Direct application of Kazdan--Warner results and variational methods.
result Determines which functions are realizable as scalar curvature functions on bundles.
Study on polyharmonic curves on spheres and space forms.
problem Classifying polyharmonic curves of constant curvature.
method Analyzing curves on spheres and space forms, deriving explicit families.
result New insights into higher order variational problems.
The Gauss-Bonnet curvature of order is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension , as the usual scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals…
We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consis…
The paper solves curvature prescription on a disk with negative Gaussian curvature.
problem Prescribing Gaussian curvature and geodesic curvature on a disk with negative Gaussian curvature.
method Variational approach, critical points of a functional, perturbation argument, monotonicity trick, blow-up analysis, Morse index estimates.
result General existence results for the curvature prescription problem.
Study optimal partitions on spheres using fractional Q-curvature and variational methods.
problem Optimal partition problem on the sphere with fractional Q-curvature.
method Variational approach, symmetry analysis, Hölder regularity results.
result Existence of a symmetric minimal partition.